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EUAN 


THE UNIVERSITY OF ALBERTA 


Design of Shallow Tunnels: in Soft Ground 
by 


pe seti og Negro Jr. 


VOLUME III 


AOTHESTS 
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES AND RESEARCH 
IN’ PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE 


OF Doctor of Philosophy 


Department of Civil Engineering 


EDMONTON, ALBERTA 


SPRING 1988 








THE UNIVERSITY OF ALBERTA 


Design of Shallow Tunnels in Soft Ground 
by 


Arsenio Negro Jr. 


VOLUME III 


KTHESTS 
SJBMiG@icDeTO THE FACULTY OF GRADUATE STUDIES AND RESEARCH 
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE 


OF Doctor of Philosophy 


Department of Civil Engineering 


EDMONTON, ALBERTA 


SPRING 1988 





963 


closer to F respectively. As the stresses at the opening 
decrease, the non-linearity of the ground response becomes 
more pronounced. 

Figures 6.48 and 6.49 show that the normalized ground 
displacements in this case, are not affected much by the 
amount of stress release. It is noted, however, that the 
ratio of maximum surface settlement to crown settlement 
increases slightly (up to 10%) as the stress release is 
increased (up to 70%). Figure 6.49 also shows that the point 
of maximum surficial distortion (at about 1.6D off the axis) 
does not change location with increasing stress release but 
that the normalized distortion increases slightly. 

The last two figures illustrate the format in which all 
results of the analyses are presented in Appendix C. 
Influencemert the Tunnel Depth 

Figures 6.5C to 6.53 show results of three analyses 
where the undrained strength ratio c,/yD, was kept constant 
and equal to 1.25 and the tunnel cover to diameter ratio 
varied from 1.5 to 6. The NGRC for the crown of a deeper 
tunnel presents a slightly 'stiffer' response at the early 
stage of stress release with associated smaller 
displacements, when compared with a shallower tunnel. This 
reflects the influence of the proximity of the ground 
surface, which is also apparent in the springline NGRC, but 
not at the floor. As the amount of stress release increases 
the NGRCs become more non-linear. This effect is more 


Pronounced as the tunnel gets deeper. Indeed, ground 


964 


0 
Nee es: 


KDei Cy/VYD=1.25 H/0=3 





¥ F s a 


——— 


= 


0,/ OQ 





=i 


ZS: 
(Rad.Str.)/(Init.Rad.Str.) 
0 
ee 
. eee pee eee eee 





0.0 O.2 0.4 0.6 0.8 120 ee A 1.6 
(Rad.Displ.*EtlL)/(DLam.*Init.Rad.Str.) 


U 


Figure 6.47 Normalized Ground Reaction Curves for H/D=3 and 


an Undrained Strength Ratio of 1.25 


965 
















oa 
“ LECENO 
OX of stress release 
10z “| 
202 
o +——+ 30x | 
. <—— X40 
i Sox 
602 
= 70% 
a) Ss 
iO 
O 
© 
uw 
& o 
ph se 
e i) 
(a) 
(= 
Gs 
° 
wu Z 
ah 
oO 
ro 
n 
So 
(se) 
{ 
° 
> 
i) 
Ss KO=1 Cy/YD=1.25 H/Ds3 
wm 


=0.3 —0..1 0.1 0.3 0.5 Ow 0.9 Jes] 
(Settl./Crown Settl.) 


Figure 6.48 Normalized Subsurface Settlements along Tunnel 
Axis for H/D=3 and an Undrained Strength Ratio of 1.25, 


Calculated for Increasing Amount of Stress Release 


966 


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970 


collapse is approached more rapidly as H/D increases, with 
the soil undrained strength kept constant. 

With respect to the normalized settlements, one notes 
in Figure 6.53 that the thicker soil cover permits a more 
rapid attenuation of the ground movements. This leads to the 
smaller maximum normalized settlement at the surface and the 
smaller normalized maximum distortion for a given amount of 
stress release. 

Influence of the Undrained Strength Ratio 

Figures 6.54 to 6.57 show the results of three 
analyses, where the relative depth of the tunnel was kept 
constant (H/D=3) and the undrained strength ratio c,/yD was 
varied from 0.625 to 2.5. Clearly as the strength decreases, 
the larger the displacements are for a given radial stress 
ratio. A special feature is also apparent in Figures 6.54 to 
6.56: the NGRCs for different strengths are nearly 
homothetic. The centre of similitude of these curves is 
their origin (0,/o,,=1 and U=0). The coordinates of 
homothetic points on these curves are defined by the 
intersections of the latter with arbitrary axes through the 
centre of similitude (shown by dashed lines). It will be 
shown in Section 6.4 that if these points were used as 
reference values and if each curve is transformed into 
another with coordinates normalized to their respective 
reference values, then these normalized curves almost 
coincide. Single curves for each point of the tunnel contour 


can thus be formulated, regardless of the value of the 


971 


0.00 


O25 


0.50 


DEPTH (Z/H) 





0.0 0.2 0.4 0.6 0.8 1.0 


SETTL./CROWN SETTL. 


NOTE: Cy/ O*1.25, Ko #l.0, 50% RELEASE 


¢ 


0.0 


0.1 


0.2 


03 


SETTL./CROWN SETTL. 


H/D0 #1.5 


0.4 





DIST. TO THE AXIS (Y/D) 


NOTE! Cy/YD#125, Ko #1.0, 50% RELEASE 


Figure 6.53 Normalized Subsurface and Surface Settlement for 
an Undrainee Strength Ratio of 1.25 and 50% Ground Stress 


Release, Calculated for Different Tunnel Depths 


972 


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975 


undrained strength ratio. This finding will give more 
generality to the projected solution as will be discussed 
later. 

Figures 6.57(a) and (b) present the distributions of 
normalized ground settlements at the surface and subsurface 
for an equal amount of stress release of 50%. As the ground 
strengths are different in each case, for a fixed radial 
stress ratio (0,/c,,) the factor of safety against collapse 
increases as the strength increases. In other words, for the 
50% release, the case with lower value of c,/yD will be 
closer to collapse (but with an FS greater than unity). 
Despite this, the distributions of normalized subsurface 
settlements almost coincide, as shown in Figure 6.57(a). The 
distributions of surface settlements are also very close to 
each other. It is apparent, however, that the weaker the 
soil is (thus, the lower the factor of safety is), the 


larger the normalized surface distortion is, as expected. 


6.3.2.2 Cohesionless Soil Model Results 
Effect of the Amount of Stress Release 

Figures 6.58 to 6.60 present typical results for a 
cohesionless soil with ¢=30°, K,=0.8 and H/D=3. The 
"stiffness" of the ground can be measured by the derivative 
of the NGRC. At the springline (Figure 6.58), one notes that 
the ground softens progressively, as the stress release 
increases. At the crown and floor, however, during the 


initial stages of stress reduction, the ground seems to 


stiffen slightly but below a certain radial stress ratio, 


Sp 


Cy/YD = 1.25 


Cy/7De2. 


DEPTH (2/0) 


Cy/Y0#1.25a2.5 





0.0 0.2 0.4 0.6 0.8 1,0 


SETTL./CROWN SETTL. 


NOTE + H/O=3, Ko# 1.0, 50% RELEASE 


¢ 


0.0 


0.1 
Cy/YD = 0.625 


Cov Osa25 
0.2 


Cy/7D=2.5 


0.3 


SETTL./CROWN SETTL. 


0.4 





0 2 4 6 8 Ke) 
DIST. TO THE AXIS (Y/D) 
NOTE ' H/D*3, Ko#=1.0, 50% RELEASE 
Figure 6.57 Normalized Subsurface and Surface Settlements 
for H/D=3 and 50% Ground Stress Release, Calculated for 


Different Undrained Strength Ratios 


377 


the response is also that of softening for increasing stress 
release. That initial stiffening response, was not observed 
for the frictionless soil, where K, was set equal to unity. 
When K, is less than unity, the ground at the crown or floor 
initially experiences an increase in the minor principal 
Stress (in the horizontal direction), which leads to an 
increase in the tangent modulus and thus a stiffer response. 
After the principal stress directions reverse so that the 
radial stress becomes the minor principal stress, a 
progressive softening is also noted at crown and floor. 

The distribution of normalized subsurface settlements 
for the cohesionless soil model seems to be more affected by 
the amount of stress release, than for the. frictionless 
model. AS indicated in Figure 6.59, as the amount of stress 
release increases, larger normalized settlements are 
observed. The same effect is noted for the normalized 
surface settlements, in Figure 6.60, when it is also seen 
that larger normalized surface distortions are observed for 
increasing stress reductions. 

Pibbuence Of the Tunnel Depth 

Figures 6.61 and 6.62 present results of analyses for 
Ke=0.8. and) $=302; carried out »for three tunnel depths, As it 
is noted, the NGRC for the tunnel crown is more sensitive to 
the relative depth than are the other points of the tunnel 
contour. In fact, the curves at springline and floor almost 
coincide. Moreover, as the tunnel gets deeper, the effect of 


the proximity of the ground surface on the NGRCs of the 


G78 


K0=#0.8 PHI=30 H/D=3 


Or / Fro 


2 





(Rad. Str.)/(Init.Rad.Str.) 
0 


ta at a Ce ee OC 





[ae ere Eee 








0.0 0.2 0.4 0.6 0.8 i520 mee 1.4 
(Rad.Oispl.*Etl)/(OLam.*Init.Rad.Str.) 
U 


Figure 6.58 Normalized Ground Reaction Curves for H/D=3, 


K,=0.8 and 9=30° 


o79 


° 

xg LEGEND 
OZ of stress release 
102 
20% 

Oo 302 


40% 
a S0z 
60% 


Distance from Surface (D) 





=0.3 °-0.1 0.1 0.3 0.5 G.7 0.9 lot 
(Settl./Crown Settl.) 


Figure 6.59 Normalized Subsurface Settlements for H/D=3, 
K,=0.8 and ¢=30°, Calculated for Increasing Amount of Stress 


Release 


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9381 


crown is reduced and these curves move down and get closer 
together. This is exactly opposite to the trend noted in the 
frictionless soil model in which the soil strength was kept 
constant. In this cohesionless counterpart, the the shear 
strength increases as the tunnel gets deeper and this 
explains the noted behaviour. As would be expected when the 
tunnel gets deeper, the normalized subsurface settlements 
become smaller as indicated in Figure 6.62(a). These 
settlements are plotted against the depth ratio z/H (solid 
lines) and against the depth ratio z/D (dashed curves). The 
effect of settlement attentuation in the soil cover is 
perhaps better appreciated through the latter plots. Figure 
6.62(b) shows that as the tunnel gets deeper, the normalized 
surface distortions decrease and the settlement trough gets 
wider. Recall that the settlements shown in Figure 6.62 
correspond to a stress reduction of 50%. Since the friction 
angle is the same in all cases, the ground strength is 
higher for deeper tunnels. Thus, for equal stress reduction, 
the factors of safety are not the same in each case, and are 
smaller for the shallower tunnel case. 
Influence of the Friction Angle 

Figures 6.63 to 6.66 present the results of analyses 
where H/D=3, K,=0.8 and three values for friction angles 
were considered (¢=20°, 30° and 40°). A stronger non-linear 
response is noted in the NGRC of the tunnel springline, 
reflecting that the soil at this point approaches failure 


earlier than elsewhere around the tunnel. This is consistent 


982 





NOTES ; 
Lap i=s30° HKD = 078 
2. C= CROWN , S# SPRINGLINE , F = FLOOR 


Figure 6.61 Normalized Ground Reaction Curves for Different 


Tunnel Depths, Calculated for 9=30° and K,=0.8 


O83 


0.00 


0.25 





Ca ray 
~ bles 
zt H/O0#=1.5 N 
oa 
= 050 “ 
a. WJ 
4 Oo 
Oma 
1.00 
(a) 
0.0 Or 0.4 0.6 08 1.0 
SETTL./ CROWN SETTL. 
NOTE : @ =30°%, Ko=0.8, 50% RELEASE 
; H/D=6 
J 
1 H/O*3 
uJ 
lp) 
= H/O#1.5 
= 
re) 
x 
oO 
~N 
sy 
= 
= 
Ww 
(op) 
(b) 





DIST. TO THE AXIS (Y/D) 


NOTE $*30°%, Ky*0.8, 50% RELEASE 
Figure 6.62 Normalized Subsurface and Surface Settlements 


Cor Ditterent TuinnelDepths, (Calculated? fori¢=30°, Kj=008 


and 50% Stress Release 





984 


with the selected K, value which is smaller than unity. 

The NGRCs for different friction angles shown in 
Figures 6.63 to 6.65 were also found to be nearly 
homothetic, with the centre of similitude at their origin 
(Z=1 and U=0). The coordinates of homothetic points on these 
curves are defined by the intersection of the latter with 
arbitrary axes through the centre of similitude. It is shown 
in Section 6.4 that if these points were used as reference 
values and if each curve is transformed into another with 
coordinates normalized to their respective reference values, 
then these normalized curves almost coincide. Single curves 
for each point of the tunnel contour can thus be formulated 
independently of the friction angle. This finding will give 
more generality to the projected solution. 

Figure 6.66 presents the distribution of normalized 
ground settlements at the surface and subsurface, for a 
stress release of 40%. The noted differences in the 
settlement distributions can be partly attributed to the 
differences in shear straining in each case. AS the amount 
of stress release is the same, but the strengths are 
different, the factor of safety against collapse in each 
case is different. Correspondingly, the displacement fields 
should differ. Higher normalized surface distortions are 
found for decreasing ¢ but the width of the surface 
settlement trough does not change much. 


Influence of the In Situ Stress Ratio 


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0 CROWN 





0.0 0.2 0.4 06 as LO 


U 


NOTE: Ko=0.8, H/D =3.0 


Figure 6.63 Normalized Ground Reaction Curves of the Crown 


for H/D=3 and K,=0.8, Calculated for Different Friction 
Angles 


986 


0 SRINGLINE 





A i RAEN CMEMGEEE 8 RERUN eA Cam Aer Ie} 


U 


NOTE: Ko=0.8, H/D #3.0 


Figure 6.64 Normalized Ground Reaction Curves of the 


Springline for H/D=3 and K,=0.8, Calculated for Different 


Friction Angles 





‘0.0 0.2 0.4 0.6 0.8 LO 


U 


NOTE: Ko=0.8, H/D =3.0 


Figure 6.65 Normalized Ground Reaction Curves of the Floor 


Pere) U=seanc hk =0.0, Calculated for DitiLerent Friction 


Angles 


987 


988 


DEPTH (Z/D) 





0.0 O:2 0.4 0.6 0.8 1.0 


SETTL./CROWN SETTL. 


NOTE ' Kg#0.8, H/O#3 , 40% STRESS RELEASE 


¢ 
0.0 


Ohl tp 


0.2 


SETTL./CROWN SETTL. 





DIST. TO THE AXIS (Y/D) 


NOTE ' Kyo *0.8, H/0#3.0, 40% STRESS RELEASE 


Figure 6.66 Normalized Subsurface and Surface Settlements 
for H/D=3, K,=0.8 and 40% Ground Stress Release, Calculated 


for Different Friction Angles 


989 


Rar leetorathestrictioniess soil model, a single value of K, 
waS considered, in the parametric analyses for the 
cohesionless model, different K, values were assumed. 

Figures 6.67 to 6.70 present the results of analyses 
for vH7D=4;)o=30° and for three in situ stress ratios, 
C20... Capel ccs Cie bees Oo)ine 

For K, smaller than unity, soil elements close to the 
Springline undergo local failure before other points at the 
tunnel contour. This fact is apparent from the NGRCs, which 
show a more pronounced non-linear aspect at the springline 
than at elsewhere (Figures 6.67 to 6.69). For K, equal to 
unity, soil elements at the springline, crown and floor 
reach local failure at approximately the same amount of 
ground stress release. 

As noted earlier in this section, when K, is smaller 
than unity, the NGRC at the crown and floor, exhibit a 
‘stiffening’ response due to the increase in the minor 
principal stress in the early stages of the unloading 
process. This effect is not noted when K,=1. 

Figures 6.70(a) and (b) show the distribution of 
normalized displacements at the surface and subsurface. It 
is evident that K, has a marked influence on the 
distributions. A lower K, value promotes a larger maximum 
normalized settlement, higher normalized surface distortions 
and a narrower settlement trough. Although the dimensionless 
displacement Hoe teeleeclrOonn tor Kk =) ois larger than U for K, 


smaller than unity, the absolute surface settlement at the 


990 


CROWN 





NOTE : H/D23, p=30° 


Figure 6.67 Normalized Ground Reaction Curves of the Crown, 
for H/D=3 and 9=30°, Calculated for Different In Situ Stress 


Ratios 


SPRINGLINE 





NOTE +: H/D=3, 30° 


Figure 6.68 Normalized Ground Reaction Curves of the 
Springline, for H/D=3 and #=30°, Calculated for Different In 


Situ Stress Ratios 


952 


FLOOR 


or/Oro 


2 





NOTE : H/D=3, =30° 


Figure 6.69 Normalized Ground Reaction Curves of the Floor, 


for H/D=3 and $=30°, Calculated for Different In Situ Stress 


Ratios 


9933 


Gentre tinem@hon ch < isi alwaysilargersthan? that. fore K,=1, for 
the same initial tangent modulus and the same amount of 
Seress tekeases Inrethemtwords efor sk. decreasing,. ithe 
average vertical strain in the ground cover above the tunnel 


axils decreases. 


6.3.2.3 Relationships Between Surface and Crown 

Settlements 

The practical interest that exists in relating the 
crown settlement (u,) to the surface settlement at the 
tunnel centreline (S) is derived from the fact that once the 
former is assessed in some way, the latter could be readily 
estimated through some relationship. Many authors 
investigated the subject and proposed different procedures 
Meio accomplish this. | 

Atkinson and Potts (1977:318) proposed simple 
expressions relating the normalized surface settlement 
(S/u,) to the relative depth of cover (H/D) and the soil 
type. Their expressions were semi-empirically developed from 
observation of displacements in tunnel model tests, before 
ground collapse (FS>1). A similar approach was followed by 
Seneviratne (1979:56) once more using model test results. 
Ward and Pender (1981:265) explored the subject further and 
pointed out some of the limitations of the above approaches. 
Moreover, they discussed the limitations of the S/u, 
relationships derived from linear elastic analyses (for 
instance, Oteo and Sagaseta, 1982:657). Resendiz and Romo 


(1981:72) developed a relationship through non-linear 


994 


DEPTH (2/D) 





0.0 0.2 0.4 0.6 0.8 1.0 


SETTL./CROWN SETTL. 


NOTE : H/D0 #3, @#30°, 50% RELEASE 


¢ 


SETTL./CROWN SETTL. 





DIST TO THE AXIS (Y/D) 


NOTE: H/D#3, $=30°, 50% RELEASE 


Figure 6.70 Normalized Subsurface and Surface Settlements 
for H/D=3, 9=30° and 50% Ground Stress Release, Calculated 


for Different In Situ Stress Ratios 





IEE 


elastic numerical modelling and suggested that the 
normalized surface settlement should be a function of H/D 
and of the soil strain at failure, as determined in an 
undrained triaxial test. Wong and Kaiser (1987:331) proposed 
a conceptual model relating the normalized surface 
settlement to the amount of stress release allowed. They 
contended that upon the reduction of ground stresses at the 
opening, the settlement ratio (S/u,) remains fairly constant 
initially, increasing rapidly with yielding and subsequent 
collapse at which stage it approaches unity. 

If general relationships to estimate the S/u, ratio 
were to be derived for pre-collapse situations (i.e., for 
good ground control conditions), and if they were to be 
based on the two soil models introduced earlier (Sections 


6.2.2 and 6.2.3), then they would have to be expressed as: 


Spy Ey cals 
Jaalige RAS "Dp? 





ea eae b6.s1¢ | 


for the cohesionless soil model, and as 
Sie ne eee 
Us’ IFp-" DD 


for the frictionless soil model. 








F=f ( ) icky hel 
The results of the parametric analyses included in 
Appendix C could be helpful in establishing these 
relationships. To assess the role of the dimensionless 
variables shown in the equations 6.14 and 6.15 on the 
settlement ratio, it seems worthwhile to examine some of 
those results. Figure 6.71, for instance, presents how the 
settlement ratio for the frictionless soil model, varies 


with the stress ratio Z, with the undrained strength ratio 


256 


and with the tunnel depth. One notes that at the early part 
of the unloading process and especially for the higher 
strength soils, the normalized surface settlement is 
virtually independent of the amount of stress release 
applied to the opening. Consequently, it does not depend on 
the factor of safety at pre-collapse stages, despite the 
pronounced non-linear response one notes even for the 
stronger soils (see, for instance, Figure 6.54). 

The above result is not new as it was noted earlier by 
Atkinson and Potts (197723180) througnestaticetunne emode! 
test results in overconsolidated kaolin, performed in 
Cambridge by Cairncross (1973) and Orr (1976). In these 
tests, the undrained shear strength ratio ranged from 0.4 to 
0.8 and the H/D ratio from 0.35 to 1.2. For deeper tunnels 
or weaker soils, however, one notes in Figure 6.71 that the 
normalized surface settlement becomes more affected by the 
Stress ratio, L. Therefore, the Atkinson and Potts (1977) 
conclusion and proposed correlation ficr™the S/upsratiouis 
valid only for the conditions they studied (for instance, 
very Shallow tunnels) and cannot be generalized to other 
Situations. Moreover, the linearity between S/u, and H/D 
suggested by the Atkinson and Potts (Op.cit.) relationship 
have not been confirmed by centrifuge model test results 
obtained later by Mair (1979:127), also in Kaolin with c,/yD 


of about 0.38 and larger cover to diameter ratios (up to 


Ze ll 





“0.0 O2 0.4 06 0.8 1.0 


SURF./CROWN SETTL. 


Figure 6.71 Variation of the Maximum Normalized Surface 
Settlement with the Amount of Stress Release and other 


Variables, for the Frictionless Soil Model 


998 


Another feature that is apparent from Figure 6.71 is that 
depending on the depth ratio, as ground failure is 
approached the settlement ratio may either increase or 
decrease. This seems to reflect the mode of collapse that 
prevails in shallower or deeper tunnels, with the surface 
settlement developing at a faster or slower rate compared to 
the crown settlements. Similar results are found for the 
cohesionless soil model. These are shown in Figure 6.72 for 
H/D=3. Once fore the normalized surface settlements seem to 
be unaffected by Z for small amounts of ground stress 
release or for larger amounts provided the soil strength is 
high. Moreover, the influence of the stress ratio K, now 
becomes apparent. While for low K, values, the settlement 
ratio seems to increase as failure is approached, for higher 
K,, that settlement ratio may in fact decrease. The previous 
criticisms to Atkinson and’ Pott's”(1977)®relationship tor 
clays are also applicable for their proposed relations for 
sand. 

The above results indicate that the surface settlement 
ratio depends on H/D, on the soil strength (c,/yD or ¢), on 
the in situ stress ratio, K,, and on the amount of stress 
release allowed (0,/o,,), as is stated by the expressions 
6.14 and 6.15. Furthermore they indicate that the S/u, 
ratio, regardless of the soil model considered (fully 
undrained or fully drained), may either increase or decrease 
with Z decreasing and ground collapse approaching. 


Therefore, the conceptual model proposed by Wong and Kaiser 





0.0 0.2 0.4 0.6 0.8 1.0 


SURF./CROWN SETTL. 


Figure 6.72 Variation of the Maximum Normalized Surface 
Settlement with the Amount of Stress Release and other 


Variables, for the Cohesionless Soil Model 


1000 


(1987:332) 1s incomplete, as it assumes that in the 
pre-failure regime, the settlement ratio is always 
increasing as yielding occurs. Im fact, some results By wong 
(1986) (not included in Wong and Kaiser, 1987) actually 
demonstrate the opposite. Wong (1986) studied the subject 
through finite element modelling using an elasto-plastic 
stress-strain relationship. For H/D=2, K,=1.3 and $=30° 
(analysis ST3 by Wong, 1986:97), sthe numerical jresults 
indicate that the settlement ratio may decrease as the 
stress ratio £ decreases. The Wong and Kaiser (1987:333) 
suggestion that, regardless of the mode of behaviour of the 
ground defined by its properties, the tunnel depth and the 
in situ stresses, the settlement ratio S/u, at collapse 
becomes unity is yet to be proven. The ratio of settlement 
increments AS/Au, may possibly tend toward unity at a 
complete ground collapse condition but the settlement ratio 
may not. 

To complete the discussion, Figures 6.73 and 6.74 were 
prepared. They represent graphically the relationship given 
by the equation 6.15, for the frictionless soil model, and 
for ground stress releases of 30 and 50% respectively. These 
amounts of stress reductions in the 2D model could 
correspond in an actual tunnelling situation, to transverse 
sections located near the tunnel face and at some distance 
behind it. Comparing these two figures, one notes that the 
normalized surface settlement is not very sensitive to the 


undrained strength ratio, or to the amount of stress 














1001 


reduction. The settlement ratio is more dependent on the 
relative tunnel depth. Note also that it has been inferred 
that as H/D tends toward zero, the settlement ratio should 
tend towards unity. Observe that in Figure 6.74, the results 
of Mair's (1979) centrifuge model test in kaolin have been 
included (tests 2DP and 2DV). The undrained strength ratio 
of—this-sorl-wes-about 0734, 

Similarly, Figures 6.75 to 6.80 were prepared for the 
cohesionless soil model. It should be noted that the data 
referring to the 50% stress release, for ¢=20°, K,=0.6 and 
0.8 were calculated after a few elements of soil at the 
Springline had failed. As explained, the numerical model was 
not designed for this condition. However, since the failed 
zone was limited to a fairly small region next to the 
springline, it 1S believed that it had negligible effect on 
the settlement above the crown. The failed zone extended to 
no more than D/4 beyond the tunnel contour and up to no more 
than 45° measured from the horizontal axis towards the crown 
or floor. The area of failed elements corresponds typically 
to less than 7% of the tunnel area. The suite of plots shown 
in the last six figures represents the function given by 
equation 6.14. They may serve as practical design charts for 
estimating the settlement ratio, for conditions differing 
from those considered in the parametric analyses whose 
results were included in Appendix C. 

To illustrate the subject further, Figure 6.81 


reproduces data from 28 case histories gathered by Ward and 


1002 


Legend 
Cu/(gama.D)=2.5 
Cu/(gama.0)=1.25 





Surf. /Crown Settl. 





Figure 6.73 Relationships between Normalized Maximum Surface 


Settlement and Relative Depth of Tunnel, Calculated for the 


oy ne —— —— 





Frictionless Soil Model, for 30% Stress Release 


1003 


Legend 
Cu/(gama.D)=2.5 
Cu/(gama.D)=1.25 


Cu/(gama.D)=0.625 
Cu/(gama.D0)=0.3125 





MAIR'S (1979) TESTS 


Surf./Crown Settl. 





Figure 6.74 Relationships between Normalized Maximum Surface 
Settlement and Relative Tunnel Depth, Calculated for the 


Frictionless Soil Model, for 50% Stress Release 


1004 








0.8 
am 
o 
) 
erei0 6 
> 
O 
O 
> #04 
— 
_ 
= 
Y) 
0.2 
0 
0 1 2 3 4 5 6 


Figure 6.75 Normalized Settlement Ratios for o=40° and 30% 


Stress Release 


1005 





Surf. /Crown Settl. 








Figure 6.76 Normalized Settlement Ratios for ¢=40° and 50% 


Stress Release 


1006 





Surf. /Crown Settl. 





Figure 6.77 Normalized Settlement Ratios for =30° and 30% 


Stress Release 


1007 





Surf./Crown Settl. 





Figure 6.78 Normalized Settlement Ratios for ¢=30° and 50% 


Stress Release 


1008 


0.8 


(@) 
o 


Surf./Crown Settl. 
= 
nm 


(@) 
N 





Figure 6.79 Normalized Settlement Ratios for ¢=20° and 30% 


Stress Release 





1009 











1- 
0.8 
= 2 
io 
® 
cs 0.6 
= . 
> e r 
e 
© 0 
—~ 0.4 Mos 
2g, a6 
2B 
Y 
OezZ iS 
eee | 
0 1 2 5 4 =) 6 
H/D 


Figure 6.80 Normalized Settlement Ratios tor $=20° and 30% 


Stress Release 


1010 


Pender (1981:267). It shows the ratios of surface to crown 
settlements measured by field instrumentation, plotted 
against the cover to diameter ratio. In most cases, the deep 
settlement measurements were taken somewhat above the tunnel 
crown. SO, in general, the S/u-srat Ou smlarger (nang: ae 
actual one. Ward and Pender (Op.cit.) tried to include only 
short-term displacements, excluding those due to drainage, 
but it is doubtful whether this has been really achieved. 
For instance, Cases 17, 18, 24 and 25'*, are respectively, 
the Mexico Siphon tunnel (see Table 5.31), a tunnel in 
Buenos Aires, a sewer tunnel in Belfast (see Section 
3.3.4.2), and the Thunder Bay tunnel (see Table 5.31 and 
Section 3.3.4.2) and may have, very possibly, included 
ground displacement caused by partial drainage and 
consolidation. In cases like these, one would expect to find 
settlement ratios larger than those experienced under time 
independent conditions, as assumed in the present study (see 
Section Bac. 42). 

Two extreme curves are Superimposed on the Ward and 
Pender data in Figure 6.81. These were obtained in the 
present study and correspond to 50% stress release, for a 
soil with ¢=20° and K,=0.6 and 1.0. It is noted that a large 
number of cases is bound by those two curves, which also 
bound a similar set of data collected by Heinz (1984:49) of 


NATM case histories (not shown). This may not be claimed as 


For identification of the cases numbered in Figure 6.81, 
the reader is referred to Table 1 in Ward and Pender, 
CT9S 13266. 2o7o 


OQ 
© 


2 
Oo 


SURF./CROWN SETTL. 





Note: See Ward and Pender (1981:266) for identification 
of the case histories. 


Figure 6.81 Normalized Settlement Ratios Observed in some 


Case Histories 


1012 


a proof of the validity of the presently proposed settlement 
ratios relationships. However, the relationships derived 
herein may be used to explain the apparently chaotic set of 
field data shown in Figure 6.81. The scatter of data can 
possibly be elucidated in terms of differences in K,, in 
soil strength and in the amounts of stress release 
associated with the different case histories. 

In Section 5.2.1.2, it was mentioned that a linear 
elastic finite element back analysis of a shallow tunnel 
case history could lead to either an overestimate or an 
underestimate of the maximum surface settlement, whenever a 
match in the crown displacement is achieved. Conversely, if 
a match between measured and backcalculated surface 
settlement is attained, then either an over or an 
underestimate of the crown settlement may be obtained. The 
reasons behind these statements are now, perhaps, clearer. 
In a linear elastic analysis, the surface to crown 
settlement ratio is always constant, regardless of the 
amount of stress release. When a non-linear response is 
taken into account this may not happen, as indicated by the 
results shown in Figure 6.71 or 6.72, or in the results 
shown in Appendix C. For a non-linear ground response, the 
surface to crown settlement ratios can either increase or 
decrease, with respect to the decreasing stress ratio, Z. It 
depends on the type of soil model being considered (the 
"undrained' - frictionless soil model, or the 'drained' - 


cohesionless soil model), on the cover to diameter ratio, on 


10173 


the in situ stress ratio and on the soil strength. These 
facts explain the above statements and the comments included 
im Section’ 5.2.4.2 "regarding Figures 5:6%and 5.7..Note that 
ine Pqures Ven 71Sandu6.7, 2). the start ing)ooi nes: at’ 2-09) wfor 
all curves shown (after the first 10% of ground stress 
release), give the settlement ratios for linear elastic 
analyses under the conditions considered. The ground 
response after the first unloading increment, as calculated 
by the present piecewise elastic numerical solution, is 
identical to the result a linear elastic analysis would 


yield. 
6.4 Generalization of the Results 


6.4.1 Opening Remarks 

In this section, an attempt will be made to achieve the 
objective established in Section 6.1: the development of a 
general procedure that would allow the ground reaction 
curves or stress release curves of shallow tunnels to be 
obtained, without the need of additional finite element 
analyses. 

The parametric analyses that were carrried out allowed 
those curves to be obtained for certain specific conditions, 
defined in terms of soil properties (for example, a 
particular soil strength) or geometry (H/D). In practice, 
conditions differing from those will normally be found and 


interpolation between the NGRCs obtained will usually be 


1014 


required. If some general expression, such as that given by 
equations BoP lin dr ’6 91 27e@could bemobtarmed mtnensthnerprebiem 
would be simplified. Moreover, if some practical situation 
is found to lie beyond the range of the variables 
considered, one may attempt an extrapolation of the results 
of the parametric analyses through these expressions. This 
could involve some degree of uncertainty and may yield 
unreliable results, if certain conditions are not fulfilled. 
Some of these conditions will be discussed later. 

The generalization of the NGRC will be developed 
separately for the frictionless soil model representing an 
undrained soil response, and for the cohesionless soil model 


representing a drained soil response. 


6.4.2 Frictionless Soil Model 

It was anticipated in Section 6.3.2.1 that, for a given 
cover to diameter ratio, the NGRCs of points of the tunnel 
contour for different undrained strength ratios, are nearly 
homothetic. This can be demonstrated taking, for example, 
the results shown in Figures 6.54 to 6.56. Assume that the 
Origin of the NGRCs, point O, defined by the stress release, 
a=1-z=0 and the dimensionless displacement U=0, is the 
centre of similitude. Through O, draw an arbitrary axis OP. 
The point P can be taken, for instance, as the extreme point 
of the NGRC for c,/yD=2.5. The line OP will intersect the 
NGRC for c,/yD=0.625 and 1.25 at points M and N. Read the 


coordinates of points M, N and P. For instance, for the NGRC 


10415 


of the tunnel crown (Figure 6.54), these coordinates are 
those shown in Table 6.10. Then use these coordinates as 
reference values and normalize each of the NGRCs to each 
corresponding reference value. In other words, replot the 
results of the numerical analysis with the normalized 
coordinates a/a,,, and U/U,,,. If this procedure is applied to 
the curves for the tunnel crown, Figure 6.54 will be reduced 
to that shown in Figure 6.82. One notes that the twice 
normalized ground reaction curves (NNGRC) virtually coincide 
(Points M, N and P of Figure 6.54 do coincide in Figure 
6.82), and it is said that the original NGRCs (Figures 6.54) 
were nearly homothetic. If they were perfectly homothetic, 
then they would coincide exactly. Rigorously, this is not 
the case but, for all practical purposes it can be said that 
they do coincide. 

Moreover, if it is attempted to find out what function 
fits best the points shown in Figure 6.82, it is soon found 
that a hyperbola can be quite well adjusted through them. 
The NNGRC could then be expressed as: 
a(1ee)/(=2",5) = Ee [6.16] 


ref 


A/ A, of 
This function presents two advantages. It has a finite 


limiting value when U/U tends to infinity, as a ground 


rer 
reaction curve should have when a tunnel collapse condition 
is approached. Secondly, the hyperbola can be transformed 
into a linear function (see Figure 5.10) and a linear 


regression analysis can be applied to the transformed 


normalized points. If this transformation of coordinates is 


1016 


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made and a linear regression is undertaken for the data 
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in equation 6.16 are: 


a 0.6444 


b Che Sys) 


and that the correlation coefficient is: 

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which is an indication of the goodness of the fit. The 
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quite well. 

If the above process is repeated for the springline and 
floor curves, then the twice normalized curves shown an 
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used for normalizing these curves are also shown in Table 
6.10. Note that for each point of the tunnel contour, the 
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all situations. 
In order to eliminate the arbitrary reference values 


appearing in equation 6.16, it can be further transformed as 


below: 
as tt U 
Ute v a ME b U/ Ue. 
Thus: 
U 
= $1 %- a | Se a ee 
a ae iat te lees tay. ee 
or: 
U 
re 1 - [ela 
where 
A = yee) Ue ee) 
B = b/a.t, 


The newly defined parameters can be calculated from Tables 
GepiOmtowG. 12) and thus Table 6.13j)i1s obtained. 

The coefficient A is independent of the strength raeio 
and Figure 6.85 shows how A varies with the relative depth 
of the tunnel, for three points of the tunnel contour. The 
coefficient B for each point of the tunnel contour is found 
to vary with both the soil strength and the tunnel depth. 
Figures 6.86 to 6.88 summarize these dependencies. In each 
of these figures, in order to facilitate the interpolation 
of B for intermediate values of strength or depth ratios, 
the coefficient B has been plotted against the strength 
ratio, for constant H/D values (solid lines) and against the 
depth ratio, for constant c,/yD values (broken lines). 

Equatacnwo.!7 and charts given in Figures 6.85 to 6.88, 


represent the proposed generalized solution for obtaining 


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H/D 


NOTE + C= CROWN , S2SPRINGLINE , F = FLOOR 


Figure 6.85 Variation of the Coefficient A with Relative 


Tunnel Depth, for the Frictionless Soil Model 


1026 


H/D 





Figure 6.86 Variation of the Coefficient By for sthestunne! 


Crown, with Strength and Depth Ratios, Calculated for the 


Frictionless Soil Model 


O27 


H/D 





Figure 6.87 Variation of the Coefficient B for the Tunnel 
Springline, with Strength and Depth Ratios, Calculated for 


the Frictionless Soil Model 


1028 


H/D 





Figure 6.88 Variation of the Coefficient B for the Tunnel 


Floor, with Strength and Depth Ratios, Calculated for the 


Frictionless Soil Model 


1029 


the ground reaction curves for three points on the contour 
of a shallow tunnel in a soil ee nee cenn sd by the 
frictionless model (equation 6.12). Both coefficients, A and 
B have some physical meaning. The above solution was 
designed for applications within the common ranges of 
variables defined in Section 6.2.5 and its validity is not 
suggested to hold beyond those ranges. However, it seems 
interesting to investigate the limits of the solution, in 
order to assess the physical meaning of the coefficients A 
and B, in simple qualitative terms. 

Prechew limit. for Ul approaching infinity, equation 6.17 
becomes: 

Zci-¢ [6.18] 

This would be the limiting stress ratio acting on the 
opening at a condition of very large radial displacements. 
One notes that 2, does not depend on A, but only on B, which 
in turn is a function of the soil strength and of the depth 
of the tunnel. Thus, B is a parameter reflecting the 
ultimate state of the tunnel at collapse. Since B is related 
to a limiting stress at tunnel collapse, it should depend on 
the ground strength and on the tunnel depth, as indicated by 
any limit analyses solution applied to shallow tunnels. 

Moreover, for constant H/D and c,/yD approaching 
infinity, Figures 6.86 to 6.88 suggest that B tends towards 
zero. Under these conditions, for U tending towards 
infinity, equation 6.18 indicates that Z, would tend towards 


an infinitely large and negative value, which indeed would 


1030 


be required in such infinitely strong ground to cause its 
collapse. On the other hand, for H/D constant and c,/yD 
tending to zero, Figures 6.86 to 6.88 suggest that B would 
tend towards infinity. Thus, for U approaching infinity, Z, 
would tend towards unity. This is again a consistent result 
for an extremely weak ground, which would collapse under any 
amount of stress release at the opening. A Similar result is 
obtained by making c,/yD constant and H/D increasing towards 
infinity. In this case, B would alsomtend to) intinitysandec) 
to unity. The extremely high in situ stress in such a deep 
tunnel, driven through a soil of finite strength, would lead 
to collapse for any amount of stress release at the opening. 

The coefficient A reflects the in situ stiffness of the 
ground. The partial derivative function of equation 6.17 is 
easily found as: 

6Z A 


x! =< = 
U (A+BU) ? 


in which the negative sign of the partial derivative has 


[6.19] 


been dropped. For zero displacement, the slope of the NGRC 
is found to be equal to 1/A. In other words, the coefficient 
A is inversely proportional to the in situ stiffness of the 
ground, as reflected by the initial slope of the ground 
reaction curves at each point of the tunnel contour. As 
noted in Sectron 6ec02ceand illustrated in Figure 6.47, 
for a constant in Situ tangent modulus profile, the initial 
Slope of the NGRC decreases from the tunnel floor towards 
the crown. This result reflects the 'stiffening' action of 


the lower boundary, affecting the floor reaction and the 


1OSn 


"softening' action of the free ground surface allowing the 
ground at the crown to move more freely towards the opening 
upon the stress release. The initial slope of the springline 
curve should thus show an intermediate initial slope. 
Accordingly, the coefficient A should not depend on soil 
strength, as it indeed does not, but should depend on the 
position of the ground surface, which is manifested by its 
dependence on H/D, as shown in Figure 6.85. Furthermore, 
from the above discussion one would expect to get a larger A 
for the crown and a smaller one for the floor, as is the 
case. 

Besides supplying components for the prediction of the 
ground reaction curves, the general solution may also 
provide a means to estimate the degree of 'softening' of the 
ground around the tunnel upon the reduction of the in situ 
Stresses. This is particularly relevant in assessing the 
‘current' ground stiffness to be considered in a 
ground-lining interaction analyses. As the support is 
activated after some tunnel closure has developed, the 
interaction process will be controlled not by the initial 
ground stiffness, but by that existing at the instant the 
lining is installed. The effect of the ground 'softening', 
as a result of unloading, on the ground-lining interaction 
process was presented and discussed in Section 2.3.5.3. 

If the ground-lining interaction analysis is to be 
performed using a ring-and-spring model, such as reviewed in 


Section 4.3.2.4, one would need to know, for instance, what 


1032 


radial spring constant (k,) should be adopted to account for 
the non-linear response of the ground and its associated 
degree of ground 'softening', taking place before the 
support is activated. Since the stress ratio is Z=0,/o,, and 


Usu_E,,/0,,.D, it) follows; that thet partie ladenivatiwve: oteiuis 


Ge ced: 
given by: 
65 oD Oe 
: du, 


But d50,/6u, is the radial spring constant (k,) as defined in 


Section 4.3.2.4. Therefore: 


Bei 
k. =2 7 
Using equation 6.19, one gets: 
A Bei 
Key ee Ty 
(A+BU) ? 
The above equation would thus solve the posed problem, 





[6.20] 


yielding different spring constants at each point (C, S, F) 
Of the tunnelw@eontour. 

Note that if no stress release is allowed and the 
radial displacements are zero, equation 6.20 would reduce to 
a linear elastic spring. Therefore: 

ey ape era [6:2] 
It was shown in Section 4.3.2.4 that if a single average 
radial spring constant (k,) is to be considered, as used in 
most ring-and-spring models, then, for a uniform E,, ground 


profile: 





ib Sie Bees 
~ +p R 


thus, 


1033 


In fact, this approximate equation is confirmed by the 
results shown in Figure 6.85 or Table 6.13. Depending on 
HAD, MAMet setLound® to vary “fromi0:635° to) 17345 whichis not 
entirely dissimilar to 0.745 obtained from (1+v)/2, for 
v=0.49, the Poisson's ratio used in the parametric numerical 
analyses. For H/D increasing, A,, gets closer to (1+v)/2, as 
it should for a deep tunnel. 

If the ground-lining interaction analyses is to be 
performed using a ring-and-plate model, such as reviewed in 
Section 4.3.2.3, one would need to know the current tangent 
Young's modulus of the ground (E,), at the instant the 
Support is installed. If it is assumed that this modulus is 
directly proportional to the spring constant (or the slope 


of the GRC), then: 
E 
E 


Substituting equations 6.20 and 6.21 into the above, one 


rte 
k 


ra: 





ti 


gets: 

gees 
A+BU 
Different current moduli would be found at distinct points 


E,= eae 


t 


[6.22] 


ti 


around the tunnel contour. The available ring-and-plate 
solutions normally operate with constant ground modulus'’. 


Thus: 


'7 If a numerical ring-and-spring model with discrete beams 
andaspriings (See®Séction®47332.4),°1s-to be used; “then 
different spring constants could be assigned at distinct 
points of the tunnel contour. This would account for the 
different degrees of ground ‘softening’ observed around the 
opening. 


1034 


Or 





Any of the above expressions can be used to estimate the 
average current tangent modulus at the instant the support 
is activated. 

It should be pointed out that the current tangent 
modulus given by equation 6.22 reflects a "weighted" average 
of all the ground mass affecting the ground response at a 
certain point of the tunnel contour. Therefore, it is 
neither equal to the modulus of the soil elements 
immediately adjacent to the opening nor to that of elements 
far away. A closer inspection of the results of the 
parametric analyses, revealed that the moduli given by 
equation 6.22;°for alcertaln’pointeotethemcunnelecontour, 
correspond approximately to the current tangent moduli of 
elements located radially away from the contour, at 
distances varying from 0.25D to 0.65D. The magnitude of this 
distance was found to vary according to the tunnel depth H/D 
and, to a lesser degree, to the amount of stress release. 

AS a reasonable approximation, it may be said that the 
moduli derived from equation 6.22 for the tunnel crown and 
floor, are nearly equal to the moduli of elements located, 
respectively, at half diameter above and below the tunnel. 
The springline modulus also corresponds to the modulus of an 
element situated at about D/2 radially measured from that 


Poloe. 


1035 


As stated earlier the solution presented is strictly 
valid only within the ranges of the variables considered in 
the numerical analyses that supplied the data for the 
generalization. Although apparently unbounded, equation 6.17 
may not be valid for large amounts of stress release (or low 
x). Large dimensionless displacements U may involve ground 
failure. The numerical model used to develop the present 
solution does not represent this behaviour properly and all 
data obtained after this event were disregarded in the 
generalization process. In fact, for large displacements, 
the solution presented can furnish values of the stress 
ratio Z that may violate the failure criteria. This subject 


will be discussed in Section 6.5. 


6.4.3 Cohesionless Soil Model 

te wasementioned) inp Section 6.3.2.2 that for a given 
cover to diameter ratio, the NGRCs of each point of the 
tunnel contour, for different friction angles, are nearly 
homothetic. This can be demonstrated considering, for 
example, the results shown in Figure 6.63 (crown, K,=0.8, 
H/D=3.0). Assume that the origin of the NGRCs, point O, 
defined by the stress release a=1-Z=0 and the dimensionless 
displacement U=0, is the centre of similitude. As for the 
frictionless soil model (Section 6.4.2), draw through O an 
aebitratryraxis, OP. Point P can be taken, for instance, as 
the point on the curve for $=40° corresponding to a=0.7 


(s2050)eeThe line OP will intersect the NGRC for #=20° and 


1036 


30° at points M and N. The coordinates of these points (a,,,, 


U...) are indicated in Table 6.14. Use these coordinates as 


ref 


reference values and normalize each NGRC in Figure 6.63 to 
each respective reference value. Replot the numerical 


results in terms of the normalized coordinates, 1 - a/a,,, 


and U/U The curves in Figure 6.63 would thus be 


ref * 


transformed into Figure 6.89. The preferred new ordinate 


variable, A, is defined as: 


fs Tlie 
Pons, Saber [6.23] 


ref 


Note that once the results of the numerical analyses 





rA=1-4 =) lake 
Gree 

for ¢=20°, 30° and 40°, are twice normalized as explained, a 
series of points are obtained and it seems possible to fit a 
Single curve through them. The twice normalized ground 
reaction curves (NNGRC) are almost coincident (Points M, N 
and P, of course, coincide), and this confirms that the 
Original NGRCs (Figure 6.63) are nearly homothetic. 

Repeating the above process to the NGRCs of the tunnel 
Springline and floor shown in Figures 6.64 and 6.65, then 
the twice normalized points of the ground reaction curves 
shown in Figures 6.90 and 6.91, respectively, are obtained. 
The reference values (points M, N, P in Figures 6.64 and 
6.65) used for normalizing the numerical resullts are also 
shown in) Table 6.14. Note that themratiosnain/Uchashown 
represent the slope of the arbitrary axes OP used in the 
reduction process. AS was seen for the tunnel crown, the 


twice normalized points of the springline and floor seem 


arranged in such a way that a single curve can be well 


1037 


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1039 


fitted through them. Some points appear to deviate slightly 
Preomatnestitted curve, but, for all practical purposes it 
can be said that all NGRC at different points of the tunnel 
contour for different friction angles, seem to be very close 
to homothetic, as single (but distinct) normalized NGRCs can 
be found for each of these points. 

The next step undertaken was to define what function 
Fits best the points shown in the last three figures. 
Hyperbolae such as were used to fit the data from the 
frictionless soil model (equation 6.16) did not yield good 
results. The main reason for this was that this function 
could not accommodate the apparent inflexion noted in the 
early part of the curve for the floor (See Figure 6.91) and 
to a lesser extent, for the crown. It has been mentioned in 
other sections in this chapter that, when K, is smaller than 
unity, elements of the ground at the crown and floor exhibit 
a stiffening response, resulting from the stress changes 
that occur at those points (an increase in the minor 
principal stress). Once the direction of the principal 
stresses are fully changed, the ground response is again 
that of softening, as for the springline. A hyperbola could 
not, therefore, fit this response, as its derivative 
function is monotonic and does not show a change in sign in 
its second derivative. 

On the other hand, it seemed reasonable to assume that 
the suitable function should have a definable limit, which 


is approached monotonically, as in the case of a hyperbola. 


1040 





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In other words, it seems reasonable to assume that a 
collapse radial stress exists at each point of the tunnel 
contour, which is approached asymptotically upon large 
displacements. 

Moreover, it was decided to select a single function 
type, that would fit the normalized numerical results found 
at the three points of the tunnel contour in all of the 
cases covered in the parametric analyses. 

Six types of functions were thoroughly investigated and 


tested to best fit the requirements. Making a/a,.,,=y and 


ref 


U/U..,=x, these functions were the following: 


x 


a) S x/(P, +5Po xX 4 Pre 4 


P, +P, KEES 2 


b) Wes x/(P, +) Pox <P se $s 


c).y = x/(P, + B5x + P3(P,tx)* es”) 


= x/(P, + Pox +e (el / (PS SP, x) 9 


aiey 
ejey = x/(P, +-P.x@ey (1/(Bs+baxt Pex? +h x) 0) 
fay = x/(P, + P2x 4 (1/7(BstPa xt Pix?) )) 


To test these functions, a non-linear least-squares 
regression technique was used, where the unknown parameters, 
P., were allowed to vary, to establish a 'best-fit' through 
the known pair of data (x,y). This was accomplished using 
the computer program BMDP3R, Non-linear Regression (Revised 
Version,) October, 1983), which Mie nar wm of Malet accu moa 
package developed at the University of California. Besides 
the data set with the (x,y) pairs, the program requires the 
function to be described by FORTRAN statements as well as 


the function partial derivatives of the function with 


1043 


veanecrero the parameters, P.. 

Some numerical difficulties were found for certain sets 
of data (especially for the floor), when applying the 
program to those functions including exponential terms, a, b 
and c (floating point overflows and exponential errors). The 
investigations therefore were concentrated on functions d, e 
and f above. Visual inspection of the fitted curves on the 
numerical data revealed that function f was the best fit for 
all sets of data tested (basically all cases analysed for 
H7D=1..5). 

Besides the visual inspection, a comparative analysis 
Ore thesquality, of the fitting for the different functions 
can be made through the residual mean square calculated by 


the program (Torgerson, 1986). This index is defined as: 


RMS EResidualesumvofl ’Squaress @2(y —"q)% 
~ Degrees of Freedom (N-p) 


where y is the observed value of the dependent variable, g 
is the evaluation of the function, N is the total number of 
samples (x,y) and p is the total number of independent 
parameters (P,) in the function. The function fitting the 
data best would furnish the lowest RMS value. Through this 
Criterion it was confirmed that function f provided the best 
fitting of the numerical results. Accordingly function f was 
selected and used to fit all normalized numerical data. This 
function expresses the twice normalized ground reaction 
curve as: 

he 1-2— = 1-- 


ref 





1044 


U7 eet 


Py +P2U/U..¢t 1/(P3tPuU/U,o,tPs (U/U,e,) ? 
The parameters P, (i=1,5)*tor the crown, springlime™ and 





[6.24] 
floor of a tunnel with H/D=3, and K,=0.8 are shown in Table 
6.14, and the equation of the best fitted curves for these 
points are reproduced in Figures 6.189 to’ 6.917 

The above procedure was repeated to all cases analysed 
and the results of these reduction process are summarized in 
Tables 6. 15," Of 10" ale Onis LOU shee orm Ole BanC ail .0) 
respectively. In all cases ,the RMS varied between 10°>* to 
10-5. All normalized numerical results and corresponding 
best fitted curves are included in Appendix D. There is not 
a rererence value for RMS that would allow an assessment of 
the goodness of fit in absolute terms. The RMS is usually 
interpreted in relative terms, for comparitive purposes. 
However, one can appreciate that quite satisfactory fitting 
was obtained in all cases. 

At this stage, one notes that the equations for the 
NNGRC's have been derived for each point of the tunnel 
contour, for each H/D and K,, and are independent of the 
friction angle. The dependence on ¢ is represented by the 
different a.., or U,., values associated with each friction 
aa But this dependence can be expressed separately. The 
relationships between the slope of the arbitrary axes 
a..,/U,., used in the transformation process, and the depth 
ratio H/D (note that this slope is constant for any ¢ value) 


can be expressed separately. 


1045 


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1048 


The same development applied to the frictionless soil model 
(Section 6.4.2) for assessing the changes in the ground 
stiffness with the reduction of the in situ stresses can be 
applied here. The partial derivative of A with respect to 
U/U.-,(A') can be relatédWtoma “Curent, gground Spring 
constant (k,), that could be used in ring-and-spring models 
to analyse the ground-lining interaction developing once the 
Support is installed, after a certain amount of stress 
release. Recalling equation 6.23 and the definitions of the 


stress ratio, L=0o,/o and of the dimensionless 


ro! 


displacement, U=u,E,,;/o,,-D, it can be demonstrated that: 


AX oO ——_ 
5U/U rer Geet Ei bu, 
Since 6a,/éu, is equal to k,, then 
a. E 
= ees oe [6.25] 


ref 


On the other hand, A' can be obtained by 


differentiating equation 6.24 


























6A U U US 
1 eee ao +1/(P3+P +P Ges 
5U/U rer ret / s “Ulet sy 
U U U U? 
-[P,+P ad WAGED a mpm ele & 1X 
Uret ; “Lee i MET “yes 
[P2-(Pyt2P.U ). (P,+Pa—+ Pg eae [6.26] 
ref ret ref 


in which the negative sign of the derivative has been 
dropped. 

Knowing the parameters, P, (i=1,5) and the reference 
values a.., and U,,, (Tables 6.15 €o°6y17)p tei sceroce: bu emer. 
define, using equation 6.24 the amount of stress release a 


associated with a certain dimensionless displacement U. 


1049 


Moreover, with equations 6.25 and 6.26, the radial spring 
constant corresponding to U, can also be determined. 

ay: instead, a ring-and-plate model is to be used for 
the ground-lining interaction analysis, one would need to 
know the current tangent modulus of the ground (E,), rather 
than the spring constant. If it is assumed that this modulus 
is directly proportional to the spring constant (or the 


slope of the GRC), then: 


E. K. 
Bei ikgf 
and, from equation 6.25: 
EE -A_z, [65027 ] 
r Dt 


remembering that a,.,,/U Enguehd~D arerconstant{ fonga 


retry Mei 
particular tunnel situation. Note that A', is the derivative 
Cha N robe Zero Stress release at the opening. 

Distinct current moduli (or current spring constants) 
would be found for the crown, springline and floor. The 
available ring-and-plate models operate with a constant 
ground modulus. Hence an operational modulus could be 
considered by averaging the moduli at those points of the 
tunnel contour (the springline modulus being counted twice). 


This modulus would then be (for a uniform E,, ground 


profile): 


Pe 
Se es en 


1 


E [6.28] 
The value of the partial derivative of A for zero 
stress release, \',, is obtained by setting U=0 in equation 

6.26. It represents the initial slope of the NNGRC and 


therefore is related to the in situ tangent modulus of the 


1050 


ground, 4Thus: 
ws. ayes [6.29] 

The equations furnished above, together with the data 
Summarized in Tables 6.15 to 6.17, represent the proposed 
solution for obtaining the ground reaction curves or stress 
release functions for three points of the tunnel contour of 
a shallow tunnel, in a soil represented by the cohesionless 
model as indicated by equation 6.11. Although these results 
could be easily programmed for calculations, it seems 
convenient to have them presented in form of charts. 
Accordingly, Figures 6.92 to 6.99 were prepared and 
represent the complete solution for K,=0.6. This sequence of 
charts is described below. 

Figure 6,92"furnmisnes@the ratio’ Of @.../U_., OF 
(AS 2es/ Ul = a function of the relative depth of the 
tunnel. It was mentioned earlier that this ratio does not 
depend on @. AS it is noted in Table 6.15, the reference 
values for the amount of stress release, a,,, or (1-Z,.,), 
were selected for the reduction process in such a way as to 
make them independent of H/D. They resulted in depending on 
the friction angle ¢ only. Figure 6.93 shows how that 
reference Sa eaees with ¢, expressed here through the 


function: 


be, elt SANDE a fe i2singd 
a '="=sing i? 1-sing [6.30] 


Note that (m-1) is the ratio of stress difference to minor 
principal stress at failure that appears in equation 6.2. 


With Figure 6092 and®6.93,; “onencanaGeteUr etorek»=040 pang 


1051 





(1-S!Gref) /Uref 





Legend 


Crown 


Springline 


Floor 





Figure 6.92 Slope of the Arbitrary Axes used to Obtain the 


NNGRC, Represented as a Function of H/D, for K,=0.6 


1052 








Legend 


° Crown 


S  Springline 


+ Floor 





Figure 6.93 Relationships between a.,, and the Friction Angle 


for K,=0.6 


1053 





c 
c 
6.6 F 
0 ‘Z 
c 
O° 2 S 
S hed ° 
3 SS 
=) =, ss 
E oS. 
ig “Sos 
—1.5 as 
ag 
<< Pao 
—2 ee 
=Z.5 
nae 
ee Ged 1 Lo 2 22.0: 5S. O19 @4° (4.95055 
U/Uref 





Figure 6.94 Twice Normalized Ground Reaction Curves for 


Tunnel Crown Calculated for K,=0.6 


1054 


Lambda 





O £05 Ml eto @2 92.5) ses Some me mS 
U/Uref 





% U/Ure¢ max = 7.89 


Figure 6.95 Twice Normalized Ground Reaction Curves for the 


Tunnel Springline, Calculated for K,=0.6 


1035 


LTT JS Jig (LS a ae | 





Lambda 






Legend 
Oo H/O0=1.5 

Baie) 3-0" 
HZDS6: 0" 













Figure 6.96 Twice Normalized Ground Reaction Curves for the 


TOnneser Toor; Caiculated for K.=0.6 


1056 


Deriv. Lambda 





H/D=1.5 


H/D=3.0_ 
H/D=6.0_ 





Figure 6.97 Variations of" N'for=the Tunnel crown) 


Calculated for K,=0.6 


1057 


‘ae T 


un 
a 











Oo 
Io} QO) 
€ 
S L 
aa 1 = 
> © 
ie L 
seb) 
ras 
0.5 ne 
) ape 
0 q 
OF* 5020 1 =) 2° @6205° ©F3* e335" 4 4.598 5 


U/Uret 





* U/Urefmax = 7.89 


Figure 6.98 Variations of A' for the Tunnel Springline, 


Calculated for K,=0.6 


1058 





125 
2) 
ne) 
QO 
= 
2) 
zl Pa 
= 
: lenders 
(=) face 
0» 
0 
0 O55 Si S185. 22 “2 3. eo ee eee 





Figure 6.99 Variations of X' for the Tunnel Floor, 


Calculated for K,=0.6 


15.9 


for any H/D and $ covered by the parametric analyses. 
Figures 6.94 to 6.96 present the twice normalized 
ground reaction curves (NNGRC) for K,=0.6, calculated for 
the tunnel crown, springline and floor, respectively. Note 
that in each of these figures, the NNGRCs for distinct 
ratios of H/D have been plotted together. With the exception 
of the crown, the NNGRC's virtually coincide. This tends to 
indicate that the effect of H/D is more pronounced at the 
crown than elsewhere. By understanding this, it seems quite 
feasible to interpolate data between the curves and to 
estimate NNGRCs for tunnel depth ratios other that those 
considered in the parametric analyses. Figure 6.95 indicates 
that for K,=0.6, failure at the springline is approached 


more rapidly for U/U increasing than at the other two 


nee 
points of the tunnel contour. The short vertical lines 
intersecting the NNGRCs in the last three figures delimit 
the regions where numerical data were available and through 
which the curves were fitted. The portions of the curves to 
the right of these regions represent extrapolations of the 
numerical results. The validity of the functions beyond the 
region delimited can not be assured. Although the assumption 
that the fitting functions have always an asymptote and that 
a good fitting of the numerical results was achieved, there 
is not the assurance that the extrapolated results can 
correctly portray the ground behaviour upon large 


displacements. This subject will be discussed further in the 


following section. 


1060 


Figures 6.97 to 6.99 represent equation 6.26 and they 
provide relationships between the first partial derivative 


OL kana UU 


ref/ 


(A'), that are needed to assess either the 
current spring constants through equation 6.25 or the 

current tangent moduli of the ground through equation 6.27. 
Note that the values of X', (equation 6.29) can be obtained 


1 


from these curves by setting U/U equal to zero. The last 


ref 
three figures were plotted like the NNGRCs and the same 
comments made regarding the latter curves are also 
applicable here. The initial rises in the A' curves for the 
crown and floor reflect the 'stiffening' response of the 
ground noted during the early part of the unloading process. 

A similar sequence of charts were prepared for K,=0.8 
(Figures 6.100 to 6.107) and for K,=1.0 (Figures 6.108 to 
6.115). The suite of plots just described represent the 
proposed generalized solution (equation 6.11) that can be 
used to estimate both the amount of stress release and the 
stiffness of the ground after a certain closure of the 
opening. The solution was formulated in a discrete form for 
H/D and K, and this may require interpolation of data for 
intermediate depth ratios and in situ stress ratio. For 
instance, this could be done graphically, through the charts 
given. Figure 6.116, for example, presents a chart for 
obtaining the initial slope oft) the NNGRC Ns tor 


intermediate in situ stress ratios or depth ratios. 


1061 


(1-SlGref) /Uref 








Legend 


Crown 


Springline 


Floor 





Figure 6.100 Slopes of the Arbitrary Axes used to Obtain the 


NNGRC Represented as a Function of H/D, for K,=0.8 


1062 








Legend 


fe) Crown 


Od Springline 


+ Floor 





Figure 6.101 Relationships between a.,, and the Friction 


Angle of K,=0.8 


1063 





Lambda 


| 
N 
EUR PUPIL Ser 





oa) 
=e, 
OF ,075 =! Ino’ @2 g275' ¢3° gon pA 4:5" 3S 
U/Uref 

Legend 
Oo H/D=1.5 
& H/0=3.0_ 
OE ab le lhe 








Figure 6.102 NNGRCs for Tunnel Crown, Calculated for K,=0.8 


1064 


Lambda 





O«fosS “1 "5 92 “Sts [3s Sa¥5 ¥4 m 455005 
U/Uref 





Figure 6.103 NNGRCs for Tunnel Springline, Calculated for 
K,=0.8 


1065 









E 

E 

oa 70-2 E 
a 

eo! : 

io} |p! 
= 





Faigucea 6ei0¢eNNGRCs (for Tunnel Floor, sCalculatedsfor K;=0.8 


1066 


Deriv. Lambda 





Figure 6.105 Variations of X\' for the Tunnel Crown, 


Calculated for K,=0.8 


1067 





Deriv. Lambda 





Oren? 5 ee OS a5 ms 4.5 5 
U/Uref 








Riguce 62106 Variations "of NY for “the Tunnel “Springline, 


Calculated efor Kk 5=0 18 


1068 


N 





—s 
Nn 
a Weft So cl ls os We te 





Deriv. Lambda 


oO 
On 
UA Ae Pag ae ALS 





OUT al PO, 2 2,0) go - O70 we 4a on mS 


U/Uref 





Figure 6.107 Variations Of A’ tor tne funneler moore 


Calculated for Ki=0.8 


1069 


(1-SlGref)/Uref 








Legend 


fe) Crown 









4  Springline 


Floor 





Figure 6.108 Slopes of the Arbitrary Axes used to Obtain the 


NNGRC Presented as a Function of H/D, for K,=1.0 


1070 





Legend 


Crown 






Springline 






Floor 


Figure 6.109 Relationships between a.,, and the Friction 


“ANG Le fore Ke =i) 


Lambda 
| 


Legend 
oO H/D=1.5 


H/D=3.0° 
B70 Ss S207 





1071 





Figure 6.110 NNGRCs for Tunnel Crown, Calculated for K,=1.0 


WO72 


Lambda 





oO Sos 1) Sits 2 SZ sk) 5 eS) ecm 
U/Uref 





Lambda 


1073 





Oo gl Tao? gt geod gone ecu) gd. 4.5m 5 


U/Uref 







Legend 
Oo H/D=1.5 
a oles, 
H/0=6.0_ 





Figure 6.112 NNGRCs for Tunnel Floor, Calculated for K,=1.0 


1074 


Deriv. Lambda 





0. “Ors: “yrs, 2 Noes. Bs: tS eee eee 
U/Uref 


Legend 
H/D=1.5 


H/D=3.0_ 
H/0=6.0_ 





Figure 6.113 Variations of A" for the Tunnel Crown, 


Calculated for K,=1.0 


1075 


2 

17-9 
2) 
aC) 
= I 
ii - 
o r 
a 

UES s ia 








Fagure 6.914 Variations of A‘. for) the Tunnel, Springline, 


Calculated for K,=1.0 


1076 


Deriv. Lambda 








0 Sots “| Sits 42) Sos. © 3 Bere ee 
U/Uref 





Figure 6. 115" Variations of x wiror tne Tunnel Floor, 


Calculated for K,=1.0 


O77 





Figure 616 Varrations of the Initial Slope of the Twice 


Normalized Ground Reaction Curves with K, and H/D 


1078 


6.5 Limits of the Generalized Solution 


6.5.1 Extrapolation of the Numerical Results 

It waS pointed out earlier that, for the generalization 
of the parametric analyses results in terms of ground 
reaction curves, all numerical output after local failure 
had developed was disregarded (Section 6.3.2). The reasons 
behind this decision were: a) the numerical model used was 
not designed to represent this type of behaviour (for 
instance, the failure criteria can be violated); b) the GRC. 
resulting after local failure of an element or group of 
elements to which a low tangent modulus is assigned ceases 
to be homothetic; c) the generalized solution was not 
intended to cover responses approaching soil failure or 
ground collapse. 

In order to achieve the projected generalization and to 
find out the functions reprsenting the GRC for the 
particular soil models investigated (e.g., equations 6.11 
and 6.12), some curve fitting techniques were used. The 
NGRCs were found to be described by equations such as 6.17 
and 6.24, obtained by applying non-linear regression 
techniques to the numerical results of the parametric 
analyses. 

The numerically derived functions that resulted were 
unbounded and they therefore, may be used to interpolate 
data between the discrete numerical results of the 


parametric analyses, and also they may be used for the 


1079 


extrapolation of these results beyond the range of the data 
investigated. Although the latter use of these functions was 
not. initially envisaged, it was found: that, for practical 
use and under certain circumstances, some limited 
extrapolation of data could be required. This may happen 
when the resulting calculated GRC was "too short" as can be 
the case in soils with low @ and under low K,, where local 
failure is attained at small amounts of ground stress 
release. An approach to this problem could be to accept the 
onset of global collapse simultaneously with the event of 
local failure. Accordingly, this would imply a horizontal 
extension of GRCs at the points of local failure so that 
very large displacements develop for a certain amount of 
stress release. 

Clearly, wae alternative presents an inconvenience: it 
may be too conservative in terms of ground stress 
displacement (and thus loads onto the support) and in terms 
of opening closure. With this problem in mind, it was 
contemplated that the use of the fitted functions to 
extrapolate the ground reaction curves could be a more 
sensible alternative, provided the implications of this 
operation are fully recognized and caution is exerted in 
interpreting the extrapolated results. 

To illustrate this point, consider the NGRCs shown in 
EiguresG.. 06,4600 5H, U=3 gao*30 seand eK 7=0 (8. 7/Theseweurves are 
reproduced in Figure 6.117 by the solid lines OR. 


R represents the end points of the curves shown in Figure 


1080 


6.58 and correspond to the numerical results obtained in the 
last unload increment, prior to the load step at which local 
failure developed (at an element next to the springline in 
this particular case). If the NGRCs are extended beyond R, 
using equation 6.24 and the corresponding parameters found 
in Table 6.16, the dashed curves shown in Figure 6.117 are 
obtained. One notes that the extrapolated portions of the 
NGRCs do not present any significant discontinuity at point 
RY, 

By now examining Figure 6.64, we identify point R of 
the springline NGRC for $=30°, as the end point of this 
curve for Z=0.4. This point reappears in the NNGRC shown in 
Figures6590 alsovas, R.gNotewrhat) in°thestatter facures” co 
the right of R there exists another point (T), which is the 
end point of the NGRC for ¢=20° (Figure 6.64). By virtue of 
the homothety, to this last point there exists a 
corresponding homothetic point (T') in the NGRC for 9=30° 
which coincides with T in the twice normalized ground 
reaction curve (Figure 6.90). This point could not be 
numerically obtained as the unloading increment used was too 
large. In this particular case, had this increment been 
about 7% instead of the standard 10%, point T' on the NGRC 
for ¢=30° in Figure 6.64 would have been obtained 
numerically. Regardless of this, it was possible to identify 
point T' from point T in Figure 6.64, by assuming that the 
NGRCs are homothetic and Bvite ate Ad the similitude axis OT 


and defining T' as the intersection of this axis with the 


1081 


o0€=¢ 
pue g°Q="M ‘€=G/H JOJ SDOYON 944 JO UOTSUaIX| L11°9 aINbry 


n 
91 b'I 2" 01 0 . i 
| 8 90 v0 Z0 g S008 


NMOUD \ 


ANIISNIYdS 


YOO 14 





1082 


extended NGRC for 9=30°. 

If the property tof homothety cansbe accepted fer the 
NGRCs of different @ values, then the extended portion RT' 
of the curves in Figure 6.117 are fully admissible. The 
terminal points T' of the extended NGRCs are easily found 
through the ratio U/Ul;, of the extreme rigntward pointer 
shown in Figure 6.90, and indicated by the right vertical 
bar’ in all A versus U/U,., curves (Figures 6.94 to’ 6796, 
6.102 to 6.104 ande6.110 sto 6. Ii2)eeinebrver, the resuics 
given by these curves up to these points can be used with no 
restrictions. However, beyond them (to the right of points 
T' in Figure 6.117), there is not assurance that the 
relationships between stresses and displacement are correct. 
In particular, stresses giving stress ratios below Z,, may 
either violate the failure criteria or may be smaller than 
the minimum required to maintain stability of the opening. 
In other words, the stresses given by the extrapolated NGRC 
beyond T' may not ensure the equilibrium of the system. 

Unfortunately, there is no simple way to check if the 
failure criteria is being locally violated. In fact, one may 
assess the global stability of the tunnel, to evaluate how 
close to tunnel collapse one is, when uSing acting stresses 
lower than that at T'. A stability calculation may give some 
indication as to whether or not the acting stresses are high 
enough to ensure global equilibrium but it will not ensure 
that local equilibrium is attained. Neither will it 


guarantee that the failure criteria is not being violated 


1083 


locally. Consequently, the displacements associated with 
these stresses are likely to be in error, and are possibly 
smaller than the correct ones. In spite of these 
limitations, if one still needs to go beyond T' in Figure 
6.117, the least that can be done is to ensure that the 
factor of safety for the acting stresses is greater than 
unity. 

The discussion just presented referred to the 
cohesionless soil model but it applies to the frictionless 
soil model as well. The extension of the NGRCs shown in 
Figure 6.47 for H/D=3 and c,/yD=1.25 is presented in Figure 
6.118. The same notation used before is adopted herein. The 
extension beyond point R is obtained using equation 6.17 and 
the corresponding Parameters found in Table 6.13 or in 
Figures 6.85 to 6.88. Similarly the reader is referred to 
Figures 6.55 and 6.83. Note that in the latter, point T lies 
Sieeide EHERE gure. *The* positions of @points T' Sin*Frigure 
6.118 are shown in the inset. The same comments made 
regarding Figure 6.117 are valid here. The main difference 
lies in how the points T' are found for the frictionless 
soil model. As the general solution for this case was 
presented ina slightly different format, some additional 
information is needed for establishing T'. This is supplied 
By@Pigures GeutS*tors.12iewhich provadewine'values “of +U,., 
for different c,/yD and H/D values. Note that U,,, was 
deliberately chosen as to result in a linear function of the 


strength ratio. In the inset shown in each of these three 


1084 


GZ°L=a4/"9 pue €=d/H 10} SOMON 94 JO uOTSUaIX| BlIL°9 aaNbry 


nN 
Ov 9¢ ce 82 vec Oe? 91 a | 80 vO O 
05 a ae = : 3 voxe 
“abel i 2 il = ee ee = = 
TC) ae ga a eae oa co 
ea eee ae ne 
~~ ~ as 
uy 4 4 
vO 
SS) 
1) 
Q 
ie 
“N 
NMOS 904 
fe) 
ANINONINdS 
4yOo15 80 
Ol 





1085 


figures, the maximum value of U/U calculated for each H/D 


ret 
is also given. This maximum normalized dimensionless 
displacement corresponds to the terminal point T identified 
by the vertical bars shown in the NNGRCs for the 
cohesionless soil. 

Another peculiarity found in the results of some 
analyses with the frictionless soil model was that tensile 
Stresses developed in some elements around the opening, 
before the development of shear failure. This was noted 
mainly in the shallower tunnel cases with higher strength 
ratios. As explained in the introductory remarks to Section 
6.3.2, all numerical data output obtained following the 


development of any tension in the ground was disregarded. 


6.5.2 Assessment of the twoaDinen svenal Stability 

Although not recommended by the reasons exposed in the 
previous section, if one is forced to use the generalized 
solutions beyond their terminal points, an assessment of the 
two-dimensional tunnel stability is required. It should be 
stressed that this does not secure that the extrapolation is 
correct, but this evaluation can provide indications of 
whether or not the calculated stresses for a certain closure 
of the opening are sufficiently high to keep it stable. 
Though the excavation stability verification is usually 
circumscribed to the tunnel face and heading (See Section 
Ger sa ET EhiS particular case the two-dimensional 


Stability evaluation also becomes important. The subject was 


1086 


CROWN 


(U/Uref_)max 





Figure 6.119 Reference Values used for Normalizing the NGRC 
of the Tunnel Crown for the Frictionless Soil Model and 


Components to Define the Limiting U of the Fitted Function 


1087 


SPRINGLINE 


(U/ Uret )mox 





Figure 6.120 Reference Values used for Normalizing the NGRC 
of the Tunnel Springline for the Frictionless Soil Model and 


Components to Define the Limiting U of the Fitted Function 


1088 


FLOOR 


(U/Ure¢ \m ox 


Uref 





Figure 6.121 Reference Values used for Normalizing the NGRC 
of the Tunnel Floor for the Frictionless Soil Model and 


Components to Define the Limiting U of the Fitted Function 


1089 


discussed and reviewed in different sections of this thesis, 
especially in Section 4.3.4. 

The possibility of violating the boundary or failure 
conditions in limit equilibrium analyses of the ground mass 
suggests the use of plasticity solutions. The available 
plasticity solutions for estimating the two-dimensional 
Stability of a shallow tunnel are based on the Limit 
Theorems (Lower and Upper Bound). These theorems can be 
proved for materials with an associated flow rule, where the 
angle of dilation, ¥, is) equal) to the friction angle,*‘¢. 
While this assumption provides a good approximation in the 
case of a saturated soil under undrained loading, in which 
w=¢=0, in general it does not properly represent the actual 
soil behaviour at ultimate failure under drained loading, 
when usually w is not equal to 9. Davis (1968:352) showed 
that the Upper Bound Theorem can be proved to be correct 
even for this pemaue don (see also Atkinson, 1981:117). No 
complete proof of validity of the Lower Bound Theorem was 
found so far for this condition. There are some indications, 
however, that it may be valid for ¢ not equal to yw and w=0 
(for instance, Cox, 1963 - quoted by Davis, 1968:345 and 
Palmer, 1966). 

Davis (1968:346) suggests that while the plastic 
deformations may strongly depend on the flow rule, the 
collapse loads may not. Atkinson (1981:118) seems to share 
this opinion and suggests that ates approximate lower bounds 


calculated for a soil with Ww not equal to ¢ by assuming w= 


1090 


are unlikely to exceed the true collapse loads. 

An additional aspect of the plasticity solutions is 
that a unique collapse load is steadily approached, and 
remains constant at its maximum once collapse is fully 
developed. This can be proved to hold (Davis, Op.cit.:346) 
on the condition that the geometry of the problem is not 
changed significantly. Regarding the tunnel problem, this 
implies that a unique collapse load can be found provided 
‘loosening' effects and associated changes in the geometry 
of the problem does not occur (see Section 3.2.3). In other 
words, plasticity solutions could be used to approach the 
stresses associated with point E in Figure 3.8. Beyond it, 
the changes in geometry would have to be considered in order 
to investigate the loads associated with point G in Figure 
3.8, which lies on the speculated sweeping upward ground 
reaction curve as discussed in Section 3.2.3. If the 
Grepiecenarcs are limited as in gocd ground control 
tunnelling conditions, which are usually required in urban 
environments, it will be sufficient to investigate the 
ground stability on the assumption of uniqueness of the 
DiesclcuryesOLut lon. 

The available plasticity solutions to the 
two-dimensional shallow tunnel problem usually represent the 
internal stress acting on the circular opening as a uniform 
pressure which, at ground ‘collapse, is denoted by p,. An 
exception to this rule are solutions that assume the cavity 


being filled by a heavy and non-permeating fluid with a 


1e3,1 


density equal to the soil density (for instance Caquot, 
1934:81, or D'Escatha and Mandel, 1974). In both cases, it 
is noted that the conditions assume@ do not correspond 
exactly to the loading conditions assumed in the 
two-dimensional numerical modelling adopted in the present 
study. As described in Section 5.2.2, in any stage of the 
unloading process, the tunnel contour is subjected to 
stresses that correspond to a fraction Z=1-a of the in situ 
stresses, which can be calculated through the expressions 
given in Figure 2.14. One notes that while in the plasticity 
solutions the opening walls are subjected only to normal 
(radial) stresses, the numerical modelling adopted herein 
assumes that both normal and shear stresses act on the 
opening contour, which are equal to Zo., and =r, 
respectively; hwithso;, andi, given pneBigured2.14,ehvengtor 
K=1 (i.e., r=0), the conditions in the numerical model do 
not fully correspond to that in the plasticity solution, 
where the tunnel is assumed to be filled with a fluid with 
density equal to y. The latter implies the existance of a 
pressure gradient across the tunnel, from crown to floor, 
equal to y, whereas this gradient is equal to Ly in the 
numerical model adopted (with = smaller than unity) « 

Even when working in terms of stresses averaged around 
the tunnel contour, there will always be a non-zero residual 
shear stress acting on the shallow tunnel walls, whose 
effect on the limiting tunnel pressure has not been fully 


assessed. Besides affecting local failure, these shear 


1092 


stresses may affect the global collapse condition and 
therefore the limiting tunnel load: For K»notscequalanco yy, 
they may lead to an increase in p,. For K=1, p, may also 
increase since Ly is less than y. Increased limiting tunnel 
pressure implies a smaller factor of safety. defined as (see 
Chapter 2): 

i hows 1] 


SP hes 


where o, is the in situ verticalsstress, ate themtunneldy ais 
and o, “is "the current internal Stress insthe st unne leeeAcmthe 
latter, in the numerical model, varies from point to point 
of the contour, it seems convenient to define o, as the 
current average radial stress acting on the tunnel profile. 
Thus: 

C= Le [6.32] 
where o,, is the average in situ radial stress, which is 
given by: 


1+K 


coe pert + 2) forse 


Ql 


Therefore: 


Ware eel.) 42) Glek:) /litKe ate, 
PS = T= EU1+K,) 72) = G=K)/Ui+K,) + a [6.34] 


with Z, equal to the ratio p,/@,. and, @,2lez,\; 

It should be remembered that the collapse pressure 
given by plasticity solutions is independent of the initial 
stresses acting in the ground before the tunnel is built and 
also independent of the sequence in which the stress release 


is applied to the opening, provided it is steadily increased 


109s 


during unloading (Davis, 1968:346,347, D'Escatha and Mandel, 
1974:46). Hence p, does not depend on K,, although the local 
failure in the ground during the unloading process does 
depend on K,. Note that FS expressed by equation 6.31 or 
6.24515 dependent on Ki, mot through. p,/but through the 
current average radial stress acting on the opening contour. 
Note, moreover, that this definition of FS is approximate 
Since the effect of the shear stresses acting on the contour 
was disregarded. 

Safe estimates of p, can possibly be made through 
solutions developed from applications of the method of 
characteristics. As it is based on a statically admissible 
stress field not viclating the failure criteria, the 
Solution it provides is a lower bound (theoretically safe) 
estimate of the—-exact Solution. In other words, the p, value 
estimated is higher than (or equal to) that actually causing 
the tunnel collapse. On the other hand, an upper bound 
(unsafe) solution will provide an estimate of p, which is 
lower than (or equal to) the correct value. 

If the flow rule is associated, with w=¢, then the 
stress characteristics line coincide with the velocity 
characteristics (or slip lines). Therefore one can assume 
that, at collapse, the soil strength is fully mobilized 
along the slip lines and the safe collapse load can be 
estimated through the method of characteristics (see, for 


example, Wu, 1966:215). 


1094 


For a purely cohesive soil, D'Escatha and Mandel (1974) 
and Seneviratne (1979:79) (reproduced in Davis et.al., 1980) 
presented lower bound solutions for the two-dimensional 
shallow tunnel problem which are shown together in Figure 
6.122. The solutions by these authors are intrinsically the 
Same and assume that the opening is supported by a uniform 
pressure (y,=0). They differ slightly (see curves for 
¢./yo=0.25 and 1.00)" because ™Senevirarness cCharactem st1c 
net was less refined than D'Escatha and Mandel's. The tunnel 
contour was divided into 16 equal segments in the former and 
into 60 in the latter. The D'Escatha and Mandel solution is 
therefore numerically more accurate. However, it does not 
provide solutions for the intermediate range of strength 
ratios which is of considerable practical interest and for 
this reason, both solutions are reproduced in Figure 6.122. 

Aso plotted in Figure 6. |22eare thesresul.sportmoome 
two-dimensional centrifuge model tests (Series I) by Mair 
(19793121). It can be appreciated that the interpolated 
plasticity solution provides a very close estimate of the 
collapse tunnel (internal) pressures observed in the tunnel 
model tests. 

For a cohesive and frictional soil, D'Escatha and 
Mandel (1974) also presented 'safe' estimates of the 
collapse tunnel pressure, once more assuming wW=$. These are 
given by the curves shown in Figures 6.123 to 6.126, for @¢ 
equal to 10° to 40° and different cohesion, c, expressed in 


terms of the ratio c/yD. Again, this solution assumes that 


16 
0' ESCATHA ANO MANDEL (1974) 
Cyu/YO = 0.25 
— — — DAVIS ET AL (1980) 
a) CENTRIFUGE TESTS BY MAIR (1979), 
Cy/YO = 0.38 
es ca elmen D' ESCATHA ANO MANDEL 
SOLUTION INTERPOLATED 
12 FOR Cy/YO*0.38 
Lors 
/ 
a 
° va 
e WA 0.38 ( INTERP) 
~e vs 4 
an vA ge 
y/ pars 7 0.50 
Y yi a“ 
4 Y 7 og 
7 
i ial ae, 
: Za 
oo eA 
MA a 
a oz t 
af | ee 
wee 
seta 
0 See 
5.00 
-4 
(3) | 2 3 4 5 6 
H/O0 


Figure 6.122 Lower Bound Solution for Tunnel Collapse 


Pressure ina Frictionless Soil 





1095 


1096 


the tunnel walls are submitted to a uniform pressure, p,. 

An upper bound solution for local collapse of a wedge 
of cohesionless soil at the tunnel crown (See Figure 3.7) 
can easily be obtained by a work calculation assuming wW=$¢. 
This solution, which gives a tunnel pressure which is lower 
than (or equal to) the actual internal pressure at collapse 
was first proposed by Atkinson et.al., (1975:84) and is 
given by: 

Sets Iggeet [6.35] 

provided that the sides of the wedge do not intersect the 


ground surface. In other words, provided 


initrd 
D sing 





=" [6.36] 
Values of the p,/yD given by equation 6.35 and the 
restriction given by 6.36 are shown in Figures 6.123 to 
6.126 by chaindotted lines. It can be observed that for any 
@ this Upper Bound solution always provides a p, value lower 
than that given by the lower bound solution (for c/yD=0). 

If the above solutions are compared to results of 
tunnel model tests carried out under drained conditions, 
mixed results are obtained. The lower and upper bound 
solutions are found to bracket the loads associated with 
local collapse as observed in static tests in sand by 
Atkinson et.al., (1975:84). However, under some 
circumstances they do not approximate the collapse pressures 
found by Seneviratne (1979:64) in drained static tests in 
normally consolidated kaolin. The lower bound solution given 


was found to underpredict the observed collapse pressure of 


i ORS 


p|/gama.D 








LOWER BOUND SOLUTION 
BY D'ESCATHA AND 
MANDEL (1974) 


——='=— UPPER BOUND SOLUTION 
BY ATKINSON ET AL(1975) 


Figure 6.123 Lower and Upper Bound Solutions for Tunnel 


Collapse Pressu re in a Frictional and Cohesive Soil (¢=10°) 


OQ 
E 
=. 0.5- 
pas 
@s 
0 
-0.5 
0 


1098 


PHI=20 
5 10 15 20 25 
1.5 
C/YD=0 
0.1820 
0.5 
~Levatc=0) 
0 
0.4$14 
. ? 0.5 
5 10 15 20 26 
H/D 





LOWER BOUND SOLUTION 
BY D'ESCATHA AND 
MANDEL (1974) 


ao) aU rene SOU NDE SOLU ION 
BY ATKINSON ET AL (1975) 


Figure 6.124 Lower and Upper Bound Solutions for Tunnel 


Collapse Pressure in a Frictional and Cohesive Soil (9=20°) 


1099 





EniSoe 
0 = 5 10 15 20 25) 
0.90 0.90 
0.75 10.75 
ie 
0.60 . + 0.60 
a 
So 0.45 L 0.45 
e Ls 
fs) i 
OD 0.30 OSS + 0.30 
ic SSS 3 t 
| 0.1443 
0 0 
0.2174 
-0.15 en 
0 o 10 ie Z0 Ps) 





LOWER BOUND SOLUTION 
BY D'ESCATHA AND 
MANDEL (1974) 


—-— UPPER BOUND SOLUTION 
BY ATKINSON ET AL (1975) 


Figure 6.125 Lower and Upper Bound Solutions for Tunnel 


Collapse Pressure in a Frictional and Cohesive Soil (9=30°) 


eeles4bG 
0 5 10 15 20 
0.20 
0.15 C/YD=0 
i lbagenee di ke 
CQ 0.10 
o} : 
fe 
D> 
> 0.05 
Ls 0.1259 
0.00 
0.140] 
-0.05 
0 5 10 15 20 
H/D 





1100 


15) 
0520 


70315 


0.10 


A iD T Noa ra ee 





0.05 


0.00 


“= 0705 
Vis) 


LOWER BOUND SOLUTION 
BY D'ESCATHA AND 
MANDEL (1974) 


—-— UPPER BOUND SOLUTION 


BY ATKINSON ET AL (1975) 


Figure 6.126 Lower and Upper Bound Solutions for Tunnel 


Collapse Pressure in a Frictional and Cohesive Soil (=40°) 


1101 


these tests conducted with small cover to diameter ratios. 
This may be an indication that, in fact, the solution does 
not provide ‘safe’ estimates of the collapse load for 
materials with a non-associated flow rule behaviour. Another 
interpretation would be that the deformations in the soil 
prior to collapse do have an effect on the collapse pressure 
(see Section 4.3.4). Nevertheless, whatever the reasons are, 
these findings suggest that caution should be always 
excercised when using the above solutions, and it is 
considered good practice to allow as much safety as possible 
or preferably to avoid any extrapolation of the NGRCs. 

If the above lower bound solutions are applied to the 
conditions assumed in the analyses that led to the results 
shown in Figure 6.117 and 6.118, the corresponding p, and 
Z,=p,/o,, are found. These are shown by the chaindotted lines 
also plotted in these two figures. Those points in the NGRCs 
below the Z=Z, line would thus imply stresses acting on the 
opening smaller than those needed to keep it ina state of 
imi equiaaorrum., Noterin=Sigurer6. tijethatythe 
dimensionless displacement, U, at the tunnel crown 
corresponding to Z=Z, is about U=1.5, which is not 
dissimilar to the limiting value of U=1.8 suggested in 
Section 2.3.4.3 for a near collapse situation. Note also 
that in Figure 6.118 the NGRCs are not intersected by the 
C= ine ana thus tieyea Ways Give stréssés which “ane 
higher than those at collapse. This feature is observed for 


some situations in the generalized solution for the 


1102 


frictionless soil model, but only when R,=1. When the 
failure ratio is smaller than unity, the NGRC becomes 


steeper and the Z=Z, line may intersect them. 


6.6 Summary and Conclusions 

The main objective of this chapter was to develop a 
procedure that would allow the ground reaction or stress 
release curves of points at the contour of a shallow tunnel 
to be obtained without the need of finite element analyses. 
In parallel with this development, it was also attempted to 
obtain the relevant ground displacements associated with the 
reduction of the in situ stresses, with the tunnel 
construction being represented by a two-dimensional 
Simulation. 

To achieve these aims, some additional simplifications 
were introduced into the 2D finite element model presented 
in Chapter 5. These simplifications led to the establishment 
of two stress-strain models: the frictionless soil model and 
the cohesionless soil model. The first would represent the 
behaviour of a saturated soil under undrained conditions, 
whereas the second would simulate the drained behaviour of 
frictional soil without cohesion. Both models originated 
from the hyperbolic model presented in Section 5.2.2.1. The 
first was obtained by setting ¢=0, K,=1 and by making the 
Janbu's exponent n equal to zero. The second was obtained by 
setting both the cohesion, c, and the exponent, n, equal to 


zero. With these simplifications, it was found that the 


1103 


resulting stress-strain curves for these two groups of soil 
present the property of homothety. This property allows the 
responses of homothetic circular tunnels (tunnels with the 
Same cover to diameter ratio) to be normalized and become 
represented by a unique response. Unique normalized ground 
reaction curves (NGRC) are thus found at corresponding 
points on the contour of homothetic tunnels. Furthermore, 
unique normalized distributions of ground settlements at the 
surface and in the subsurface are also found. These findings 
facilitate the generalization of the numerical results since 
they become independent of the scale of the problem. 

It was shown that the normalized responses calculated 
for these models are invariant with regard to the unit 
weight of the soil. It was also demonstrated that the 
normalized responses calculated for a certain failure ratio 
R, are eaSily extended to different R, values. This enables 
the projected solutions to be developed for a fixed R, value 
equalstotinity : VForeReidifferenttftrom unity) werther ithe 


undrained strength, c orethesiraction angle, would, be 


ul 


transformed into an equivalent strength parameter (c,, or ¢,) 


ue 
using equations 6.3 and 6.7. By implementing the equivalent 
strength parameters an identical solution is obtained. 

It was demonstrated that the normalized responses are 
invariant to the Janbu's modulus K. Both models set Janbu's 
modulus n equal to zero, therefore assuming that the in situ 


tangentemodulus is®constant@withidepth.*This is restrictive 


since the normalized ground response was shown to be 


1104 


sensitive to the increase of the in situ modulus with depth, 
notably at the crown and at the floor. A simple expedient 
was devised to make the NGRC at those points less sensitive 
to the modulus increase with depth. It consists of 
normalizing the displacements at the contour to the in situ 
stiffness of the ground at points located a half diameter 
radially away from the tunnel. This artifice provided 
favourable results and one can approximate the increase of 
the in situ modulus with depth in both soil models. 

The use of the cohesionless soil model for soils in 
which the cohesive component of strength is not zero was 
investigated. The normalized ground response was found, of 
course, to depend on the soil cohesion. By neglecting its 
influence an unduly conservative ground response may be 
obtained. To bring the results of the cohesionless soil 
model closer to the correct one, another approximation was 
proposed and provided reasonable, yet safe, results. It 
consists of increasing the friction angle in the 
cohesionless soil model, so that the in situ strength at the 
Springline elevation is equal to the actual strength, 
defined in terms of the principal stress difference at 
failure. The adjusted friction angle to be used in the c=0 
model is given by equation 6.10 and is found to depend on 
the actual c and ¢ values and on the in situ minor principal 
stress at the tunnel axis elevation. 

In order to further reduce the number of variables, the 


effect of Poisson's ratio on the normalized ground response 


L105 


was investigated. For the frictionless model, the assumption 
Siiaaratlouglose 0.5 was justified as representing the 
behaviour of a saturated soil under undrained conditions. 
For the cohesionless model, the ground response for two 
typical values of Poisson's ratio was studied, which lead to 
the adoption of a Poisson's ratio of 0.4. 

After the introduction of the above simplifications, 
general equations describing the NGRC for the two soil 
models were presented in terms of dimensionless and 
independent variables (equations 6.11 and 6.12). Parametric 
finite element analyses were the modelling tool used to 
establish the relationships among these variables. It was 
decided to investigate these relationships within ranges of 
variables which would include the most common situations 
found in practice. These ranges were established by 
investigating 53 case histories of shallow tunnels. Most of 
these were identified as having been built under good ground 
control conditions where collapse and pronounced time 
dependent responses were not entailed. The most typical 
ranges in situ stress ratios, of cover to diameter ratios 
and strength parameters (¢ and c,/yD) were identified. As a 
consequence of this investigation, the variables for the 
Parametric finite element analyses were finally defined as 
Ghy/ Dei Setomoped=0. ton t0e eceiyDe0ws 125 eto 205k 5..0..6 0 
a0 

Details of the 2D parametric finite element analyses 


were presented, including the number of unloading increments 


1106 


used, the assessment of its effect on the numerical results, 
the finite element mesh design and particularly the effect 
of the fixed lower boundary. The latter was found to have a 
significant effect on the numerical output, mainly because 
of the constant modulus profile assumption. Unrealistic 
numerical results in terms of the ground settlements and 
ground reaction curves were obtained depending on its 
position. In order to select the most convenient location 
for this boundary a very large number of cases were 
processed. A distance of 1.5D below floor elevation was 
selected as being a position close enough to the tunnel 
invert to minimize excessive ground heave and far enough 
away to minimize its effect on the stiffness expressed by 
the ground reaction curves. 

The generalization of the parametric analyses data was 
limited to pre-failure conditions because the numerical 
model is unable to simulate the soil failure. These results 
are included in Appendix C and D in compact graphic form, in 
terms of the distributions of normalized ground surface and 
subsurface settlements and of the NNGRCs for three points of 
the tunnel contour. 

Some of these results for both soil models were 
presented and discussed in detail. The influence of each 
dimensionless variable on the ground response was 
individually assessed. This included the effects of the 
amount of stress release, the relative tunnel depth, the 


undrained strength ratio, the friction angle and the in situ 


110% 


stress ratio. Conclusions regarding these assessments can be 
Founc@inespecifycisections e@fethis chapters Of particular 
interest was the normalized ground reaction curves for 
different strengths (either expressed in terms of c,/yD or 
@) that are homothetic with the centre of similitude at 
their origins (Z=1 and U=0). Therefore, if the NGRCs are 
normalized once more, unique twice normalized ground 
reaction curves (NNGRC) are found irrespective of the soil 
strength. This finding had a major impact on the 
generalization of the numerical modelling results, as the 
projected solution could be extended to soil strengths other 
than those used in the parametric analyses. 

Relationships between surface and crown settlements (S 
and u.) were derived from the results of the numerical 
analyses. These relations were found to depend on the 
relative tunnel depth (H/D), on the strength parameters 
(en yiitorieg Stand toni ithe sin isituystress: ratiojs;Moreover it 
was found to be a function of the amount of stress release 
allowed. The ratio, S/u,, was found to either increase or 
decrease depending on the stress relief, on K, and on the 
geometry of the problem. Results obtained by other authors 
regarding this aspect were critically assessed and 
previously proposed relationships for the settlement ratio 
S/u, were analysed. The relations proposed herein were found 
to bound field observations, and it is suggested that they 
could be used to define the maximum surface settlement above 


a shallow tunnel whenever the crown settlement is assessed. 


1108 


The results obtained were used to explain the limitations of 
linear elastic analyses which, when used to back-analyse 
actual tunnel performances, lead to either over or 
underestimates of surface settlement, whenever a match in 
the crown displacements is achieved. 

The results of the numerical modelling were further 
generalized in an attempt to define expressions relating the 
variables controlling the ground response. Such expressions 
would allow one to obtain the ground reactions for 
conditions or variables other than those considered in the 
modelling stage. It would serve as a tool to furnish the 
ground reaction curves for any tunnel size, at any 
intermediate depth, in soils described by any in situ 
stiffness or strength. This generalization process was 
undertaken in parallel for both the frictionless and the 
cohesionless soil models. 

A single expression (6.17) relating the radial stress 
ratio Z£ and the radial dimensionless displacement U, with 
two parameters (A and B), was found to be a good 
representation of the normalized ground reaction curve for 
the frictionless soil model. Parameter A was found to depend 
on the point of the tunnel contour being considered and on 
the relative tunnel depth. Parameter B was also found to 
vary around the contour but to a lesser degree, and depends 
largely on the undrained strength ratio (c,/yD) and the 
relative depth of the tunnel (H/D). The first parameter was 


Shown to reflect the in situ stiffness of the ground and the 


1109 


second, the ultimate collapse state of the ground. Charts 
were prepared, which allow A and B to be estimated for three 
points of the tunnel contour (crown, springline and floor), 
for any cover to diameter ratio or any undrained strength 

re tee, 

The partial derivative of the function expressing the 
NGRC for the frictionless soil model is easily related to 
the ground stiffness at any stage of the tunnel unloading 
process. Therefore, it can provide estimates of the 
'current' radial spring constant (k,) for any amount of 
tunnel closure (equation 6.20). These constants vary from 
point to point of the tunnel contour, and could be used as 
input parameters for a ground-lining interaction analysis 
using conventional ring-and-Spring models such as reviewed 
in Chapter 4, 

If instead, ring-and-plate models were to be used for 
the interaction analysis, the 'current' tangent Young's 
modulus (E,) could also be derived from the derivative of 
the general NGRC function (equation 6.22). 

These derivations would enable an approximation of the 
degree of ‘softening’ experienced by the ground around a 
Shallow tunnel and the amount of stress release associated 
with a certain degree of opening closure taking place before 
the lining is installed or activated. This would serve to 
improve the approximations for the actual mechanisms 
involved in the ground-lining interaction process and 


perhaps lead to better estimates of tunnelling performance. 


1110 


Even though more difficult, similar reasoning and 
development were applied to the cohesionless soil model. The 
homothetic responses of the ground in terms of the NGRC for 
different friction angles, allowed an additional 
normalization of the NGRC. The twice normalized ground 
reaction curves (NNGRC or A - curves) could then be fitted 
by a single five parameter function (equation 6.24}. Best 
fit parameters were then found through a non-linear 
regression technique using a statistical program package, 
for each point of the tunnel contour, for each cover to 
diameter ratio and for each in situ stress ratio. The A 
curves obtained for each of the three points of the tunnel 
contour were found to be relatively insensitive to the cover 
to diameter ratio (except for the tunnel crown) but more 
sensitive to the in situ stress ratio. The first finding 
assures that the definition of the ground reaction curves 
for intermediate H/D values could be easily achieved by data 
interpolation. Interpolation of data for intermediate in 
Situ stress ratios could also be made, but with a larger 
degree of uncertainty. Estimates of the ground reaction 
curves for any tunnel size, depth, soil stiffness, strength, 
or in situ stress ratio can thus be obtained, either using 
the basic general expression and appropriate parameters or 
through the prepared design charts. 

Once more, estimates sof S’current sradialgsprang 
constant (k,) or 'current' tangent Young's modulus (E,) 


could be obtained by partial differentiation of the basic 


TE 


function giving the NNGRC. Through equations 6.25 to 6.27 
the ground stiffness could be assessed for any amount of 
Stress release prior to ground failure, or for any amount of 
tunnel closure. To facilitate this assessment, a suite of 
convenient design charts was also prepared and presented. As 
before, through these derivations, one would be in position 
to approximately account for the degree of 'softening' and 
the degree of stress release associated with a given amount 
of tunnel closure. These factors could be used as input for 
traditional ground-lining interaction solutions, using 
either ring-and-spring or ring-and-plate models. Since most 
of these models operate with a constant spring constant or a 
constant ground modulus, average stiffness values would have 
to be estimated from the values found for the crown, 
springline and floor. 

Although the solutions obtained were developed for use 
within the ranges of variables considered, one may be 
tempted to use them beyond the ranges of stress release 
covered by the parametric analyses. In some instances, the 
NGRC may be "too short" and some limited extrapolation may 
be needed for certain practical applications. The limits of 
validity of the solutions were then explored and defined. 
The terminal points, T, of the functions were identified 
both for the cohesionless and for the frictionless model 
solutions. The validity of extrapolations beyond those 
points cannot be assured, as the corresponding stresses 


eventually obtained may violate the failure criteria or they 


ANZ 


may not be sufficient to keep the opening stable. The 
associated displacements will likely be in error (possibly 
smaller than the correct values). The generalized®solutions 
are not recommended beyond their terminal points, but if 
required, an assessment of the 2D tunnel stability is 
needed. The limitations of plasticity Solutions ior this 
purpose were assessed and discussed. Lower bound solutions 
such as those by D'Escatha and Mandel (1974) or Davis 
et.al., (1980) were presented and the shortcomings of their 
application to the posed problem were discussed. The factor 
of safety assessed using the collapse tunnel pressures that 
these solutions provide are interpreted, at best, as a crude 
estimate of the actual factor of safety. While these 
solutions seem to operate quite satisfactorily for the 
frictionless soil model (¢,=0), they may not provide good 
approximations of the collapse loads in soils under drained 
conditions (c and ¢ not equal to 0). Unsafe estimates of the 
collapse tunnel pressure may be obtained. It was then 
advised to allow as much safety as possible or, preferably, 
to avoid any extrapolation of the NGRC beyond its terminal 
point. An additional criteria that could be evoked is the 
limiting dimensionless crown displacement, U=1.8, derived in 
Section 2.3.4.3 from observations in tunnel model tests. 
Recall that above this value a ground collapse condition was 
noted in the physical models. 

A compact generalization of the numerical results in 


terms of ground settlement, similar to that undertaken for 


413 


the ground reaction curves, was not attempted, although it 
could possibly be implemented. In fact, Resendiz and Romo 
(1981) succeeded in doing so, using a model similar to the 
frictionless model used herein. Nevertheless, the charts 
included. in-Appendix-C, with normalized surface and 
subsurface settlements, do present sufficient generality for 
practical use, simply requiring interpolation of results for 
conditions other than those considered in the parametric 
analyses. An example of the interpolation procedure required 
was presented in terms of the relations between the maximum 
surface and crown settlements discussed earlier. 

Finally, the approximate nature of the generalized 
solution must be emphasized. It reflects the simplifications 
introduced, the very approximate nature of the numerical 
model used, of the simplified constitutive models employed, 
the approximations related to the finite number of 
increments used to describe the degradation of the soil 
stiffness in the pre-failure regime, etc. Although these 
approximations have been introduced with some degree of 
discernment and judgement, which in turn requires equal 
consideration when using these solutions, it is important to 
test them against actual field cases. This is the last step 
in the development of the projected design procedure. It 
would include the validation of the proposed method, 
comparing its results with measured tunnel performance. The 
proposed method needs to be calibrated by assessing the 


deviation of predictions and by identifying the reasons for 


1114 


these deviations. This is the next natural step in the 
generalization process, following the modelling, stepejust 
completed in the present Chapter. Accordingly, this will be 
one of the goals of the next chapter. 

However, before this undertaking, an additional and 
important aspect of the tunnel design should be addressed. 
This refers to the ground-lining interaction phase of the 
tunnel construction representation, only briefly discussed 
in this chapter. The development presented so far attempted 
to portray the ground response through a 2D representation, 
up to the instant the support is activated. The first part 
of the next chapter will present and discuss procedures to 
represent the interaction process, taking into consideration 
the main features of the ground response prior to the 
Support installation. With this, a design procedure for 


shallow tunnels would have been completed. 


7. DEVELOPMENT AND VALIDATION OF A DESIGN PROCEDURE 


Pernt Loauet: on 

In Chapter 6, an approximate procedure was developed 
through which relationships between radial stresses and 
radial displacements at a point of the contour of a shallow 
tunnel could be obtained without the need of finite element 
modelling. These relationships, expressed for two idealized 
soil models in both algebraic and graphic forms, allow the 
amount of ground stress release in a two-dimensional 
representation to be determined from the knowledge of the 
tunnel closure. Other relationships were derived from which 
the change in the ground stiffness could be assessed by also 
relating it to the amount of tunnel closure. The first type 
of relationship expresses the stress transfer, or 
two-dimensional 'arching' process around a stable opening 
upon its closure, by a reduction of the stresses acting on 
its contour. The second portrays the response of the ground, 
in terms of changes in its stiffness associated with this 
Stress transfer and of the tunnel closure. 

Concurrently with these developments, relationships 
were established between the amount of stress release 
allowed at the opening and the settlements induced at the 
ground surface and subsurface. All relations were presented 
in a non-dimensional and scale independent form to allow 


generality. 


SEIPIRS) 


1116 


The above derivations assumed that the opening was 
Supported by internal radial and shear stresses which had 
been gradually and continuously reduced =tora tracrion:c of 
the in situ stresses. This was in order to simulate the 
ground stress transfer process in an advancing tunnel. A 
simplification in this simulation process was to disregard 
the action of an actual lining support (see Sections 2.3.5 
and 2.3.6). Both in a simplified 2D representation or ina 
3D simulation, it is known that the lining does affect the 
ground response, as it interacts with the soil by inducing 
changes in stresses that may alter its stiffness. 

In the present chapter the soil-lining interaction will 
be discussed and a simple model representing the process 
will be presented and evaluated. The effects of the lining 
action in the 2D ground response will be addressed" An 
approximate procedure to account™ for theses erfeceseinucne 
tunnel design will be proposed and evaluated. Within this 
procedure, the ground stress reduction by arching and the 
changes in ground stiffness taking place prior to lining 
activation (in a 2D representation) will be taken into 
consideration. Subsequently, a complete sequence for design 
will be proposed, in which the ground settlements and lining 
loads will be obtained simultaneously. This newly proposed 
method is verified against results of simulations of shallow 
tunnels through numerical and physical modelling. Some of 
the limitations of the proposed procedure will be discussed. 


Finally, the method will be validated by comparing its 


PUL 


predictive capabilities against observed field performances 


in some case histories. 
7.2 Soil-Lining Interaction 


7.2.1 Choice of the Model: Hartmann Solution 

The basic aspects of the soil-lining interaction 
process were discussed in Sections 2.3.5 and 2.3.6. The 
available models representing this process were reviewed in 
Section 4.3.2. The most simple lining design methods that 
take into account the interaction between the ground and the 
Support in the lining loads calculation, are those reviewed 
tresectivons 7403. 2. 3 Fand 6473.02. 45\ri se 8%, othe bring=and-plate 
and the ring-and-spring models, respectively. It was shown 
in Section 4.4.2.2 that the latter is perhaps the most 
popular type of statical system used for assessment of 
lining loads in Shallow tunnels. Any of the above models 
could be used in connection with the design procedure being 
developed in this thesis. If fact, the ring-and-spring 
models, with ground reaction represented by discrete springs 
or bars, could be the most convenient one to be used. These 
models could allow different ground stiffnesses to be 
considered at distinct points of the tunnel contour. This 
would permit, as suggested in Section 2.2, taking into 
account a 'weakened' or ‘softened embedment' condition 
whenever applicable. Certain criteria would have to be 


established and tested in order to assign different spring 


118 


constants (radial and tangential) for points at intermediate 
locations between crown, springline and floor. 

The effect of such differential degradation of the 
ground stiffness around the tunnel could not be considered 
if other soil-lining interaction models were chosen. An 
averaged ground stiffness would have to be defined, as these 
models assume a constant modulus of deformation, to 
represent the linear elastic behaviour of the ground. The 
accuracy lost with this assumption is in turn compensated by 
a gain in the compactness of the solution, particularly if a 
closed form analytical solution is selected. The latter, 
furthermore, would offer considerable ease in the design 
procedure since it permits simpler handling of parametric or 
sensitivity analyses which are sometimes requinedsateinitial 
design stages. Additionally, it is believed that if such an 
analytical solution could be coupled with the design 
cromenten, the further simplifications it requires would not 
be discordant with the overall approximate nature of other 
components in the design procedure. 

These aspects lead to favouring a closed form 
ring-and-plate solution to analyse the soil-lining 
interaction phase of the design procedure and to serve as a 
tool for the lining load prediction. The consequences and 
limitations of this choice will be further discussed in this 
and in the following sections. A number of options are 
offered with this choice but by inspecting the solutions 


reviewed in Section 4.3.2.3, not many are found that would 


1119 


not introduce further approximations. In fact, Hartmann's 


(1970,1972) approach is the only one that makes full 


allowance for the non-uniform stress field existing ina 


Shallow tunnel, which is generated by in situ stress ratios 


different from unity sandeby the action of gravity. 


Therefore, 


the problem at hand, 


the features of a shallow tunnel situation; 


gravitational stress gradient 


neglects, however, the effect 
the ground is approximated by 
a gravitational stress field. 
that the effect of the ground 


can be considered as neglible 


that the influence of the gravity 
the interaction analysis for most 


Hartmann's original solution 


lining as linear elastic, 


materials. The opening is assumed to be circular, 


this solution seems to be the most convenient for 


in that it partly accounts for one of 


the 
across the opening. It 
of the ground surface, since 
an infinite plate subjected to 
It was shown in Section 2.2 
surface on the lining response 
for H/D greater than 1.5 but 
cannot be disregarded in 
shallow tunnels. 


treats both soil and 


isotropic and homogeneous 


the ground 


mass is represented by an infinite plate and the lining by a 


weightless thin cylindrical shell of constant thickness. The 


lining is assumed to be installed in the opening before any 


displacement occurs in the ground. Moreover, 


lining and 


ground are assumed to be in full contact, so a non-slip 


condition at the interface is admitted. More recently 


(Hartmann, 


1986: unpublished report) a full slip solution 


was derived but will not be discussed herein for reasons 


1120 


presented in Section 2.3.5. 

A zero stress horizontal surface represents the ground 
surface, so that the principal in situ stresses are vertical 
(yz) and horizontal (Kyz), where z is the depth to a point 
measured from the ground surface, y is the unit weight of 
the soil and K is the in situ stress ratio. 

The lining is activated by an excavation loading 
condition (see Section 4.3.2.3). The solution is formulated 
for both plane strain and plane stress, the former being 
more relevant to the tunnel problem and is presented herein. 
The equilibrium and compatibility conditions are formulated 
in terms of polar coordinates. The internal forces in the 
lining are computed using Flugge's (1962:134) differential 
equations relating lining thrust, shear and bending moments 
to the lining displacements. 

The relative stiffness of the lining-ground system is 
expressed in terms of two coefficients, referred to as the 
compressibility (a) and the flexibility (8) ratios. These 


are defined as: 


@ Ezra te) 
TE. 1-H,?) Fer. 
E,1,(1+u) 
E [7.2] 


Taipei 
where xr, 15 ther tunneljradius( gt) ou, anders nnuatanewelactar 


constants for the ground and the support, A, is the cross 


$s 
sectional area of the lining per unit length of the tunnel 
(A,=d=lining thickness) and I, is the moment of inertia of 


the support per unit length of the tunnel (I,=d°/12). 


1424 


Note that these ratios are related to the corresponding 
Batios-usedeinethesBinstein and Schwartz (1979) derivation 


(Grand#?-eseeSectionv2. 2. Sa2yebys 


1 
C(1-yn) 
1 
B= F(1-y) 


Typical flexibility and compressibility ratios can be 


a= 


found in Table 7.1. These were calculated using both the 
Hartmann and Einstein and Schwartz definitions. The ratios 
were calculated for six soil types and three common lining 
systems. The assumptions made are given in the table. Note 
that the deformation moduli given correspond to initial 
tangent values at a confining stress of 0.1 MPa. As 
indicated in Section 2.3.5.3, different stiffness ratios 
would have to be considered if the effect of the global 
"softening' of te ground on the decrease of the in situ 
stresses with tunnel advance, was to be considered. 

‘The notation and conventions used in Hartmann's 
solution are shown in Figure 7.1. Note that stresses are 
positive in compression. A summary of the solution is 
presented in Figure 7.2, where the equations for lining 
stresses, displacements and internal forces are given. It 


should be noted that v' ana Va 


ro gO 


given by equations 7.7 and 
7.9 are the components of the total radial and tangential 
displacements, to which the overall heave component v",, 
given by equation 7.10, should be added in order to obtain 
the total lining displacements v,, and v,,, as shown in 


equations 7.6 and 7.8. Note moreover, that the heave v",, is 


eat useage 


a 





Sour Steel sera bets Precast Cerone ce Shotcrete ‘4? 
lagging segments 
Type E, | tN B F B F 8 F 
(MPa) 
Soft clay 8 0.030 40 0.140 10 0.040 35 
Medium clay 25 0.010 125 0.040 30 0.012 110 
Sit re cr ay, 100 0.003 500 0.010 120 0.003 450 
Loose sand 40 0.007 200 0.030 50 0.008 170 
Medium sand 60 0.004 300 0.020 70 0.005 250 
Dense sand 120 0.002 600 0.009 150 0.002 550 
$ = esse 7 = = 1,200 : 2 a = 2,800 
Notes: (1) Typical initial tangent values at a confinement pressure 


of 0.1 MPa [ee at 10 m depth in a soil with 
Y¥- 20174 KN/mitand Ky = ORaa) 


(2) Lagging stiffness disregarded. Influence of joints in 8 
disregarded. Joint compressibility partly accounted. 
Eg 200 Gpayt a/r p= Mien Pre Mon 


(3) Influence of joints in 8 disregarded. Joint 
compressibility pares accounted. Es = 10 GPa, 
G/F = 0-1; ao ai altos 

(4) Hardening effect partly accounted. Es = 10 GPa. 
Presence of ribs or lattice girders disregarded. 


djre =60 0650 T= 47/12. 


(5) In all cases p = iQ. 


Table 7.1 Typical Relative Stiffness Ratios for Common 


Linings and Soils 


iipass 


dependent on r;, which is the radius of an arbitrary circle 
at which the vertical heave displacement is set equal to 
ZerOemo Let iS) boundary is. located at infinity, an infinite 
heave would be calculated. Therefore, for practical 
applications, r, is to be chosen according to the position 
of a stiffer horizon below the tunnel floor or to the 
location of points below the floor, where no displacements 
are noted. From what was shown in Table 6.9 (Section 
Ovswleciye ane fatio t2/r; 19 likely to be larger than 1/4 or 
1/3. It should be pointed out that the shear force Q, given 
by equation 7.13 was not included in Hartmann's original 
solution, but it could be obtained by differentiating the 
bending moments (Q,=(1/r,)0M/d9). 

Hartmann's solution includes second-order terms that 
are frequently neglected in similar derivations (for 
example, see Schwartz and Einstein, 1980:367). The first 
term in the expression for the bending moments (equation 
7.12 in Figure 7.2) represents the influence of the lining 
perimeter reduction, caused by the action of the thrust 
forces. A decrease in the lining diameter produces an 
increase in the lining curvature that leads to bending 
Stresses. These stresses will be present even under an ideal 
Psotropie loading conditions (K= land zee 1.) in most 
cases this term is of small magnitude. The second term in 
thesexpresstonaiorsMegives thes imtluence’ of the overburden 
stress yz, on the bending moment and the third describes the 


influence of the gravitational stress gradient across the 


1124 





N—>0 700?9 
Org)?O 
Vig 1 Vag: DISPLACEMENTS Mg: BENDING MOMENT 
Oros 999: NORMAL STRESS Ng: THRUST FORCE 
Tego: SHEAR STRESS Qg: SHEAR FORCE 


Figure 7.1 Notations and Conventions used in Hartmann's 


Solution 


ie 


(at wey ay) 


Wirt Woy) } 


(turd) 


(ore) 


(6°) 


(8°72) 


Cee} 


(970) 


(S" es 


(eo) 


(ea) 


uoTINTOS Ss,uueWw_IeH Aq UaATH S|ad10¥4 


Teujaquy pue sjuswaoe{tdstq ‘sassaiqs Hbututy z*y¢ aanbty 








0, (dl2("p-cdeste-c) Bew(te-s)e4 0.0_, (9 (2("p-e)pstg-slesm(nz-c) +t) 4 
oc)juye x a ih Sa ts 2 ee = A Se -« 
gOS ae J (e+) (Fe-€) O-E SEE ba d(ozet) (Tp-e) CH-1E 2 


g d (2 (4p-€ 6418-2) 84D (TP-S) +4 
o¢) 805° 3A ae : (tes) f 
Sau c Oo (meet) (Fp-8) (r-0) + (¢7)809 








0,0., (9 (2(1e-C)¥-t9-s]E+D(NZ-C) 44) Zz 0,0,, (+D+tdz ¢ 
z 


=a i IES Te 
(Oz +t) (Tp-€) (I-LE 2 dare) 
d( V(Tp-0) ) 





























0 (d(o(ty-c)esta-c)e+o(rp-s)ei}ey 0 0, {9[ 2("¥-C)p4ttg-s c4+v(NZ-C) +t] Z O., (241) .0:0_, (d+m4t)z © 
(9¢)809 2A o uo -/" —~ (@7)609 1 CaN =~ @s05> 24224 aah E. - 
cee z (dooc+dp+n) (Mp-€) (a-1) oz) (doz1 +0 e+) (TV e-€) (4-0) ea ames TS ESO Weary 
t 
ON a(t-t)y Ozn 
OL) 2 Tipe) (Tat) 
0,,(8(2(4e-€ )6+te-2] 84+D(49-S Jet) B 0,0,, (B(R(4y-C) oH 9-s) E+ (NZ-C) 4) Z 0 (D+1)8 a oo 
oc )uye 3A = eee @z)uys 1a uw c quye aA m=) =e 
(toc rute, (doredei) (¥y-0) (a1) * (Jeeps) (pe) GH-) ez Tame) We 
gute Orn + of), 3M, 
0 {(d(2(tp-¢)6—1e-c] 8+9(1b-S) 41) 8 0.0_, (P(9(tp-c)eto-slem(tz—-C) +t} Zz 0 (D4+1)8 | 0.0 (d+D4+1)7%), @ 02 
soo sh ae = x= OT ee 909 3A4 ee RPA ese = Sim A 
(te) zt (9e+t) (Fp-0) 0-0) “ee 7 (074+) (4p-c) (4-1) anP° Ze Ot | wrt ) Wet 
e805 OFA + Pn aes 
- 0 {U(0(ty-C)6+4 6-2) 84D(4e-S)41) © 0_,d(p(tp-c)pstg-sle+m(tz-€) +t 0 (D+t)e 0o2 
oc )yuye 4A so) = se / = t@ uye zAv_ = outs 2A S22 2 = 1 
ee D( Pez (Mee) (X-WIE af ca Dido+1)(Tp-C)ia-t) D( MeL) 
0 (d(o(Ne-c)esda-claro(rp-s)+i}o 0 (U(V(Tp-E) pet g-slEe+m(mtz-C) 4} Z 0 (peur (t-u)e 0 (J+D40)Z oe 
SA ee oe ie ee ee Se eee = (SDSS A a a ae ee ee SA ee a ee ey ay Bhs... age 5 
(C)802° INT tye lerTor—sileewttpacan) Oacay ~ (29°? **rgToite-cowatei ui Tem (Tanded) OID On hese Cattwiaese * Weezer) 





0 (U(p(tp-e)este-cleaso(tp-sleij)e 0, (P(2(Mp-E)o+ta-slEe+w(NZ-C)sijz 0 (v4ide 0 (d+v41)z 04 
goo 1 Lp SS ek) ER IR ey. pp a ee ee OD» ZA. o> as eS - e059 414 = 2h 2 eee = — 
eae (Jeze-dze-0) (4 e-6) 1-0) eas (Woz ae) (Mp-e) Or-0) v(MeL) (J+) O61) 
taunts 4 Gaur ea a 
(wo) —---3e-c aro ke = 2 


- 
(tet) 1g (M41) wa 


1126 


opening. Correspondingly, the first and third terms in the 
expression for N, (equation 7.11) represents the influence 
of the overburden stress, while the second and fourth terms 
describe the influence of the gravity. Moreover, in the 
latter equation, the first and second terms reflect the 
effect of the mean normal in situ stress, while the third 
and fourth terms reflect the effect of the mean in situ 
stress difference. 

Provided the thickness of the lining is small in 
comparison with the tunnel radius, the second order moments 
are small. Moreover, if the tunnel is deep (a large z,/r, 
ratio), then Hartmann's solution becomes equivalent to other 
deep tunnel solutions. In fact, it yields results which are 
basically identical to Windels (1967), Curtis-Muir Wood 
(1976), Einstein and Schwartz (1979) and Ahrens et.al., 
(1982) closed form solutions, as discussed in Section 
4.3.2.3. It isesimplemeo Sifb¥ that iffthe liningwmis 
disregarded, (i.e., a=f=0), Hartmann's expression for the 
tangential stresses at the opening contour reduces to 
Mindlin's solution for an opening in an infinite plate under 
gravity with lateral restraint (Mindlin, 1940:1136 - 
equation 58). 

In order to assess the consequences of the infinite 
plate assumption in Hartmann's solution, a comparison 
between this solution and the results of a finite element 
analysis of a shallow tunnel would be required. 


Unfortunately, the lining representation adopted in the 2D 


112g 


finite element model presented in Chapter 5 does not permit 
a direct comparison to be made, particularly regarding 
bending moments. Ranken (1978), however, performed a few 
finite element analyses of circular shallow tunnels, whose 
results can be used for this comparison. Five linear elastic 
analyses were carried out, for different cover to diameter 
ratios that varied between 0.5 and 4.5. Details of this 
Study were presented by Ranken (Op.cit.:96) and will not be 
repeated herein, except for essential information. In all 
analyses, the in situ stress ratio in the ground was kept 
constant andwequal—to+0,5- The ratioyvof<-the-soil stoglining 
Young's modulus was set equal to 0.0205. The Poisson's ratio 
of the soil was 0.25 and the lining was 0.1562. The lining 
thickness was 10% of the tunnel radius. The resulting 
relative stiffness-ratios were a=6.25 and £=0.0052. As 
indicated in Table 7.1, these values would be typical for a 
rib and lagging lined tunnel, in a medium to loose sand. The 
lining was installed before any release of ground stresses 
took place. A fixed lower boundary was set up at three 
diameters below the tunnel floor. A gravitational field 
stress condition was considered, so the results incorporate 
both the action of the ground surface and of the in situ 
stress gradient across the opening. Although both full and 
no slip conditions were considered for the lining-ground 
interface, only the results of the latter case will be 


examined herein. 


1128 


The solution by Hartmann, as Summarized in Figure 7.2, 
was applied to these cases and some representative results 
are shown in Figure 7.3. The results of the closed form 
solution are presented by continuous lines, whereas the 
numerical results by Ranken are shown by discrete points. In 
the upper part of Figure’ 7.3, the calculated thrust forces 
and bending moments at the crown of the tunnel lining are 
plotted together. The lower part shows the calculated crown 
vertical displacement and some diameter changes. All results 
were conveniently normalized to allow the comparison. Fairly 
Similar results were found at other points of the contour. 

Apart from inaccuracies that could be attributed to 
numerical approximations, the differences in the results by 
the two approaches can be ascribed to the influence of the 
ground surface, included in the finite element analyses. As 
expected, these differences seem to increase as the tunnel 
becomes shallower. Hartmann's solution seems to furnish 
results which are either approximately equal to or greater 
in magnitude than the finite element solution. Moreover, it 
is noted that both solutions tend to yield very similar 
results for cover to diameter ratios greater than 1.5. This 
tends to confirm that the effect of the ground surface on 
the lining response seems indeed negligible for H/D greater 
than 1.5 (see Section 2.2). Moreover, even for smaller 
ratios, the analytical solution seems to furnish results 
which are greater in magnitude than the numerical solution. 


Therefore, the Hartmann solution may provide conservative 


08 


Oo 
o 


HARTMANN 


Ny/Y Zo fo 


& 


HORIZONTAL DIAMETER 


Lal a 





THt29 


Myp/¥Zor 


(INCREASE ) 


CROWN 
DISPLAC. (Vo) 
HARTMANN 


(AD/D,)E/YZ 


VERTICAL DIAMETER 





4 


(DECREASE) 


Figure 7.3 Comparison of Lining Responses Calculated for 


Different Tunnel Depths, 


wWitheanaswitnout Account ot* the 


Influence of the Ground Surface 


1130 


estimates of the lining response for H/D smaller than 1.5. 
These findings are,.valid not only (fon theatunnelac pownbut 
for other points of the contour. 

The largest discrepancy noted refers to the magnitude 
of the crown displacement. This result is not surprising 
since it is the one that involved a larger degree of 
approximation. AS can be noted in Figure 7.2, the radial 
displacement v,, (equation 7.6) depends on the ground heave 
vv". given by equation 7.10. Two approximations are involved 
in estimating this component of the crown displacement. The 
first refers to r,, defined earlier as the radius of an 
arbitrary circle at which the heave is equal to zero. To 
Simulate the conditions involved in the finite element 
analyses, r; was set equal to 3.5D (thus r,/r,=1/7), so that 
Such a circle is tangent to the lower boundary of the finite 
element mesh. This correspondence is, however, not exact and 
differences in the displacement calculation may result. 
Secondly, as) noticed an equation 7210, themheavesw ~: 
depends only on the ground modulus E, and is independent of 
the lining stiffness. Except for the soil to lining modulus 
ratio, Ranken (1978) did not provide the value of soil 
modulus used in his analyses. A modulus of 20.09 MPa has 
been chosen in the present calculation, which may differ 
from that selected by Ranken. 

In summary, it may be concluded that, subject to the 
assumptions made (notably that of a linear elastic behaviour 


for soil and lining), Hartmann's solution seems to provide a 


eiesal 


good approximation for the two-dimensional ground-lining 
interaction process, despite not taking the influence of the 


ground surface into account. 


7.2.2 Soil-Lining Interaction Analysis for Delayed Lining 
Installation in a Non-Linear Ground Mass 

Hartmann's solution, like other two-dimensional closed 
form solutions, assumes the lining to be in. place before the 
excavation loading develops. It was shown in Sections 
4.3.2.7 and 5.2.1.1 that there is no rigorous procedure to 
account for the effects resulting from delaying (in space) 
the installation of the lining, which involves 
three-dimensional stress changes in a simpler 
two-dimensional representation. It was also explained 
earlier in this thesis why the approximate procedure of the 
gradual reduction of the in situ stresses around the tunnel 
was favoured in the present work, to simulate the delayed 
placement of the lining in a 2D model. 

In essence, this approximation mimics the actual stress 
changes or arching process that leads to smaller ground 
loads being transferred to the lining as a result of the 
interaction process within the soil. In Chapter 5 it was 
shown that, although approximate, this procedure may indeed 
furnish sensible results in terms of estimates of the tunnel 
performance. 

If Hartmann's solution is to be used to analyse the 


soil-lining interaction phase of the tunnelling process, 


1132 


instead of the finite element simulation used in Chapter 5, 
then that ground load reduction should be introduced” into 
the solution. Provided that in this analysis the lining 
response, in terms of its loads and displacements, is the 
only matter of interest, then the reduction of ground loads 
can be easily implemented. It can be represented by a 
reduction in the ground stress field, which can be 
introduced by a reduction in the unit weight of the ground. 
This in turn, can be related to the amount of stress release 
(a) taking place up until the instant the support is 
installed. The reduced unit weight of the soil would then 
be: 

Yea TY (loa) = yo C714] 
where y is the actual in situ weight of the ground. 

It should be remembered that in the two dimensional 
representation adopted in this study (see Section 5.2.2.1) 
which led to the generalized results presented in ENED Lerten 
the same stress reduction factor is applied uniformly to all 
points of the tunnel profile and reduces both radial and 
shear stresses in the same proportion. Due to this 
assumption, the current stresses acting on the perimeter of 
the 2D opening after a certain ground stress release a, and 
immediately before the support installation against the 
ground of “unity welght "vy and@in Sltusstressmratlomnegmacre 
exactly equal to the in situ stresses acting on the contour 
of a tunnel with same geometry, yet to be excavated in an 


undisturbed ground mass, which has the same in situ stress 


Liss 


ratio K, but with a reduced unit weight given by equation 
7.14. Provided that a linear elastic behaviour can be 
ensured, the ground-lining response will not depend on the 
stress and strain changes developing in the ground prior to 
the support installation. The magnitude of the lining loads 
and lining displacements furnished by Hartmann's solution, 
with a field stress reduced through a reduced soil unit 
weight, will be correct, despite the stress changes and 
displacements induced within the ground mass being 
incorrect. The reduced unit weight approximation does not 
account for the stress changes and displacements induced 
prior to lining activation. Apparently there is no formal 
impediment to adapt Hartmann's solution for this new 
condition, which would explictly include a stress release 
factor (1-Z) in the derivation, and take into account the 
stress and strain changes in the ground prior to installing 
the support. In fact, the Schwartz and Einstein (1980:393) 
derivation, incorporating a ‘soft core' region (see Section 
4.3.2.7) that leads to 'pre-support' ground movements is an 
example of how such a solution could be worked out. 
Nevertheless, such an undertaking was felt to be out of 
scope of the present work and therefore was not attempted. 
If the non-linear behaviour of the ground is taken into 
consideration, then the above approach is merely an 
approximation. The stress changes in the ground prior to the 
lining installation do have an impact on the ground-lining 


interaction, as the ground stiffness is inevitably changed. 


1134 


This point was raised and discussed earlier in Section 
2.3.5.3. Moreover, the separate analyses of the pre-support 
ground response and of the soil-support interaction phase 
require a superposition of effects for the final equilibrium 
condition to be found. The validity of such superposition in 
a non-linear problem is debatable. Furthermore, there is 
nothing to support a linear elastic interaction analysis in 
a soil that exhibits a non-linear response. Perhaps the only 
argument that could be raised in its favour, is that, 
provided the increments of ground displacements taking place 
after lining installation are small in comparison with those 
develoning in the finite unloading increment imposed to the 
ground before that, then the linear elastic approximation 
may not be entirely discordant with the non-linear, yet 
piecewise elastic model used herein. But this argument may 
not be valid under certain practical situations. Therefore, 
if the linear elastic interaction analysis using Hartmann's 
solution is used in connection with the design procedure 
being developed herein, it must be regarded as an 
approximation whose consequences should be assessed 
accordaingiy. 

For this end, the numerical analyses described in 
Section 2.3.5.3 can be helpful. Three analyses were carried 
out there, with different amounts of ground stress release 
at the circular opening prior to lining, installations, 0%, 
40% and 80%. An additional calculation was performed 


allowing a full stress release. The ground properties were 


1135 


as given earlier in that section. The soil strength 
including both frictional and cohesive components and the in 
Situ tangent modulus increasing with depth, were maintained 
in all analyses. The tunnel with a 4m diameter and 6.2 m 
cover was intentionally lined with a very soft and flexible 
Support, so that substantial displacements could develop 
after its installation. This condition is therefore a 
Critical one in terms of enhancing the effect of the 
non-linear response of the soil on the ground-lining 
interaction process. For the reasons exposed in Chapter 5, a 
fairly thick lining with 0.5 m thickness was used with 
constant elastic properties, which led to a compressibility 
ratio arot 0.9.(Ce1.8 and-aapflexi bil tteyeratioss nof <0 i007 
(F=86). These were calculated with respect to the in situ 
elastic modulus of the soil at tunnel axis elevation. Such 
values could correspond to a flexible and compressible 
shotcrete lining in a medium clay. 

The equilibrium points in terms of radial stresses and 
radial displacements at crown, springline and floor, as 
obtained by the finite element analyses, are shown in 
Figures 7.4, 7.5 and 7.6, respectively. The continuous solid 
curves shown are the ground reaction curves for these three 
points, as calculated for a full stress release without 
installing the lining. The dashed lines link the points 
representing the equilibrium to the starting point of lining 
activation. The abscissae of the latter points give the 


radial closures of the opening when the support was 


1136 


activated. Unlike the radial displacements that were 
calculated at nodal points, the radial stresses at 
equilibrium were obtained by extrapolating the calculated 
element stresses to the tunnel contour. 

Following this, an attempt was made to obtain the 
equilibrium points (L and E as explained later), using 
Hartmann's solution coupled with the generalized solution 
presented in Section 6.4 for obtaining the ground reaction 
curves. In order to achieve this, a few additional 
assumptions had to be made and these are discussed in the 
following. 

As stated earlier, the closed form interaction analyses 
require an assessment of a reduced unit weight, y,.4,, for the 
soil. It is necessary to evaluate the ratio of current 
stress to initial stress (Z) at tunnel contour when the 
support is installed, so that y,,g can be determined through 
equation 7.14. If the GRC given by the generalized solution 
presented in Section 6.4 coincided with the GRC calculated 
in the finite element analysis, then there would be no need 
to calculate Z at lining installation, as it would be equal 
to that imposed in the analysis. However, since the GRCs do 
not coincide, as the generalized solution is an 
approximation of the correct GRC, that need is justified. 

To assess Z, one could assume that the radial 
displacements at the instant of lining installation are 
known and equal to those given by the finite element 


calculation. In a more general case, these displacements 


1137 


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1138 


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1140 


could be estimated by the approximate procedure presented in 
Section 5.3.5.2 for the assessment of tunnel closure at 
sections behind the face. With the radial displacements 
developing before lining installation, it is possible to 
determine the value of a or Z at crown (C), springline (S) 
and floor (F). This is done through the NNGRC or A curves 
given by equation 6.24. Before that, however, one has to 
define a,,,=1-Z,,,9 20° C, S andi PF, wwhicheis (agtuncwon of 
(m-1), which, as indicated by equation 6.30, is a function 
of the friction angle, %. The friction angle to be used for 
this purpose is the one obtained after adjusting the actual 
@ to account for the non-zero cohesion, as indicated by 
equation 6.10. In this case, there is no need to amend the 


friction angle since the failure ratio, R is equal to 


nekes 
one. Otherwise # would have to be adjusted according to 
equation 6.3. After defining the a,.,, values, the U.,., values 


should be obtained@through’ the functiontselacing tae 7/Us eco 


ree 
H/D. This requires interpolating the reference values 
between the solutions given by two consecutive K, curves 
which bracket the K, used in the finite element analyses 
(0.75). Alternatively, one could simply assume the solution 
set with K, closer to the required value. This alternative 
was favoured herein and the reference values for a and U 
were taken from the solution set for K, equal to 0.8. 

Note also that since a variable in situ modulus profile 


has been assumed in the numerical analysis, the 


dimensionless radial displacements U should be normalized to 


1141 


the in situ tangent moduli of the soil at points located 
half diameter radially away from the opening profile, as 
explained in Section 6.2.4.2. 

The resulting parameters obtained following the above 
sequence of calculations are given in Table 7.2. One notes 
that different 2 ratios are obtained at different points of 
the tunnel contour, thus a mean Z should be considered to 
get Yea, by averaging the calculated ratios (springline 
being counted twice). For the 40% stress release considered 
in the finite element analysis, the approximate solution 
yielded 38.2% for the same radial closure of the opening. 
For the 80% stress release, the approximate solution 
furnished 75.4%. In both cases, the calculated amount of 
stress release for the same radial displacement at lining 
installation, is slightly smaller than the correct one, 
indicating that the approximate solution furnishes 'safer' 
estimates of the current ground stresses at the instant the 
Support is activated. 

It should be noted that only for the springline in the 
80% stress release case, the generalized solution for the 
GRC was used beyond its limits, as defined in Section 6.5.1. 

In summary, the reduced unit weight of the soil which 
Simulates the reduction of the ground loads for the 
lining-ground interaction analyses, was obtained through the 
solution presented in Section 6.4.3. It was assumed that the 
eeeresl closure is known at the instant the support is 


activated. The reduced unit weight is calculated directly 


1142 


GEOMETRY: H = 6.2 m 0 = 4.0 m 
PROPERTIES: y = 16 kN/m?, Kj, 80.79). 1 Cuwe39°KPaly Oe a aneh 
Re = 1.0, K = 330, n 20.23, p= 0.3 
Thus: 
(Bugle 2 23-53 MPa, (EB, )g = 24.68 MPa, 
(Ey y)p 7 25-22 MPa 
And: 


6, Se, (tu ee U)) oe orth) 





REFERENCES VALUES: H/D = 1.55, Ko = 0.8 and (m- 1) = 2.278 
Thus; 
@ £ @ /U € U € 
(@3 OsS72 0.860 0.665 
s 0.528 te WS} 0.469 


sy 0.448 2.020 0.222 


REDUCED UNIT WEIGHT: 
a. For FE with 0% stress release: ae le 16 kN/m? 


b. For FE with 40% stress release: 


u (mm) U U/U of nN 2 
before lining 
ic Siirae 0.336 0.505 0.400 0.657 
Ss 4.35 0.267 0.570 0.330 0.646 
F 6.55 0.248 Viet 16) -0.070 0°521 
= = 3 
Thus £ = 0.618 and Yr oaieant I = 9.88 kN/m 
c. For FE with 80% stress release: 
u_ (mum) U U/U . ef x t 
before lining 
Cc 15.98 0.930 1.399 -0.200 0.314 
17.91 W102 2.350 -0.380 0.271 
F 15.04 0.570 2.568 -0.950 0.126 





5 = 3 
Th = 0.24 = = 6 
us £[ 246 and ne B yrG 3.93 kN/m 


Table 7.2 Calculation of the Reduced Unit Weight of the Soil 


for the Lining-Ground Interaction Analyses 


1143 


from the NNGRC or A curves. 

The other key aspect to be assessed 15 the ground 
modulus to be used in the interaction analyses. Had a fairly 
stiff lining been used, one could use an average modulus 
calculated at the instant the lining is installed, following 
the procedures set up in Section 6.4.3 (equation 6.26 and 
6.29). The lining presently considered, however, is very 
deformable and large displacements are thus expected to 
occur. Therefore, the ground stiffness when the equilibrium 
with the liner is eventually achieved, is likely to be very 
different from that at the instant the lining is installed. 

Regardless of what ground stiffness is to be considered 
or how it should be considered, an important assumption has 
to be made. It will be assumed that the tangent stiffness of 
the ground is uniquely related to the tunnel radial closure 
(U), through the derivative function of A (A') presented in 
Section 6.4. In other words, it will be assumed that the 
ground stiffness does not depend on the action of the 
lining. This is obviously a simplification which actually 
may not be correct. It was shown in Section 2.3.5 that the 
presence of the lining does affect the ground response, as 
it induced stress changes in the ground which in turn may 
cause changes in its stiffness. This effect will be 
disregarded by assuming that the A' curves are not affected 
by the interaction between lining and ground. 

This assumption also implies that the A' curves can 


provide the ground stiffness either upon unloading or 


1144 


loading. That is to say that the stiffness of the ground can 
be assessed through these curves both for increasing or 
decreasing tunnel radial closure. The latter could occur, 
for instance, at the springline of a lined tunnel squatting 
and forcing the soil to move outwardly. Note that the fully 
reversible stress-strain behaviour assumed in the 2D 
numerical model used in this work (Section 5.2.2.1) does not 
necessarily ensure the reversibility of behaviour in terms 
‘of the X" curves. 

An unstated assumption related to the use of both A and 
A\' curves for each point of the tunnel contour, is that they 
have an independent existence. In other words, the sequence 
of loading or closure at different points of the opening 
profile, does not affect the response noted at a particular 
point of the contour which is always given by a unique A or 
A’ curve. It was shown that the twice normalized ground 
reaction curves were derived from the parametric analyses, 
where a gradual stress reduction was simulated by applying a 
uniform amount of stress release (a) at the opening. Thus, 
the A and A' curves were obtained for this particular 
unloading condition and other unloading sequences could have 
led to different responses in terms of the A and X' curves. 
Therefore, in using the generalized solution developed, one 
may disregard the loading condition originally imposed and 
May Operate with the curves for each point of the tunnel 
contour, as if they were unaffected by responses at other 


points of the contour. This is equivalent to assuming that 


1145 


the ground response could be as that given by a series of 
radial springs, one for each point of the contour but not 
connected to each other. Obviously this is not true, as the 
responses expressed in terms of the A and X' curves do 
reflect the interactive responses of all points of the soil 
mass. The error involved in this assumption will be reduced 
considerably when the stress ratio, Z, or tangent stiffness, 
E., obtained independently for each point of the tunnel 
profile are averaged and single = or B, values are defined. 

The idea of using a reduced ground stiffness compatible 
with the tunnel closure in a non-linear ground mass, for a 
lining-ground interaction analysis was proposed earlier by 
Kaiser (1981:265) for uniform stress field conditions. In 
this case, a Simpler approach was favoured by approximating 
the ground response by a bilinear elastic model. 

The average tangent stiffness of the ground at the 
instant the lining is installed can be readily estimated. 
Wathythes¥7U-y divensin| Table 7.2, EE; isgobtainedgthrough 
the A' curves (or equation 6.26) and equation 6.27. The 
tangent moduli caculated for the tunnel crown, springline 
and floor are then averaged as indicated by equation 6.28. 
The calculations and results obtained are summarized in 
Table 7.3. As it is noticed, the delayed installation of the 
lining in the second finite element calculation (40% stress 
release) caused a drop in ground stiffness of about 27% and 


of about 75% in the third analysis (80% stress release). 


a. F.E. with 08 stress release 
1 alae Sa EXP= 45 
i 2 c 
~ iL 
= + + 
Thus E. (0 ge 2 Feds Beker 


b. F.E. with 40% stress release 


Ee = ( fry) aa 


With U/U of at lining installation (see 


U/U ref As 
ic 0.505 1.000 
Ss 0.570 0.940 
F 1.116 0.840 


c. F.E. with 80% stress release 


1146 


/4 = 24.53 MPa 


previous table) 


re 
i E 


(MPa) 
1.278 18.41 
1.425 16.28 
1.006 21207 


Thus E = 18.01 MPa 


With U/U et at lining installation (previous table) 





UV ef x 
Cc 0.930 0.420 
S 1.102 0. 160 
F 0.570 0.440 


- cnt) 
1.278 7.73 
1.425 AT 
1.006 41.04 
Thus E,: 6.08 MPa 


Table 7.3 Calculation of the Reduced Average Ground 


Stiffness at Lining Installation 


1147 


At this point, the lining-ground interaction analysis can be 
performed. For each case (0, 40 and 80% release), the 
analytical solution summarized in Figure 7.2 was applied, 
uSing the data presented in Tables 7.2 and 7.3 as input. The 
radial stresses at the crown, springline and floor were 
found from equation 7.3, after calculating the relative 
stiffness ratios from equations 7.1 and 7.2. These are the 
equilibrium stresses acting on the lining at those points. 
The incremental radial displacements at these points 
resulting from the lining-ground interaction can be 
calculated from equations 7.6, 7.7 and 7.10. Note that the 
value of the in.situ stress ratio (K, or K) to be 
considered, is the value used in the finite element 
calculations (0.75). The value of r, to be used in equation 
7.10, for assessment of the ground heave, v",,, as explained 
in Section 7.2.1, is the distance between the tunnel centre 
and the lower (fixed) boundary of the finite element mesh. 
Lipthistcase x, Gistequal .toei3..5 motThesfinal (totalyeradial 
displacements are obtained by adding the incremental 
displacements to the displacements that took place before 
lining installation. 

The values of final radial stresses and displacements 
thus obtained furnish the equilibrium points indicated by 
LeLtecAlLininarigiresa7..4), 74 5eandy /.n6.Q0ne, notes! thatrinn 
aimost all.easestandsateallapointsiofpthegtunnelecontour ;j 
the stresses at L are smaller than the radial stresses at 


equilibrium calculated in the corresponding finite element 


1148 


analysis¢liThi sresultes: not'supra'sing, as it) ls a 
consequence’ of the criteria’ used’ to- define the* ground 
stiffness Ejvadopted-in “this"analysis? Tt*has™been™~assumed 
that the soil modulus during the lining-ground interaction 
remained equal to the soil modulus immediately prior to 
Support activation. Since the increments of ground 
displacements after Support installation in this particular 
case, are quite substantial due to the very compressible 
lining used, there is also a Substantial change in the 
ground stiffness during the interaction process. Tunnel 
closure increases during this process and ground stiffness 
decreases accordingly. Hence at equilibrium the average soil 
modulus is likely to be appreciably less than that when the 
Support was activated. Therefore, if no allowance is made to 
account for the additional 'softening' of the ground after 
the support is installed, then the calculated lining loads 
will likely be less than the correct ones as the support 
will be more compressible and more flexible relative to the 
soil than it should be. 

A possibly better and safer assumption regarding the 
lining loads is to perform the interaction analysis, by 
assigning the ground its final stiffnesses, defined at the 
point of equilibrium. This would take into account the 
additional ground stiffness degradation due to the increment 
of tunnel closure after the lining is installed. However, 
this incremental closure is not known beforehand. An 


iterative procedure would have to be devised to solve the 


1149 


problem. The lining-ground interaction analysis would be 
Started by assigning the ground a stiffness compatible with 
the tunnel radial closure at the instant the support is 
activated, as it was done before. An incremental tunnel 
closure would then be calculated and a new ground stiffness 
would thus be estimated. The process would then be repeated 
until convergence is obtained, when the estimated ground 
stiffness at equilibrium is equal to the assigned ground 
stiffness used in the interaction analysis. 

This algorithm was applied to the three cases being 
Studied. For the 0% stress release case, five iterations 
were needed for solution convergence. An average modulus, &, 
of 13.37 MPa was finally found, which represents 45% 
additional degradation of the ground stiffness resulting 
from the lined tunnel eee closure. For the 40% 
stress release case, four iterations were needed and the 
resulting average ground modulus was equal to 9.79 MPa, 
representing a decrease of about 46% relative to the modulus 
at lining installation. For the 80% stress release case, 
only three iterations were needed and the resulting average 
ground modulus was 4.58 MPa, thus 25% lower than that at 
support activation. 

The resulting equilibrium points found through these 
calculations are those indicated by letter "E" in Figures 
7.4 to 7.6. The reduction in the adopted ground stiffness 
resulted, as expected, in higher radial stresses than at 


corresponding points L. More importantly, however, the 


1150 


equilibrium stresses at E are systematically higher than the 
equilibrium stresses given by the finite element 
calculations. This indicates that, the adoption of the 
ground tangent stiffness calculated for the total tunnel 
closure at the final equilibrium situation is a safer 
assumption, though not excessively so, with respect to the 
ground loads acting on the tunnel. Conservative estimates of 
bending moments and thrust forces, as well as larger 
eccentricities are obtained with the latter assumption. 

With respect to the radial displacements at 
equilibrium, mixed results are obtained. At the crown and 
floor (Figures 7.4 and 7.6), the displacements at both 
points L and E are larger.than those found in the finite 
element analyses. While at the crown, the movements given by 
the approximated procedure are not excessively greater than 
the correct one, at the floor the displacements are 
substantially overestimated. It seems that this results from 
an overestimation of the ground heave component calculated 
by equation 7.10. The heave has been estimated using the 
same ground modulus used in the interaction analysis. This 
modulus seems to represent well the soil around the opening 
which is participating in the lining-ground interaction 
process. However, it may not approximate the stiffness of 
the ground below the tunnel, which would be underestimated 
by the procedure used. 

At the springline, the displacements at both points L 


and E tend to be smaller than that calculated by the finite 


Si 


element analysis. The degree of underestimation, however is 
not excessive. 

With respect to displacements, those calculated at 
tunnel crown are of primary importance in the design 
procedure being developed. This is because all settlements 
developing above the tunnel are related to the crown 
displacement, through the normalized settlement 
distributions. As indicated in Figure 7.4, the approximate 
method for ground-lining interaction tends to furnish 
slightly conservative estimates of the crown displacements, 
which is likely to lead to conservative estimates of surface 
and subsurface settlements. 

The arrows indicated in the last three figures give the 
directions followed by the equilibrium points, in the 
proposed analysis, during the iterative calculations. 
Although depicted as straight vectors, the path from L to E 
in the iterative analyses is, in fact, a non-linear one. 

The above results indicate that, despite the 
approximate assumptions introduced, a very reasonable 
estimate of the equilibrium points is possible using the 
generalized solution developed in Chapter 6, coupled with 
the closed form solution presented in Section 7.2.1. 
Moreover, the procedure, derived to account for the delayed 
installation of a lining in a non-linear elastic ground, 
seems to furnish safe estimates of lining loads and 
displacements at the opening which in turn lead to safe 


estimates of the ground settlements. It should be noted that 


TiS32 


the generalized solution for the ground response was used 
without restrictions and, indeed, at certain points and 
Situations, it was used beyond its limits of strict validity 
CSEG@E1ON sO. owls). 

For design purposes, it seems preferable to use a 
ground modulus compatible with the final tunnel closure for 
the tunnel closure. As this is not known beforehand, a 
Simple algorithm was developed to generate the solution, 
which is found after 3 to 5 iterations. 

In solving the problem at hand, a calculation sequence 
emerged. This sequence will be summarized and discussed in 
Section 7.3 aS a guideline for practical use. Other 
sequences or assumptions could have been introduced, and 
were in fact, attempted with end results that did not differ 
entirely from those shown. The one presented herein, 
however, seemed to be the most convenient and easiest for 
practical use. 

There were two basic assumptions in the development of 
the above solution. Firstly, the effect of the delayed 
lining installation is accounted for by reducing the ground 
stress field, through a reduction of the unit weight of the 
soil. The latter is obtained from an average amount of 
stress release, calculated using the A curves (normalized 
ground reaction curves), for an estimated tunnel closure at 
the point the support is installed. Secondly, the reduced 
tangent stiffness of the ground to be considered in the 


interaction analysis, is that given by the A' curves or the 


1153 


derivative of the normalized ground reaction curves. It is 
assumed that the ground stiffness is uniquely related to the 
tunnel closure through these curves, which are assumed to be 


unaffected by the lining action. 


7.2.3 Influence of the Lining Presence on the Ground 
Settlements 

The parametric analyses presented in Chapter 6 
considered the tunnel to be unlined and, therefore, the 
resulting normalized settlement distributions shown, for 
instance, in Figures 6.48, 6.49, 6.59 and 6.60 (see also 
Appendix C) disregard the influence of a lining on the 
ground movements. If a fairly rigid lining is installed 
after a substantial amount of stress release, the increment 
of ground movements during the soil-lining interaction may 
not be very significant compared to the magnitude of 
movements developed in the pre-support phase. Hence, the 
settlements developing in the pre-support stage will tend to 
dominate. However, if the lining is flexible or 
compressible, then the settlement after lining installation 
can also be significant. This component of the total ground 
movement is the one affected by the presence of the lining 
in a two-dimensional tunnel representation. 

The results of the finite element analyses discussed in 
Section 7.2.2 are helpful for the assessment of the lining 
influence on the ground settlements. The solid curves in 


Figures 7.7 and 7.8 represent the final distributions of 


1154 


subsurface and surface settlements, calculated for the cases 
with ground stress release of 40 and 80% prior to lining 
installation. The broken curves represent the distributions 
calculated for an unlined opening, with crown settlements 
approximately equal to the final crown displacement 
calculated for the two lined tunnel cases. Since the 
geometries and soil properties are the same in all cases, 
the differences between solid and broken curves reflect the 
influence of the lining on the ground settlements. In the 
first case, the settlements for the unlined tunnel 
correspond to a stress release of 55%, to which a crown 
settlement of 8.97 mm was calculated (against 8.88 mm 
obtained in the lined case). In the second case, the 
settlements for the unlined case correspond to a stress 
release of 80%, and a 16.60 mm crown displacement (compared 
with 16.10 mm obtained in the corresponding lined case). To 
obtain a match of crown settlements for the lined and 
unlined cases, interpolation of displacements between 
consecutive unloading steps in the unlined case would have 
been required, but this was not attempted. 

For equivalent crown displacements, one notes that the 
related settlements in both lined cases are smaller than 
those obtained for the unlined situation. Although the 
ground surface distortions are not much different for the 
lined and unlined cases, the latter tend to be marginally 


higher. 





Elevation (m) 





Legend 
o FE Lined 
+ FEUatined 











+ 


Settlement (mm) 
4 i) 


Legend 


o FELIned 


Tae leo 





> FE Unlined 











onal y aes ete ide, CO 20789 240 28 
Distance to the axis (m) 


Figure 7.7 Distribution of Final Settlements Calculated for 
a Tunnel Lined after 40% Stress Release, Compared to the 


Distributions for an Unlined Tunnel Equal Crown Displacement 


1456 





Elevation (m) 





a le Legend 
° 


FE Lined 


4 + FE Unlined 


Odie A lA eeu 16 
Settlement (mm). 














Settiement (mm) 











) 4 Sh Gizt\ add? w2bhuw2hianz8 
Distance to the axis (m) 


Figure 7.8 Distribution of Final Settlements Calculated for 
a Tunnel Lined after 80% Stress Release, Compared to the 
Distributions for an Unlined Tunnel with Equal Crown 


Displacements 


iPhone? 


One notes, furthermore, that as the amount of ground stress 
release before supporting the tunnel increases, the 
differences in settlements for the lined and unlined cases 
decrease. The pre-Support displacements tend to dominate 
over the after-support movements as the lining installation 
is increasingly delayed. The incremental movements after the 
liner is installed tend to reduce as the ground stress 
reduction increases, but also as the support becomes stiffer 
relative to the ground with increasingly delayed 
installation of the lining. At the instant the lining is 
installed for the 40% stress release case, the relative 
Stiffness: ratiositare: 8=0).005 and a=0.96s Forethe 80% stress 
release case, they are B=0.015 and a=2.85. The additional 
ground softening resulting from the extra. 40% ground stress 
release, caused the same lining to behave three times 
stiffer. 

The presence of a lining inhibits the 'flow' of soil 
into the tunnel which is unrestrained in the unlined case. 
Thus, the volume of soil being lost into the opening is 
reduced. Moreover, if the lining squats, soil elements 
adjacent to the opening will be pushed outwardly and this 
May generate an opposing displacement field in the ground. 

This mechanism was found to develop for other ground 
and lining conditions, and it may be said that by neglecting 
the presence of the lining, a conservative estimate of the 
ground settlement distributions above the tunnel will 


normally be found. 


1158 


Accordingly, the normalized settlement distributions 
obtained in the parametric analyses and presented in 
Appendix C, can be used in practice, since it is sufficient 
to calculate the amount of stress release in the unlined 
tunnel solution, which causes the same crown settlement 
obtained in the lining-ground interaction analyses. This 
amount of stress release is easily obtained from the 
normalized ground reaction or A curves defined for the 
tunnel crown by inputting the calculated dimensionless crown 
displacement found at equilibrium. As the sets of normalized 
settlement distributions were obtained for particular values 
of H/D, K,, @ or c,/yD, and amounts of stress release, some 
data interpolation may be needed. Alternatively, the sets of 
data corresponding to the ground properties nearest the 
actual one are selected and the data interpolation is 
restricted to finding the normalized settlements for 


intermediate values of H/D or stress releases. 


7.3 Guidelines for Using the Proposed Design Procedure 

The purpose of the present section is to suggest and 
discuss a sequence of steps to be followed when applying the 
proposed design procedure. 

Following the general scope of the present work, 
emphasis is given to some of the geotechnical aspects of a 
shallow tunnel design. No attempt is made to address, for 
instance, the structural design of the lining, although, the 


output of the present procedure may serve as input for this. 


159 


Geological and geotechnical investigations, though playing a 
Paramount role in underground project designs (see, for 
example, U.S. National Committee on Tunnelling Technology, 
1985) are discussed just briefly with reference to what 
directly concerns the proposed procedure. 

Considering the relative simplicity of the procedure, 
and in spite of its identified limitations which are derived 
from the assumptions adopted in its development, the 
proposed method seems to be useful for sensitivity studies. 
These are frequently performed in feasibility projects and 
in basic designs, regarding alignment optimization, 
selection of construction procedures, the assessment of 
influence of parameter variability, etc. Moreover, its 
Simplicity offers considerable attraction for design 
reevaluation during tunnel construction, as part of an 
observational design approach. The method can be applied and 
calibrated simultaneously Wit et eld monitoring, tis 
serving aS an auxiliary tool for design feed-back and for 
decisions being made during construction. All calculations 
involved are simple and easy to perform. The entire 
procedure can be implemented in a small micro-computer. 

The guidelines presented herein were developed for the 
cohesionless soil model described in Chapter 6. However,it 
can be easily adapted to the frictionless soil model which 
provides a better representation of the behaviour of 


Saturated soils under undrained loading conditions. 


1160 


Figure 7.9 shows a flow chart that summarizes the main 
sequential steps of the proposed procedure for shallow 
tunnel design. The geometry of the problem is initially 
assessed. A range of tunnel covers along the alignment is 
defined or critical sections are selected. If the tunnel 
contour is non-circular, and provided it does not deviate 
too much from a circle, the diameter of a circular profile 
with equal excavation area is defined. 

The geological conditions are then assessed. The design 
procedure was developed for time independent conditions, so 
it is important to identify the groundwater conditions, soil 
permeabilities and coefficients of consolidation. The method 
asks for uniform ground conditions, at least in the 
subsurface profile from half diameter above to half diameter 
below the tunnel. One has to verify whether or not ground 
uniformity can be assumed for this horizon. Mixed face 
conditions, for instance, cannot be handled by the proposed 
procedure. Typical ground properties, or their ranges are 
then defined. If triaxial test results from good quality 
undisturbed samples are available, the input parameters for 
the hyperbolic model can be obtained. Undrained or drained 
Parameters are defined according to the type of analyses to 
be carried out. A profile of an in situ tangent modulus is 
carefully defined, and possibly adjusted according to 
available results ofeingsitu tests (for@instances 
pressuremeter tests). A reliable estimate of the in situ 


stress conditions is also needed. This is an important issue 


GEOMETRY Ranges of cover 
Equivalent diameter 








GEOLOGY AND 
GROUND 
PROPERTIES 


Groundwater condition 
Y. Koy c, , Ree E 
Variabilities 


a Mg k, cy 


c-- 


Construction procedures 
Support system 

Support activation 
Variabilities 


CONSTRUCTION 
TECHNOLOGY 


Check on: 

Ground control conditions 

Risks of ground collapses (local?) 
Soil type (strain weakening?) 

Soil volume changes? 

Drained vs undrained response? 


APPLICABILITY 
OF 
DESIGN METHOD 


Y 
PRE-SUPPORT 
GROUND 
RESPONSE 


LINING-GROUND 
INTERACTION 
ANALYSIS 





--—-7-----y7--- 


MeERSS2 


Tunnel closure 
Amount of stress release ——————_ 
Ground stiffness change 

Limits of validity 


Stability check (3D,2D) 
Expected ranges of responses 











Iterative procedure 

Ground heave assessment 
Distributions of M, T 

Lining distortions 

Expected ranges of lining response 





Final tunnel closure 
Subsurface settlements —————» 
Surface settlements 
Maximum ground distortions 
Expected ranges of displacements 


PREDICTION OF 
GROUND 
OLSPLACEMENTS 










CONSTRUCTION 
FOLLOW-UP AND 
MONITORING 


Cee ee ee A OS a er es 





Figure 7.9 Suggested Sequence of Steps for Application of 


the Proposed Procedure for Shallow Tunnel Design 


1162 


in the analysis, since both soil and lining performances are 
very dependent on these conditions. Unfortunately, 
measurement of K, in soils is not a simple task. Recent 
development in push-in stress cells in soft to medium clays 
have shown promising results (Chan and Morgenstern, 1986). 
If specific site data is not available, information and 
field evidence from excavations in nearby areas can provide 
some help. Additionally, the variability of strength 
properties, deformation parameters, etc, should be carefully 
assessed in order to define parameter envelopes. Special 
attention should be paid to geological features that may 
likely control local stability conditions. This includes 
fissures, bedding, sand lenses, etc. Local collapses or 
cave-ins typify poor ground control conditions under which 
the present design procedure ceases to be valid. Such 
occurrences however, can be entirely avoided by appropriate 
construction techniques or properly chosen construction 
methods. 

Regarding the latter, the design procedure requires a 
complete knowledge of the method of tunnelling to be 
undertaken. As originally developed, the present design 
method is strictly applicable to full face tunnelling, or to 
tunnelling with minor face excavation staging. In section 
7.5, however, the procedure is tested against a few cases 
that deviate from these conditions. A key issue is the 
identification of the location where the support is 


activated and this does not necessarily coincide with the 


Vre:S 


point of lining assembly, especially in grouted or expanded 
lining systems. In shotcrete linings, the activation point 
seems to correspond to the section where the Support ring is 
closed. The proposed method is also strictly valid to good 
lining-ground contact conditions. While this is normally 
ensured in shotcrete linings, it may not prevail in some 
prefabricated linings installed in sections with large 
overbreaks ‘caused by excessive overcutting or local 
collapses, or in poorly grouted or backfilled supports. The 
support characteristics and properties should be defined, 
and their variability assessed (particularly for a shotcrete 
Support). The expected ranges of lining installation delay 
also have to be estimated. 

The proposed design procedure should be used preferably 
within its range of applicability, avoiding for instance, 
extrapolation of results beyond the established limits (see 
Section 6.5) of the stress release and stiffness reduction 
normalized curves or corresponding equations (Section 6.4). 
Accordingly, interpolation of data should be performed 
within the ranges of variables used in the parametric 
analyses (Section 6.2.5). Similarly, the estimates of tunnel 
closure using the approximate solution developed in Chapter 
5 should be made having in mind the limitations and 
restrictions discussed in Sections 5.3.5.2 and 5.3.6.1. 
Broadly speaking, the method should be used only when good 
ground conditions are ensured, so that near collapse 


Situations are precluded. A tentative criterion to identify 


1164 


such situations was set up in Section 2.3.4.3. Results of 
tunnel model tests indicated that a limiting value for the 
dimensionless crown displacement associated with near 
collapse Laraxelonetesn be defined. Values of U in excess of 
1.8 were generally indicative of near collapse conditions in 
those tests. A good ground control condition would 
necessarily mean U values at the crown smaller than that 
figure and possibly less than 1.0. The interpretation of 
field data in a number of case histories (see Section 
5.3.6.1) seemed to confirm the proposed criteria. 

The lining-ground interaction analysis used in the 
proposed design procedure is strictly applicable to good 
lining-ground contact, aSedefined) in Sectionn2, 355,08 The 
qualityvof this contact, dependsponidi tferentatactors: as 
discussed in that section. In prefabricated lining systems 
it depends to a large degree on the size of the space left 
unfilled behind the support. As suggested in Section 
2.3.5.4, the maximum allowable overbreak at the tunnel crown 
can be calculated from a limiting increment of dimensionless 
crown displacement, which was estimated to be 0.5 to 0.65. 

Moreover, the proposed design procedure is not to be 
used for soils with stress-strain behaviour departing 
appreciably from that described by a hyperbolic 
relationship. Conditions involving appreciable ground volume 
changes (dilation, consolidation) also cannot be handled by 
the proposed procedure. The type of analysis to be 


performed, which can be either undrained or drained, should 


1165 


be assessed independently, according to the ground profile, 
soil type, itS properties and the construction scheme to be 
used (lining type, rate of advance, etc). The simplified 
criteria set up in Section 3.3.4.5 can be helpful for this 
assessment. 

The next step in the design sequence is the evaluation 
of the pre-support ground response. This requires an 
estimate of the tunnel closure, which can be made through 
the procedure developed in Section 5.3.5.2. The 
dimensionless radial displacements at three points of the 
tunnel contour can be estimated once the distance (x) behind 
the face where the support will be activated and the in situ 
stress ratio K, are both known. If a drained analysis is 
made using the cohesionless soil model, then the amount of 
Stress release and the ground stiffness changes at the point 
the support is activated can be assessed, using the 
solutions derived in Section 6.4.3 and as explained in 
Section 7.2.2. Adjustments of the friction angle will be 
needed for soils with a non-zero cohesive strength component 
and with failure ratios different from unity. For a variable 
in situ deformation modulus profile, the radial 
displacements at the tunnel contour are normalized to the in 
situ moduli at points located half diameter radially away 
from the opening, as explained in Section 6.2.4.2. Reference 


values for the amount of stress release (a,..,) and for the 


ref 


dimensionless displacements at tunnel contour (U,,,) are 


ref 


found for the particular geometry (H/D), the in situ stress 


1166 


ratior(K,) "and "forithe adjusted inictionvangle (mei) taine 
reduced unit weight of the soil and reduced ground 
stiffnesses are found by averaging the stress ratios (Z) and 
the current stiffnesses (E,) at the point where the support 
is activated. These are obtained through the twice 
normalized ground reaction curves and their derivative 
functions A and A' curves, see Section 6.4.3). 

The three dimensional stability condition of the tunnel 
is verified, both at the face and at the unsupported 
heading, through some of the methods discussed in Section 
4.3.4. A check of the two-dimensional tunnel stability is 
also made, using for example, the solutions presented in 
Section 6.5.2, taking into account the reduction of the 
ground stress calculated at the point the support is to be 
activated. Provided the calculated factors of safety ce 
acceptable (greater than or equal to 1.3 to 1.6; see data 
and discussions on Sections 2.3.4.3 and 4.2.3), the 
construction procedure is applicable and the proposed design 
method can be used. Otherwise, changes in the construction 
method have to be considered or additional ground control 
procedures, such as those discussed in Section 4.2.3, should 
be implemented. 

Expected ranges of pre-support ground response are thus 
defined and used as input for the lining-ground interaction 
analysis. The latter is carried out using the solution 
presented in Section 7.2.1), following@themuteratuve 


procedure described in Section 7.2.2. Attention should be 


ons 


paid to the evaluation of ground heave, which tends to be 
overestimated by the solution given in Section 7.2.1. 
Distributions of lining loads So ONC including 
bending moments and thrust forces, calculated for the ranges 
of expected ground responses. They are used in the 
independent structural design of the lining. Lining loads 
and distortions are checked for acceptability. Total tunnel 
closure is calculated and checked if admissible. If the 
estimated lining-ground response is not acceptable the 
construction technology has to be reviewed accordingly, and 
may require changes to the lining system or in the 
construction procedures. 

The total crown displacement is then obtained. The 
associated amount of stress release is calculated through 
the A curve for the tunnel crown, as explained in Section 
7.2.3. Finally the subsurface and surface ground settlements 
are calculated using the distributions of normalized 
settlements included in Appendix C, and may require some 
data interpolation. If the calculated ranges of surface and 
Subsurface settlements are not acceptable or if the risk of 
damage to existing structures is high (see Section 4.2.2), 
then again, the construction procedures have to be 
reassessed or additional ground control measures considered. 

Finally, during the construction follow up and field 
monitoring, the anticipated ground and lining performances 
can be verified. Reasons for possible departures from the 


predicted behaviour can be assessed and back analysis of 


1168 


performance can be made, calibrating the proposed procedure 
to a particular site condition. This feedback process may 
provide further insight on the tunnelling activities and may 
help in reevaluating the design and construction, as well as 
Support new decisions to be made. 

To facilitate some of the calculations involved in the 
application of the design sequence described, Figures 7.10 
to 7.14 were prepared. As explained earlier, they were 
developed for applications using the cohesionless soil 
model, but they can be easily adapted for the frictionless 
model. The present calculation sheets include the geometry 
of the problem, the ground properties and the tunnel closure 
at lining activation (Figure 7.10), reference values used to 
assess the amount of stress release and the ground stiffness 
at the section the support is activated (Figure ye Ge et 
two dimensional ground stability verification and input data 
for the lining-ground interaction analysis (Figure 7.12), a 
Sheet for the iteration calculations (Figure 7.13) and 
finally a Sheet for the subsurface and settlement 
calculations (Figure 7.14). For easier reference, some of 
the equations that are used were reproduced in the 


calculation sheets. 


7.4 An Example of the Use of the Proposed Design Procedure 
The Alto da Boa Vista Tunnel built in Sao Paulo, 
Brazil, is used as an example of the application of the 


proposed design procedure. This case history was described 


1169 


|. GEOMETRY 
2. GROUND PROPERTIES 
2 : /c) tang + 
gq =arc sin | Sees ons ei = SA pw) 
de = arc sin (1-Ry + Re csc) a onen os 
2 sin d 


2c cos® +2oazsing 


Fee ee eS ee 


ieee eee cacti 


at S 
3. TUNNEL CLOSURE AT LINING ACTIVATION 


1/2 Obelow F 
Tee [eee [eae Pe 


Figure 7.10 Calculation Sheet with Geometric Ground 


2 
Re (1-sing)( o)- a2) 
Eon = K Pg (23) pe 










Properties Data and Tunnel Closure at Lining Activation 


1170 


4) REFERENCE VALLES 
ee el ar 





5. STRESS RELEASE ANDO STIFFNESS CHANGE 
AT LINING ACTIVATION 





Ey = (Epc+ 261g + Etp)/4 = feet pas] 


Figure 7.11 Calculation Sheet for Estimates of the Stress 


Release and Ground Stiffness at the Section the Lining is 


Activated 


1472 


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1173 


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1174 


in detail in Section 5.2.4.2, where it was back-analysed 
through the two-dimensioal finite element model presented in 
Chapter) Ss. In the present section jaas well as in the 
following, no attempt will be made to best fit the observed 
performance. The case history data are used as a design 
case, where the most likely tunnel response in geotechnical 
terms, is the matter of interest. For consistency, however, 
and to allow a comparison with the results of the numerical 
back-analysis performed earlier, the same parameters then 
used will be applied here. 

The tunnel had a soil cover of 6.2 m and a near 
circular excavation profile with 3.9 m height and 4 m width. 
The equivalent diameter of a circular excavation with an 
equal area is 4 m. 

The tunnel was driven above the water table through the 
variegated silty sand and a drained analysis using effective 
stress parameters and zero pore pressures is justified. The 
ground properties and geometric data are reproduced in 
Figure 7.15. The justifications for using these soil 
Parameters were given in Section 5.2.4.2. An in situ stress 
ratio of 0.8 was selected in the present analysis (as 
opposed to the 0.75 ratio back analysed earlier), in pfded 
to simplify the application of the generalized solution. 
Interpolation of data regarding the K, value was thus 
eliminated. Note that since the soil cohesion is zero and 
its failure ratio is equal to one, there is no need to 


adjust the friction angle using equations 6.3 and 6.10 which 


Vig S 


are reproduced in Figure 7.15. The variation of the in situ 
tangent modulus with depth is given by the last equation in 
Biouress.di,lusing thescatculated inesitueprincipal 
stresses. The in situ modulus at half a tunnel diameter 
above the crown, at the springline elevation and half a 
tunnel diameter below the floor are thus calculated and will 
be used to normalize the radial displacements at the 
corresponding points of the tunnel contour. The moduli 
obtained are marginally higher than those used in the 
numerical back-analysis, as a slightly higher K, was assumed 
here. However, they are not discordant with the in situ 
deformation modulus profile, as obtained through 
pressuremeter tests (see Figure 5.14). 

In a routine design application, ranges of geotechnical 
Parameters, such as those given in Table 5.9 for the 
variegated soil, would be used instead of the single 'most 
likely' set of parameters being considered herein. 

As described in Section 5.2.4.2, the tunnel 
construction was performed under good ground control 
conditions, that were provided both by the favourable ground 
and by the good construction quality achieved. The shotcrete 
Support and the face excavation were installed in stages. As 
suggested earlier in different sections of this thesis, it 
seems reasonable to allow in such a support type, that its 
activation takes place when the shotcrete ring is closed at 
the floor. Therefore, the radial closure of the opening, at 


the point the lining is activated, is calculated at a 


1176 


LP GEOMETRY 


2. GROUND PROPERTIES 


K =| 400 n= 


scoring [0] os (247) = [Fe | 
k, | + (03/c) tang A 
Qa = arc sin ceca 5 024 Saal) 
de = arc sin (1-Ry + Re csc$) | = 
Cd eee Re. 2 7 
i-singde 


po = [101.33kf) x 3 ('.5, wae) 


R¢ (I-singd)( a )- 3) 


Q 
fN 


Oz n : ‘ai 
E ti = K po (2S) [ : = 30477 682 (%/10/,93) 


1/2 D above C 
GtesS 


1/2 Obelow F 


2c cosd + 2a02sing 


67.20 26.0) 
130, 40 20,70 
193, 60 33,89 
















av, 30,326 


3. TUNNEL CLOSURE AT LINING ACTIVATION 


G 0, $87 0.01IS2ZS§ 
s 0, 469 
F 0,352 


0.01385 F2 
Figure 7.15 ABV Tunnel - Input Data and Tunnel’ Closure at 













0,019026@ 





Lining Activation 


17 


section located 2.6 m behind the heading face (x/D=0.65). 
Possible variations in the depth of heading advance or 
distance of lining closure behind the face, would have to be 
considered in a real design calculation and thus a range of 
tunnel closures at the lining installation point would have 
to be considered. For the present, it is sufficient to 
contemplate the most probable delay of support application 
in this case history. Being a shotcrete support, the 
lining-ground condition can be defined as good. 

Tunnel closure at crown, springline and floor are 
estimated using the approximate solution given in Section 
5.3.5.2 (Figures 5.94 to 5.96). If the design procedure was 
being used during construction, the actually observed tunnel 
closure would possibly be known. The measured radial 
displacements would, thus, be used instead of the estimated 
values. As shown in Figure 7.15, the dimensionless crown 
displacement is smaller than 1.0 and this is consistent with 
the good ground control conditions met in this case history. 

The stress-strain curves of the variegated soil are 
very closely represented by a hyperbolic relationship. In an 
actual design activity, the risk of local ground collapse 
would have to be assessed through a close inspection of the 
locally occurring geological features and through available 
Stability solutions which can provide the means for 
assessing the face and unsupported heading stability. In the 
present analysis this is not needed since the tunnel was 


already built, and no local or global instability process 


1178 


was detected during construction. Except for some 
contraction likely to occur in this soil upon shearing, the 
overall volume changes expected in this ground were small, 
and this is partly due to the good ground control conditions 
implemented. No appreciable volumetric changes associated 
with changes in the mean normal stresses were expected, 
partly because the overconsolidated nature of the deposit. 
For all these reasons, the proposed design method seemed to 
be fully applicable to the case history considered. 

The next step consists of the evaluation of the ground 
Stress release and of the ground stiffness at the point of 
Support activation. The corresponding calculations are shown 
in Figure 7.16. The solution set for K,=0.8, summarized in 
Figures 6.100 to 6.107, is used. With the calculated 


strength factor (m-1) (Equation 6.30), a is obtained from 


ref 
Figure 6.101 and the ratio a,,,/U,., is determined through 
Figure 6.100, for the cover to diameter ratio being 
considered. The reference dimensionless displacements at the 
crown, springline and floor are thus obtained. The 
corresponding ratios U/U,,, are then calculated and checked 
if they are greater than the limiting ratios defined in the 
solution generalization. In this case, the U/U,,, values lie 
to the left of the terminal points of the fitted function, 
represented by the right vertical bars in Figures 6.102 to 
6.104, and so no data extrapolation is involved. Using the 


last three figures, the values of A are obtained and the 


corresponding stress ratios (Z) at the crown, springline and 


VETS 


4. REFERENCE VALUES 


a re Ce 


0.825 0,62 0.603 
0.46S 4 0. 4s 
0.435 ZO2 0.21¢ 


BS oMmecoenelen oe AND STIFFNESS CHANGE 
AT LINING ACTIVATION 


ond 













0,587 | 0.973 No 
0.469 L130 No 
0,382 | 1.637 Ne 








Pree ew ee 
C 0.97 0,44| 
Los OS le 






T+ F02E5+5,00 =[2.471 | 


145 hikes [ose] 
ie ais [7536] kW/m? 


STIFFNESS 7 6, 2(X/)i) Ey, 





Ey =(Erc+ 2615+ Ere )/4 =| 13,962 | Mf 


Figure 7.16 ABV Tunnel - Stress Release and Ground Stiffness 


at Lining Activation 


1180 


floor are calculated using the definition given by equation 
6.23. The stress ratios are averaged and the reduced unit 
weight of the soil (y,,,) is calculated according to the 
definition given by equation 7.14. 

Similarly, the derivative of the twice normalized 
ground stress release curve (A') are found, for the 
calculated U/U,,, values at lining installation, using the 
curves; given in Figures 6.105 to 6.107. The current ground 
moduli are then calculated using equations 6.27 and 6.29 (or 
Figure 6.116). They are then averaged and E, is obtained. 

It should be noted that since the ratio H/D lies 
between two values considered in the parametric analyses 
(1.5 and 3.0), some interpolation of data may be needed to 
obtain A and A'. Since the solutions for those two values 
lie very close together, a linear interpolation is 
sufficient. As an alternative, the nearest available 
solution (H/D=1.5) could be used, as the error involved is 
insignificant. If the analysis is done through a programmed 
solution, A and A' would be calculated from equations 6.24 


and 6.26, using the parameters, P,, shown in Table 6.16. 


is 
One can note from Figure 7.16 that at the section the 
lining was installed, the ground stress had been reduced by 
52.9% and the average ground modulus had been also reduced 
but by 55.6%. The former ground stress reduction is higher 
than that found in the back analysis of this case history 


presented in Section 5.2.4.2 (40%) and also higher than the 


amount given in Section 5.3.7 (59%). This is because a 


1181 


Slightly higher K, value has been adopted herein, as well as 
considering an increased distance between face and point of 
lining activation (x/D=0.65), and other approximations 
involved in the proposed calculation method. 

A two-dimensional stability check would then be made. 
This is done here using the lower bound solution presented 
in Section 6.5.2. For instance, one could say that $=30° and 
the tunnel collapse pressure, p,, is obtained from Figure 
6n125,eWitnec=0 and H7ZD=1°955% Thecfactor*of safety *of the 
ground is calculated using equation 6.34 and ve 1S °founduto 
be about 1.4 (see Figure 7.17). This value is totally 
acceptable, thus the construction procedure and the proposed 
design method are applicable. 

The next design step is the ground-lining interaction 
analysis. The values of y,,, and E, are used as input in this 
analysis. The depth to the tunnel axis is taken as the soil 
cover plus the half tunnel diameter. The analytical solution 
assumes the lining thickness to be small. Thus, an average 
Mininguradius jof 95eme(for aeshoteretethiningel0Vemethick) 
is considered. Following the discussions presented in 
Section 5.2.4.2, a reduced shotcrete modulus of 10 GPa is 
adopted (instead of the 8.65 GPa modulus back analysed in 
Seet 2ong5e2 5402). The aradiusper FpeofAthetarbitrary circle at 
which the vertical ground heave displacement is set equal to 
zero (see Section 7.2.1), would be chosen as that tangent to 
a stiffer horizon below the tunnel. According to the 


description of the local geology (Section 5.2.4.2), it would 


1182 





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1183 


be at 11.85 m. This is however, more than the depth of the 
tunnel axis, and would likely lead to excessive ground 
heave. Instead, r, is selected so that the arbitrary circle 
becomes tangent to the ground surface (r,=8.15 m). The 
coefficients of relative stiffness can then be calculated 
(equations 7.1 and 7.2) and are found to be a=52.7 and 
B=0.0116 (Figure 7.17), which are typical for a shotcrete 
lining in a medium clay (Table 7.1). 

The overall ground heave is calculated using equation 
7.10 (Figure 7.2) and it is found to be 2.54 mm (see Figure 
7.17). The analysis is conducted using the solution shown in 
Figures? ja) Cequations-7.3 and 7.7). Increments of radial 
displacement Au, are found. Using equation 7.6, these are 
added to the heave displacement. The final radial 
die placements are found after adding those calculated at 
lining installation (Figure 7.15). New U and U/U,,, values 
are found {or) the three points of the contour. New A" and 5, 
are thus obtained and a new £&, is found to be equal to 13.73 
MPa, which is to be compared to that formally calculated 
(13.482 MPa). The new modulus is found to be 1.8% greater 
than the previous one. A second iteration analysis is thus 
performed, using the new modulus as input. (Figure 7.18). A 
new ground heave is calculated and the calculation is 
repeated. Another E, is found (13.623 MPa), which differs 
from the old value by less than 0.6%. Therefore, no more 
iterations are needed. The final radial stresses and 


displacements are thus found, as shown in Figure 7.18. As 
\ 


1184 


explained in Section 7.3, bending moments and thrust forces 
in the lining are also obtained in this calculation but are 
not discussed. Note, moreover, that the normalized final 
dimensionless displacement (U/U,,,) at the tunnel floor 
exceeded the limit of the fitted A and A' functions for this 
point (see Figures 6.104 and 6.107), unlike at the other 
points of the tunnel contour. Notwithstanding this, the 
extrapolation of data at the floor did not impair the 
quality of the results obtained, as will be seen later. 

A total crown settlement of 8.1mm is obtained, which 
leads to a U/U,,, value of 0.882. Using the A curves shown in 
Figure 6.102, it is found that this crown displacement 
corresponds to A equal to 0.1 and, therefore, to a Stress 
release at the crown of 47.25%. As recommended in Section 
7.2.3, the induced ground settlements are obtained through 
the results of the parametric analyses (Appendix C) for the 
unlined tunnel case, considering the amount of stress 
release (=47%) which causes, in the unlined tunnel, the same 
crown settlement obtained in the lining ground interaction 
analysis (Figure 7.19). 

To obtain the distribution of subsurface settlements, 
the normalized subsurface settlement plots, such as the one 
shown in Figure 6.59, are used. The corresponding sets of 
normalized plots which bracket the case in hand are selected 
in Appendix C. These are taken as the plots for K,=0.8 and 
gea0, and for H/D=1.5 and 3.0. Curves for these two cover 


to diameter ratios, for 47% stress release are obtained by 


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1186 


linear interpolation (see Figure 7.19), from the curves 
which bracket that amount of stress release (i.e., the 40% 
and 50% curves). Curves similar to those shown in Figure 
6.62(a) are thus obtained. This is followed by another data 
interpolation, at equal normalized depth to cover ratios 
(z/H), so that the normalized settlement curve for H/D=1.55 
is obtained (see Figures 6.62a and 7.19). The distribution 
of subsurface settlements is immediately calculated as the 
crown settlement is known. Note that to find the 
distribution for the actual friction angle (¢=29°), the 
process would have to be repeated once more for g=20°, and 
the normalized displacements would have to be interpolated 
from those| found. for the 20% and 30° @ruictaon sangles.))c4s 
Simpler, as well as sufficient, to select the set of 
normalized plots for the friction angle which is closer to 
the actual one, or to assume the angle that leads to a more 
conservative settlement eat ima tet The same comments apply to 
intermediate values of the in situ stress ratio (K,). The 
above calculations can be relatively tedious to perform if 
done manually. However, they can easily be programmed and 
performed in a small micro-computer. 

The distribution of surface settlements are obtained 
through an identical procedure, except for using the 
normalized surface settlement plots, such as the onewencen 
in Figure 6.60. Similar interpolation procedures are 
undertaken, after selecting the sets of normalized plots 


from Appendix C which bracket the case being investigated. 


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The interpolations are performed at equal normalized 
distances to the tunnel axis (Y/D). The resulting 
calculations are summarized at Figure 7:19, where the 
calculated maximum surface distortion is also indicated. 

If it was only necessary to know the maximum surface 
settlement, then the above calculations would not be needed. 
It would be adequate to use the relationships between 
surface and crown settlements presented in Section 6.3.2.3. 
For the present case, the solution would be found using the 
relationships given in Figure 6.77 to 6.80. 

The calculations above complete the basic geotechnical 
design. Obviously, the lining design is an activity in 
itself, which will not be covered herein. The structural 
design of the support can, however, make use of the lining 
internal forces’ (N,, M,, Q,) calculated through equations 
7.11 to 7.13, (Figure 7.2) with the input parameters used in 
the second and final iteration (Figure 7.18). 

It was suggested in Section 7.2.2 that the ground heave 
as calculated through equation 7.10 tends to be excessive. 
In fact, Hartmann's solution for this aspect of the shallow 
tunnel entails an indeterminate degree of approximation, 
represented by the value of the arbitrary radius r,. Before 
a comparison between calculated and measured behaviour is 
made, it seems convenient to explore the influence of the 
calculated heave on the overall tunnel response. For this, 
the calculations presented in Figure 7.17 were repeated, 


assuming, that v",, was equal to zero. The resulting 


1133 


calculations are depicted between parentheses in Figures 

Ju lJje and 7.185 AS expected, the total»yradialsdisplacement at 
the crown increases, as the floor displacement decreases, 
while the springline radial displacement remains unchanged. 
The average ground modulus Ee, calculated at the end of each 
iteration, however, does not change by any significant 
degree. This seems to reflect the compensating effect of the 
increased crown displacement against the decreased floor 
heave. As a result, the radial stresses onto the support at 
equilibrium, do not change significantly. In essence, the 
effect of the overall ground heave given by equation 7.10 is 
mostly restricted to the vertical displacements at the floor 
and crown. Therefore, it should affect the subsurface and 
surface settlements which are normalized to the crown 
settlement. Settlement distributions were calculated with 
the zero heave assumption (not shown in Figure 7.19), and it 
was found that the amount of stress release in the unlined 
tunnel solution, which produces the same final crown 
settlement obtained in the interaction analysis (10.61 mm), 
was about 56%. 

Figure 7.20 presents a comparison between the observed 
and calculated final subsurface settlements at the ABV 
tunnel. The calculated ground movements neglecting the heave 
component given by equation 7.10 are larger than the 
measured values, showing that this assumption leads to 
conservative settlement predictions. The calculated 


settlements including the heave component are, on the other 


1190 


hand, very much in agreement with the field measurements. 

A similar comparison regarding surface settlements is 
presented in Figure 7.21. Again the zero heave assumption 
leads to conservative estimates of the surface movements. 
The larger number of measurements includes a greater scatter 
of data so that both the heave and the no heave assumptions 
lead to results are contained well within the measurements. 

The calculated and measured radial stresses acting on 
the lining are shown in Figure 7.22. Note that the 
calculated stresses are not sensitive to the heave 
assumption. The proposed design procedure seems to 
overestimate the ground loads onto the lining, except at the 
floor, where a good match is obtained. Bearing in mind the 
known capitan tes associated with the installation of 
contact pressure cells notably at the upper arch of the 
lining, the agreement between measured and calculated stress 
seems fairly reasonable. 

Finally, Table 7.4 summarizes the comparisons between 
the measured and calculated performance for the ABV tunnel. 
Note that the tunnel radial closure at the face was 
calculated using the approximate solution presented in 
Section 5.3.5.2 (aspects 3 and 6). The maximum horizontal 
convergence of the lining (aspect 5) was calculated by 
discounting the springline movements that took place ahead 
of the tunnel face (not measured). The total settlement of 
the lining roof (aspect 8), measured by internal levelling 


in the field, was calculated by discounting the crown 


SH 


Cr rr OOOO eee 


Pe) 


Performance aspect Measured Calculated 
Heave'*? Heave (>) 
Included Excluded 
1. Maximum final surface 
settlement mm 5), (0) fal. 4 Bq 2 


2. Maximum slope of final 
settlement trough at 


surface - 169) pSO@) 1:2,000 1:1,800 
3. Floor heave at tunnel face mm. 11657? Doh Do 2 
4. Final floor heave Praia 5-914) Wo 8.5 
5. Maximum horizontal 

convergence of the 

lining mm 0.85 25.4 DG 
6. Crown settlement at 

tunnel face mm 1.569) 3.9 B09 
7. ‘Final crown settlement mm ao) 8.1 106 
8. Total settlement of 

I Seb Uele mb colosa mn 4-5 (ihn, 2 Sa F 
9. Pressure on lining at 

the crown kPa 30 44 44 
10. Pressure on lining at 

springline kPa 35 58 58 
11. Pressure on lining at 

the floor kPa 55 S77 3)// 


Notes: (1) Average values. 
Taking into account the ground heave in the 
lining-ground interaction analysis. 
(3) Disregarding the ground heave in the lining-ground 
retraction analysis. 
(4) Measured at 0-2 m and 1.2 m below tunnel. 
(5) At 0.2 m above tunnel. 


Table 7.4 Performance at the ABV Tunnel as Measured and as 


Calculated by the Proposed Procedure 


Te 








€ 
wa -47 
S 
no 
9 
row 
Lid 
-8- 
pal 
-12 






WITHOUT HEAVE 


WITH HEAVE 


10 0 10 20 
Settlement (mm) 


30 





+ -2 


ey 





pee) 


T1e 2 


Legend 


° Weasured 


Predicted 





Figure 7.20 Calculated and Measured Final Subsurface 


Settlements at the ABV Tunnel 


ive 


Settlement (mm) 








=i29=10-8 6 -=4 -24 0 2 4 16. 8 
Distance to the axis (m) 


Legend 


O° Measured 


oF Predicted 





Figure 7.21 Calculated and Measured Final Surface 


Settlements at the ABV Tunnel 


1194 


(KPa) 
200 








150 IN SITU RADIAL STRESS 


lOO 
PROPOSED METHOD 


MEASURED 


150 200 (KPa) 


150 


200 
(K Pa) 


Figure 7.22 Calculated and Measured Radial Stresses on the 


ABV Tunnel Lining 


FESS 


settlement that also developed ahead of the face. 

Regardless of the assumption made with respect to the 
ground heave, the comparisons shown speak for themselves. If 
the results of the present calculations are compared with 
those of the best fit finite element analysis shown in 
Secure 0o)..2 P22 BCR gure 05 $9 tho 1S 22. eeand “Table 5.242.) y cone 
May appreciate that they are similar, if not better. In 
fact, the introduction of a procedure, although approximate, 
to estimate the tunnel closure prior to lining installation 
which takes into account the three dimensional nature of the 
problem, did improve some of the predicted aspects of the 
performance, particularly aspect numbers 3, 5, 6 and 8, 
which are those more affected by the 3D nature. On the other 
hand, the approximations and simplifications introduced in 
developing the proposed method did not seem to deteriorate 
the predictions of the remaining aspects. 

The time required for someone to use this design 
procedure for the first time, with the help of a 
programmable calculator, should not exceed an hour, or less 
with a microcomputer, provided the input data is fully 
digested and well defined. 

The agreement between calculations and measurements 
might have been unintentionally biased by the familiarity 
with this case study that the writer gained earlier through 
its numerical back analysis. The purpose of the present 
section was to illustrate the use of the proposed design 


procedure, more than to validate it. For the latter purpose, 


1196 


additional comparisons between measured and predicted tunnel 
performances are needed, and are provided in the next 


section for a representative range of practical situations. 
7.5 Validation of the Proposed Method 


7.5.1 Verification Against a Tunnel Model Test 

Some tunnel model tests conducted: in Cambridge in the 
seventies, present almost ideal conditions to test the 
validity of the proposed procedure to estimate the ground 
response around a tunnel. A number of verification 
calculations were undertaken using these experimental test 
results. One of them is presented in this section. It refers 
to an undrained centrifuge test in kaolin, carried out by 
Mair (1979), in which the ground reaction curve of the 
tunnel crown was obtained by Simultaneous measurement of the 
applied internal tunnel pressure and the associated 
displacements at a point immediately above the crown. The 
measured ground reaction is compared with that calculated 


through the procedure proposed herein. 


‘7.5.1.1 Model Test Procedures 
The experimental result being focussed on, refers to 
test 2DP (series II), described by Mair (Op.cit.:62). Very 
briefly, a plane strain model test was conducted at 75g 
acceleration in the Cambridge Geotechnical Centrifuge (see 
Schofield, 1980), using Spestone kaolin as the modelling 


material. The two dimensional apparatus, housing the model 


1197 


test, was completely sealed, so that it allowed the 
modelling of a saturated soil under undrained conditions. A 
Special grease and grease application technique ensured a 
plane strain condition to be met. A kaolin slurry with a 
water content twice the liquid limit was placed in the 
apparatus, and was fully consolidated under a final vertical 
Stress of 171 kPa. After unloading and removing the entire 
front of the apparatus conveniently tilted back, silvered 
perspex balls at 10 mm spacing were pressed into the clay 
surface, as markers for displacement measurements. Pore 
pressure transducers were also inserted. The model apparatus 
was then assembled in the centrifuge and an equilibrium 
stage achieved at 130 r.p.m, which corresponds to 75 g ata 
4 m radius. Water was sprayed to the top surface of the 
model to ensure saturation and a water level coinciding with 
this surface. After a few hours, pore pressure equilibrium 
was noted. The gravity scaling factor and model dimensions 
were such that, at equilibrium, the vertical effective 
stresses everywhere in the soil were less then the 
consolidation pressure applied earlier (171 kPa). The kaolin 
was therefore overconsolidated, with the degree of 
overconsolidation increasing towards the top surface. 

The centrifuge was then stopped, and a circular opening 
representing the tunnel with a 60 mm diameter, was cut using 
a specially designed tool. Test 2DP was prepared in such a 
way that the cover to diameter ratio was equal to 1.67. A 


greased rubber bag was then inserted into the tunnel. The 


1198 


centrifuge was restarted and equilibrium conditions restored 
at 75 °q. Concomitant with® centrifuge ™ speed burtd-uepetiec 
tunnel internal pressure was steadily increased to maintain 
it equal to the vertical overburden stress at the tunnel 
axis. This was ensured by a pressure line connected to the 
tunnel bag. 

Once the design acceleration was reached the centrifuge 
was kept at constant speed and the tunnel internal pressure 
immediately reduced in rapid increments, until ground 
collapse was observed. Simultaneously, photographs of the 
model were being taken, for each applied internal pressure 
reduction. The changes in the position of the silvered 
perspex balls could be monitored through the photographs. 
Ground strains and displacements could thus be obtained 
using specially developed techniques and data reduction 


procedures. 


7D slon2 sSOUL-Conditions im jthée fTest 
If the model is scaled back to gravitational 


conditions, then the prototype tunnel dimensions are found: 


D 7S, XeiOve06, sop4;. Sum 

H =). 267 9D = 57 <5:5 em 
The depth of the tunnel axis is equal to 9.675 m. 

Through laboratory testing and the measurements taken, 
Mair (1979) was able to define the profiles of the 'in situ' 
effective stress, of.K,,/ OCR andythebundrained strength; ver, 


with depth. These were shown in Figure 5.17 in Mair 


(Op.cit.) and are not reproduced here. The overconsolidation 


1799 


ratio was found to decrease from 5 at half diameter above 
Phevtunmelpetor2 atdhalfediameter belowiit sisoithatean 
average OCR would be 2.8 at the tunnel axis elevation. The 
value of K, was found to vary from about 1.3 to 0.85 between 


those two points, and, at axis elevation, K, was found to be 


° 
approximately equal to 1.0. Since the 'ground water level' 
waS Maintained at the top surface, the resulting in situ 
total stress ratio also equal to one. The undrained strength 
varied from 22 to 26 kPa between those two points with an 
average c, of 24 kPa at the axis. These values corresponded 
to the undrained strength of the kaolin in plane-strain 
extension (Mair, 1979:99,103), consolidated to the effective 
stress acting at those points. 

At tunnel axis elevation and assuming full saturation, 
the soil water content was found to be about 61%, 
corresponding to a void ratio of 1.59 and a specific gravity 
of 2.61. The unit weight of this saturated soil was equal to 
16 kN/m?. 

The initial tangent modulus for the kaolin can be 
estimated from parameters determined by many research 
workers who investigated this type of soil at Cambridge. 
According to critical state concepts, the response of a 
lightly or overconsolidated saturated kaolin under undrained 
loading, is close to linear elastic, until the stress path 
reaches the state boundary surface. With the usual Cambridge 
notation (see Atkinson and Bransby, 1978, for example), the 


undrained elastic modulus for this soil at stress levels 


1200 


below yield, is calculated by: 
(agli 

I aaepragy eo Dede hy) 
as the shear modulus of a soil is the same in effective or 
total stress terms. In this expression, E' and uw" are the 
Young's modulus and Poisson's ratio in terms of effective 
stress and uw, is the undrained Poisson's ratio (0.5). 
According to Seneviratne (1979:Table 2.1), u' for this soil 
is equal to 0.33. The modulus E' can be calculated from the 
bulbk=modulus4k', through: 

E' El eee ails Dida Pee 
and K' of an overconsolidated soil can be obtained from the 
Slope k of the swelling curve of a isotropic compression 
ere th 

K' =P Baie 
where v is the specific volume (1t+e) and p' is the 
octahedral normal effective stress invariant ((o', + 
2K,0',)/3). The swelling curve of clay in the v x ln(p') 
representation of the isotropic compression, is approximated 
by a straight line in the Cam-Clay model. The slope, k, of 
this line for kaolin was found to vary from 0.04 to 0.06 
(Mair, 1979:124). As one is interested in obtaining the 
initial tangent modulus, the lower k seems to be the most 
respresentative value. 

Combining the equations above one gets: 


ae 1 
Ey prec kc aloes lea diese ey 
With p' and v calculated at a point half diameter above 


the’ tuntel®crown,”one* obtains Bl. Maic (Op -cit.cr mgure 


2011 


5.16a) indicated that the soil water content at this point 
was equal to 62% and from it, the specific volume can be 
Calculated, ends smeduaimio 2.62 mMair (Op.cit.:Figure 5.17) 
indicated that p' at this point was about 40 kPa. The 
undrained Young's modulus is, thus, found to be equal to 
about 3.0 MPa. This value is the initial tangent modulus 
that will be used to normalize the displacements at tunnel 
crown. 

Aefo? buregratio,;gRa,’ of about, 0.9 is calculated -from 
undrained triaxial test results on this soil, presented by 


Roscoe and Burland (1968). 


7.5.1.3 Predicted and Measured Ground Responses 

The generalized solution derived for the frictionless 
Soil model (Section 6.4.2) will be used to estimate the 
ground reaction curve for the tunnel crown. It should be 
pointed out that the solution derived for the frictionless 
Soil model does not completely represent the model test 
condition. In the latter, a uniform pressure was applied 
inside the tunnel, which was a fraction of the overburden 
vertical stress at the axis elevation. The derived solution 
on the other hand, assumes a non-uniform stress applied over 
the opening contour, which results from the initial 
gravitational stress field. However, if in both cases, the 
acting stresses onto the tunnel are normalized to their 
respective initial values, a comparison between the observed 


and calculated responses can perhaps be made. 


1202 


For K,=1 and ¢=0, the normalized ground reaction curve 


is given by equation 6.17: 


U 
A+BU 


The coefficient A for the tunnel crown can be obtained 





pa i Leeiy | 
from Figure 6.85. For a cover to diameter ratio of 1.67, A 
is found to be equal to 0.89. The coefficient B, also for 
the tunnel crown, can: be obtained from Figure 6.86. The 
undrained strength of this soil has to be adjusted, 
according to equation 6.7, since its failure ratio is less 
than unity. The equivalent cohesion is thus calculated as 


Cie = C,/Rp*26.67 kPa. Therefore, the equivalent undrained 


ue 


strength ratio becomes c,,/yD=0.37. In Figure 6.86, B is 


found to be equal to 1.74. Thus: 


ss pine AOS ie 
0.89+1.74U 


This equation relates the ratio of radial stress at the 


x =1 [7.19] 
crown to the normalized crown displacement. In the model 
test, however, the displacements were not measured exactly 
at tunnel crown but at some distance above it. From Mair 
(1979:Figure 5.26), one may suggest that this distance could 
have been about 20 mm or D/3. The distribution of the 
normalized subsurface settlements for H/D=1.5 and 
c,/yD=0.3125, included in Appendix C, shows that at that 
normalized distance above the crown, the normalized 
settlement is almost independent of the amount of stress 
release and is equal to 0.72. For H/D=3, the settlement 
ratio is about 0.66. Hence for H/D=1.67, it should be 0.713. 


Thus, the resulting crown settlement (u.) is equal to 1.402 


1203 


times the settlement at the displacement marker (u). The 


normalized crown displacement is: 
O..E,, 
D0. 


where u.=1.402u, E£,,=3 MPa, D=4.5 m and o,,=120.24 kPa. 


U 


Therefore: 

U=34.98 = [7220] 
Table 7.5 shows the displacement ratios u/D measured at the 
point above the crown by Mair (1979:Figure 7.8), for 
different tunnel internal pressure ratios Z,. The radial 
stress ratio, Z, at the crown was calculated by equation 
7.19, for the normalized crown settlement calculated from 
equation 7.20. The predicted and observed response are shown 
in Figure 7.23. A certain degree of uncertainty exists in 
the calculated response, since the exact position of the 
point where the displacements were measured is not exactly 
known. Accordingly, the calculations were repeated, assuming 
now that this point was located at 10 mm above the tunnel 
crown. The dimensionless crown displacement is then: 
U=30.81 u/D. 

The ground reaction curve calculated with this 
assumption is also included in Figure 7.23. One notes that, 
regardless of the assumption made regarding the position of 
the marker used for measuring the displacements above the 
crown, the agreement between predicted and observations is 
good. 

The limiting U beyond which the solution represented by 


equation 7.19 ceases to be formally valid, can be calculated 


1.000 


0.513 


Notes) (1) 


(2) 


(3) 
(4) 


1204 


a) ncad 5 (4) 
0 0 1.000 
0.009 0.315 0.781 
OOM 0.525 0.709 
0.036 1.259 ORS oil 
0.083 2.903 0.511 


Ratio of applied internal pressure in the model test 
tunnel. 

Displacement ratio measured in model test at a point 
above the crown. 

Normalized settlement at the crown. 

Predicted radial stress ratio at crown for (3). 


Table 7.5 Measured and Calculated Ground Response in the 


Centrifuge Model Test 2DP by Mair (1979) 


= AND Sy 





‘205 


CALCULATED 
MEASURED 


DISPL. AT |Omm 
7 ABOVE CROWN 


DISPL. AT 20mm 
ABOVE CROWN 


»y, 


0 0.02 0.04 0.06 0.08 0.10 


u/D 


NOTE: TEST 2DP BY MAIR (1979) 


Figure 7.23 Measured and Calculated Ground Response at 


Tunnel Crown for the Centrifuge Model Test 


1206 


through Figure 6.119. This is founds towpe equa Comin cu, 
which corresponds to u/D equal to 0.052 or 0.059, depending 
on the assumed position of the meaSuring point (20 mm or 10 
mm above the crown respectively). To the right of point T, 
the dashed curves represent numerical data extrapolation, 
and nothing ensures that the ground response predicted by 
the theoretical solution beyond T is correct. 
Notwithstanding this, the curves bound the experimental 
observation even beyond their limits. 

Figure 7.23 also indicates the ratio of the collapse to 
initial tunnel pressure, Z,. This was calculated through the 
lower bound solution shown in Figure 6.122. For a strength 
ratio c,/yD=0.33 (not adjusted) and H/D=1.67, the collapse 
pressure is estimated as 73.2 kPa, while the initial 
internal pressure applied was equal to the overburden stress 
at axisjelevation, i-ce., 154.8 kPa.) Hence, 2, resulvedein 
being 0.473, a value that seems to agree with test and 
calculated results. 

Finally, Figure 7.24 compares the measured and 
calculated surface settlement profiles at two internal 
pressures (107 and 92 kPa), corresponding to Z=0.71 and 
0.61. The experimental results were presented by Mair 
(1979:Figure 5.18) and Mair et.al., (1981:326). The crown 
displacements were calculated through equation 7.19, and 
were found to be equal’ to 0583) x) 10e-sande4t 27a xe Ue moana 
the opening diameter. The profiles were obtained through the 


distributions of normalized surface settlements included in 


1207 


Appendix C, using the procedure given in Section 7.3 and 
exemplified in Section 7.4. While it provided settlement 
magnitudes close to those observed, the shape of the 
settlement troughs furnished by the proposed calculation 
method did not fully agree with the measurements. This is 
particularly true for the larger amount of stress release, 
when Z gets closer to the collapse stress ratio (0.473). 
Figure 7.24 shows that the surface distortions are 
underestimated by the proposed method. This result, however, 
is not Surprising and the reasons behind it were discussed 
in different occasions in Chapter 5. It was shown there, 
that this type of result is a common feature in finite 


element modelling of shallow tunnel behaviour. 


7.5.2 Verification Against a Three-Dimensional Finite 
Element Analysis of a Shallow Tunnel 

Katzenbach (1901) performed a fairly refined three 
dimensional finite element analysis of a shallow tunnel, 
where the ground was represented by a non-linear elastic 
Stress-strain relationship using the orginal hyperbolic 
model formulation (Duncan and Chang, 1970). Some results of 
this study were published by Katzenbach and Breth (1981) and 
some features of this analysis were reviewed in Section 
5.3.3 of this thesis. While in Section 7.5.1, the new design 
method was verified against an ideal plane strain situation, 
this finite element analysis may serve to check it against 


an idealized three-dimensional condition. 


1208 


O.0l 


CALCULATED The abeees 


0.02 


SERIE. 0 


0.03 > = 0.7! 





Ve i 0 | fa 


DIST a hO gE sAXiSiy7D) 


0.0l 


0.02 


CALCULATED 
0.03 > =0.6l 


SET ite/ D 





04 
s ae sl O | 2 


GIST, STOUT Memaxto U7 UO) 


NOTE: TEST 20P BY MAIR (1979) 


Figure 7.24 Measured and Calculated Surface Settlement - 


Profiles for the Centrifuge Model Test ac twortnternas 


Pressures 


1209 


In the analysis, the ground mass was assumed to be 
uniform, with the following parameters: y=18.5 kN/m?, 
Ko=U,8, czucU kPa, ¢=20°, 8 =0..9., wanbi.'s, modulusiK=225,, 
Janbu's exponent n=0.6, and w=0.45. The tunnel was assumed 
to be circular, with an excavated diameter of 6.7 m and with 
a soil cover of 11.35 m. The support was represented by 
cylinder, 0.2 m thick, with elastic parameters given by 
E=12.6 GPa and u=0.2 (Katzenbach, 1981:39). An incremental 
construction simulation was performed with a full-face 
stepwise excavation and delayed support application. Some 
uncertainty existed in the definition of the distance from 
the face at which the lining was installed. The reason for 
it was that, in the numerical simulation, this distance was 
not kept constant throughout the analysis. When the support 
ring immediately behind the control section was activated 
(see LS13 in Katzenbach; Op.cit.:Figure 37b), the distance 
between its middle point and the vertical tunnel face was 
6.25 m. The support ring immediately beyond the control 
section, on the other hand, was installed at 7.25 m behind 
the face. For calculation purposes, it will be assumed that 
lining installation occurred at 1D behind the face. 

The proposed calculation method was applied to this 
problem, using the same parameters and considering the same 
conditions found in the 3D analyses. At lining activation, 
the reduced unit weight of the soil was found to be 9.46 
kN/m?, corresponding to about 49% stress release. The 


averaged soil modulus at this point was calculated to be 


1210) 


9.24 MPa, representing a 68% decrease in the ground 
stiffness, resulting from the delayed application of the 
support. The lining-ground interaction was carried out 
neglecting the overall ground heave developing after lining 
installation, otherwise calculated by equation 7.10. A final 
crown settlement of 55.3 mm was obtained, which corresponded 
to 54.4% stress release at the crown in the 2D unlined 
tunnel solution. 

The distributions of final subsurface settlements along 
the tunnel axis are compared in Figure 7.25. A fairly good 
agreement between the results of the 3D finite element 
analysis and of the proposed calculation method is observed. 
A good agreement was also found for settlements at sections 
between the face and the point of lining activation. Poorer 
agreement was noted for sections at or ahead of the face, 
where the three-dimensional effects, not L1UllysaccOunteduron 
in the proposed method, are more pronounced. 

In Figure 7.26, the profiles of final surface 
settlements are compared and, once more a reasonable 
agreement is noted. The zero settlement noted in the 3D 
finite element analysis at the furthermost point from the 
tunnel axis, resulted from the imposed zero displacement 
condition adopted by Katzenbach (Op.cit.) at the lateral 
boundary of his finite element mesh. The latter was located 
closer to the tunnel axis, than was the case in the 


parametric analyses that generated the present solution. 


Subsurface Settlements 


=3 0-20 —10° 0 





SN 

e 4 

Sd 4 

S 4 

Sle inh 

Oo 4 

> 4 
o 
LJ 

-12 = 

| 

4 





= X0, 
=i 2 Ol a0) 


10920530 40° 50 60 70 
4 


oO 


ios le on a ee 
! 
> 


° 












12 
Ihe 
b 
—16 Legend 
° 30FE 
New Wethod | 
=20) 


10° P209430).40> 50 60, 7.0 


Settlement (mm) 


NOTE: 


30 FE RESULTS BY KATZENBACH (1981: 100) 


Figure 7.25 Subsurface Settlement Distributions Calculated 


in a Three-Dimensional Analysis and by the Proposed Method 


Wiz 


Unfortunately, Katzenbach (1981) did not present either the 
resulting ground loads onto the Lining orn the epeningeradial 
displacements. However, the displacements and ground 
stresses he provided at nodal points and elements located at 
some distance away from the tunnel contour, seem to be 
consistent with those presently calculated at the 


lining-ground interface. 


7.5.3 Verification Against Actual Case Histories 


7.5.3.1 Foreword 

In order to complete the validation of the proposed 
design procedure, it was tested against a number of case 
histories. They included both shielded and non-shielded 
driven tunnels, mainly through fairly firm ground. The case 
histories were selected according to the availability of 
adequate or sufficient published field instrumentation data. 
The comparison between predicted and observed performances 
was restricted to cases where good ground control conditions 
were met. Some of these cases, however, included conditions 
that deviate from those originally covered by the proposed 
calculation method. These were included to test the 
applicability of the new method, for practical situations 
outside its formal range of validity. The calculations were 
performed without attempting to best fit, or bracket the 
observed performance. The input parameters were selected by 
the writer, on the basis of available information and to 


best represent the most probable conditions found in each 


1213 


Surface Settlements 
Ou 5 iO etSe 200 a 25a 508 35 wdOw 451150 





aia = 
E7104 rp —10 
E 
re =15~ emis 
ee-204 b -20 
3 


Se Pee 


Legend 


| 
-30-4 2 Se o 30FE | 
+ New Method | 

| 


ie we eaiere Second (Find iftctai.ea. 1 --- 
0 5 10 lo ecm 2S SOUS oO ee 4O. 4550 50 


Distance to the axis (m) 








NOTE: 30FE RESULTS BY KATZENBACH (1981! 103) 


Figure 7.26 Profile of Surface Settlements Calculated in a 


Three-Dimensional Analysis and by the Proposed Method 


1214 


case. A general appraisal of the results obtained are 


inclLuaed im Section 7.04. 


7.5.3.2 A Large Shielded Tunnel in Marl: The Frankfurt 
Baulos 23, Domplatz Tunnel 

This case history refers to a subway tunnel built in 
1970-1971, north of the Frankfurt Cathedral and described by 
Chambosse (1972). A twin tunnel system (13.2 m centre to 
centre) was built, with 6.7 m excavated diameter and cover 
of 11.8 m. The analysis here is concentrated to the south 
tunnel which was driven first. 

The subsurface soil profile included a 9 m thick 
Surficial layer of quaternary sand and gravel, covering a 
tertiary clayey marl, locally known as Frankfurt clay. The 
tunnel was entirely driven through this layer, which is 
intercepted by occasional limestone bands (up to 2 m thick), 
which are usually water bearing. The ground water level was 
Originally located at about 5 m below the surface, but it 
was lowered by an extensive dewatering program, initiated 
one year before excavation started, so that tunnel 
construction was undertaken through a fully dewatered 
ground. 

The Frankfurt clay is an overconsolidated fissured soil 
(CH) which was fairly intensively investigated by laboratory 
testing at the Univeristy of Darmstadt. Katzenbach (1981) 
summarized the results of these investigations and prepared 
a table with typical properties for this clay (also 


reproduced by Heinz, 1984:239). As input data for the 


PALS 


analysis of this case history, ground parameters equal to 
the mean between the maximum and minimum properties provided 
by Katzenbach (Op.cit:Table 4) were selected and are shown 
in table ./.6, 

An open face shield 6 m long, with working platforms at 
the face, was used for the construction, which proceeded at 
a 5 m/day rate of advance. The lining system included a five 
segment precast concrete ring, 90 cm wide and 35 cm thick 
with tongue and groove longitudinal joints, which were 
assembled under the shield tail protection. The fifth 
segment was a smaller key segment located at 30° from the 
tunnel floor. The external diameter of the concrete lining 
WaS,0.55 M, SO that a nominal void space of “775"em" had*to be 
filled, all around the ring, once the shield was advanced. 
Chambosse (1972) did not provide details of the grout 
filling operation, but it is believed that it took place 
typically at every second ring installed. 

Tunnel construction was carried out under fairly good 
ground control conditions, with no localized ground 
collapses or instabilities having been reported. The 
presence of fissures in the marl could have been a point of 
concern regarding the applicability of the proposed design 
procedure. However, recent theoretical and experimental 
evidence on other fissured overconsolidated soils 
(Costa-Filho, 1984) seem to suggest that the effect of 
discontinuities on the pre-failure initial portion of the 


stress-strain curve is reduced, provided they are (and 





kN/m> 


Unit weight Y 
In situ stress ratio Ky 
Effective cohesion intercept ce 
Effective friction angle o' 
Railunes rato Re 
Janbu's modulus K 
Janbu's exponent n 
Poisson's ratio u 
Notes: (1) Suggested by Katzenbach (1981)., 


(2) Adopted value. 


1216 


(3) All other parameters are mean values calculated from 
extremes provided by Katzenbach (1981: Table 4). 


Table 7.6 Typical Properties of Frankfurt Clay used in the 


Analysis 


T2177 


remain) closed. A fully drained behaviour was assumed, 
consistent with the assumption that the ground water level 
was lowered below the tunnel invert. The ground settlements 
associated with the changes in pore pressure induced by the 
dewatering had, in fact, almost stabilized before tunnel 
construction commenced. Accordingly, both the predictions 
and measurements, to be presented later, refer only to those 
additional ground movements caused by the tunnel excavation. 
Although the Frankfurt clay cover at the instrumented 
section was a little less than half tunnel diameter, the 
analysis assumed that the soil profile could be approximated 
by a uniform layer of marl, and the presence of the 
limestone bands was disregarded. 

An adjusted friction angle of about 30° was found and 
an average in situ modulus of 21.79 MPa was estimated for 
this soil. It iS uncertain when the lining was brought in 
full contact with the e501, as the amount of overcutting 
over the shield as well as the details of the grouting 
Operation were Sparse. It was, thus, assumed that lining 
activation took place at the middle point of the second ring 
leaving the shield tail and that the soil did not close over 
the shield body. This results in a distance of 7.35 m behind 
the face or at about 1.1D. The calculated amount of stress 
release at this section resulted in being equal to 56% and 
the current average tangent modulus at this point was 
estimated as 9.61 MPa. The two-dimensional ground stability 


verification for this condition furnished a factor of safety 


1218 


of 1.66, indicating that the opening was stable and the 
ground ‘far from collapse. The lining=-ground interaceion 
analysis was then conducted. No allowance was made for the 
possible increase in lining flexibility, resulting from the 
presence of joints. In fact, Muir Wood's (1975) correction 
for the moment of inertia of a jointed lining is only 
applicable when more than four equal segments are included. 
In this case, the fifth segment is the key element, which 
was much smaller than the others. A Young's modulus of 29.4 
GPa was assumed for the lining, as this was the value 
adopted by Chambosse (1972:77) on the lining measurements 
data reduction. This modulus is likely to be slightly high, 
knowing that the longitudinal joints normally increase the 
compressibility ratio of the support. This is particularly 
true, for instance, when bitumen strips or neoprene sheets 
are used as seals, which however, was not the case herein. 
Had these materials been used, the lining compressibility 
would have been entirely controlled by their stiffnesses. 
The flexibility and compressibility ratios defined in 
Section 7.2.1 were calculated as 0.5 and 468, respectively. 
The ground heave after the support was activated was assumed 
to be zero. Only one iteration. was needed to define the 
equilibrium conditions, which reflects the very stiff nature 
of the lining. 

The final crown settlement was estimated as about 59 mm 
(U=0.657) and the associated amount of stress release at the 


crown was calculated as 56.7%. The distributions of final 


42,19 


subsurface and surface settlements were immediately obtained 
and are shown in Figures 7.27 and 7.28. A maximum surface 
settlement of about 30 mm was calculated. In these figures, 
field measurements are included for comparison. The 
settlements due to dewatering prior to the drivage were 
subtracted from the measurements (Chambosse, Op.cit.:Figure 
74). One notes that the predicted settlements agree quite 
well with the measurements. 

The predicted distribisionstofethrust ‘forces (N,) and 
bending moments (M,) in the three upper segments are shown 
in Figure 7.29. The notation and sign convention used is 
that given in Section 7.2.1. The position of the 
longitudinal joints are indicated by the letter 'J', and the 
Grown and springlinée by ‘C’.an@™'S", respectively. 
Unfortunately no load measurements were undertaken in the 
South tunnel lining. Lining instrumentation was introduced 9 
months later in the second driven North tunnel. Thrust 
forces and bending moments were calculated through strain 
measurements in seven niches cast in the inner sides of the 
three upper lining segments. Each niche housed six pairs of 
steel reference pins, fixed against the niche's transverse 
wall, so that the strain distributions across the half 
thickness of the lining could be obtained using a DEMEC type 
mechanical strain gauge. A linearized stress distribution 
across the lining thickness was calculated, assuming a 
Young's modulus of 29.4 MPa, and from it, the thrust forces 


and bending moments were obtained. 


1220 


Subsurface Settlements 
= 30-20 —10" 0 0) S20 eS ORS ORS Om 6ONN7.0 











04 SsTSer= 0 
= = 
| 
— ~44 | -—4 
pay. na 
Sguctil | bs 
i | | 
> 1 
a) | 
we =124 -12 
en —16 Legend 


° Measured 


Predicted 





= 20) 2.0) 
= 50=-20'= 10520 Oe 202550) F405 0> o08 7.0 


Settlement (mm) 


NOTES : 
|. SETTLEMENTS CAUSED BY THE FIRST TUNNEL MEASURED AT EXTENSOMETER 
T4, FROM CHAMBOSSE (1972! FIGURE !9) 


2. THE SETTLEMENTS DUE TO GWL LOWERING WERE SUBTRACTED FROM 
MEASUREMENTS. 


‘Figure 7.27 Measured and Calculated Final Subsurface 
Settlements over the South Tunnel in the Frankfurt Baulos 23 


_(Domplatz) Subway 


T2721 


Surface Settlements 























-30 -25 -20 -18 -10 -5 0 5 10 1$ 20 25 30 
0 0 
E | 
: l 1 
- L le) - 
£ ot | 4 
° -40 | --8 <> 
ae ! £ 
at 610) r-12 £ 
= a 
7) o 
-80 Legend --16 O 
4 ° Weasured 
-1004 + Predicted | fF -20 
SS | 
-120 -24 
-30 -25 -20 -15 -10 -S 0 5 10 15 20 25 30 
Distance to the axis (m) 
NOTES: 


|. MEASUREMENTS AT DOMSTRASSE INSTRUMENTED SECTION AFTER FIRST TUNNEL 
CONSTRUCTION, FROM CHAMBOSSE (1972! FIGURE 19) 


2. THE SETTLEMENTS DUE TO GWL LOWERING WERE SUBTRACTED FROM MEASUREMENTS. 


Figure 7.28 Measured and Calculated Final Surface 
Settlements over the South Tunnel in the Frankfurt Baulos 23 


(Domplatz) Subway 


1222 


so@ 
SOUTH NORTH 





Pn SOO 
E 
AY 
FL, nd (eo) 
Zz o=m— MEASURED ( 29 TUNNEL) ; 
= PREDICTED (It TUNNEL) 
8 
a 
-45° -90° =1s oo 1g8Q° IS5< 90° 45° 
MEASURED (2 TUNNEL) 
ie 
NS 
E 
FS 
x 
a 
= 





100 
-45° -90° =i35¢ \g0° 135° 390° 45° 
J S J Sc J S J 
(innerside) g (outerside) 


Figure 7.29 Predicted Lining Loads in the South Tunnel (ist 
driven) Compared to Measurements in the North Tunnel (2nd 


driven): Frankfurt Baulos 23 (Domplatz) Subway Tunnels 


1223 


The width of the central soil pillar separating the two 
tunnels was about one tunnel diameter. The lining 
measurements taken in the second tunnel were very probably 
affected by the first tunnel already built. Notwithstanding 
this, the measurements recorded in the second, are presented 
together with the lining loads predicted for the first in 
Figure 7.29. Although an immediate comparison is not 
formally valid, one notes that the results are not entirely 
‘discordant. The differences between predicted and measured 
loads could be partly attributed to the changes in ground 
stiffness and in ground stresses induced by the first tunnel 
construction. The transverse arching process triggered by 
the first drivage may partly explain the higher thrust 
forces measured at the springline in the second tunnel, as 
well as the higher bending moment at the crown. This 
interpretation is supported, in qualitative terms, by the 
numerical results obtained by Ranken (1978:219, case C) who 
Studied the lining responses in two parallel tunnels 
consecutively driven, assuming a linear elastic response for 
HOLT Sot) andes ining. 

The overall agreement between predicted and observed 
performances in this case history was quite satisfactory, 
despite no attempt having been made to best fit the 
observations. In a real design situation, it is believed 
that the observed performance could have been totally 
bracketed, had the variabilities of the ground properties 


and construction operations been taken into consideration. 


1224 


7.5.3.3 Twin NATM Tunnels in Marl: The Frankfurt Baulos 

25, Romerberg Tunnel 

This case history refers to a twin subway tunnel built 
in 1970, at the Romerberg, Frankfurt, that was described by 
Chambosse (1972), Edeling and Schulz (1972) and Schulz and 
Edeling (1973). More recently, this case history was 
reviewed by Heinz (1984:236). It was in this project that 
the so called New Austrian Tunnelling Method was first used 
for the constructron*ofmaneurbanmtunnel ingsoi) Tne ewe 
parallel tunnels were advanced simultaneously at a distance 
of 12.7 m from centre to centre. bach tunnel haduanctrculec 
profile with a 6.48 m excavated diameter. The soil cover at 
the instrumented sections was 11.5 m. Although the proposed 
design procedure was developed for a single and isolated 
tunnel, it seems interesting to assess the influence of a 
parallel and simultaneous tunnel construction. 

The ground conditions were similar to that found at the 
Frankfurt Domplatz Tunnel (Section 7.5.3.2), built nearby. 
The tunnels were entirely driven in the Frankfurt clay marl, 
which is overlain here by a 4 m thick layer of sand and 
Gravel which, in turn isscovered by a.2.0 m thick=11.0 one 
marl is also interbedded with limestone bands although 
thinner than at Domplatz (less than 1 m thick). The clay 
marl cover at the instrumented section was about 5 m thick. 
The ground water level, originally located at 4 m depth, was 
also lowered through deep wells to elevations below the 


tunnel invert. Dewatering was initiated six months before 


4225 


tunnel construction started (Chambosse, 1972:48). The ground 
settlements associated with dewatering had stabilized before 
construction commenced, so that it was possible to separate 
these movements from those resulting from the tunnel 
construction proper. The ground mass, as it appeared during 
tunnel advance was free of water (Chambosse, Op.cit.:6), so 
that a drained behaviour could be assumed in the analysis. 
The relevant ground properties for the Frankfurt clay were 
already presented in Section 7.5.3.2. The same parameters 
used for the Domplatz Tunnel will be considered herein. 

The simultaneous tunnel construction was carried out 
with sequential excavation of the face in three stages: 
heading, bench and invert. The shotcrete lining was also 
applied in stages, immediately after the ground was exposed. 
An average rate of advance of 1.2 m/day was achieved. The 
excavation round length was about 1.2 m, which corresponded 
to the spacing of light segmented steel ribs, with a channel 
profile (TH 48, 16.5 kg/m). A steel mesh (Q188) was also 
incorporated into the support system. The total shotcrete 
thickness varied between 15 and 18 cm. When the tunnels 
passed the instrumented section II, the support ring closure 
at the floor took place typically at 5.7 m behind the face 
(Edeling and Schulz, 1972:Figure 11). The steel ribs were 
fixed to the ground by 3 to 4 m long anchors which were 
found later to have contributed nothing to the opening 
support (Schulz and Edeling, 1973:251). Tunnel construction 


was undertaken under good ground control conditions and no 


1226 


ground instabilities were reported. 

The analysis was conducted neglecting the presence of 
the second tunnel, and assuming a uniform layer of clay 
marl, An adjusted friction angle of about 30° was found and 
adopted for the cohesionless soil model. The average in situ 
tangent modulus was estimated as 21.49 MPa, with the 
parameters shown in Table 7.6. The lining activation was 
assumed to have taken place when the shotcrete lining ring 
was closed at the floor, 5.7 m behind the face. The 
calculated amount of stress release at this point was 55.5% 
and the current average tangent ground modulus at this 
section was estimated as 9.63 MPa. The ground stability 
verification yielded a factor of safety of 1.67, which 
confirmed that the opening was stable. 

The lining-ground interaction analysis was carried out 
neglecting the action of the light steel ribs, but 
considering a nominally increased shotcrete thickness of 20 
cm. A lining modulus of 10 GPa was considered as well as a 
Poisson's ratio of 0.2. The. flexibility and.compressi bidity 
ratios were estimated as 0.03 and 90, respectively, which 
are typical for a shotcrete lining in a soft to medium clay 
(see Table 7.1). A single iteration was required to define 
the equilibrium condition. The ground heave after the 
support was activated was neglected. 

The final crown settlement was estimated as 59.1 mm 
(U=0.694) and the associated amount of stress release at the 


crown waS calculated as 57.8%. The calculated distributions 


V2) 


of final subsurface and surface settlements are presented in 
Figures 7.30 and 7.31. Included in these figures are the 
measured displacements that were caused by the simultaneous 
construction of the two parallel tunnels. One notes in 
Figures 7.30 that, while the displacements of points closer 
to the tunnel are well approximated by the proposed 
procedure, those at the surface are substantially 
underestimated. This result was indeed expected, as the 
prediction did not take into acount the second tunnel 
construction. Ranken (1978:197) studied the development of 
surface settlements over twin tunnels built simultaneously, 
through linear elastic finite element analyses. He showed 
that the interaction between two shallow tunnels (H/D=1) is 
small provided the centre to centre distance is greater than 
2 diameters. In other words, for this condition, the 
settlement profile obtained by superimposing two single 
tunnel settlement distributions is approximately equal to 
the actual profile from the two parallel tunnels 
Simultaneously built. In the present case history, the 
tunnels were built at slightly more than two diameters 
apart. However, unlike the conditions focused on by Ranken 
(Op.cit.), the tunnels were deeper and the soil response was 
definitely non-linear. Nevertheless, if the superposition of 
surface settlements is accepted, then the profile found is 
that shown in Figure 7.32. One observes now, that a better 
agreement between measured and predicted responses is 


attained, at least with regard to the maximum surface 


1228 


Subsurface Settlements 
— 3 O—2O5— 10880), Om 20040  S5O0n 6O= 70 








Elevation (m) 
| 


: 
3 
| np rie 
| | 
Legend 


© Measured | 


Predicted 








-30-20-10 0 10 20 30 40 SO 60 70 
Settlement (mm) 


NOTES: 
|. MEASUREMENTS AT EXTENSOMETER 02 OVER THE NORTH TUNNEL, AT THE 
INSTRUMENTED SECTION IL, FROM CHAMBOSSE (1972:52) 
2. THE SETTLEMENTS DUE TO GWL LOWERING WERE SUBTRACTED 
FROM MEASUREMENTS. 
3. MEASURED SETTLEMENTS CAUSED BY SIMULTANEOUS CONSTRUCTION OF 
THE TWO PARALLEL TUNNELS. 


Figure 7.30 Measured and Calculated Subsurface Settlements 


over the Frankfurt Baulos 25 (Romerberg) Subway 


1229 


Surface Settlements 


0 S 10 15 20 figs) 30 
Oe eg 











° eg } 
10 ° aie 
Coxe 
E F 
£ - 205 a 20 
c 
E aal| 
2. -30- + -30 
eee 
Legend 
-40-4 O Weasured --—40 
fe) + Predicted 
oY) iets Se Ey Seen eS Seo SaaS ame — epee) 
0 5 10 15 20 29 30 
Distance to the axis (m) 


NOTES 
|. CALCULATED SETTLEMENTS FOR A SINGLE TUNNEL CONSTRUCTION. 
2. MEASUREMENTS AT THE INSTRUMENTED SECTION I, OVER THE SOUTH TUNNEL, 
FROM CHAMBOSSE (1972 '52) 
3. THE SETTLEMENTS OUE TO GWL LOWERING WERE SUBTRACTED FROM MEASUREMENTS. 
4. MEASURED SETTLEMENTS CAUSED BY SIMULTANEOUS CONSTRUCTION OF THE TWO 
PARALLEL TUNNELS. 


Figure 7.31 Measured and Calculated Final Surface 
Settlements at the Frankfurt Baulos 25 (Romerberg) Subway 


Tunnel 


1230 


settlements. The maximum surface distortion is however, 
underestimated by this approximate calculation procedure. 
Figure 7.33 presents the calculated and measured radial 
stresses acting on the upper arch of the shotcrete lining, 
after equilibrium was achieved. The stresses were measured 
by Glotzl contact pressure cells (15 x 25 cm). Note that, 
while the prediction was made for a single tunnel, the 
measured stresses include the effect of the twin tunnels 
Simultaneous construction. Note, moreover, that the 
measurements were taken at both tunnels, and for easier 
comparison, they were plotted together by making the 
relative position of the measuring points with respect to 
the central soil pillar to coincide. Accordingly, the 
measurements at the springlines (S) facing the central 
pillar (@anererce laws plotted together at the +90° position 
(measured from the floor), whereas those at the outer 
abutment springlines (outer side) are plotted at the -90° 
position. The predicted radial stresses seem to be closer to 
the average stress measured, except at the springlines, 
where the measured stresses are underestimated, notably at 
the inner springline, on the pillar side. If it is admitted 
that the readings are correct, one could speculate that the 
interaction between the two tunnels forced them to squat 
more and to mobilize higher radial stress at the springline 
elevation. This interpretation, however, is not clearly 
Supported by Ranken's (1978:194) numerical results, which 


suggested that shallow twin tunnels built simultaneously, at 


123) 





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4 = Ny = NS 
aot) A { } jodi 


200 







na 





@ 


I50F 





PREDICTED 
(SINGLE TUNNEL) ° 


° 
° 


0, (kPa) 





100 


a | —_@-® 


50 MEASURED 


(TWIN TUNNELS) 





—— 


fe) 


=90° =135¢ 180° is52 90° 
S € Ss 
(outerside) 6 (innerside) 


NOTE' MEASUREMENTS TAKEN BOTH IN NORTH ANDO SOUTH TUNNELS, 
DATA FROM CHAMBOSSE (1972 '!72) 


Figure 7.33 Predicted Lining Radial Stresses for Single 


Tunnel and Measured Stresses in Twin Tunnels at the 


Frankfurt Baulos 25 (Romerberg) Subway 


(232 


L230 


one diameter apart (centre to centre), exhibit lining loads 
that do not differ greatly from those for a single tunnel. 
However, the generality of Ranken's suggestion is debatable, 
Since he assumed linear elastic responses for both soil and 
liner, and more importantly, he assumed that the linings in 
both tunnels were installed before soil excavation (no 
Stress relaxation before support application). 

A special device, designed by Interfels, installed from 
the surface, allowed the measurement of horizontal 
displacements in the ground at the springline elevation, 0.5 
m away from the tunnel contour. A maximum horizontal 
movement towards the opening of about 19.5 mm was measured 
at Section II, in the outer abutment of the tunnel, just 
prior to the closure of the shotcrete ring at the invert 
(Chambosse, 1972:Figure 75). This value is compared with the 
31.9 mm of inward radial displacement calculated at the 
springline of the single tunnel contour at lining 
installation. 

Horizontal diameter changes were also measured in three 
shotcrete rings in the North Tunnel (Edeling and Schulz, 
1972:357). An increase of the horizontal diameter of 12 to 
13 mm after completion of the invert was recorded (10.5 to 
13 mm according to Chambosse, 1972:66). The lining-ground 
interaction analysis indicated that this increase amounted 
to 11.1 mm, which compares favourably with the measurements. 

The comparison Hue, eh calculated and observed 


performances in this case history should be carefully 


1234 


interpreted, since the proposed procedure does not account 
for the interaction of two parallel tunnels. However, one 

can observe that the resulting calculations are consistent 
with the observations and that the noted differences could 


be attributed to the interaction between the tunnels. 


7.5.3.4 A Small Shielded Tunnel in Till: The Mississauga 

(Ontario) Tunnel 

This case history refers to a small sewer tunnel, built 
in 1972 in Mississauga, about 29 km from central Toronto 
(Ontario), and described by Seychuk (1977) and by DeLory 
et.al. (1979). The tunnel was bored with a 4.27 m diameter, 
under a soil cover of 11.5 m at the installed instrumented 
section. 

The soil profile included a 3.5 m superficial layer of 
recent alluvial sands and gravel, covering a very dense 
12.5 m thick, sand-clay till layer. Underneath it, there was 
another till layer, of 4 m thickness, comprising a dense 
Silty sand. The shale bedrock was found at about 20 m below 
the surface. The ground water level was about 6 m below the 
surface before construction started. 

According to DeLory et.al. (1979), the tunnel was 
driven entirely through the very dense upper sand-clay till 
(SPT in excess of 100), which was found to be a relatively 
Uniform. Sot lym wip hee oe See. 36% silt and 44% sand and 
gravel. The till liquid limit was about 22%, the plastic 
limit 15% and the natural water content 8%. Its unit weight 


was 24 kN/m? and the in situ effective stress ratio was 


W235 


estimated as 0.9 to 1.0. The effective friction angle was 
measured in the laboratory and was found to be 35°. The 
effective cohesion intercept was not given by those authors, 
but data from Nipawin and Edmonton Till (Wittebolle, 1983), 
suggest that it could be greater than 30 kPa. These data 
also indicated that the failure ratio from drained triaxial 
Pesteerors.i1S1S01ly may vary from.0.7 to 3.0, 

Menard pressuremeter tests carried out in boreholes 
adjacent to the tunnel gave values of the in situ modulus 
from 55 to 160 MPa. Assuming that the tests were conducted 
rapidly and that some disturbance might have been caused by 
boring, a value of 140 MPa can be taken as typical for the 
undrained tangent in situ modulus of this till at the tunnel 
axis elevation. If a drained Poisson's ratio of 0.3 is 
assumed for the soil, then eee anes modulus of about 120 
MPa is calculated, provided that (at small strains) the soil 
behaviour is approximately linear elastic. No details of 
those tests or of the variation of the pressuremeter moduli 
with depth are given. It was then assumed that the drained 
soil modulus varied linearly with the square root of the 
vertical in situ effective stress (See Lambe and Whitman, 
1969:159). In situ drained moduli of 105 and 134 MPa, were 
therefore estimated at a half diameter above and below the 
tunnel respectively. 

Piezometers installed from the surface above and near 
the tunnel, showed that there was a lowering of the water 


pressure near the tunnel after construction. The piezometric 


1286 


head at the axis elevation, at a point 2.1 m away from the 
springline decreased from 7 m to 1.4 mor less (DeLory 
et.al. 1979:193), after the tunnel was built. The original 
water pressures around the tunnel were not re-established 
after tunnel completion, which seems to have acted as a 
drain. If an option had to be made between an undrained and 
a drained analysis, one could suggest that the observed 
performance might have been closer to the latter condition 
rather than to the former. Accordingly, the selected ground 
parameters, defined in terms of effective stresses, that 
were considered for the analysis of this case history are 
those given in Table 7.7. 

An open face shield, 5.18 m long, was used for 
construction. Excavation was by an Alpine miner road header, 
with the face being undercut some 0.6 to 1.5 m ahead of the 
Shield hood (Seychuck, 1977). The cut profile was about 75 
to 100 mm less than the shield outside diameter. The shield 
shaved off the remaining soil annulus as it was advanced. 
The rate of advance was about 5 m/day when passing the 
instrumented section. The primary lining system consisted of 
steel set rings (H section, 100 x 100 mm) in four segments 
and concrete planks (75 x 200 x 1200 mm), assembled inside 
the shield in 1.22 m lengths. As the primary lining emerged 
from the tail, it rested on the tunnel floor and a space of 
about 50mm between planks and ground was left at the sides 
and of about 100 mm at the crown. According to DeLory et.al. 


(1979:192), this void was filled with pea gravel, and was 


1237 


Unit weight Y kN/m? 24 
In situ stress ratio Ky - 1.90 1) 
Effective cohesion intercept er kPa 30 ch! 
Effective friction angle o' G2) RO 
Failure ratio Re = 0.8 GP) 
E, (drained) 1/2 D above crown MPa 105 ie 
- at springline MPa 120 
1/2 D below floor MPa 1a4anbe? 
Poisson's ratio (drained) u = 0.3 ah 
Notes: (1) Assumed value. 
(2) Derived from pressuremeter tests. 


Table 7.7 Selected Ground Parameters for Drained Analysis of 


Tunnel in Dense Sand-Clay Till 


Ween 


later grouted with cement. While the filling and grouting 
operations took place, for most of the tunnels Vengen yy at 
distances greater than 12 m behind the face, at the 
instrumented section it is estimated that these operations 
happened after the face had advanced about 7.5 m past the 
section. It is unlikely that the grout completely filled the 
gravel voids, particularly at the crown. An excessively 
robust unreinforced concrete liner, 460 mm thick, was 
installed about 3 months after passing the instrumented 
section. 

Thanks to the good ground conditions rather than to the 
quality of the construction, no instability process was 
observed. The ground response was good, despite some poor 
construction procedures, such as face undercutting, the use 
of : long shield or the delay in filling and grouting the 
void behind the liner, to name a few. The ground drainage 
caused by the tunnel construction, induced changes in pore 
pressures which, in itself, may have induced ground 
displacements. The magnitudes of these were likely to be 
small, as the till is very dense, has a low void ratio 
(about 0.22) and a high bulk modulus. The soil volume 
changes due to consolidation induced by the dewatering were, 
thus, probably small. Probably the volumetric strains caused 
by the shear and mean normal stress changes induced by the 
tunnel excavation were also small. A factor of safety in 
excess of 1.7 was calculated in the two dimensional ground 


stability verification, for the instant the support was 


1239 


activated. From what was shown in Section 2.3.4.4, for this 
Safety level (load factor smaller than 0.6), the volumetric 
expansion in this soil was probably small. 

Since the proposed design procedure does not take into 
account the consequences of pore water pressure changes, it 
was conveniently assumed that the ground was entirely 
dewatered, so that those pressures were equal to zero. 
However, the in situ ground moduli considered, (Table 7.7) 
are drained moduli corresponding to effective stress levels 
existing in situ, prior to dewatering. In other words, the 
possible gain in stiffness, which results from the increase 
in the effective stress level that would follow the ground 
water drawdown (either the hypothetical or the actual one) 
was neglected. 

Having in mind the above discussion and considering the 
fairly uniform nature of the ground, it was felt that the 
proposed design procedure could be applied to this case 
history, and a comparison between the predicted and observed 
performances could be made. However, as noted above, it 
should be remembered that the latter may reflect not only 
the response resulting from the tunnel excavation, but also 
that resulting from changes in the pore water pressures. 

An adjusted friction angle of about 40° was calculated, 
on account of the cohesive strength component and of the 
non-unity failure ratio. A certain degree of uncertainty 
existed regarding when the lining was brought in full 


contact with the soil. The exact grouting procedures were 


1240 


not clear, as well as the amount of overcutting over the 
shield. It was, thus, assumed that lining activation took 
place 7.5 m behind *the*face, as) mentioned wearlier .A smaller 
distance would have to be considered if the shield had been 
advanced without an overcutter head. But this is unlikely, 
since high thrust forces are required to propel the shield 
in such a strong material, without some overcutting rim. The 
amount of stress release at lining activation was about 57% 
and a reduced average tangent modulus of 70.13 MPa was 
obtained (41% less than the original value). 

Although an attempt was made to jack the concrete 
planks together circumferentially (DeLory et.al., 1979:195) 
it is very unlikely that they all came in full Comeacteml ta 
minor gap of 0.5 mm had been left on average between each of 
the 64 or so planks used, a uniform convergence of more than 
10 mm in the lining diameter would be required to put all 
the planks in tuUllecourace tn each other. This convergence 
is an order of magnitude higher than the expected lining 
convergence. Therefore, it is reasonable to accept that, 
besides not being able to carry any bending in the plane 
transverse to the tunnel, the concrete plank lagging was 
very compressible under uniform transverse loading, and thus 
unable to carry tangential thrust. If a fully compressible 
contact between the planks is conceded, then they operate 
Simply by transferring the ground loads in the longitudinal 
direction, to their supporting ribs. Accordingly, the 


relative stiffness of the lining would be that given by the 


1241 


SS eemericswacel.22 Mi spacing only. Thus, a Lining with a 
Momentecoreanertia Of 3.615 x 10°* m*/m and a cross sectional 
area of 2.016 x 10-2? m?/m was considered for the 
lining-ground interaction, with elastic properties of steel 
(E,=200 GPa and u=0.2). The interaction analysis was 
conducted with the reduced unit weight of the soil and the 
reduced ground modulus mentioned earlier. Two iterations, in 
which the ground heave after support activation was 
neglected, were needed to define the equilibrium condition. 
An average ground modulus of 60.49 MPa was found for this 
condition, and the flexibility ratio (8) and compressibility 
(a) ratios resulted in values of to 1.92x 10°-* and 4.44 
respectively. As can be observed in Table 7.1, these values 
are typical for this type of support in dense sand or stiff 
clay. The ground modulus at equilibrium is roughly half of 
the in situ modulus value. 

The final crown settlement was estimated as about 7.3 
mm (U=0.648) and the associated amount of stress release at 
crown was calculated as 60%. The distributions of the final 
subsurface and surface settlements were then obtained and 
are presented in Figures 7.34 and 7.35, together with the 
results of the field measurement program given by DeLory 
et.al. (1979:194). Seychuk (1977) made reference to problems 
regarding survey accuracy and thermal effects affecting the 
readings. Notwithstanding this, the predicted values broadly 
conform to the measurements, but it is clear that the 


maximum surface distortion is underpredicted by the 


1242 


Subsurface Settlements 
-8 -6 -4 -2 QO 2 4 6 8 


Elevation (m) 


Legend 


(oe) Measured 


+ Predicted 





-8 -6 -4 -2 0 2 4 6 8 
Settlement (mm) 


NOTE: MEASUREMENTS RECORDED AT POSITION C( TUNNEL AXIS) 
BY DELORY ET AL (1979 :194) 


Figure 7.34 Measured and Calculated Final Subsurface 


Settlements over the Mississauga Sewer Tunnel 


1243 


Surface Settlements 
-8 -6 -4 -2 0 Zz 4 6 8 


Settlement (mm) 


Legend 


e) Measured 





+ Predicted 





@oueGn fe? 20g 2 46 8 
Distance to the axis (m) 


NOTE* MEASUREMENTS PRESENTED BY DELORY ET AL (19797194) 


Figure 7.35 Measured and Calculated Eanal.Surtace 


Settlements over the Mississauga Sewer Tunnel 


1244 


calculations. 

The predicted distribution of radial stress acting upon 
the lining is shown in Figure 7.36. The measured radial 
stresses, also shown in this figure, were recorded by 250 mm 
diameter flat contact cell pressures, installed between the 
primary lining and the ground. The cells were placed in the 
void behind the lining and were packed on each face with a 
soil-bentonite-cement mixture, to fill the space between the 
soil and support. The mixture was designed to have 
mechanical properties similar to the surrounding till. 
Twelve cells were used but one at springline (270°) ceased 
to operate some time after installation. Four of them 
installed in the upper arch (three at the crown and one at 
45° off) seem to have had faulty installation. As explained 
by DeLory et.al. (1979:195), "the cells near the crown acted 
as hard spots taking more than their share of the load". In 
fact, stresses more than 100% in excess of the overburden 
stress were recorded in these four cells. Their results are, 
thus, unreliable and therefore not included in Figure 7.36. 
The measurements indicated correspond to final long term 
readings, one year after installation, although they do not 
differ to any substantial degree, from the short term 
readings taken a few weeks after the cells were installed 
(DeLory etal. ,.Op.citw mics 

A substantial scatter in the cell pressure measurements 
is noted, possibly reflecting the effects of the 


installation procedure adopted. However, if an average 


1245 


240 


200 





160 
= PREDICTED 
a 
S20 
o 

80 iY wgiae 

90 

Ge 45° 90° 1559 180° 620s 270° S|5% 360° 
F S Cc S F 


6 


NOTE' LONG TERM MEASUREMENTS BY O€ LORY ET AL (1979'196). 


DATA FROM FIVE FAULTY CELLS WERE EXCLUDED. 


Figure 7.36 Measured and Predicted Radial Stress 


Distributions on the Lining of the Mississauga Sewer Tunnel 


1246 


distribution of radial stresses were to be defined, it would 
not liesfarefromethe predictedadistrapurtions 

A slope indicator was installed at 0.91 m away from the 
tunnel springline. DeLory et.al., (Op.cit.:Figure 3), 
presented a plot of radial movements in the ground, at the 
springline elevation, with tunnel advance. If an average 
curve is fitted through the points shown in that plot and if 
the displacements are extrapolated to the tunnel contour 
following the procedure used in Section 5.3.6.1, one would 
find that, at the point of lining activation, the radial 
closure at the springline would have been equal to 6 mm and 
equal to 7.5 mm at final equilibrium. These values are to be 
compared with the calculated closures of 6.3 and 7.4 mm 
respectively. The total increase of about 2.3 mm in the 
horizontal lining diameter, after its activation (i.e., 
after grouting), is in agreement with the convergence 


measurements taken in the field (DeLory et.al. Op.cit.:193). 


7.5.3.5 A Large NATM Tunnel in Granular Soil: The 

Butterberg Tunnel, in Osterode, near Hannover 

This case history refers to a large road tunnel built 
in 1977-1979 in Osterode, about 50 km southeast from 
Hannover (Germany), that was described by Duddeck et.al., 
(1979 and 1981). The initial portion of the job, completed 
earlier in 1975-1976, served as a test tunnel designed to 
supply information on tunnelling performance for the main 
tunnel. The investigations carried out in this experimental 


length were reported by Meister and Wallner (1977). The 


1247 


tunnel had a non-circular profile of 10.11 m high and 11.70 
m wide. The area of excavatiion was about 100 m?/m, 
corresponding to an equivalent diameter of 11.5 m. At the 
main instrumented section the ground cover was 13.6 m. 

The ground profile at the instrumented section included 
a thick Quaternary terrace deposit (Pleistocene Epoch) 
consisting of a well graded sandy-silty gravel (GM) with a 
of silt component of 10% and with boulders of diameters up 
to 1m. The tunnel, at this section, was entirely driven 
through this granular soil. Underneath it, some 12 m below 
the tunnel floor, there was a limestone bedrock. The 
groundwater level was not identified by the exploratory 
boreholes drilled down below the tunnel floor. Some local 
seepage was detected at the portal region, associated with 
rainfall infiltration. Otherwise the quaternary deposit was 
water free. 

Laboratory and in situ direct shear tests were 
conducted on the granular soil and showed that the effective 
shear strength parameters were c'=20 kPa and ¢'=33°. This 
soil showed some strength loss after peak, with residual 
values given by c',=5 kPa and ¢'.=30°. The soil deformation 
behaviour was investigated by plate bearing tests in test 
pits and inside the tunnel. From these tests, an average 
constrained modulus for the 'intact' material of 330 MPa was 
estimated (Duddeck et.al., 1979:208). For a Poisson's ratio 
OrU ss eaevOuUng SumOcuLusS Of e245 MDa is.found.for,.this.soid 


at a depth of about 24 m. If that value is taken as the in 


1248 


situ tangent modulus, and if it is assumed to be 
proportional to the square root of the vertical effective 
stress, in situ moduli of 141 MPa, 217 MPa and 279 MPa are 
estimated at points located half a diameter above the tunnel 
crown, at the springline elevation and half a diameter below 
the tunnel floor, respectively. 

The failure ratio (R,) for this material is not known, 
but results from triaxial tests on similar materials when 
compacted (Duncan et.al., 1980), suggest that R, should be 
close to 0.7. The unit weight of this soil is 22 kN/m? and 
an in situ stress ratio of 0.5 was suggested by Duddeck 
et.al. (1979:208). A drained analysis is justified for this 
case, and the corresponding parameters adopted are 
Summarized in Table 7.8. Peak strength parameters are 
assumed on the grounds that the opal around the opening was 
not strained to failure. 

The tunnel construction was carried out with the 
segmented excavation of the face in stages: heading, bench 
and invert. The bench and invert excavations were undertaken 
in steps, in such a way that a central core was always left 
supporting the tunnel face (see Figure 7.37). The heading 
was advanced in two successive rounds of one metre each. 
This was followed by a 2 m advance of the bench, floor 
excavation following this in 4 m sections. An average rate 
of advance of 2 m/day was maintained during construction. 
The excavation at the crown was carried out under the 


protection of Snore torepoling (io emmlong 


1249 


Unit weight y kN/m? 22 
In situ’ stress iratio Ke - 0.5 
Effective cohesion intercept ch kPa 39 te) 
Effective friction angle o! (°) 33 (2) 
Failure ratio - Re = 0.7 (3) 
E, (drained) 1/2 D above crown MPa 141 (4) 
. at springline MPa 217 (4) 
1/2 D below floor MPa 279 (4) 
Poisson's ratio Pe ee ee ee 
Notes: (1) Average values provided by Duddeck et al (1979:208). 
(2) Peak strength parameter. 
(3) Estimated. 
(4) Derived from the average constrained modulus from plate 


bearing tests by Duddeck et al (Op.cit). 


Table 7.8 Selected Ground Parameters for Analysis of Tunnel 


in Well-Graded Sandy-Silty Gravel 


1250 


The primary lining consisted of a shotcrete layer 30 cm 
thick, steel wire mesh and light steel ribs at 1 m spacing 
(GI profile, 130 mm high, 37 kg/m in the instrumented tunnel 
length). Steel ribs were not installed at tunnel invert. At 
the instrumented section, the shotcrete ring closure at the 
floor took place when the face was 9.5 m past the section. 
This was the distance between the middle point of the 
shotcrete section at the invert and the heading face 
(Duddeck et.al., 1979:Figure 16). 

A very careful construction procedure was undertaken. 
The eae tai weneiienay ground stability verification at lining 
closure, furnished *a “factor ofMabouts tioz ijeewhiche seen 
fact, slightly low. However, no ground instability was 
observed during construction. Such a low factor of safety 
may indicate that some shear stress concentration in the 
ground might have happened. If this is so, the development 
of some shear dilatancy in thie fairly dense granular soil 
could be expected. The proposed design procedure does not 
take into account this type of ground response, as well as 
the effects of some strain weakening that might have also 
developed around the opening, possibly at the tunnel 
shoulders. Notwithstanding this, the analysis was carried 
out neglecting such likely occurrences. 

An adjusted friction angle of 41.5° was calculated on 
account of the cohesive strength component and of the 
failure ratio different from unity. At lining closure (0.826 


D behind the face), the amount of stress release was about 


1250 


64% and a reduced tangent ground modulus of 115.82 MPa was 
calculated (46% less than the original average value). In 
these calculations, the generalized A and A' curves for 
K,=0.6 were used for simplicity and to avoid the need of 
extrapolating the numerical results for the selected in situ 


stress ratio. At lining closure, the U/U values for the 


ref 
crown and the springline did not exceed the limits of the 
fitted solutions. For the floor, however, it was necessary 
to use the solutions beyond their formal limits. 

The lining-ground interaction analysis was carried out 
neglecting the contribution of the light steel ribs and the 
shotcrete reinforcement. A lining modulus of 10 GPa was 
considered, as well as a Poisson's ratio of 0.2. These are 
the same parameters used for the Romerberg Tunnel (Section 
7.5.3.3). Meister and Wallner (1977:930) also used this 
Young's modulus for the lining in their numerical analysis 
Gietiis case Mistory. Four iterations, ingwhich thesground 
heave after support activation was neglected, were needed to 
obtain the equilibrium condition. An average ground modulus 
of 144.78 MPa was found for this condition which was higher 
than that found at lining closure. This modulus increase 
resulted from the mode of deformation where the lining 
squatted and pushed the soil outwards at the springline 
region. The radial confinement increase in this area caused 
the soil to become locally stiffer and this led to an 
increase in the average ground modulus, which at 


equilibrium, was now only 32% lower than the in situ average 


AZ3¢2 


Valle. Por this*condition? the™tlexibrlity (pf) and 
compressibility (a) ratios were found to be about 5 and 
0.0012, which are typical for shotcrete linings in dense 
sand (See Table 7.1). 

The final crown settlement was estimated as 22.1 mm 
(U=0.906) and the associated amount of stress release at the 
crown was about 70%. The resulting distributions of 
subsurface and surface settlements were calculated for 
K,=0.6 and =40°, which are not far from the values which 
that should actually have been considered. An inevitable 
numerical extrapolation was required regarding the 
normalized settlements, with respect to the cover to 
diameter ratio. In this case study, this ratio was equal to 
1.183, whereas the parametric numerical results (Appendix C) 
furnishes distributions for H/D=1.5 and 3.0. A linear 
extrapolation was made, and the distributions obtained are 
those shown in Figures 7.37 and 7.38. Included in these 
figures are the settlements measured in the instrumented 
section MS3 (Duddeck et.al., 1979:214). 

From Figure 7.37, one notes that, although a reasonable 
agreement was obtained between predictions and measurements, 
the proposed method underestimated the ground movements 
close to the tunnel. One could speculate that this is due to 
the unaccounted for influence of the volumetric expansion of 
the soil above the tunnel (see Figure 2.9, d and e). Plastic 
dilation in the ground is known to increase the tunnel 


convergence, for a given amount of stress release (e.g., 


1253 


Subsurface Settlements 


-50-40-30-20-10 0 10 20 30 40 50 





| 
04 | Lo 

=6 2) _ 
& 
> 104 =10 
o 
= 
thse pale 
uJ 


-~20- 





=25-1 —25 Legend 


| ° Measured 





+ Predicted 








= S58) SSE 
= 054.0. 5.0. 20 = 10, Ol Oe 2 Oitr dS Oiee4 04-50 


Settlement (mm) 


NOTE . MEASUREMENTS AT SECTION MS3, BY OUDOECK ET AL (1979 +214) 


Figure 7.37 Measured and Calculated Final Subsurface 


Settlements over the Butterberg Tunnel 


1254 


Total Surface Settlements 
=1210<8- 6 sro a 6S UNI 








fe) 
YY 
2 
Se 
5 
=O 
5 
om 104 
feb) — 
mM -50- = 
-60 & 
o 
I bjt 
-80- + ~20 
4 
wmf -22 
ae rr 
; 28 Measured 











—110 -28 Predicted 
“2, + te pee mee perayORRe ETI 
=—12-10/=6i—6 —4t = 20M) SaeGeSed0 12 


Distance to the axis (m) 


NOTE: MEASUREMENTS AT SECTION MS3, BY OUDOECK ET AL (1979! 214) 


Figure 7.38 Measured and Calculated Final Surface 


Settlements over the Butterberg Tunnel 


ies} 


Lombardi, 1973, Ladanyi, 1974). The non-circular shape of 
the tunnel cross section could also be evoked to explain the 
underestimation of the crown displacements. From Figure 
7.38, one notes that the magnitude of the maximum surface 
settlement observed at surface, was reasonably estimated by 
the proposed method, but once more the distortions were 
underestimated. 

Predicted and measured radial stress onto the lining 
are shown in Figure 7.39. The measurements were taken with 
Maihak pressure cells (Duddeck et.al., 1981:181), but 
details of their installatien were not given. One set of 
measurements was taken in the experimental tunnel length, 
while the other was taken during the main tunnel 
construction under similar conditions. One notes that the 
shapes of the predicted and measured stress distributions 
show some sort of broad agreement. However, the predicted 
Stresses are typically twice the stresses measured. 
Although, it was shown that the proposed method may tend to 
slightly overestimate the final lining loads (see Section 
7.2.2), this fact does not explain the result obtained in 
this case. Had a smaller shotcrete modulus been assumed a 
slightly better agreement could have been obtained. A 
smaller modulus would imply, however, a poorer agreement 
regarding the lining convergence results. A horizontal 
diameter increase of 3.96 mm was meaSured in the field 
(Duddeck et.al., 1981:180) after the shotcrete ring closure. 


The predicted increase in the horizontal diameter was 3.74 





PREDICTED 
re) 
a 
a4 
—— a 
o MEASURED AT 
EASURED AT MAIN re 
f TEST TUNNEL 
e 
Oe 45° 90° 1352 Igo° Ceo 270° S152 360° 
F S Cc S F 
6 
NOTE: 


MEASUREMENTS BY OUODECK ET AL (1979: 213 ANO 215) 


Figure 7.39 Measured and Predicted Radial Stress 


Distributions on the Shotcrete Lining of the Butterberg 


Tunnel 


1256 


(29 


mm, which agrees very well with the observed value. Had a 
"softer' lining been considered, this agreement would not 


have been so good. 


7.5.3.6 A Large Shield in Till: The Edmonton LRT Tunnel 

This case history was described in detail in Section 
5.2.4.3, where it was back analysed through the two 
dimensional finite element model presented in Chapter 5. 
Basically, the same parameters used in the best fit case 
were considered in the present analysis. 

The tunnel was driven with a 6.172 m diameter under an 
8.9 m cover of soil. A uniform till layer was considered 
with a friction angle of 40° and zero cohesion (R,=1). For 
an in situ stress ratio of 0.75, in situ tangent moduli of 
45.35 MPa, 65.18 MPa and 80.22 MPa were calculated, at 
points half a diameter above the crown, at the springline 
elevation and half a diameter below the tunnel floor, 
respectively. 

Assuming that the lining was activated at one diameter 
behind the tunnel face, the amount of stress release 
immediately before liner installation could be calculated. 
To avoid data interpolation, the A curve for K,=0.8 was used 
for this purpose. The average stress release was about 60% 
and a reduction of about 43% in the ground modulus was 
found. The former value is higher than that found in the 
back analysis presented in Section 5.2.4.3 (50%), which 
results from the approximation above and from others which 


are built into the proposed calculation procedure. The two 


1258 


dimensional ground stability verification at the section the 
lining was activated furnished a factor of Satety sofia zo. 

In shielded driven tunnels, it 1S convenient to check 
if the soil stood up over the TBM body, in order to assess 
if the assumed distance between the tunnel face and the 
section of lining activation is correct. This verification 
could not be performed in the Domplatz and in the 
Mississauga tunnel cases, since no information was available 
regarding the amount of overcutting used. In the LRT tunnel, 
the ground was bored with a diameter 19 mm larger than the 
outer diameter of the TBM body (Branco, 1981:23). 

By uSing the approximate solution for tunnel closure 
presented in Section 5.3.5.2 and the in situ ground moduli 
estimated earlier, one can calculate the increments of 
radial displacement at the crown and floor, from the tunnel 
face to the shield tail. The latter extended to 5.5 m behind 
the face (0.89D). From this, one finds that thelwertical 
diameter of the bored opening reduced by 14.5 mm from the 
face to shield tail. This value is less than the clearance 
provided by the overcutting (19 mm). The reduction in the 
horizontal diameter was also less than the above clearance. 
Therefore, one could say that the soil did not close over 
the shield body. This, in fact, happened whenever the tunnel 
was bored through uniform till. In some locations, however, 
(see Section 2.3.5.4), sand pockets were cut at the crown 
and localized instabilities were observed. Some soil ran 


into the unsupported heading, over the cutting wheel, and 


NW Ore ne) 


some soil blocks may have rested over the TBM. But the 
remainig till mass, otherwise stable, did not come in 
contact with the TBM body. In other words, in the present 
case, it is valid to assume that the opening was virtually 
unsupported until the lining left the shield and was 
expanded against the opening profile, about one diameter 
behind the tunnel face. 

For the lining-ground interaction analysis, it was 
assumed that the wooden lagging did not contribute to the 
fining. Stitiness. Its action was, thus, limited to 
longitudinally transferring the ground loads to the steel 
EL bse eee 20aGPa yp =0).2 Melee 1.82 x, 10°* m*/m) ae .t..22 m 
Spacing. Two iterations were needed to find the equilibrium 
condition, which yielded an average ground modulus of 34.85 
MPa, and relative flexibility (8) and compressibility (a) 
lining ratios of 0.0055 and 10.74, respectively (typical for 
this type of support in stiff clays or medium sands). 

The final crown settlement was calculated as 17.5 mm 
(U=0.754) and the associated amount of ground stress release 
at the crown was about 65%. The final subsurface settlements 
could not be measured in this case history, since the 
magnetic extensometer installed at the tunnel axis was lost 
when the TBM reached the instrumented section (Branco, 
$981:77,a110). Accordingly, only the final surface 
settlements were calculated and are compared to the 
measurements in Figure 7.40. Note, however, that while the 


settlements were measured at points located 3 m below the 


1260 


surface, the calculated movements refer to points on the 
actual ground surface. If the calculation method allowed the 
estimate of settlement distributions in horizontal planes 
below the surface, a better agreement with the observations 
would have been found, with a larger maximum displacement 
and a narrower settlement trough. In fact, the settlement at 
the-tunnel axis, 3 m below surface, was calculated as 9.7 
mm, while the actually measured settlement was 9.65 mm. 
Nevertheless, the agreement between calculations and 
measurements is reasonable, although it is again noted that 
the proposed procedure tends to underestimate the ground 
distortions. 

The calculated thrust forces in the steel ribs are 
shown in Figure 7.41. Thrust forces were measured at rib 
joints by loads cells, as decribed in Section 5.2.4.3. Ncte 
that the calculated and measured thrusts have been reduced 
for the tunnel unit length, on a 1.22 m rib spacing basis. 
Note also that the lining self weight was discounted from 
the readings. The predicted thrusts seem to very closely 
bound the long term load cell meaSurements (taken 293 days 
after installation), while the short term reading (16 days, 
tunnel face 36.4 m ahead of the instruments) tend to lie 
below the calculated levels. 

In general terms, the tunnel performance predicted 
herein, agrees well with the back calculated tunnel 
performance presented in Section 5.2.4.3 using the two 


dimensional finite element method, as well as with the field 


126] 





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1263 


measurements. 


e525. /77AN-NATM Turinel in-Marl?°The “Munich U' = Bahn - 
Line -8/1,; Baulos 18.2 

This case study refers to a subway tunnel built in 
1975-1976 in Munich (W. Germany), that was described by 
Laabmayr and Pacher (1978) and Laabmayr and Weber (1978). 
Two parailel tunnels were driven consecutively, each one 
wren fa ear tcirevlar’cross *sectron, 6.98 "m high and 6.32 m 
wide. The area of excavation of each tunnel was 37.50 m?/m, 
corresponding to an equilvalent diameter of 6.91 m. At the 
instrumented section (MQ4), under the Mathilde Covent (by 
Mathildenstrasse), the ground cover was 22 m and the tunnels 
were driven almost 19 m apart (centre to centre), i.e., with 
a central pillar width of almost 2 diameters. The analysis 
and instrumentation results to be discussed refer to the 
first tunnel driven only. 

The geological and geotechnical conditions found in 
Munich, and especially at the site were described by 
Laabmayr and Pacher (1978), Gebhardt (1980) and Krischke and 
Weber (1981) and were summarized by Steiner et.al., (1980) 
and Heinz (1984:267). At the instrumented section, the 
subsurface profile included a 10 m thick superficial layer 
of quaternary sands and gravels, underlain by a 7 m thick 
layer of tertiary dense sand rich in mica (Flinzsande). 
Below this layer, there was a thick layer of stiff to hard 
calcareous clay marl (Flinzmergel), also from the Tertiary, 


which is locally intercepted by other tertiary sand layers. 


1264 


A perched water table was found at about 15 m below surface, 
within the upper tertiary sand layer. The other sands 
layers, within the marl, were found;to~bewsaturatedsand 
under pressure. A deep dewatering system was installed and 
operated for more than three months before tunnel 
construction commenced. The ground settlements due to 
dewatering had been completely stabilized before the tunnel 
was driven through the stiff marl, and, during excavation, 
the ground was found to be almost water-free. A drained 
analysis could, thus, be considered. 

It will be assumed that the ground response was 
controlled mainly by the marl, although the tunnel cross 
section was intercepted by a tertiary sand layer between the 
haunches and the floor. The geotechnical properties of the 
marl summarized by Steiner et.al. (1980:170) and by Krischke 
and Weber (1981:11) were used as a basis for the definition 
of the input parameters for the present analysis (see Table 
7.9). A coefficient of earth pressure at rest slightly 
higher than that suggested by these authors (0.8) has been 
assumed, perhaps more consistent with the higher 
overconsolidation of this soil, compared with the Frankfurt 
marl. No indications were found regarding the variation of 
the deformation modulus of the Flinzmergel with depth. 
Accordingly, a constant initial tangent modulus was assumed, 

corresponding to the upper bound value given by those 
(AuGihoes, Since no information was provided regarding the 


failure ratio, the same R, used for the Frankfurt marl was 


Unit weight Y kN /m ou 
In situ stress ratio Ky = 1.0 
Effective cohesion intercept cr kPa 40 
Effective friction angle o' (i) 258 
Failure ratio Re = 0.8/4) 
Initial tangent modulus Ey MPa 200 
Poisson's ratio u - Ole 25 
Notes: (1) Parameters based on data published by Steiner et al 


(1980:170) and Krischke and Weber 
(2) Assumed equal to that for the Frankfurt marl. 


Table 7.9 Selected Parameters for Munich Marl 


(USES WAL 


1285 


1266 


adopted herein. 

The tunnel construction was carried out with the 
sequential excavation of the face in three stages, heading,. 
bench and invert, with immediate application of shotcrete 
over the exposed ground. The rate of advance when the tunnel 
face passed the instrumented section was about 4 m/day. The 
excavation round length was 1 m at the heading and Benen 
and 2 m at the invert. The primary lining also included 
pereetnee shaped segmented steel ribs with a channel profile 
(TH 21/58), at one metre spacing, which were not closed at 
the floor. The shotcrete total thickness was 16 cm from 
crown to haunches, and was increased to 20 cm in the invert. 
The support ring was closed, at the instrumented section, 7 
m behind the face (Laabmayr and Weber, 1978:89). Similarly 
to the Romerberg Tunnel in Frankfurt, ground anchors were 
also installed here, but again, their effectiveness in 
controlling the ground behaviour was questioned (Laabmayr 
and Weber, Op.cit.:82). 

The tunnel construction was undertaken under good 
ground control conditions, wtih no ground instabilities 
being reported. 

An adjusted friction angle of 31° was calculated. At 
lining activation the amount of stress release resulted was 
about 51% and the average ground modulus was 97.69 MPa, 
which corresponds to a 51% decrease from the in situ value. 
The 2D ground stability verification at lining closure 


furnished a factor of safety of 1.91. 


1Z67 


The lining-ground interaction analysis was carried out 
neglecting the action of the steel ribs, and considering a 
shotcrete lining 0.16 m thick, with a modulus of 10 GPa and 
a Poisson's ratio 0.2. Two iterations were needed to define 
the equilibrium condition, neglecting the ground heave after 
the support was activated. A ground tangent modulus of 79.82 
MPa was found at equilibrium, from which flexibility and 
compressibility ratios of 0.0015 and 7.73 were calculated, 
respectively. 

A final crown settlement of about 9.9 mm was calculated 
(U=0.616), for which a stress release of about 55% resulted 
at tunnel crown. At the instrumented section, the surface 
settlement profile could not be determined due to 
interference from existing constructions. The ground 
movements were measured only at two points, as shown in 
Figure 7.42. Here, only those movements observed during 
tunnel construction are considered: the displacements due to 
dewatering were discounted. Figure 7.42 shows also the 
calculated final subsurface settlement distribution. It 
seems to agree with the sparse field data available. 

Figure 7.43 presents the calculated and measured radial 
stresses acting around the lining contour. The stresses were 
measured by Glotzl contact pressure cells and they 
correspond to the final readings, taken 7 months after the 
instruments were installed. The measurements indicated some 
load increase with time. It is not known when the second 


tunnel was DUTLt ana therefore, 1t 1S not known if these 


1268 


Subsurface Settlements 


=12=10.- 8.26 ~44-2i9Q, 2 AneGme'Sr 10812 








YZ 


Elevation (m) 





Legend 


O tdeasured 








+ Predicted 





“1210-8 -6-4-2 0 2 4 6 8 10 12 
Settlement (mm) 


NOTES 


|. MEASUREMENTS AT EXTENSOMETER E3, FROM LAABMAYR AND WEBER (1978'85) 
2 SETTLEMENTS OUE TO DEWATERING OISCOUNTED. 


Figure 7.42 Measured and Calculated Final Subsurface 


Settlements over the Munich Line 8/1 Tunnel 


1269 


readings were affected by the parallel tunnel construction. 
If they were, the effect may have not been very pronounced 
as the second tunnel was built almost three diameters apart 
from the first one. A substantial scatter was observed in 
the field data. The predicted stresses tend to bound the 
measurements. 

No convergence measurements were reported at the 
instrumented section MQ4. However, they were taken at 
another section (MQ1), some 400 m from MQ4, where not much 
dissimilar conditions were encountered (Laabmayr and Weber, 
Op.cit.:81). A fairly uniform lining closure was observed, 
confirming the in situ stress ratio adopted in the analysis. 
Horizontal diameter decreases of up to 2.3 to 3.9 mm after 
the shotcrete ring closure at the floor, were measured at 
two different convergence bases (H1 and H2). These values 
compare well with the 3.2 mm horizontal diameter reduction 


predicted in the lining-ground interaction analysis. 


7.5.3.8 An NATM Tunnel Built under Compressed Air: The 
Munich U-Bahn-Line 5/9, Baulos 7 
The present case history refers to a subway tunnel 
built in the early eighties in Munich (W. Germany), that was 
described by Baumann et.al. (1985). It represented one of 
the first experiences in using shotcrete combined with 
compressed air (Weber, 1984). 
Two parallel tunnels were driven consecutively, each 
one with a near circular cross section, 7.0 m high and 6.5 m 


wide. The excavation area of each tunnel was 38 m?/m, which 


129.0 


350 
300 

PREDICTED 
250 


200-F 


0; (kPa) 


150 


MEASURED 





fe) 45° 90° FSS cetB0S, 225° 1270S S| Somme coe 
° S Cc SS F 


NOTE! MEASURED AT MQ4, TAKEN 7 MONTHS AFTER INSTALLATION 
(LAASMAYR ANO WEBER, 1978! 87) 


Figure 7.43 Calculated and Measured Radial Stress 


Distributions on the Shotcrete Lining of the Munich Line 8/1 


Tunnel 


i203 


corresponds to an equivalent diameter of 6.95 m. At the 
instrumented section MQ7, located West of the Maximilianeum, 
not far from the Isar River bank, the ground cover was 20 m 
and the tunnels were driven about 13 m apart (centre to 
centre), leaving a central pillar of slightly less than one 
diameter between the two tunnels. The analysis and 
instrumentation results to be presented refer to the first 
tunnel driven only. The second tunnel passed the 
instrumented section MQ7 4 months after the first tunnel had 
passed it. 

The geological and geotechnical conditions found at 
rhasmsSitenarelsimidari tosthatadescribedeingSectiont7<5.327% 
At the instrumented section, the subsurface profile included 
a 4.5 m thick superficial layer of quaternary sands and 
gravels, underlain by a thin marl layer (2.5 m thick). Below 
it, a 10 m layer of fine to medium dense tertiary sand was 
found. Underlying this, a thick layer of stiff to hard 
tertiary marl was encountered, through which the tunnel was 
driven. The marl cover above the crown was slightly less 
than half tunnel diameter. At floor elevation the marl was 
cut by a metre thick sand layer. 

A perched water table was found at the quaternary 
granular soil. Another one was found 15 m below the surface 
within the thick sand layer. The thinner sand layer below 
the tunnel was also saturated and under pressure. A 


compressed air pressure of 60 kPa was used in this section. 


Voye2 


For the analysis, it was assumed that the ground 
response at the crown was controlled by the stiffness of the 
sand layer located above the tunnel, with an in situ drained 
modulus estimated as 144 MPa from data presented by Krischke 
and Weber (1981:111). For the springline and floor, a 
tangent modulus of 200 MPa was assumed for the stiff marl, 
as for the U-Bahn-Line 8/1. 

Based on data from Krischke and Weber (1981:111) and 
Steiner et.al., (1980:170), the coefficient of consolidation 
of the marl was estimated as 2 x 10°? cm?/s. The likely 
ground response in terms of pore pressure development, can 
be estimated through the tentative criterion proposed in 
Section 3.3.4.5. If the pore pressure changes in the 3 m 
marl cover were solely caused by the change in the hydraulic 
boundary condition at the tunnel contour, one could use the 
criterion given in Figure 3.23. It would then be found that 
over one week, the marl cover would experience partial 
consolidation, but over a time span of a month, appreciable 
consolidation (more than 90%) would have developed. If an 
average rate of advance of 3 m/day is admitted (Weber, 
1984:26), then the 'short term' ground response, 
corresponding typically to an advance of the face from a 
section, say, 2D behind the instrumented section, to a point 
4D past this section, would take place in a two week time 
interval. The 'short term' response would thus involve some 
amount of ground consolidation, and could not be defined as 


‘undrained'. Accordingly, a drained analysis was favoured, 


1273 


and the same 'drained' parameters used for the Munich 
'Flinzmergel' in Section 7.5.3.7, were adopted presently. 

The tunnel construction was similar to that used in the 
U-Bahn-Line 8/1, except that here the shotcrete ring was 
closed at a shorter distance from the heading face (5.5 m 
typically). The minimum shotcrete thickness used was 15 cm, 
and at some locations it was increased to 18 cm. A single 
steel mesh (Q188) was laid on the inner side of the 
shotcrete lining. Lattice girders at one metre spacing were 
installed instead of steel ribs. 

The tunnel construction was carried out under good 
ground control conditions, and no ground instabilities were 
reported. 

An adjusted friction angle of .31.2° was calculated on 
account of the cohesive strength component and of the 
failure ratio equal to 0.8. In calculating the tunnel 
closure at the instant of lining activation (0.8D behind the 
face), allowance had to be made regarding the compressed air 
pressure being applied. To take it approximately into 
account, the in situ radial stresses at crown, springline 
and floor were reduced by 60 kPa, the magnitude of the air 
pressure. This approach was used earlier by Deere et.al. 
(1969:87). This is equivalent to reducing the soil cover 
Septnebysen cam tore arsoileunit. weight of 21 kN/m?)> In so 
doing, the amount of stress release at lining activation was 
about 49% and the mean ground modulus was 95.03 MPa. The 


latter corresponded to about a 49% decrease of the original 


1274 


in situ ground modulus. The 2D ground stability verification 
of lining closure furnished.a, factor of saletveotac. Dil, 

The lining-ground interaction analysis was carried out 
neglecting the action of the steel mesh or of the lattice 
girders. A nominal 16 cm thickness for the lining was 
considered. As on other occasions, a shotcrete modulus of 10 
GPa and a Poisson's ratio of 0.2 were assumed. Two 
iterations were required to define the equilibrium 
condition, neglecting the ground heave after support 
activation. A tangent modulus of 80.18 MPa was found for the 
ground at equilibrium (about 43% if the in situ value). 
Flexibility and compressibility ratios of 0.0014 and 7.67 
were calculated, respectively. 

A final crown settlement of 10.05 mm was calculated 
(U=0.578), which corresponds to about 52% of the stress 
release at the tunnel crown. The field instrumentation at 
section MQ7 did not include the measurement of the 
subsurface settlements along the tunnel vertical axis. Only 
surface settlements were measured, as indicated in Figure 
7.44. Two measured profiles are indicated, one corresponding 
to a 'short term' condition (10 days after the face passed 
the section) and another corresponding to a ‘long term' 
condition (140 days after passage, immediately prior to the 
second tunnel drivage). Both at the short or long term 
conditions, the tunnel was being submitted to the same 
compressed air pressure. The calculated settlement profile 


is included in Figure 7.44 and one notes that it agrees 


W2Ue 


Total Surface Settlements 

















=1$ = 10 =5 0 S 10 15 20 25 
a te) 
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ae : Predicted = 
4 OE 
o 
| 
ie rite 
| 
Be | Pres 
aaeu8) =350 








“15 -10 <5 0 5 10 15 20 25 
Distance to the axis (m) 


NOTE: MEASUREMENTS AT SECTION MQ7, DATA FROM BAUMANN ET AL (1985! 336) 


Figure 7.44 Measured and Calculated Surface Settlements over 


the Munich Line 5/9 Tunnel 


1276 


roughly well with the 'short term' profile. This agreement, 
however, may be claimed as fortuitous, since there is not a 
complete correspondence between the ‘partial consolidation’ 
condition, that may have actually developed in the short 
term, and the completely drained behaviour (with "zero" pore 
pressures) assumed in the calculations. On the other hand, 
it 1S not unexpected that the measured ‘long term' 
settlements are larger than those calculated ihe proposed 
calculation method does not take into account the ground 
volume changes and associated consolidation settlements 
resulting from porewater pressure changes and ground water 
drainage into the tunnel (see Section 3.3.4.2). 

The tunnel convergence was also measured at the 
instrumented section. Reference points at the springline 
were installed right at the face, so that the changes in the 
horizontal tunnel diameter before and after shotcreting the 
invert could be observed. The radial displacements at the 
face were calculated using the approximate solution 
presented in Section 5.3.5.2. The reduction in the 
horizontal diameter, from a section at the face to the 
section where the shotcrete lining ring was closed, was thus 
calculated to be 9.36 mm. This compares favourably with the 
10 mm reported by Baumann et.al. (1985:336). After closure 
of the shotcrete ring, the lining-ground interaction 
analysis furnished an additional reduction in the horizontal 
diameter of 2.66 mm, which also agrees well with the 2.5 mm 


measured. 


Vea 


Specially designed load cells were installed to measure 
thrust forces and bending moments in the shotcrete lining 
(Baumann et.al., 1985:450). The measuring system was 
developed by Philipp Holzmann AG Munchen. Details of this 
patented measuring gauge were not disclosed. Apparently it 
consisted of a 2 m long, 0.2 m high and 0.12 m wide steel 
box, installed at the lining haunches and shoulders, and 
interrupted the whole shotcrete lining thickness. It is not 
clear how the system measured the bending moment (or normal 
force eccentricities). The calculated thrust forces exceeded 
the forces actually measured both in the short and the long 
terms AS noted in -Sectton 2.3.6.3,saeanonminitorn thrust 
force distribution was observed, along the 2 m long embedded 
gauge. For a given position of the face, the highest thrust 
force was recorded at the cell end closer to the face. The 
measured values indicated in Table 7.10 correspond to the 
mean values provided by Baumann et.al. (1985:336). The 
eccentricities measured ranged between 1 and 3 cm, while 
those calculated were one order of magnitude smaller (about 
OF0G*cm)y That= could "be~an indication that “the in™sieu 
stress ratio is not equal to one, as assumed. But this would 
be conflicting with the mode of lining deformation, which 
was well estimated by the calculations. Nothing further can 
be advanced, as the reliability of these measurements cannot 
be assessed due to the paucity of the information made 


available. 


Performance aspect 


Unit Calculated 


1. Maximum surface settlement mm B00 
2. Horizontal diameter change 

from face to the section 

the support ring is 

closed mm 9236 
3. Final horizontal diameter 

change of shotcrete 

lining after invert 

closure mm 2.66 
4. Thrust force at haunches : 

(45°) kN/m 693 
5. Thrust force at shoulders 

(WSS ¥ kN/m 617 
Notes: (1) Measurements at Section MQ7. 


(198 523316))r 


1278 


Measured 
Shor Lon 
MELE pao 
Boll 4.9 
10.0 = 
= Xo SMG) 
476 550 
305 400 


Data from Baumann et al 


(2) Short term: 10 days after the tunnel face passed the 


section. Long term: 


140 days after passage. 


Table 7.10 Measured and Calculated Performances at the 


Munich Line 5/9 Tunnel 


L273 


7.5.3.9 An NATM Tunnel Driven Through Heterogeneous 
Ground: The Bochum Baulos A2 Double Track Subway Tunnel 

This case history refers to a double track subway 
tunnel that was described by Jagsch et.al. (1974), by 
Hofmann (1976) and reviewed by Heinz (1984:69). It was built 
in 1973 in Bochum (W. Germany) and was one of the first NATM 
cases with large cross-sectional area built in an urban 
environment. 

The tunnel cross section waS non-circular being 8.3 m 
high and 10.1 m high. It had an excavation area of 64 m?/m, 
which corresponds to an equivalent diameter of 9.03 m. At 
the instrumented section MQI, the ground cover was 12.2 m. 

A fairly heterogeneous subsurface profile was found at 
EDISeSTLG ml cine ludedsa gemetnick superficial fili layer. 
which was the embankment cf an existing railway line. Below 
it there was a sloping layer of a fairly hard chalk-marl 
about 7 m thick at the tunnel axis. Underlying this, there 
was a softer sandy-marl, 4 m thick, that rested over a 
succession of sandstone, shale and coal. The ground water 
level lay below the tunnel floor. A mixed face condition was 
thus encountered, with a stiffer marl at crown, a softer 
marl down below the springline and sedimentary rocks at the 
DLO las 

Wittke and Gell (1980:113) presented the geotechnical 
properties of some of these materials, found however, at a 
different location (Bochum Baulos B3). While the stiffer 


marl had an "elastic" modulus of 100 MPa, the softer one had 


1280 


a modulus of 40 MPa. The proposed calculation method was 
clearly not designed to be used in such heterogeneous ground 
conditions. However, it would be interesting to test it 
under such conditions. A constant in situ tangent modulus of 
70 MPa was thus assumed for the entire ground mass. The 
drained friction angle of these deposits did not vary much 
and it was assumed to be equal to 25°. The cohesive strength 
of these materials varied appreciably, within almost one 
order of magnitude. For the analysis, it was decided to 
consider the lowest cohesion value, corresponding to that of 
the softer sandy-marl, which was equal to 64 kPa. An average 
unit weight of 20 kN/m*® was estimated for the ground mass. 
No data was available regarding the failure ratio of these 
materials. An arbitrary value of 0.9 was thus selected. No 
indications were provided regarding the in situ stress ratio 
(K,) in the ground. It is believed that it should not differ 
much from other marls found in the Frankfurt or in the 
Munich areas. A stress ratio of 0.8 was thus liberally 
assumed. Wittke and Gell (Op.cit.:113) suggested that the 
Poisson's ratio for this soil should be about 0.3. 

A very careful construction was carried out, with the 
face excavated in a variation of the heading-bench-invert 
scheme, classified as type T3 by Eisenstein et.al., 
(1985:711). The depth of each excavation round was 0.8 m at 
the heading and bench and 1.6 m at the invert. A 25 cm thick 
shotcrete lining was installed as well as steel sets at 0.8 


m spacing. The average rate of advance was 2 m/day. The 


1281 


closure of the shotcrete lining at the floor took place 6.4 
m behind the face. Some 3 m long ground anchors were 
installed to hold in place the steel ribs during their 
assemblage. 

The adjusted friction angle was calculated and was 
about 35°. At lining closure, a stress release of about 57% 
was eStimated and an average ground modulus of 37.58 MPa was 
obtained (almost 54% of the assumed in situ value). 

The lining-ground interaction analysis was carried out 
neglecting the steel ribs, and considering a shotcrete 
modulus of 10 GPa and a Poisson's ratio of 0.2. Two 
iterations were required to find the equilibrium conditions, 
neglecting the ground heave after support activation. A 
36.31 MPa ground modulus was found at this condition, which 
enabled the flexibility and compressibility ratios of 0.0057 
and 21.24 to be calculated, respectively. These values are 
typical for shotcrete in bh: clays or medium sands. 

A final crown settlement of 22.4 mm resulted (U=0.712), 
for which a stress release of 60% at the crown was 
estimated. The calculated final subsurface settlements are 
compared to those measured (Jagsch et.al., 1974:13) in 
Figure 7.45. The predicted magnitudes of the ground 
movements are comparable to the measured but the estimated 
shape of the settlement distribution does not agree with the 
observations, which involved less vertical straining. The 
departure could be attributed to the crude simplifications 


introduced (e.g. the uniform modulus profile) as well as to 


1282 


the non-circular shape of the excavation. 

Similar results were obtained regarding the surface 
settlement profile (Figure 7.46). As in other case 
histories, the surface distortions were underestimated by 
the proposed calculation method, although the magnitudes of 
the displacements are closely estimated. 

Finally, Figure 7.47 shows the predicted and measured 
final radial stresses acting on the shotcrete lining. The 
measurements do not reveal any clear trend. If a mean 
measured stress distribution could be defined, the 
calculated radial stress distribution seems to approximate 
it. The non-circular tunnel contour, as well as the 
heterogeneity of the ground around the tunnel could partly 
explain the disagreement between the calculated and observed 


stress distributions. 


7.5.3.10 An NATM Tunnel with Staged Installation of the 

Lining: The Sao Paulo North Extension Double Track 

Tunnel 

This case study refers to a large subway tunnel built 
in 1984 in Sao Paulo (Brazil), that was described by Cruz 
et.al. (1985), Negro et.al. (1985a & b) and by Eisenstein 
et.al. (1986). The double track tunnel was built with a 
non-circular cross. section, with, an. ateawoliy 6 get.) 5. oem 
high and 11.4 m wide (9.84 m equivalent agence cis At the 
instrumented sections $1-3 and $1-9, the ground cover was 


Vote ioe ile 


Subsurface Settlements 














=24—20= 165-12 =Se=—4 Ors) 667712 16 20° 24 
2S oe es ee es ee ee en ee ee 


2 2 
07S SMeEn= feo: et 
-2?- -—2 
ay fe) -—4 
ax SSS, 5 -=6 
8 
-8-+4 - —§8 
cc 
ee 10H L-10 
fe) [ 
a 1S pees = 2 
iia g 
~16 4 + 16 
-18- + -18 
-20- Sj eo + —20 
-22 -22 





-24-20-16 -12 -8 -4 0 4 8 12 16 20 24 
Settlement (mm) 


NOTE! MEASUREMENTS AT EXTENSOMETER E€3, FROM JAGSCH ET AL (1974'1/3) 


1283 


Legend 


° 


+ 





Figure 7.45 Measured and Calculated Final Subsurface 


Settlements over the Bochum Baulos A2 Tunnel 


Weasured 


Predicted 


1284 


Surface Settlements 
~12 210} 31 Berea? PF oto Flies Beacon 













: ~10 —2 
~ og -4 
o 
eS wil ' -6 
av | 
Feat , ae 
Yn 1 | 
-50- =10 tira 
| = 
25 - er. 
an 8 
-80-4 -16 
- 904 r ~18 Legend 
: knee OQ Measured 
-100 fF -20 + Predicted 








£80 pp pe 22 
—(2— 108s — 6a 4 ae 2 Oe 6 


Distance to the axis (m) 


NOTE' MEASUREMENTS AT THE INSTRUMENTED SECTION MQI; DATA FROM JAGSCH ET AL (1978113) 


Figure 7.46 Measured and Calculated Final Surface Settlement 


Profiles over the Bochum Baulos A2 Tunnel 


F265 


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1286 


The tunnel was driven through an overconsolidated stiff and 
fissured clay layer, about 12 m thick, which is a sediment 
of the Sao Paulo Tertiary formation. This soil underlay a 
superficial fine clayey sand layer, about 9 m thick, and 
overlay a 4 m thick fine clayey sand layer, below the tunnel 
floor. A perched water table was found 3 m below the 
surface, in the upper sand layer. The piezometric level in 
the lower sand layer showed that a nearly uniform pore water 
pressure distribution with depth existed in the stiff clay 
layer (in situ pore pressures of about 60 kPa). Some 
dewatering was undertaken at the portal area, about 80 m 
away from the measuring section, and caused a 30 kPa 
reduction in the porewater pressures in the lower aquifer, 
prior to the tunnel excavation. The settlements associated 
with this operation were separated from those that developed 
during tunnel advance. 

Negro et.al. (1985:57) showed that the tunnel acted as 
a drain, and that the 2 to 3 m cover of stiff clay above the 
crown resulted in being appreciably consolidated from this 
drainage during the drivage process. Accordingly, in the 
analysis of this case history, it will be assumed that the 
ground response was controlled by the drained parameters of 
these soils. Triaxial tests on the stiff clay furnished a 
friction angle of 25°, a 3 kPa coheésionvand a) talluresracio 
of 0.75. The in situ effective stress ratio of this soil was 
estimated as 0.9 and an average unit weight of 20 kN/m? was 


determined for this sedimentary deposit. The in situ 


1287 


(drained) tangent modulus for the stiff clay at the tunnel 
Springline elevation was estimated also from triaxial tests 
and resulted as 124 MPa. The modulus of the soil located 
half a diameter above the crown (the fine sand) was 
estimated as 47 MPa and that half a diameter below the floor 
(also the fine sand) was 64 MPa. 

The tunnel construction was carried out in two main 
phases: heading excavation with a temporary shotcrete invert 
and bench excavation. At the instrumented sections, the 
temporary inverted arch was demolished and bench excavation 
took place when the heading face was about 19 m past the 
section. The heading face was always advanced leaving a 
central supporting ground core. The face was excavated in 
one metre round depths, and the temporary invert was 
shotcreted in 2.5 to 3 m lengths. The average distance 
between the mid-section of the temporary invert being 
shotcreted and the heading face was Weim at the 
instrumented section. The depth of bench advance varied 
between 2.5 and 3 m. The final shotcrete invert was 
installed immediately after bench excavation. 

Besides a 0.25 m thick shotcrete, the support system 
included two steel wire meshs (4.48 kg/m?) and polygonal 
steel sets (I 8") at one metre spacing. The shotcrete lining 
thickness was increased to 0.40 m, after tunnel completion, 
and an additional steel mesh included. This tunnel is quoted 
as being the first subway tunnel in soil where shotcrete was 


used as the final support. The average excavation progress 


1288 


was 3 m/day for the heading and 6 m/day for the bench. The 
overall rate of completed tunnel construction was about 1 
m/day. 

The conditions involved in this case history, depart 
considerably from those envisaged for the application of the 
proposed calculation procedure. The tunnel was driven below 
the ground water level, with minor dewatering. The tunnel 
cross-section was non-circular, and, more importantly, the 
staged application of the lining, with a temporary invert at 
the heading, respresented a condition not considered in the 
development of the calculation method. 

The analysis was conducted for a circular opening with 
the same excavated area as in the Bochum case history. The 
ground parameters were defined for the effective stress 
levels prior to tunnel construction. However, the stress 
release at the opening was calculated assuming that the pore 
pressures were zero in the ground mass, as for the 
Mississauga tunnel case. To account, at least partly, for 
the particular sequence of construction, it was assumed that 
the delay distance of lining activation was equal to the 
distance between the heading face and the mid point of the 
temporary invert section (7.7 m), plus the distance between 
the bench face and the mid point of the final invert section 
being shotcreted (about 1.4 m). In other words, it was 
assumed tht the lining deformations developed only after the 
shotcrete ring was closed at the floor of the completed 


tunnel section. Although this approximation may lead to 


1289 


unrealistic estimates of the displacements at sections 
between the heading face and the bench face, it was felt 
that it could be reasonable in estimating the final 
equilibrium condition. 

An adjusted friction angle of 30° was calculated. At 
lining activation, 0.925D behind the face, a ground stress 
release of about 53% was estimated and an average ground 
modulus of 36.71 MPa was calculated (almost 60% reduction 
from the in situ value). The 2D ground stability 
verification at this instant furnished a factor of safety of 
is Se 

The lining-ground interaction was analysed for the 
completed tunnel section, neglecting the existence of the 
steel sets and meshes, and considering the 0.25 m shotcrete 
with a 10 GPa modulus and a Poisson's ratio of 0.2. Two 
iterations were required to find the equilibrium condition, 
neglecting the ground heave after support activation. 

A final crown settlement of 29.6 mm was calculated 
(U=0.631), for which at stress release of about 50% resulted 
at tunnel crown. The comparison between calculated and 
measured performances was restricted to surface and 
subsurface settlements, as lining loads were not measured in 
this case history. While the measured settlements shown in 
Figures 7.48 and 7.49 do not include the ground movements 
due to dewatering at the portal areas prior to tunnel 
excavation, they do include those settlement components 


resulting from ground drainage and consolidation, which were 


1290 


Subsurface Settlements 


-5040-30-20-10 0 10 20 30 40 S50 


0-—T=m=EM 7 PO 








Elevation (m) 


Legend 
Meas. $1-9 
Meas. S1-3 








Predicted 





-50-40-30-20-10 0 10 20 30 40 SO 
Settlement (mm) 


NOTE: MEASUREMENTS AT SECTIONS SI-3 & SI-9. DATA FROM NEGRO ET AL (1985) 


Figure 7.48 Measured and Calculated Final Subsurface 


Settlements over the Sao Paulo North Extension Tunnel 


ero 


Total Surface Settlements 
10 15 


= 15 =O a) @) =) 


Settlement (mm) 

















Legend 

Oo Meas. S!1-9 

Meas. SI-3 
+ Predicted 

-100 25 
=15 = 10 —5 0 5 10 15 
Distance to the axis (m) - 
NOTE 


MEASUREMENTS AT SECTION SI-3 & SI-9. DATA FROM NEGRO ET AL (1985) 


Figure 7.49 Measured and Calculated Final Surface Settlement 


Profiles over the Sao Paulo North Extension Tunnel 


1292 


not accounted for in the calculations. As it is noted in 
those figures, the settlements were underpredicted by this 
approximate analysis. Also underestimated were the ground 
surface distortions. 

The shape of the cross section of this tunnel deviates 
considerably froma’ c¢irrcular™profive. In fact, trom al. 
cases investigated, this is the one that presents the lowest 
opening height to width ratio (h/w=0.746). This can partly 
explain the underestimation of the magnitude of the crown 
and of the maximum surface settlement. The former was 
underpredicted by about 8% and the latter by about 14%, 
using the calculation, where a circular cross section of 
equal area was assumed. 

In order to assess the influence of the cross section 
shape on the crown settlement, the solution numerically 
derived by Negro and Kuwajima (1985) may be of some help. 
This solution was obtained from 2D linear elastic parametric 
bounday element analyses of deep unlined tunnels, with cross 
section shapes similar to the present one. The dimensionless 
crown displacement is given by: 

U =1.058 - 0.179% - 0.385K, [en 
where the symbols have the usual meaning. When the height to 
width ratio is reduced from 17 to 0.746, as sinsunlsecace Eton 
K, equal to 0.9, one finds that the crown displacement 
increases 8.5%. This value is of the same order of magnitude 


of the underprediction quoted earlier. 


Vea3 


7.5.4 Analysis and Interpretation of the Results Obtained 

The application of the proposed calculation procedure, 
to simulate shallow tunnel performance obtained through 
three-dimensional numerical modelling, through centrifuge 
model testing and through field observation of full scale 
prototypes, revealed that it does yield sensible results. 
With the above exercise, the Proposed method was tested and 
its use exemplified for a variety of conditions that may be 
found in actual practice. 

In only one case, the centrifuge model test, the 
calculation procedure made use of the frictionless soil 
solution. In another few cases, although the ground was 
Saturated in situ, the analyses were carried out assuming a 
fully drained soil behaviour. While in the former, an 
Gideeined condition could be assumed, in the latter, 
evidence showed that partly drained conditions had 
prevailed, so that the use of the frictional soil solution 
was preferred. 

In the applications of the method to actual tunnel 
cases, no attempt was made to best fit the observed 
performances. These tests were not back analyses, since it 
was assumed that all of the parameters governing the tunnel 
response were known. Accordingly, these parameters were 
selected, and in a few cases, assumed, so that they would 
represent the most probable conditions found in each case. 
The departures noted in each case study were discussed and 


tentatively explained. In the present section, a general 


1294 


appraisal of the results is attempted, with minor references 
to particular cases. 

In some cases, the proposed method was tested to 
conditions that deviate considerably from those originally 
set up in the development of the method. As mentioned 
earlier, some cases investigated (Mississauga, Munich 5/9 
and Sao Paulo North Extension) involved constructions below 
the groundwater level, under partly drained conditions. In 
these cases, drained ground parameters were estimated for 
the in situ effective stress conditions. However, the ground 
stress release and lining-ground interaction analyses were 
carried out assuming the ground mass was fully drained, with 
zero pore pressures. Obviously, this does not correspond to 
reality but it was necessary to use this approximation since 
the proposed calculation method does not allow an effective 
Stress analysis to be performed with proper consideration of 
the pore pressure effects. Notwithstanding this, the results 
obtained with that simplifying assumption compared well with 
the observed performances. The agreement partly resulted 
from the fact that, in those cases, the ground was 
overconsolidated, stiff or dense, and less susceptible to 
volume changes induced by porewater pressure changes. 

In some cases, the method was applied to construction 
conditions that also departed from those originally 
admitted. In the Munich 5/9 tunnel, compressed air was used 
during excavation. In the Sao Paulo North Extension Subway, 


the tunnel was advanced in stages, using a temporary invert 


1295 


in the heading excavation, which was later removed. These 
Singular operations were accounted for in the calculations, 
through simplified, yet conscious approximations. In the 
former, the ground cover was reduced proportionally to the 
compressed air pressure. In the latter, the support 
activation delay was calculated assuming that the first 
Stage lining (with the temporary invert) did not deform: 
lining activation was assumed to occur only after the invert 
of the completed tunnel section was installed. In the 
Romerberg tunnel (Frankfurt), the simultaneous construction 
of the twin tunnels was partly accounted for by 
Superimposing the settlement solution of single isolated 
tunnel, and neglecting the interaction between the tunnels, 
on the assumption they were sufficiently far apart. The 
approximations introduced in these cases were found to be 
sufficient to make the calculated performance closer to that 
observed. 

In three cases (Butterberg, Bochum and Sao Paulo North 
Extension), the tunnel profile deviated considerably from 
the circular section which was assumed in the development of 
the calculation method. Some of the noted disagreement in 
these cases (e.g., the underestimation of final crown 
settlement) could be attributed to this fact. 

Having in mind these peculiarities, the overall results 
obtained can be better assessed through a table summarizing 
the analyses conducted (Table 7.11). In all calculation 


results, the overall ground heave component resulting from 


1296 


the trend the opening exhibits to 'float' after the lining 
is activated (equation 7.10), was neglected. Of course, the 
heave component prior to lining installation is included in 
the displacement field provided by the proposed method. The 
reasons and consequences of this assumption were presented 
and analysed in Section 7.4. 

One notes that most of the cases analysed involved 
fairly favourable ground conditions (stiff or dense soils), 
and in all of them fair to good construction quality was 
ensured. In most cases, the tunnel was driven above the in 
situ water table (A) or above the lowered phreatic surface 
(LO). In a few cases, discussed earlier, the tunnel was 
advanced below the groundwater table (B), without 
dewatering. 

In a large number of cases, the tunnel was built 
following the so called NATM, using shotcrete as the main 
supporteelement:aInfall-cases;withwasnassumed) thatuthes 
lining was solely represented by the shotcrete ring 
activated at the section where it was closed at the floor, 
and always considering a Young's modulus of 10 GPa anda 
Poisson's ratio of 0.2. In the shield driven tunnels, steel 
ribs and lagging or segmented concrete linings were used. 
The support systems used in the case studies investigated 
ranged very widely, from very stiff to very flexible. The 
flexibility ratio, 8, was found to vary, in these cases, 
from about 0.5 to 0.0014. The compressibility ratio a, on 


the other hand, varied from about 468 to 4.44. These ranges 


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1298 


entirely cover the spectrum of relative lining stiffness 
likely to be found in practice (see Table 7.1). 

The calculation procedure was tested in tunnels with 
ratios of cover to diameter’ ranging trom de14 toes. 1o7eand 
with opening height to width ratio (h/w) ranging from 0.75 
to 1.08. The lining activation was assumed to have taken 
place at 0.65D to 1.75D behind the tunnel face. 

The adjusted friction angles of the ground involved 
varied from 29° to 41.5°. Only in the centrifuge model test 
a ¢=0 condition was assumed, and an undrained strength ratio 
of 0.37 was considered. The in situ stress ratios varied 
EROMkO;S TOtNTes 

The calculated amount of ground stress release at 
lining activation (L.A.) varied from 0.29 to 0.64. Excluding 
the model tests, the average amount of stress release found 
in the case histories was 56%. Incidentally, this value 
compares very favorably with the arbitrary 50% stress 
reduction proposed by Muir Wood (1975:124) for lining 
design. At lining activation, it was found that the ratio of 
average current to in situ tangent ground moduli varied from 
0.41 to 0.59. This ratio, at the equilibrium condition, was 
estimated to vary from 0.40 to 0.68. These findings indicate 
that™ fora quick. lining-ground interaction analysis, it may 
be sufficient to assume a 50% reduction in both the ground 
stresses and in the ground in situ stiffness, provided that 
the ground control conditions are comparable to those of the 


present case histories. 


packs: 


The two dimensional stability verifications, at the 
sections where the linings were activated, yielded factors 
of safety varying from 1.21 to 2.01, with the median value 
being about 1.6. It could be said that the proposed 
calculation method can be successfully applied to predict 
the performance of a tunnel, whenever this factor of safety 
is within or above that range. As-discussed in Section 
2.3.4.3, for this safety range, the ground displacements are 
small, and high shear strain concentrations in the ground 
are minimized or avoided. On the other hand, it was shown in 
SGcUsgehe 25.5, stiat) for thiswrange of factors of safety, a 
non-linear ground response is to be expected. In fact, 
ground stiffness reductions in the order of 50% were, 
indeed, calculated in these case histories, despite the good 
ground control conditions met. 

Only in three case histories was there a need to use 
the extrapolated part Sane twice normalized stress release 
Or ground reaction curves. The extrapolation beyond the 
formal limit of the generalized solution was needed mainly 
for the tunnel floor. Despite this, the results obtained in 
these cases were comparable to those where no extrapolation 
of results was needed. 

The final dimensionless crown displacement (U) was 
found to vary between 0.5 to 1.0, approximately. The latter 
figure was suggested, in Section 2.3.4.3, as a reasonable 
bound for crown settlements in tunnels built under good 


@eoundecontrol conditions, Thus, quick estimates of the 


1300 


final crown displacement, under these tunnelling conditions, 
could be performed assuming U=i1.0. 

The results of the Bone seen between measured and 
calculated performances are presented in Table 7.11 in 
simple qualitative terms. The same criteria used to evaluate 
the prediction of performance by the finite element method 
(Sect ion 15. 82551 72% Table 5.5) was adopted herein. The 
calculated value is said to approximately equal the measured 
value (=), whenever the latter is not more than 20% 
different than the former. Otherwise the calculated value is 
Said greater (>) or smaller (<) than the measured value. The 
Spatial distributions of displacements or loads were 
arbitrarily defined as good (G), regular (R) or poor(P), 
after a liberal comparison between prediction and 
measurements was made. This qualification is likely to be 
more subjective and may vary among individuals. 

A quick inspection of these results reveals that the 
proposed calculation procedure consistently yielded unsafe 
estimates of the maximum surface distortion (the maximum 
slope of the surface settlement trough). For the other 
performance aspects, the calculation method provided either 
a good or a safe estimate. If the calculated maximum 
distortions (y,) are plotted against the observed maxima 
(y,), then the results shown in Figure 7.50 would emerge. If 
the calculated and observed distortions were equal, the 
points would be accommodated along the full continous line 


shown. However, the points that were actually obtained were 


1.301 


By eaeee 


1:10,000 


CALCULATED DISTORTION Y¢ 
tl, 





:100 


~ 1:50 1:100 I:1,000 1:10,000 
OBSERVED DISTORTION Yo 


CASE HISTORIES: 


|. ABV ~SAO PAULO 6. MISSISSAUGA 

2. CENTRIFUGE MODEL(Q@=29%) 7. BUTTERBERG 

3. CENTRIFUGE MODEL (Q=39%) 8. EDMONTON LRT(Yo below surface) 
4. DOMPLATZ — FRANKFURT 9. MUNICH 5/9 (short term) 

5. ROMERBERG - FRANKFURT 10. BOCHUM 


. SAO PAULO-NORTH EXTENSION 


Figure 7.50 Calculated Maximum Surface Distortions Compared 


to the Observed Maxima 


1302 


found to lie close to the broken line. This line yields 
observed distortions which are on average, 40% greater than 
those calculated. This result was not unexpected, as it 
reflects a common trend shown by most finite element 
calcilations (see “Section 5.2.1.2). 

In order to compensate for this effect, it seems 
advisable to introduce an empirical correction into the 
calculated distortion (y,). The corrected distortion (y,,) 
would, thus, be obtained through: 

Yoo bet Ye [wees 

If the above correction was applied to the cases 
investigated, a better agreement between calculations and 
observations would be obtained. A possible exception would 
be case 6 in Figure 7.50. It corresponded to the Mississauga 
Tunnel, where, as noted in Section 7.5.3.4, some 
difficulties were experienced in levelling the settlement 
points in the field. 

A pictorial representation of results is given in 
Figure 7.51. One notes that the calculation method, in the 
majority of the cases, provided a close estimate of the 
maximum surface settlement (within a +20% margin). In only 
one case (No. 10, Table 7.11), the maximum surface 
settlement was underestimated. This case (the Bochum Tunnel) 
involved, however, very peculiar ground conditions (mixed 
face, heterogeneous profile), that deviates appreciably from 
the conditions idealized for the application of the 


calculation method. 


IsO3 


The surface distortions were underestimated in 73% of 
the cases and were never overestimated. In more than half of 
the cases, the predicted overall distribution of surface 
settlement was regular or poor. The calculation method tends 
to furnish settlement troughs which are wider than the 
observed. This does not seem to be a matter of serious 
concern, as the theoretical result is safe. Though the 
criticisms on the use of the error function or normal 
probability curve are known, it appears that this curve 
tends to fit actually observed surface settlement profiles 
better than ae distributions proposed by the present 
calculation» method. With this: fact.in mind, one could 
Suggest a correction on the width of the calculated surface 
settlement profile, as follows. 

The distance (i) between the point of inflexion of the 
settlement trough and the tunnel axes could be estimated 


assuming that the maximum surface settlement (S_..) is that 


max 
given by the proposed calculation method, and that the 
maximum slope of the settlement trough is given by equation 


7.22. If the shape of the settlement trough follows a normal 


probability curve, then: 


max 





i =0.606 facre'34 


cc 


The width of the settlement trough (w.) is ‘defined as 


Ss 


the distance between the tunnel axis and the point beyond 
which the settlements are insignificant (i.e., smaller than 


AaB > OLE S Boraapnorinaleprobabila tyecurve ,) this pointsds 


aE 


given by: 


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The above semi-empirical correction was tested in the case 
histories investigated (see Table 7.12) and was shown to 
give reasonable results. Note that the "measured" trough 
width is actually an estimated value, based on the 
measurements available, which in most cases were not 
extended to a sufficient distance from the tunnel axis to 
adequately define w.. 

As is also shown in Figure 7.51, the magnitude of the 
maximum observed subsurface settlement was always closely 
estimated with the proposed calculation method. The 
distribution of settlement with depth was good to regular in 
75% of the cases. 'Poor' distributions were noted in Case 5 
(Table 7.11), which corresponded to twin tunnels 
Simultaneously built and in Case 10, where a heterogeneous 
ground condition was encountered. For these two particular 
conditions, it would be expected that the calculated 
distribution could not compare favourably with the 
measurements. 

Figure 7:5 “indicates*further, “that ‘the thorizontal 
convergence of the lining after its activation was very well 
estimated in 71% of the cases available. Moreover, the 
Support convergence was never underestimated by the 
calculation procedure. These results seem to indicate that 


the assumption that the activation of a shotcrete lining 


1306 


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1307 


takes place at the section where it is closed at the tunnel 
floor, is a reasonable one. Additionally, they tend to 
confirm that, under this assumption, the arbitrary shotcrete 
modulus of 10 GPa adopted, combined with the assumption of 
no contribution of the steel ribs to the relative stiffness 
of the lining, are also reasonable approximations. 

It 1s difficult to perform an accurate assessment of 
the ability of the proposed method to estimate lining loads. 
This is mainly due to the fact that in most cases, contact 
pressure cells were used to measure the ground loads acting 
on the support. Depending on the installation procedure, 
this type of instrument is known to either overestimate or 
underestimate the actual contact stresses (see, for example, 
Hanna, 1985:205). A special class of problem arises when 
pressure cells are used in concrete - soil interfaces, and 
is related to the heat generated during the setting of fresh 
concrete (Hanna, Op.cit.:444). Due to these facts, it was 
decided to compare "average" lining loads, in the hope that 
the over-reading and the under-reading errors would balance 
each other. In terms of these qualitatively estimated 
average lining loads obtained from field measurements, it 
was noticed (Figure 7.51) that the calculation procedure 
either matched these loads or overestimated them. Large 
average load estimates (up to 50%) were noted in the 
Butterberg and in the Munich 8/1 Tunnels, where Maihak and 
Glotzl cells were used in the soil-shotcrete contact. This 


result may be associated with the formation of gaps between 


1308 


the cell and the concrete after the concrete temperature 
dropped (the full contact might have been lost at the 
interface). On the other hand, this effect may have not been 
present at the Munich 5/9 tunnel, where specially designed 
load cells as embedded in the shotcrete. Even so, the 
lining thrusts in this case, were also overestimated by 20 
ey OVA. 

Actually, the proposed calculation procedure was 
developed in such a way that it may indeed, lead to 
overestimation of the lining loads. Firstly, as explained in 
Section 7.2.2, the use of the final equilibrium ground 
stiffness, defined at the point of equilibrium, will 
normally lead to higher loads, unless the in situ stress 
ratio is low and the outward lining displacement at 
Springline is appreciable. This was the case in the 
Butterberg tunnel, where the overall ground stiffness 
increased from the instant of lining activation to 
equilibrium (see Table 7.11). Secondly, if the tunnel 
closure at the instant of lining activation is large, the 
combination of the 3D elastic solution for closure estimate, 
with the 2D non-linear elastic solution for the stress 
release estimate, may also lead to estimating higher lining 
loads. With regard to this aspect, an improvement on the 
closure estimate solution would be beneficial, particularly 
for cases where the delay in the lining activation is large. 
For activation at sections more than 1.5D behind the face, 


the 3D closure solution furnishes radial displacements which 


1309 


are basically independent of that distance. Hence, for 
distances greater than 1.5D, the tunnel closure is virtually 
constant and the resulting amount of stress release is also 
Gon Sita nite 

Ln ineatitymetoremostascilsfethertunnel chosure will 
increase if lining activation is delayed more than 1.5D. The 
failure of soil elements around the opening will increase 
the radial displacements and further reduction of the ground 
stress is attained, before a collapse condition emerges. 
This additional reduction in the ground stresses cannot be 
represented by the present calculation procedure, unless 
another solution for the tunnel closure is developed 
(including the ground non-linear response) and is coupled to 
the 2D ground stress release solution. In its present form, 
the proposed procedure will tend, for these conditions, to 
estimate smaller tunnel closures and higher lining loads. 
This limitation of the proposed method was discussed earlier 
in Section 5.3.7, where the problem was exemplified through 
the Edmonton Experimental Tunnel case, where the lining was 
activated at a section more than 2D behind the face. 
Fortunately, in most cases (See Figure 6.37) and under good 
gnoundicontsol conditions yhthisedustance wiselessathan of. 5Ds 

Also not assessed in the lining load estimate 
evaluation are the effects of the tunnel shape, and the 
differential soil stiffness degradation around its contour. 
It seems interesting to explore these two aspects in future 


research. The second factor could, perhaps, be investigated 


13:10 


using a ring and spring calculation model, where the spring 
moduli would vary from point to point along the tunnel 
profile, according to the stiffness provided by the 
generalized solution developed in Chapter 6 (the A' curves, 
for instance). 

Finally, it should be pointed out that the overall 
results obtained in this chapter with the proposed 
calculation method, are equivalent to those obtained by the 
finite element modelling exercises reviewed in Section 
5.2.1.2. In fact, if the diagrams shown in Figure 7.51 are 
compared to those presented in that section, it would be 
noted that the proposed procedure presented improved 
results. Using the same point rating criterion referred to 
in Section 9.2.1.2, tO evaluatemtnergqualveyvecocmcne 
prediction, some of the analyses summarized in Table 7.11 


top ranked and none showed results below the average. 


7.6 Summary and Conclusions 

In this chapter, the soil-lining interaction phase of 
the shallow tunnel problem was addressed. A simple linear 
elastic analytical model was selected to study this 
interaction in a two dimensional representation. The 
preference towards a ring-and-plate solution was justified, 
and of these, Hartmann's (1970, 1972) closed form solution 
was favoured as it is the only one that makes full allowance 
for the non-uniform stress field existing in a shallow 


tunnel. 


I3d4 


The assumptions involved in the soil-lining interaction 
Solution were described and the main aspects of its 
derivation were discussed. The relative stiffness of the 
lining to the ground mass was expressed through two 
coefficients, the compressibility ratio (a) and the 
flexibility ratio, (8). Typical values for common linings 
and soils were given. The closed form solution was presented 
in an abbreviated form (Figure 7.2) and its significance was 
assessed. 

The selected analytical solution assumes the ground to 
be represented by an infinite plate under a gravitational 
stress field. The consequences of this assumption were 
investigated, by comparing the results of the analytical 
solution with that seed from finite element analyses of 
a shallow tunnel. Since in both cases the action of gravity 
1s included, the differences in the results could be 
attributed to the influence of the ground surface, if the 
numerical inaccuracies are ignored. Numerical and analytical 
solutions were found to yield similar results for cover to 
diameter ratios greater than 1.5, which confirms the 
discussions presented in Section 2.2. Moreover, the 
analytical solution furnished conservative estimates of the 
lining response for smaller H/D ratios. Some of the 
discrepancies noted were attributed to the procedure adopted 
in the analytical solution used to account for the overall 
ground heave (the trend exhibited by a shallow tunnel to 


EL Oaite wit « 


13 tz 


In order to account for the effects of the delayed 
lining installation*in the sorl-Pining@interaction analysis, 
the use of the analytical solution had to be adapted. The 
first effect resulting from a delayed lining activation is 
the reduction of ground loads and this was introduced 
through the use of a reduced unit weight for the soil 
(equation 7.14). Though this artifice may be valid for 
linear elastic materials, for a non-linear elastic ground it 
is understood to be an approximation. The second effect, 
resulting from a delayed support installation, refers to the 
degradation of the soil stiffness, associated with the 
tunnel closure developing before the lining is activated. 

The reduction in the unit weight of the soil, can be 
assessed through the twice normalized ground reaction curves 
(r eee. deeivea in Chapter 6, once the radial 
displacements of the tunnel contour at the instant of lining 
activation are known. 

The degradation of the soil stiffness can be assessed 
through the derivative functions of the twice normalized 
ground reaction curves (A' curves). An important assumption 
then had to be made: the tangent stiffness of the ground was 
assumed to be uniquely related to the tunnel radial closure 
(U), through the derivative functions presented in Section 
6.4. The implications of this assumption were assessed and 
discussed. 

To evaluate the ground stress and stiffness reductions 


associated with the delayed support activation, the twice 


Pans 


normalized support activation, and their derivatives, at 
different points of the tunnel contour, are assumed to have 
an independent existence. In other words, the sequence of 
loading or closure at different points of the opening 
profile, is assumed to not affect the response noted at any 
other point of the contour. The approximation involved in 
this assumption is minimized when the stress ratios (LZ) or 
the tangent stiffnesses (E,) obtained indepently for each 
point of the tunnel profile, are averaged. 

To assess the consequences of the preceding simplifying 
assumptions, the proposed procedures for ground stress and 
stiffness reductions were tested against the results of 2D 
non-linear elastic finite element analyses, where different 
amount of stress release, representing varying degrees of 
delaying the support activation, were imposed prior to the 
lining installation. Accordingly, the closed form solution 
was used, coupled with the generalized solutions for ground 
reactions derived in Chapter 6. Through these solutions, an 
average tangent stiffness of the ground, at the instant the 
lining is installed, could be estimated. This value and the 
reduced soil unit weight, were used in the lining-ground 
interaction analysis. The ground loads onto the support at 
equilibrium, were found to be underestimated by this 
calculation procedure. This resulted from the assumption 
that the ground stiffness remains unchanged during the 
interaction process. The additional soil stiffness 


degradation, upon further closure of the now lined opening, 


1314 


would necessarily imply that increased lining loads would 
exist on the support. 

A better and safer assumption would be to assign the 
ground its stiffness as defined at final equilibrium. This, 
however, is not known, as the incremental closure of the 
opening after lining installation is unknown. To solve the 
problem, an iterative procedure was devised, in which the 
average ground stiffness is updated after each iteration. 
The process is repeated until convergence is obtained, which 
required not more than five iterations. With this algorithm, 
the calculation procedure was repeated and new ground loads 
acting on the lining were found. As expected, loads higher 
than those previously obtained were calculated. More 
importantly, these resulting ground loads were slightly 
higher than those given by the finite element analyses. 
Moreover, the radial displacements of the lining were 
closely estimated by the approximate analyses, except at the 
tunnel floor where the displacements were overestimated. 
This was attributed to an overestimation of the ground heave 
at points below the tunnel in the analytical approach. 

Briefly, the approximate solution developed by coupling 
the generalized solution for the ground response developed 
in Chapter 6, with the closed form solution for the 
lining-soil interaction analysis, furnished reasonable (and 
safe) estimates of the lining-ground equilibrium condition, 
both in terms of final stresses acting on the lining as well 


as final tunnel® closure’ 


TSS 


The parametric analyses presented in Chapter 6 
considered the tunnel to be unlined and, therefore, the 
resulting normalized settlement distributions disregard the 
influence of a lining on the ground movements. The influence 
on ground displacements that a lining has, when installed 
after allowing some stress release in the ground, was 
evaluated through the results of the non-linear elastic 
finite element analyses just discussed. The final settlement 
distributions obtained for the lined and unlined tunnel 
cases, were compared by estimating the amount of stress 
release required in each case to produce the same crown 
settlement. It was found that the unlined tunnel analysis 
causing an equal crown displacement, furnished settlement 
distributions similar to those obtained for the lined cases, 
and slightly higher settlement magnitudes. In other words, 
by neglecting the presence of the lining, conservative 
estimates of the ground settlements above the tunnel are 
found. Accordingly, the normalized distributions presented 
in Appendix C, can be used in practice, since it is 
sufficient to calculate the amount of stress release in the 
unlined tunnel solution, which causes the same crown 
settlement obtained in the lining-ground interaction 
analysis. That amount of stress release is easily obtained 
through the normalized ground reaction curve for the tunnel 
crown. 

As a result of the above analyses and verifications, a 


calculation sequence emerged. It was possible to produce a 


1316 


flow chart, to guide the application of the presently 
proposed design procedure (Figure 7.9). Its main sequential 
steps involve: 
a) The assessment of the geometric conditions of the 
problem; 
b) The evaluation of the geological and geotechnical aspects 
involved, including the definition of ground parameters; 
c) The assessment of the construction technology to be 
applied, including details of the support system; 
d) The assessment of the applicability of the proposed 
design procedure, including the anticipation of likely 
ground conditions to be encountered (risk of collapse, 
drained or undrained ground responses, soil volume changes, 
etc); 
e) The evaluation of the pre-support ground response (tunnel 
closure, amount of stress release, ground stiffness change, 
ground stability verification, etc); 
f£) The analysis of the lining-ground interaction including 
the calculation of thrust forces and bending moments, lining 
distortions, etc; 
g) The prediction of the ground movements, including final 
tunnel closure, subsurface and surface settlements, and 
maximum ground distortions. 

The results obtained in each step, from (e) to (q), 
should be interpreted accordingly and their acceptability 
verified. If the results obtained after any step are not 


acceptable (e.g., unstable pre-support response, excessive 


PC 


lining loads, excessive ground movements), then the 
construction technology should be reviewed and modified. If 
the anticipated tunnel performance is acceptable, then 
construction follow-up and monitoring are carried out, and 
the data collected are used for feed-back in each design 
step, aS an on-going process, as part of an observational 
design approach. 

Details of each design step were discussed and 
analysed. Special attention was given to the formal range of 
applicability of the proposed calculation method: absence of 
ground heterogeneities (mixed face conditions, for 
instance), absence of ground collapse, good lining contact, 
time independent ground response, limited ground volume 
changes, etc. Whenever possible, quantitative criteria were 
proposed to define the conditions for application of the 
method. | 

When the proposed method is applied to actual design 
problems, sensitivity analyses should be undertaken, so that 
the variability of the ground conditions are assessed, 
ranges of variation of soil properties are covered and the 
variabilities of the construction procedure are accounted 
for. The method can therefore be used in connection with a 
probabilistic design approach, and expected ranges of ground 
and lining responses can be defined. 

All the calculations involved are simple and easy to 
program. The entire procedure can be implemented in a small 


micro-computer, allowing quick on-site re-evaluation of any 


1318 


design step during the tunnel construction. The method can 
be applied and calibrated concurrently with field 
monitoring, thereby serving as an auxiliary tool for 
decisions being made during construction. 

In order to facilitate some of the calculations 
involved in the application of the design sequence 
described, a set of calculation sheets was prepared. A 
complete example of the use of the proposed design procedure 
waS presented. Each design step was covered in detail, for a 
particular case history (The Alto da Boa Vista Tunnel). The 
use of the normalized design charts presented in Chapter 6 
was exemplified. The lining-ground interaction analysis was 
conducted using two separate assumptions regarding the 
overall ground heave taking place after lining installation. 
Firstly, the ground heave was calculated and incorporated 
into the method, according to the original formulation of 
the analytical solution. It was shown that the heave 
estimate entails an indeterminate degree of approximation, 
normally leading to an excessive estimation of this ground 
movement. Secondly, the design procedure was applied 
assuming that this ground movement component was negligible. 
It was then shown that the ground heave does not influence 
the lining loads, but it does affect the radial 
displacements at floor and crown, and consequently the 
settlement distribution above the tunnel. A comparison 
between the calculated settlements using the two assumptions 


and the ground displacements measured in the field, revealed 


VERS 


that the zero heave assumption leads to slightly 
conservative estimates of field settlement. 

The proposed calculation method was verified against 
the results of a plane strain centrifuge model test of a 
shallow tunnel under undrained conditions, carried out by 
Mair (1979)ie for tthat, ethe frictionless tsoil ‘model solution, 
- developed in Chapter 6, was used. The calculated responses 
in terms of the ground reaction curve and surface 
settlements compared very favourably with the test results. 
The proposed calculation method proved, however, to furnish 
progressively poorer predictions of the maximum distortion 
of the surface settlement trough, as collapse was approached 
in the test. Nevertheless, as noted in Chapter 5, this 
result was expected. 

| The proposed procedure was then compared with the 
results of a 3D finite element analysis. The latter was 
carried out by Katzenbach (1981), who used a hyperbolic 
stress-strain relationship to represent the soil behaviour. 
Details of the 3D analysis were summarized and the results 
of the comparison were discussed. Again, the proposed method 
yielded sensible results which compared very favourably with 
the 3D analysis results. 

Finally, the proposed method was verified against a 
large number of actual case histories. The latter were 
selected according to the availability of adequate or 
sufficient field instrumentation data. Intentionally, the 


case studies selected always exhibited good ground control 


1320 


conditions, reflecting either good construction practice 
and/or good quality ground. Ground instabilities or local 
collapses were not reported in any of the cases 
investigated. To test how the method would perform in actual 
practice required the inclusion of some cases with complex 
conditions which deviated considerably from those originally 
set up in the method development. These conditions were 
represented by tunnelling below the water table with partial 
consolidation of the soil mass, Contr ueeuon involving the 
use of compressed air, staged excavation of the tunnel face 
with a temporary invert in the heading excavation, 
non-circular tunnel profiles, etc. Additional assumptions 
and approximations had to be incorporated in order to. deal 
with these distinct features. Despite this, the calculation 
method yielded sensible results, in terms of both the ground 
and the lining responses. 

It should be remembered that in all applications of the 
method, no attempt was made to best fit or bound the 
observed performances. These tests were not back-analyses of 
case histories, as it was assumed that all parameters and 
variables governing the tunnel response were known and would 
represent the most probable conditions found in each case. 

The results of the verification tests were put together 
(Table 7.11) and the overall ability of the proposed 
calculation procedure to predict actual tunnel performance 


was generally appraised. 


1321 


A global inspection of the range of conditions covered 
in these tests was undertaken. The tests covered both 
shielded and unshielded tunnel construction, different 
Support systems and lining stiffnesses, a wide range of 
geometric conditions and a fairly wide range of soil types. 

The output revealed that the amount of stress release 
at lining activation ranged from 29 to 64%, with an average 
of 56%. This value compares favourably with Muir Wood's 
iNo7 Ss 24)earbitrary reduction by 50%) of the? in situ ‘stress, 
suggested for lining design. The reduction in the ground 
stiffness resulting from the ground stress relaxation was 
found to vary from 32 to 60%. These findings indicate that, 
for a quick lining-ground interaction analysis, it may be 
adequate to assume a 50% reduction for both the ground in. 
situ stresses and the in situ (tangent) ground stiffness, 
provided the ground control conditions are comparable to 
those found in the cases studied. 

The two-dimensional stability verification, at the 
section where the linings were activated, yielded factors of 
safety varying from 1.2 to 2.0, with the median value being 
about 1.6. It was suggested that the proposed calculation 
procedure seems to provide reasonable predictions of the 
tunnelling performance, whenever this factor of safety is 
within or above that range. Tunnel model test results 
reviewed in Section 2.3.4.3 indicated that, for this range 
of safety factors, the ground displacements are usually 


small and that high shear strain concentrations in the 


1322 


ground are minimized or avoided. 

The final dimensionless crown displacement (U) was 
found to vary between 0.5 and 1.0. The latter figure was 
suggested, in Section 2.3.4.3, as a reasonable bound for the 
crown displacement in tunnels built under good ground 
control conditions. Hence, quick, and possibly, conservative 
estimates of the final crown displacement under these 
tunnelling conditions, could be performed by making U=1.0. 

The results from the comparisons between calculated and 
measured performances were presented in simple qualitative 
terms, following a similar criterion introduced earlier in 
Section 5.2.1.2. The worst result obtained was-related to 
the maximum ground surface distortions, which were, in more 
than two-thirds of the cases, unsafely estimated by ‘the 
proposed calculation procedure. Using the results found, a 
correction factor was empirically introduced, which 
indicates that the calculated maximum distortions should be 
increased by 40%, in order to make them closer to the 
measured values (see Figure 7.50). 

A similar trend was noted regarding the width of the 
surface settlement profile since the calculation method 
usually led to troughs wider than the observed. Once more, 
in order to make the estimated widths closer to the 
observed, a semi-empirical correction was suggested 
(equation 7.24). This was tested in the case histories 


investigated and furnished reasonable results. 


hee 


The results of this comparative study were then 
Summarized in graphic form (Figure 7.51), where it was shown 
that the calculation method did furnish good estimates of 
the maximum surface and subsurface settlements and of the 
lining convergence. Some difficulties were found in 
interpreting the results of lining loads. To a large degree, 
this was due to the fact that most of the lining load 
measurements were taken with contact pressure cells, and the 
difficulties associated with this type of instrument are 
fairly well known. However, it was shown that the proposed 
calculation procedure, furnished lining loads which either 
matched the average lining loads measured or overestimated 
them by up to 50%. Although this figure cannot be taken as 
definitive, as the accuracy of the measurements can be 
questioned, especially in shotcrete-soil CGOntact,etheanoted 
trend of the calculation method to overestimate lining loads 
was discussed and explained. Possible improvements on this 
aspect were discussed and suggestions for future research 


work on the subject were addressed. 


8. CONCLUSIONS 

The present research work dealt with the design of 
shallow tunnels in soil. In an attempt to avoid redundancy, 
this chapter reviews only the main points and conclusions of 
the work and the reader is referred to the summaries 
presented at the end of each chapter for a condensed, yet 
comprehensive review of all studies undertaken. 

The aims of the research were’ presented in Chapter 1. 
They included attempts to: 

a) identify unsolved or partly solved problems that 

affect existing practice; 

b) more adequately appreciate the mechanisms involved in 

the soil and in the behaviour of the supporting 

Structures, identifying the controlling parameters, 

whenever possible; 

c) develop procedures to solve these problems and 

consequently to propose a new design method; 

ad) summarize the results of the research work ina 

comprehensive manner and in a way that may be used by 

the practitioner; 

e) validate the results of the research by application o 

practical prsbwens and defining ranges of validity. 

Point (a) above was briefly addressed in Chapter 1 and 
more thoroughly in Chapter 4. The surveys presented in 
Chapter 4 revealed that there was no adequate and simple 
method to couple lining loads and ground settlement 


predictions. They also revealed that no satisfactory 


1324 


1325 


procedure was available to sensibly account for the effects 
associated with delaying the activation of the support. 
These effects are mainly represented by the ground stress 
relaxation and by the ground stiffness degradation. The 
importance of such effects seems to be recognized by many. 
However, since no sensible criteria were available to 
account for them, the practitioner was compelled to adopt 
extreme assumptions. Examples of these include the ‘full 
overburden' and partial ground embedment assumptions in 
lining design, or the assumption of full ground stress 
release for settlement predictions. With these conflicting 
assumptions, the design ceases to be an exercise of 
anticipating the 'most probable' performance, with allowance 
for adequate safety margins. Instead, it becomes an activity 
of bounding possible tunnelling performances, which makes 
the assessment of the safety margins more difficult. 

Point (b) was comprehensively addressed in Chapters 2 
and 3, where the soil tunnel behaviour was studied through 
idealizations exemplified by available theoretical and 
experimental modelling tools and supplemented by 
observational evidence from prototypes. Through this 
activity, some of the parameters controlling the behaviour 
were identified. Soil tunnels in urban environments are 
designed under requirements that lead to conditions where 
collapse mechanisms are usually avoided. Accordingly, most 
of the discussion was confined to the behaviour in 


pre-collapse stages. 


1326 


Although urban tunnels are usually designed for and 
built under ‘good ground control' conditions, it was shown 
that the factor of safety against ground collapse is fairly 
low, typically f.2 to 1.7. Moreover, it was shown tnarteccr 
these low safety factors, a non-linear response in terms of 
ground stress and strain relationships should always be 
expected. Considerable departures from linear responses were 
noted in model and prototype behaviour whenever the factor 
of safety was less than 2 to 3. 

A reasonably clear relationship appears to exist 
between the factor of safety and the dimensionless 
displacement (U) of the tunnel crown. An interim criterion, 
based on results of plane section model tunnel tests, was 
proposed to define a 'good ground control condition', for 
which the ground movements are generally acceptable and 
ground collapse is precluded. The test results indicated 
that near collapse conditions are generally met when U is 
greater than 1.8 and that values smaller than 1.0 would 
represent a 'good ground control condition'. Although 
tentative, this criterion proved to be sensible and to have 
potential use in practice. 

Another tentative criterion was derived from the above. 
It refers to the limiting dimensionless crown displacement 
increment for ground movement into a void space behind the 
lining, which might produce a near collapse ground 
condition. This dimensionless movement was suggested as 


equal to 1.0. From this, one may define a maximum crown 


132 


overcut in a TBM excavated tunnel. Once more, this is a 
tentative criterion which may be improved in the future, by 
further data collation and interpretation and by 
experimental and theoretical modelling. 

It was illustrated that if the overcutting is excessive 
and the ground displacements are large, 'gravity loads' may 
act on the lining and that its response may not be 
predictable. To avoid this condition, a 'good' lining-ground 
contact should be ensured and ground movements should be 
limited to levels below those associated with a near 
collapse stage. A speculative discussion was attempted to 
analyse the consequences of the development of uncontrolled 
ground deformations and. the associated ground 'loosening' 
process. A conceptual model was proposed and the need for 
experimental or theoretical investigations to confirm it, 
were suggested. 

Although of some value at the design stage, the above 
criteria of limiting displacements may be of little help 
during construction. For this, a criterion for ground 
Stability assessment in the field was suggested, which is 
based on the interpretation of longitudinal ground 
distortions from displacements measurements at points above 
the tunnel crown. 

While the present work focuses on the ground response 
under time independent conditions, it was felt necessary to 
analyse the role of groundwater on the ground behaviour. 


Among others, the effect of pore pressure generation and 


1328 


dissipation on tunnel stability was addressed for idealized 
conditions. The difference in the soil response to that 
observed in other geotechnical structures was explained by 
the existence of a different intervening parameter: the 
degree of stress relaxation. Moreover, this analysis 
illustrated the practical need to identify the ground 
response in either undrained or drained terms. Examples of 
how to anticipate these responses were given through 
approximate criteria derived from simplified theoretical 
analyses. The need to extend these approaches was suggested. 

Another feature studied, still within point (b), was 
the three-dimensional arching and ground-lining responses 
associated with the tunnel advance. The stress-displacement 
responses for points around the tunnel contour were 
reviewed, as well as some evidence of the three-dimensional 
Stress transfer effects. The role of some of these effects 
in controlling the tunnel lining design has not been clearly 
established, for instance, load concentrations or 
longitudinal bending of the support. Further investigation 
of these aspects would be of considerable interest, bearing 
in mind their potential consequences. 

After assessing points (a) and (b) described earlier, 
point (c) was addressed mainly in Chapter 5. To solve this 
problem, detected in current practice, it was decided to 
develop an integrated procedure that would allow lining load 
predictions and settlement estimates taking into account 


some of the most important factors controlling the tunnel 


L329 


behaviour. These are: 

1. the effect of the relative position of a horizontal and 
stress free ground surface; 

2. the action of gravitational body forces generating a 
stress gradient in the soil across the tunnel profile; 

3. the non-uniformity of the in situ stresses generated by 
a horizontal to vertical stress ratio different from 
unity; 

4, the non-linear response exhibited by soils in terms of 
their stress-strain relationships, including the 
dependence on hydrostatic and deviatoric stress levels; 

oe Ce delayed activation of the lining generating ground 
movements and associated stress and stiffness changes in 
the ground, prior to support application; 

6. the interactive nature of the load transfer process 
developed between the soil and the support. 

The decision to tneinde these factors in the 
development of the projected design procedure lies in the 
fact that they have an important role in most, if not all, 
shallow tunnel cases. Those factors, that are sometimes 
influential in one instance but not in others, had to be set 
aside, and, perhaps studied separately. 

Accordingly, in order to render the problem tractable 
and yet to ensure some generality, it was decided to limit 
the development to full face excavated circular tunnels, 
under time independent conditions, in situations not 


involving ground failure, shear dilatancy and post peak 


1330 


softening, among others. 

The finite element method was selected as an 
appropriate modelling tool to derive the projected method. A 
fairly extensive review of previous studies on shallow 
tunnel modelling using that method was undertaken. Through 
it, the ability and limitations of this method to portray 
shallow tunnel performance was critically assessed. This 
review led to the adoption of a simple two dimensional 
finite element code developed and tested earlier in this 
University for analyses of retained excavations. The program 
was adapted and implemented for the present project, where 
the soil behaviour is described by a non-linear elastic 
stress-strain relationship (the hyperbolic model). 

Through this program, tunnel construction is 
meorecentect n stages and the delayed lining installation 
and associated stress transfers are represented by a partial 
release of the ground in situ stresses. This two dimensional 
Simulation mimics the three-dimensional stress changes by 
the introduction of an additional variable, the amount of 
Stress release, a. Three well documented case histories in 
which good ground control conditions were present, were used 
to test this numerical solution. It was proved that good 
estimates of the final displacement field and lining loads 
are obtained whenever the factor, a, is appropriately 
selected. Like similar codes, this program seems useful only 
for the prediction of pre-failure responses. It is evident 


that there is a need to develop numerical modelling 


13321 


techniques that portray the ground response for conditions 
approaching collapse with the formation of shear strain 
concentrations. 

The main limitation of the selected simulation 
technique is the need for an independent estimate of the 
amount of stress release taking place before support 
activation. However, if this stress release was univocally 
related to the ground displacement, then an estimate of the 
latter, including the three-dimensional effects of an 
advancing tunnel, could be sought as an alternative approach 
to the solution of this problem. 

The need to develop a procedure to estimate the amount 
of tunnel closure at the section at which the support is 
activated was thus justified and undertaken through 
parametric three-dimensional finite element analyses. An 
approximate solution for estimating the closure of an 
unlined shallow tunnel in an elastic medium was propcsed. It 
allows estimates of radial displacements at three points of 
the tunnel contour, for different sections at or behind the 
face. These analyses also allowed simplified solutions to be 
obtained for estimates of the maximum horizontal 
displacement of the tunnel face and of the maximum 
longitudinal distortions for the ground surface and the 
Subsurface ahead of the face. 

This approximate method for estimating convergence was 
used to predict radial displacements for comparison with 


observed values in a large number of case histories. It was 


1332 


shown that it can yield sensible results whenever good 
ground control conditions are specifed and implemented in 
tunnel construction. Furthermore, it was shown that the 
method can be successfully applied to NATM construction 
schemes regardless of the soil type (but provided the face 
is stable), and to TBM or shielded schemes in firm grounds 
or even tn less stable soils, whenever the overcut is small 
and the lining is activated in full contact with the ground 
at a short distace from the face. For conditions different 
from the above, the simplified method yielded poorer 
results. Improvements to its predictive capabilities could 
be attempted in the future by incorporating the non-linear 
ground behaviour and by accounting for the support 
installation, both of which are presently negiected. 

To validate the proposed approach, the above 
approximate method was tested against results from the two 
dimensional modelling of the three case histories studied. 
By combining the estimates of the radial displacements at 
the tunnel contour from this method, with ground reaction 
curves obtained by the two dimensional modelling, reasonable 
estimates of the amount of stress release at the instant of 
lining activation were obtained. 

The above approach requires the use of a 2D finite 
element program to generate stress-displacement curves from 
which it is possible to assess the amount of stress release. 
The survey of practice presented in Chapter 4 showed that 


this may represent an inconvenience since this type of 


e618) 


analysis is not always used in routine practice. 
Accordingly, an attempt was made to develop a procedure 
which would allow these stress-displacement relationships to 
be obtained without the need for further finite element 
analyses. Parallel to this development, an attempt was made 
to obtain the relevant ground displacement associated with 
the reduction of the ground stresses, representing the 
tunnel construction in a two-dimensional simulation. 

To achieve these aims, some additional simplifications 
were introduced into the 2D finite element model presented 
in Chapter 5. These simplifications led to the establishment 
MVeiChdprer GNor twounon linear elastic istress~strain models: 
the frictionless model representing an undrained soil 
response, and the cohesionless soil model representing a 
drained response. These models exhibit stress-strain | 
relationships presenting the property of homothety, which 
allows these relationships to be normalized into unique 
stress-strain curves. Moreover, this property causes the 
response of geometrically homothetic tunnels to become 
unique when this response is conveniently normalized. These 
findings facilitated the generalization of results through 
the similitude they show, as they become independent of the 
scale of the problem. 

Through parametric numerical modelling, the role of 
each variable affecting the response of unlined tunnels was 
investigated. Finally, generalized normalized ground 


reaction curves (NGRC or A curves) were obtained for points 


1334 


at the crown, springline and floor of a shallow tunnel, and 
were presented as equations and charts. The limits of this 
generalization were identified. The partial derivatives of 
the NGRC (A' curves) were related to the ground stiffness at 
any stage of the tunnel unloading process. The two sets of 
curves (A and \') thus allow estimates of the amount of 
ground stress relaxation and ground stiffness degradation 
for any amount of tunnel closure defined by the 
dimensionless radial displacement, U, taking place at the 
section where the support is activated. These solutions 
allowed such estimates to be made for ranges of geometric 
and geotechnical conditions that were shown to cover most of 
the cases likely to be encountered in practice. The 
solutions also permitted some extrapolation of data beyond 
the ranges of conditions which had been focused on, though 
considerable care should be taken in this regard. 

Parallel with the development of these generalized 
solutions, normalized subsurface and surface ground 
displacement distributions were obtained for different 
amounts of stress relaxation associated with a given amount 
of tunnel closure. 

The soil-lining interaction phase of the shallow tunnel 
problem was addressed in Chapter 7. Once more, to avoid the 
need of finite element modelling in the development of the 
projected design procedure, preference was given to the use 
of a linear elastic, closed form solution that treats this 


problem, with full account of the non-uniform stress field 


1335 


existing in a shallow tunnel, including the effect of 
gravity. This analytical solution was adapted to take into 
consideration the effects of the delayed lining 
installation, represented by the stress relaxation and 
ground stiffness degradation. These were accounted for 
through the use of a reduced unit weight of the soil anda 
reduced ground stiffness, calculated from the normalized 
ground reaction curves and their derivative functions. The 
tangent stiffness of the ground was assumed to be uniquely 
related to the tunnel radial closure. This and other 
Simplifying assumptions were carefully assessed and their 
consequences evaluated through 2D finite element 
Simulations. 

It was demonstrated that adequate and safe estimates of 
final lining loads and displacements can be obtained by 
combining the generalized solution for non-linear ground 
response, with the analytical solution for soil-lining 
inweraction..«F ors this, eit. is: sutfacient, to estimate..the 
ground stress relaxation prior to lining activation and the 
ground stiffness at the final lining-ground equilibrium 
condition. Since the equilibrium condition is not known 
beforehand, a simple iterative procedure was devised to 
solve this problem. 

The present development could also be of some help, if, 
alternatively the soil-lining interaction analysis were done 
through the use of ring - and - spring solutions reviewed in 


Chapter 4. With them, the non-uniform degradation of the 


4336 


ground stiffness around the tunnel contour could be taken 
into consideration. 

Regarding the effect of the lining on the ground 
movements, parametric finite element studies revealed that 
the unlined tunnel analyses, with a crown displacement equal 
to the final crown settlement calculated in the lined tunnel 
case, furnished settlement distribution similar, yet 
Slightly conservative, to those found in the latter case. 
Thus, the normalized settlement distribution obtained in the 
parametric analyses can be used for settlement prediction. 
It is sufficient to calculate the amount of stress release 
in the unlined tunnel solution, causing the same crown 
displacement obtained in the lining-ground interaction 
analysis. This is easily assessed through the generalized 
ground reaction curves for the tunnel crown. 

With these developments, the aims described in point 
(c), given earlier in this chapter, were fulfilled. The 
following point, (d), required that the findings accumulated 
so far be organized in a comprehensive manner to enhance 
their use in practice. Appropriately, a flow chart was 
prepared to serve aS a guide for using the presently 
proposed design procedure (Figure 7.9). Each step of the 
procedure was presented and discussed, with emphasis being 
given to the assessment of its applicability. 

Moreover, to facilitate the calculations involved in 
the design sequence suggested, a set of calculation sheets 


was prepared, and an example was worked. It was shown that 


To 0 


all calculations are simple and easy to program, therefore 
allowing quick on-site re-evaluation of any design step 
during tunnel construction. Thus, the method can be applied 
and calibrated simultaneously with field monitoring serving 
as an auxilary tool for decisions being made during 
construction. Furthermore, at the design stage, the method 
allows sensitivity analyses to be undertaken, where the 
variability of ground conditions, construction procedures, 
geometric conditions, etc. can be taken into account. 
Accordingly, the expected ranges of ground and lining 
responses could be defined. The last point to be addressed, 
(e), referred to the validation of the proposed design 
procedure. This was also undertaken in Chapter 7, where the 
method was verifed against results of plane strain 
centrifuge model tests of a shallow tunnel. It was also 
tested against available results of a 3D finite element 
analysis in which the ground was represented by a hyperbolic 
Stress-Strain relationship. In both cases, very favourable 
results were obtained. 

Finally, the proposed method was verifed against a 
large number of actual case histories. These always involved 
good ground control conditions, reflected either by good 
construction quality or simply good ground quality. In order 
to test the performance of the method in actual practice, 
some of the case histories included conditions that deviated 
considerably from those originally set up in the development 


of the method. These conditions were represented by 


1338 


tunnelling below the water table with partial consolidation 
of the soil mass, construction involving) ehemusesct 
compressed air, staged excavation of the tunnel face with a 
temporary invert in the heading excavation, non-circular 
tunnel profiles, mixed face conditions. Additional 
approximations were introduced to account for these distinct 
features. Despite this, the calculation method yielded 
sensible results, both in terms of ground and lining 
responses. Noted discrepancies were duly explained in terms 
of the limitations of the method. 

It should be pointed out that in all applications, no 
attempt was made to best fit or to bound the observed 
performances. These tests were not back analyses of 
prototypes, as it was assumed that all parameters and 
variables governing the tunnel response were known and 
Supposedly represented the most probable conditions found in 
each case. 

A global assessment of the results obtained was 
undertaken and the ability of the Re ee procedure to 
predict tunnel performance was carefully investigated (Table 
iw ana Figure vot). 

This comparative exercise covered both shielded and 
unshielded tunnels, different support support systems and 
lining stiffnesses, a wide range of geometric conditions and 
a fairly wide range of soil types. This study revealed that 
for a quick lining-ground interaction analysis, it may be 


sufficient to assume a 50% reduction in the ground in situ 


1339 


Stresses and in the in situ tangent ground stiffness, 
provided the ground control conditions are comparable to 
those found in the cases studied. 

It was suggested that the proposed calculation method 
seems to provide reasonable predictions of the tunnel 
performances, whenever the factor of safety of the ground at 
the section the support is activated, is within or above the 
calculated range factors found in these cases (from 1.2 to 
2.0). The final dimensionless crown displacement, U, was 
found to vary between 0.5 and 1.0, and according to the 
Criterion set up earlier in Chapter 2, this confirms that 
ground control conditions were good and high shear strain 
concentrations in the ground were minimized or avoided. It 
was suggested that quick, and possibly, conservative 
estimates of the final crown displacement under these 
tunnelling conditions, could therefore be performed by 
making U=1.0. 

Except for the width of the settlement trough or the 
maximum ground surface distortion, the proposed design 
procedure yielded performance predictions that either 
matched or overestimated the observed behaviour. The surface 
distortions were underestimated in the majority of cases. To 
compensate for this not unexpected and unsafe trend, a 
correction factor was empirically introduced. Accordingly, a 
semi-empirical correction was also suggested for the 
assessment of the width of the surface settlement profile. 


In both cases, the proposed corrections improved the 


1340 


predictions, making them closer to the observed 
performances. 

With this, the aims of the present research project 
have been fulfilled. Suggestions for future research work 
have been presented in this chapter and in the summary 
sections of other chapters. Moreover, it is believed that 
the present approach could be extended to other conditions 
not covered by this work, such as twin tunnels and 


non-circular tunnel profiles. 


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1402 


18th Rankine Lecture. Geotechnique, Vol. 28, pp. 
ia ae WTA Lic 


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DD mec Oa er 


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1403 


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1404 


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1405 
Edition. McGraw Hill, London. 
QBSYIAWA TWSMAIE ATIHIG G-£ JO eTavese ~ A Xiaus 


294 
Ueterne *: ite ane ie 
-$ « “2 ? 
ee — — 
- 2 


_ 
E | 
al 
car 
a 
Lae 
| 
Ge ia 
se 
-Vvewt PT > aa 
26 a os = ane 
tne iv be 
- - Qiewrce te 
8 4 ii 
—_—— — —_— its _— 
| : 
* 
oO 
; 
© eeilied 
mee 
ee 





APPENDIX A - RESULTS OF 3-D FINITE ELEMENT ANALYSES 


1406 


1407 


Orstance to the Face (m) 
-5 -4 -3 =2 -| 0 | 2 3 4 5 








oe ( kPa) 
On, 

nl Re 2 yee 

soe I a emer: 
O2+ ee ; =| ae: 
ly yi | LC 
LC Lining Closure (postion of tne linng leading edge) 
0.0 fe) 


-10 -0.5 (e) 0§ 10 
Oistance tothe Face (0) 


Distance to the Face (m) 


Note: 
LC: Lining Closure( position of thelining leading edge) 





-10 -0.5 (0) 05 10 
Distance to the Face (0) 


Figure A.1 Radial Stresses and Displacements at the Crown 


Elevation (L/D=1/2) 


1408 


Ostance to the Face (m) 


-5 aA +3 -2 -| 0 2 3 4 5 
= 100 
LO -—— —— :—— .—_; 
eg 
Paes 
a = 80 
OW) j= - 











(kPa) 
or, , 40 
04 - LS 
Yo 
O2- Se 20 
LC Lining Closure (position of the lining leading edge) [uc 








Distance to the Face (0) 


Distance to the Face (m) 





Note: 
LC: Lining Closure( position of thelining leadng edge) 





-10 -0.5 fe) 0.5 10 
Distance to the Face (0) 


Figure A.2 Radial Stresses and Displacements at the 


Springline Elevation (L/D=1/2) 


1409 


Distance to the Face (m) 
25 -4 -3 -2 -| (@) | 2 =) 4 5 














Or 0.6F 7120 
Ore . > Or 
f ee (KPO) 
O4- 37 & 
ae - : 
o2} 4 40 
Note 
LC: Lining Closure (position of the lining leading edge) | uc 
00 ie) 
- 1.0 =IOro 0 0.5 1.0 


Distance to the Foce (0) 


Distance to the Face (m) 
-5 -4 -3 -2 -| (@) | 2 3 4 5 


LC Lining Closure (position of 
the lining leading edge) 








-10 =0'5 fe) 0.5 LO 


Distance to the Face (0) 


Figure A.3 Radial Stresses and Displacements at the Floor 


Blevation (L/D=1/2) 


APPENDIX B - LIST OF RESPONDENTS TO THE QUESTIONNAIRE 


AUSTRIA 


Prof. Franz Pacher and Mr. N. Ayaydin 
Franz Josef-Strasse 3 
A-5020 Salzburg 


Mr. Klaus Mussger and Mr. G. Schicktanz 
c/o Geoconsult Consulting Engineers 
Sterneckstrasse 55 

A-5020 Salzburg 


CZECHOSLOVAKIA 


Mr. Josef Novak 
Vodni"stavbyy o.p. 
Delnicka 12 

Praha 7 


ENGLAND 


Mr. D. Buckley 
c/o Sir William Halcrow and Partners Ltd. 
Vineyard House 
44 Brook Green 
London W6 7BY 


FRANCE 


Mr. Marc Panet 

c/o Simecsol 

115 Rue Saint Dominique 
75007 Paris 


WEST GERMANY 


Dr. @. Baumann 

c/o Philip Holzmann AG 
Herzog-Heinrich-Strasse 20-22 
8000 Munchen 2 


Prof. Heinz Duddeck 


1410 


IS 


14a. 


Consulting Engineer 
Greifwaldstrasse 38 
3300 Braunschweig 


Dr. Fritz Hartmann 
Consulting Engineer 
Bahnhofstrasse 39 
2248 Hemmingstedt 


Dr. S. Babendererde 

c/o Hochtief AG 
Rellinghauser Strasse 53-57 
4300 Essen 


Dr. F. Blennemann 

c/o Studiengesellschaft STUVA 
Mathias-Bruggen-Strasse 41 
5000 Koln 30 


Mr. J. Weber 

c/o U-Bahn-Referat 
Viktualienmarkt 13 
8000 Munchen 2 


HUNGARY 


Dr. Laszlo Rozsa 
c/o UVATERV 
JvozsetlAttilaruss8 
1051 Budapest 


Mr. G. Greschik 

c/o FTV Consulting Engineers 
Reviczky utca 4 

1088 Budapest 


SPAIN 


Mr. J.M. Gutierrez Manjon 
Cardinal Belluga 21 
28028 Madrid 


Deegce Sagaseta 
c/o E.T.S. Ingenieros de Caminos 


EGr 


20. 


Zale 


Avda. de Los Castros, s/n 
39005 Santander 


SWITZERLAND 


Dr. E. Andraskay 

c/o Basler and Hofmann AG 
Forchstrasse 395 

8029 Zurich 


CANADA 


Mra Dede enelps and Dr2vuU.k...ebnagde 
c/o UMA Engineering 

17007-107 Avenue 

Edmonton, Alberta 

RSSialGs 


Mr. E.W. Brooker 
c/o EBA Engineering 
14535-118 Avenue 
Edmonton, Alberta 
TSR 0OB9 


Mr. John Yan Egmond 
CY OnTrOW tds 
1595)Clarkersivd. 
Brampton, Ontario 
L6T 4V1 


Mr. M. Walia 

10180 - Shellbridgeway 
Richmond, British Columbia 
V6X 2W7 


USA 


Dr. Birger Schmidt 

c/o Parsons Brinckerhoff 
1625 Van Ness Avenue 

San Francisco, 
California, 94109 


Dr. Gerard Sauer 


1412 


23's 


24. 


Zo 


26. 


Zils 


20. 


or 


Consultant Engineer 

11403 Orchard Green CT. 
Reston (Washington, D.C.) 
Virginia, 22090 


Mohamad Irshad 

c/o DeLeuw, Cather and Co; 
600 Fifth Street, NW 
Washington, D.C., 20001 


Drwau. Re eMcCreatn 

c/o Golder Associates Inc. 
4104 148th Avenue, NE 
Redmond 

Washington, 98052 


MEXICO 


Mead. och: tLer 
CfOuso.m S.A, devc.y, 
Rio Becerra 27 5.° Piso 
Col. Napoles 

03810 Mexico, DF. 


BRAZIL 


Mr Claudio Casarin 

c/o THEMAG Engenharia Ltda. 
Rua Beija Flor 34, Vilage 1 
Lagoa da Conceicao 

88000 Florianopolas) Sec. 


Dr. Carlos Eduardo Moreira Maffei 
Consulting Engineer 

Al. Austria, 772, Alphaville Residencial I 
06400 Barueri, SP 


Mr. Mosze Gitelman and Mr. Roberto Kochen 
c/o Figueiredo Ferraz Ltda. 

Rua Conde de Iraja, 118 

04118 Sao Paulo, SP 


Messrs. Hamilton G. de Oliveira, Alexandre Verski and 
Sergio E.D.M. Cesar 
c/o Hidroservice (MS) 


307 


coh 


32% 


33 


34, 


1414 


Rua Afonso Celso, 235 
04119 Sao Paulo, SP 


Me. Dulsee, SOZ10 

c/o Promon Engenharia 

A.V. Juscelino Kubitschek! 830 452 
04543 Sao Paulo 


VENEZUELA 


Mr. Roberto Centeno Werner 

Consulting Engineer, Metro de Caracas 
Avenida Sojo, Quinta Chichi 

El Rosal 

Caracas 106 


EGYPT 


Mr. M.E. Abdel Salam 

National Authority for Tunnels 
56 Riad Street 

Elmohandiseen 

Carre 


JAPAN 


Mr. Yoshio Mitarashi 

c/o Kumagai Gumi Co. Ltd. 

Institute of Construction Technology 
M1 tSURUGCO-CHO, ShinguRu-Ku 
Tokyo, 162 


Dr. Keiichi Fujita 
c/o Hazama-Gumi, Ltd. 
2-5-8, Kita-Aoyama 
Minato-Ku 

Tokyo 107 


APPENDIX C - NORMALIZED GROUND SETTLEMENTS 


1415 


1416 











LEGEND | 
r OL of stress release 
- wz 4 
eal se Ht | 
Aull —= | 
f a: 
Cs ane é 
ome lee HOt ] 
Peet 
rs! / | 
= 
a " 4 
: | 
cae | 
U) a | 4 
O 2a 
(= 1 
oO 
~ 
ii [ 
4 
a c 
lt ] 
7 4 
[ee 
w 
. 4 
N 
i) 





=0.5) oO. fl Q.1 0.3 0.5 0.7 0.9 ten 
(Settl./Crown Settl. 








ao 
a SRT Ce Sees se po Ge eee 
{ 
a 0 = ee 4 
=~ 2 
= ae. | 
-~ Oo 
~ ' 4 
u 
gt 
= | 
2 4 
: f 
ee (=) 
(3) ~ 
ae 
&. 
iS LEGEND 
wu OS of stress release 
os 10% 
oe w 
S ig 
et 





-0.50 


-10 -8 -6 -4 -2 0 2 4 6 8 10 
Distance to Tunnel Axis (D) 


Figure C.1 Normalized Settlements: K,=1, c,/yD=2.5, H/D=1.5 


1417 











n 
es i | 
r LEGEND 4 
f Ss} OX of stress release 
10% ; 
2 | errs 
2) SK Ht 1 
f Ss Gh 
= | 
= [ | 
vu wp I 1 
hd OIL los 601 
ep) Fr 9 4 
Ee li 4 
o ‘a 
= 2 | | 1 
u . 
: | ] 
fe L 
Sf | 1 
a | a e a | 
2 f 
7 P | 
r 4 
gt | 
x 


ee ee ee er ean 
=0.3' <0. Bink 0.3 0.5 mA) 0.9 ef 
(SettL/Crown Setth) 


10 


0. 





-0.05 





-0.20 


-0.35 
ee ae ae 






LEGEND 


OZ of stress release 
10% 
20% 


(Surf. Settl.J/(Crown Settl.) 








-0.50 


-10 -8 -6 -4 -2 0 2 4 6 8 10 
Distance to Tunnel Axis (D) 


Figure C.2 Normalized Settlements: K,=1, ¢,/yD=1.25, H/D=1.5 


1418 


B45) 


gee 
: 


50 
3 


Sh 








4 
ae ] 
uy } 

2 a | Lov 608 ] 
wo g =) 4 
[oS 1 | 
B - 9 4 
e r ; 
a 
ie 
=i ! 
oF i} 
cle st | | 
a 
= b q 
a” i | 
w 
~ 
a 
Fr 
i 








-2.50 


=O)g Oa Ola 0.3 0.5 C.7 0.9 lot 
(Settl./Crown Settl.) 


10 





0. 


re) eere 


-0.05 
Ta ay 


-0.20 





(Surf. Settl.)/(Crown Settl.) 


-0.35 








-0.50 


=i -8 -6 -4 =2 0 2 4 6 8 10 
Distance to Tunnel Axis (D) 


Figure C.3 Normalized Settlements: K,=1, ¢,/yD=0.625, 


H/D=1.5 








w 
N 
me | 
- 
7 
8 
o : 
= o—— 
S 

fa} t 

=> 

on 

fs “ 1% 50% 

= 1 

5 

(op) 

= 

° 

ea 

=o 
3 

vu A 

OO =- 

ies 1 

oO 

~ 

n 

“ 

a od 
“ad He 
~ 
Ce od 
a | 

b 
t 
oO} 
wn 
N 
' 


SoS oe -Q.1 Ot 0.3 OFS het 
(Setth./Crown Settl.) 


-0.100 0.0S0 


Tale eae eee 


SettlJ/(Crown Settl.) 
-0.250 





(Surf. 
-0.400 


-0.550 


-10 -8 -6 -4 -2 0 2 
Distance to Tunnel Axis (D) 


Figure C.4 Normalized Settlements: K,=1, 


H/D=1.5 





Ce te ae a nn a aa 
| 
LECENO 1 
OX of stress release 
10% 
20% 
30% 
40% 
Sot 
0.9 Load 








LEGEND 
OZ of stress release 


c,/yD=0.3125, 








1420 


w 

2 [ 
r LEGEND 
} OX of stress release 
Ib 10% 
at oe 

So 

els x*—~ «Ot 

So} o—) So 
- 

a [ Son LON 

wi = 





Distance from Surface (DO) 





{5 7) 


-1.00 
yy ay pep 
yee! We ta oe 





-2.50 


‘ 
(2) 
° 
w 

6 
Q 
° 
~ 


0.1 Gas 0.5 O.7 0.9 Wal 
(Settl./Crown Settl.) 


.10 


0 


Ce ooo 











aa 

° So 

“ . 

~ Oo 

EST 
u 

ep) I 

‘e i } 
ee 
(& oe } 4 
(SP a 

SS if | 
at f [ 4 
_— 

oF LEGENO 1 
wu OL of stress release 

2) 10% 

Ges w 20% 4 
= +—+ 30 | 
= o — 40% 

BQ 7 o—) so 4 


-0.50 


-10 -8 -6 -4 -2 0 2 4y 6 8 10 
Distance to Tunnel Axis (D) 


Figure C.5 Normalized Settlements: Kj=1, ¢=20°, H/D=1.5 


1421 

















Ve) 
NN 
saab 
| 
a 
b 
a nt 
’ 
u 
By 8 ; 
es 
= s.{ 4 
3 | 
YO  - “4 
= of ] 
2 et - 
Ke 6 
° 
y =f | 
UO = 
[= 4 
fac} 
~ < 
a | 
nl } 
(cal 4 4 
a 
~ 
1 
oO} 
Va) 
“ 
1 


#03 =O... Q.1 0.3 Oe5 On 0.9 lent 
(Settl./Crown Settll 


10 


0. 





Toate 8 





-0.05 


-0.20 
rt a ae 












——\— 


LEGEND 
Aa of stress release 


(Surf. Settl.J/(Crown Settl.) 
-0.35 





=O 50 


-10 -8 -6 “4 -2 0 2 4 6 8 10 
Distance to Tunnel Axis (D) 


Figure C.6 Normalized Settlements: K,=1, ¢=30°, H/D=1.5 


1422 


-1.00 =0525 0.50 ere) 


Distance from Surface (D) 


-1.75 








=2.90 


=0.3 -0.1 0.1 0.3 0.5 5 7 0.9 vet 
(Settl./Crown Settl.) 


-0.05 0.10 


-0.20 


LEGEND 
O% of stress release 
10% 


(Surf. Settl.)/(Crown Setti.) 
-0.35 


-0.50 








-10 -8 -6 -4 =e 0 2 4 6 Q 10 
Distance to Tunnel Axis (D) 


Figure C.7 Normalized Settlements: K,=1, ¢=40°, H/D=1.5 


1423 











wo 
N 
= | 
ia LEGEND 4 
= Fes stress release | 
[ AX 204 7 
° +——+ 30 1 
es <—— 40 4 
e £ ] 
= 
' | 
’ 4 
wn } | 
N 
S 4 
1 { 
| 4 


— 








Distance from Surface (D) 





=1Vo7S 


-1.00 
ete ae ee a ae ae 





=2.90 


-0.3 -0.1 0.1 0.3 0.5 0.7 0.9 hal 
(Settl./Crown Settl.) 













oa 
So 
° 
i 
a a 4 
= 4 
o 
° 4 
= q 
' 
a 4 
aol | 
~ 
~~ 
& 8 | 
™ 4 
5 © 
[o} 1 
5 | 
= | 
= 4 
ae LEGEND 
a ° OX of stress release j 
‘at oy 10% 4 
ae Ss ott 
va <— 401 4 
= 4 
(89) 4 
oS) 
ros) 
w 
ry 
j 
° 
wn. 
7) 
3 
! 


-10 -8 -6 “4 -2 0 2 4 6 8 10 
Distance to Tunnel Axis (D) 


Figure C.8 Normalized Settlements: K,=0.8, ¢=20°, H/D=1.5 


1424 


22 


LECENO 

Ct. of stress release 
10% 

20% 


OF 
oe es 


ae 


< 
SS SS | 


-0.25 50 
Seana aparece Teel 
& 
wv 
Tes en 


Distance from Surface (D) 
-1.00 





jf, ah oe 
Cd 


=e via 


Se eee ee 





5 aaa 


-2.50 





‘ 
Q 
Ww 

' 
Q 


a Q.1 0.3 0.5 Oe 0.9 isi 
(SettL/Crown Sett.) 


0.00 


=0.15 





LEGEND 

OZ of stress release 
10% 

20% 





(Surf, Settl.J/(Crown Settl.) 
-0.30 


-O.NS 


-0.60 





Distance to Tunnel Axis (D) 


Figure C.9 Normalized Settlements: K,=0.8, $=30°, H/D=1.5 


1425 














wo 
N 
fs LEGENO 4 
} OX of stress release } 
fb 10% 
+—+ 58 
nil % 
BA ——— ee 4 
° soz 
f 602% j 
om 70% 
So 80x { 
le 4 
Uw} | . } 
8 ei 10% BON ] 
woo on 
i | 
n 4 4 
s 
fa) 
G 
wes ° \ ‘ 
5 it 
ay | 
oO ian 
~ r ’ a] 
n 
= \2 9 q 
Q ‘ 4 
2 f ] 
mt | 
ori 
w 
% Oe ae eee 
i) 


-0.3 =0.1 gat 0.3 0.5 Gen 0.9 en 
(Settl./Crown Settl.) 












° 
5 
pe eae ] 
° Oo 
= . — 
~~ oOo 
~ ( 
y ! 
5 
= 
S al 
& = r } 
ea : 
zt 
~ LEGEND 
Co) OL of stress release | 
h ' 108 
ae ++ xs | 
Ase as x<— 40 4 
Sp) 1 [ Sot q 
— 60% a 
in 70% 
r 80% | 
o 
ee A { 
wi 








-10 -8 -6 -4 -2 0 2 4 6 8 10 
Distance to Tunnel Axis (D) 


Figure C.10 Normalized Settlements: K,=0.8, ¢=40°, H/D=1.5 


-0.050 


-0.350 -0.200 
[ieee Sa aL 


a Ng a a ee 


(Surf. Settl.J/(Crown Settl.) 


-0.500 


-0.650 


Figure 





Distance from Surface (D) 


=O725 0.50 eo 


-1.00 


aS} 


=2.00 





LEGENO 
oF of stress release 
10% 


— 





-0.1 0.1 0.3 0.S 0.7 0.9 {.1 


(Settl./Crown Settl.) 


—— 
| 
gl 
Pane ve Meare Y 


s—i- 





LEGEND 

OX of stress release 
10% 

20% 








-2 ) 2 4 6 8 
Distance to Tunnel Axis (D) 


10 


C.11 Normalized Settlements: K,=0.6, ¢=20°, H/D=1.5 


1426 


1427 











1 
L LEGEND 4 
t OX of stress release | 
be 10% 
(=) zt 
a <x ios | 
=) oO St | 
Q 4 
a OG | 
3 a 
3 ° 
' 
10% 50% 
n 
é es | 
a) 
Ss 
= 
oO 
wu A 
YU - < 
(= ! 
fac} 4 
a 4 
=o 4 
(=) 
S| 
a | 
a 4 
' 








-2.50 


= Onde Olel 0.1 0.3 0.5 O67 0.9 Vo 
. (Settl./Crown Settl.) 


0.05 
a 


5 i cae SS aS SSeS LS ea 
ee eee eee | 


-0.10 


-0.25 


SS ee! ee ees tl 






LEGENO 
OZ of stress release 


-0.40 
a 
= So - 


(Surf, Settl/(Crown Settl.) 





ae 


-0.55 
i ac 


| aaa 








-0.70 





-10 -8 -6 -4 -2 0 2 4 6 8 10 
Distance to Tunnel Axis (D) 


Figure C.12 Normalized Settlements: K,=0.6, $=30°, H/D=1.5 


1428 











LEGENO 
[ = 0% of stress release 
10% 
i ae 
+ pen P4 
aL ~— x 
° Sox 
= Gg 60% 
Jb 70% 
6 b 
iE 
ea 108 708 { 
ae ag / 1 
‘= ' r 4 
S) 
(ep) a os 
= | 
a ] 
bd l=) ut 
$ ] 
wu o > 
Bie Gr at 
a q 
~~ 
wn 
“4 
Wop 4 
re 
es | od = 
r i | 
ar 1 
% OP Pe ee 
! 


S@be) SOs il Ouat 0.3 0.5 0.7 0.9 iat 
(Settl./Crown Settl.) 












S 4 

S a nn = | 

° 4 
ai | 
ee 
a ape | a 
ee 
Cael 
2 | al =~ 
o 
(& b 4 
OQ - 4 
a: | 
— ve) 
fo LEGEND 4 
ou OL of stress release 1 
é z 
we t a5 are! | 
S - x*—~ «Ot 
ep) Wek SOt 4 
ae 60% 

3 70% 

' 

° } 

wo 

3 

i} 


-10 -8 -6 “4 -2 0 2 4 8 8 10 
Distance to Tunnel Axis (D) 


Figure C.13 Normalized Settlements: K,=0.6, ¢=40°, H/D=1.5 


1429 











me 
~ LEGENO 4 
OL of stress release 
L 10% | 
ae 
f t 
a | XK it ; 
a4 Sot q 
60% 
ir 70% zs 
2 al | 
ro) | 
iS 2 
8 [ 108 708 | 
=> of ' } 
wn = r 
= ' | 
=} 4 
= 
—_ 6S [ ! { 
8) 7 4 
a | 
fa r i + 
uF } | 
oO 
Q o [ i 4 
J 4 
& it 
=k | 
' r 
ei ir 4 
0 eet ne Se eee een Re re ee or ad ms | 
' 
-0.3 -Q.1 Lave 0.3 Gos Oey 0.9 Non! 


(Settl./Crown Settl.) 


-0.05 
a 
FSS ee 


zens hs 


So a | 


LEGEND 
OZ of stress release 
10% 
/ 20% 
70% oH ch 
40% 


S02 
60% 
70% 


SettlJ/(Crown Settl.) 
-0.20 





(Surf. 
-0.35 





5 Le | 
eee en ee Sey) a eer 





=O'a00 





‘ 
io) 
‘ 
@ 
‘ 
ren) 


-4 -2 0 2 y 6 8 
Distance to Tunnel Axis (D) 


_ 
oO 


Figure C.14 Normalized Settlements: K,=1, ¢c,/yD=2.5, H/D=3 


1430 








° 
ale LEGENO ‘ 
OZ of stress release 
10% p)| 
[ 202 ] 
+——+ 30 
. ts — 408 
SOL ] 
fl 20% 
eo: 70% 5 
a So f 4 
uw ° ’ 4 
a 104 70% 
te 3 
= oO 
Oy = ; if 4 
oo | 
+ 
ca | 1 
a 
ros] 4 
asd = 
n 
a 
a 
wo 
i) 


mn 


0 -2 
Te darr va ae oe ee ala eT 
ecooe 


— hf 








D's 


=0.3 =-0.! 0.1 0.3 0.5 Qe7 
(Settl./Crown Settl.) 


Q 
wo 
- 
- 











fos) 
on 
- a a 4 
= geet 
5 Oo 
el e < 
~ a 
. 
Ww L 
= f 1 
2 mek 1 
5 8! 
ee 4 
Sexe i 
a ; 
S LECENO 
0) OX of stress release { 
(ep) 10% 4 
gf so ) 
_— o = cs 
ee meg ag ee 4 
oe) S0% | 
= 60% 
70% 1 
r { 


Ooo 


' 
(oe) 

‘ 
ao 

t 
n 
@ 
~ 
oO 


“4 -2 0 2 u 6 
Dustarce to Tunnel Axis (D) 


Figure C.15 Normalized Settlements: K,=1, ¢,/yD=1.25, H/D=3 


oO 

roy 

wt 
esl | 
—s «+ 
eS 
hy 
wu 
(3p) 
Cc 
=z 
Oo 
(= aS! 
eg MN 
Sco 
ath lb 
+ 
~ 
~~ 
wu 
on 
“om 
= Oe 

So 
On 

ic 

w 

3 

1 
Figure 





Sp eee 





Distance from Surface (0) 


—+~ +} 


5 ae | 


| 
LEGENO 7 
OX of stress release { 
10% = 
— & | 

» 4 
[ rs - 
ema hla 
- 60% | 
L : | 
f 1 

le 60% 408 

r 4 { 
10% 4 
] 
4 =| 
f 4 
a { ; 
L } 50% 1 
} ] 
aE 3 
r q 
r 4 
= Il | 
a ee a VS a a a re res eee eae |e en a Le ee | 











‘ 
Cc 


58) eligi 0.1 0.3 0.5 0.7 0.9 
(Settl./Crown Settl.) 


roy 


LEGENO 


10% 
20% 

—puameeriet oUs 

as mmr 408 


Ss 


-8 AS “4 Ee 0 2 4 6 


Distance to Tunnel Axis (D) 


OL of stress release 





SSS eet ae ee eee ee ee eee eae | 





C.16 Normalized Settlements: K,=1, c,/yD=0.625, H/D=3 


(Surf. Settl./(Crown Settl.) 


-0.35 -0.20 -0.0S 0.10 


-0.50 


7 3 


LE Aa ae aL a inf 





Distance from Surface (0) 





-10 


1432 








o 
eal LEGENO | 
} O% of stress release ] 
fe 10% 4 
| a 
% 
= “x 403 | 
oa o— 50% 4 
4 
e [ j | 
SS 4 
i 50% 108 i) 
[ 4 
S ‘) 
3 |e | 
in 9 
bet ] 
a = | 4 
1 
I 
L ] 
ort : q 


-¥ 


=8) 
SS 





=S 


Mae) la {h 0.1 0.3 0.5 0.7 0.9 ile A 


(Settl./Crown Settl.) 





LEGEND 
O% of stress re.esse 
10% 


20% 
\ ——+ ie 
10% <-— 40 
o—S so 
-8 -6 <4 -2 0 2 4 6 8 10 


Distance to Tunnel Axis (D) 


Figure C.17 Normalized Settlements: K,=1, ¢=20°, H/D=3 


10 


0. 


— pl at ol) aS 


Pe ah 
eee) 
“4 «6 
Hae 
~~ 4 
Yu 
ep) 
es 
2a 
Oo 
i=) 
CS cu 
= © 
a 
es, 
~ 
= 
vu 
wn 
roa) 
Ss 
OQ 7 
o 
wo 
i=) 

1 
Fagure 


Distance from Surface (0) 


0 
ete lS ee ee ee 


> Enh Sinus seunc! lic SUAS acai 











LEGENO 

OX of stress release 
10% 

20% 


70% 10% 


38S 
ve va be 
eee er ae a eee es eR: 








0.3 0.5 Oh 
(Settl./Crown Settl.) 








C.18 Normalized Settlements: 





—_ ESS 


=> 


LEGENG 
OZ of stress release 


10% 


ee ee ee a ee el ee ee Cn ee 


Po ee 





-4 -2 0 2 4 6 8 10 
Distance to Tunnel Axis (C) 


K,=1, ¢=30°, H/D=3 


1433 


1434 











° 
a t LEGEND 
f OZ cf stress release j 
10% 4 
an 
x ] 
2 — 40 4 
ze S0z 
60% J 
le 70% 
DQ of d 30% 4 
mule AF Tl 
8 ie 90% 108 | 
Qa 
s o f 6 | 
RQ Sle >| 
ee ae 1 
fe im 4 4 
ed a 
o>) 4q 
a nw e ry 
{S p 
© - i 
ag 
So 
(ol us L t 
oe 
_ 
i=) 
> 
(=) 
wn 
1 
Seisel Ola it On! SORE) 0.5 Ona 0.9 Holt 


(Settl./Crown Settl.) 


10 


o— —_—__—_———_ 





w 

o 

i=) 
‘ 


L 


-0.20 





LEGENO 
OX of stress release 
10% 


-0.35 





(Surf. Settl./(Crown Settl.) 


ES ee ee ee es ee ae ee eee eee Ears 


<a al eal eT 
i 
[o} 
oo 
vr 





-0.50 


‘ 
— 
eG 

4 
@ 

‘ 
(os) 


“4 -2 0 2 4 6 
Distance to Tunnel Axis (0) 


@ 
- 
Q 


Figure C.19 Normalized Settlements: K,=1, 9=40°, H/D=3 


° 
roy 
a L, 
oe =) 
oe 
vay te} 
nm 1 
vu 
ep) 
S 
=z 
o 
=. S 
(By Sat 
ee 
=4 
re 
~ 
u 
n 
em 
3 6 
Coie! 
o 
w 
ro) 

' 
Figure 


Lat al aa a a el 





-10 


C3202 Normalized Settiliements:: Kj=0.8),. ¢=20°, H/D=3 


Distance from Surface (D) 


— 6 9 5 ge = 





0 


r LEGEND 7 
[ OX of stress release 
10% 
|  o 
i dx 
+ 
r ’ 
, 


= FE (ese Sl feast 
Fa eh IT Seay te 


+ 








i ae 
coe 


4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
| 








=Oea" =O 0.1 Ona 0.5 O57 0.9 hom 
(Settl./Crown Settl.) 


LEGEND 


+——+- 30% 
x KSOC~ 
403 
-6 “4 -2 0 2 4 6 


Distance to Tunnel Axis (D) 





Be O% of stress re.ease 
10% 
20% 


oe es) eet 


ee ene ee eres eet 





SS ESS Eee 


1435 


1436 








So | 
ee ie LEGENO 
} OZ of stress release 
i 10% 
+—+ ts 
i 
os —X 
[ $e hh 
SI ; 
“yO 
Q L 108 608 
Qe 
5S of ; 
O 2b 4 
en & 
te | | 
(eR L 4 
Oo 
Uw , 4 
eer 4 
© 4 
ce 
oo } 
or 5 
[ ] 


-4.0 
ee 
eee Camere 








vf Cee ee 
=0.3 =0.1 0.1 0.3 0.5 0.7 0.9 let 
(Settl./Crown Settl.) 


10 


0. 


-0.05 





-0.20 





(Surf, Settl.)/(Crown Settl.) 
-0.35 








-0.50 


-10 -8 -6 <4 -2 0 2 4 6 Q 10 
Distance to Tunnel Axis (D) 


Figure C.21 Normalized Settlements: K,=0.8, 9=30°, H/D=3 


oO 
3 
=~ ow 
oo © 
“4 =. 
~~ Oo 
i 
wu 
ep) 
= 
Oo 
eS SS} 
> kN 
x 6 
—. 2 
ts 
- 
~ 
wu 
ep) 
Lp euf'si 
— 
bh 
@ a 
oO 
w 
oO 

1 
Figure 














eu 
ee LEGEND 1 
55) OX of stress release 
ia 10% 
20% ] 
ape eS 1 
et ——X «4 
mk SOx 
60% 
= 70% 
aS Box | 
u ° | 
QO i 
8 L 10% o% ] 
San 808 ; 
ae ’ 
€ ! 4 
o 
ae ' “ 
oO 
soa ft 1 
c yp J 
A t 
ae 
ee : 
an) |r 
* 
=) | | 
> 4 
(eo) L 
4 


Selesh Sela Al Ont Ons 0.5 (l= 7 0..9 toot 
(Settl./Crown Settl.) 


~~ ____e--_______ --___«____e-___e—____e—__—__ 9 __ +. -» 2 ___# —--—_ + —- —___e _____-__ + 


(ln 






LEGEND 
OX of stress release 
10% 


a = =a ee eal ane es 


BER ER EL 





-10 -8 -6 -4 -2 0 2 4 6 8 
Distance to Tunnel Axis (D) 


C.22 Normalized Settlements: K,=0.8, ¢=40°, H/D=3 


1437 


| ae Gere ame 





SS) ae ee) eae Eres See Sant 


o 
3 
a 
.« OO 
—~“ e 
tt e3) 
; ee | 
vu 
9) 
S 
z 
o 
(SS) 
(Bi Gy 
XS Ge 
oe: 
= 
= 
= 
u 
Sp) 
cen 
See 
Ge 
i} 
w 
i=} 

i) 
Figure 





LEGENO 


O% of stress release 
10% 
20% 


103 20% 


pe ee ae eee! (See Ne! Gee! Ee 





Distance from Surface (D) 


oS a ee es es Len 


0 
Saree mig ee ae a oe oe Np me lm ae ee hee gos we fet ae 
—e 








.0 


-4 


-$.0 


= Qiao) = Ort 0.1 0.3 0.5 Q.7 0.9 hank 


(Settl./Crown Settl.) 


'—l "see ge 


LEGEND 


OX of stress release 
10% 
20% 


i i 





-10 -8 -6 -q =2 0 2 4 6 8 
Distance to Tunnel Axis (D) 


C.23 Normalized Settlements: K,=0.6, ¢=20°, H/D=3 






1438 


a | 


a Sarto | eer 





10 


1439 


TT 














oa t 1 
ay LEGEND F 
5 OL of stress release ] 
I 103 4 
° t 
= L —_——K «401 
o—5 sa ; 
Bit | 
. PF ° 
wu oO 
& [ 108 50% ; 
S o| 
OD t+ i 4 
= t 
é | 
ee ' 
yor | 
og | | 
5 
Oa 3 ] 
5 | 
= = 
: 4 
f=) 
0 Ce ee ee 
[ 


=O73) =0%.1 Oot Or GES OR 0.3 vest 
(Settl./Crown Settl.) 


0.10 


; Sac [en ce cl a 


-0.05 




















=: 
re 
~ 
wu 
Yn 
C | 
ate | 
On 
= cs 3 
barr ' {a 4 
S LEGEND 4 
a OL of stress release 
w t ae | 
— & 30% 
— . 40% 4 
A Si } Sot 1 
r | 
i) 508 1 
Oo 
! 
-10 -6 -6 “4 -2 0 e Q 6 8 10 


Distance to Tunnel Axis (C) 


Figure C.24 Normalized Settlements: K,=0.6, $=30°, H/D=3 


1440 











=) |] 
: LECEND 1 
a [ = as of stress release { 
10% 
ie zt 4 
li ee t 
eal x~—~ «O40 
— | soz 
: 
b 
eit 
° ) 
aw SS [ ; 
a L 108 708 } 
Ne 
ale) | $ y | 
Coe mes a 
1 
§ 
ae 
So 
2 eh 
o ’ | 
~ | ea 4 
24 } { 
=| 
Oo nt t | 
| 
r = 
oul 1 
> = 
' 4 
a 4 
rr 
1 


=0.3  -0.1 O.t 0.3 0.5 0.7 0.9 test 
(Settl./Crown Settl.) 














° 
SOK ares sae ame eae ST AR RN SARS NL A PRT SS JE] TT Sf SR 
©) 
f q 
5 4 
p 4 
ae tay ie } 
“4 - 4 
vo! i=) 
_ te 43 
i t 
oO Ie | 
as 
com! 
cua 
iS [ 4 
. ' q 
= ie LEGENO 7 
a) OZ of stress release 5] 
op) 10% 4 
7 > 2 } 
Coen 
SS x<-—<< 40% 1 
wn 7 Sot 
<2 60% A 
70% 
q 
7 
9 08 ; 
FI 





-10 -8 -6 -4 -2 Q 2 4 6 8 10 
Distarce tc Tunnel Axis (0) 


Figure C.25 Normalized Settlements: K,=0.6, ¢=40°, H/D 


" 
w 


(Surf, Settl.)/(Ccown Settl.) 


10 


0. 


-0.05 


-0.35 -0.20 
nn an nd 


-0.50 


Gee fi 





Distance from Surface (0) 





-10-9 


LEGENO 


10% 
20% 
30% 
401 
Sot 
60% 
70% 


i Ee «Sealy ae 


70% 10% 


an) an Pa fe 


7 


aa aa 





= Oise a= Glouk Q.1 0.3 0.5 0.7 


(Settl./Crown Settl.) 


70% 
a 


wl) 


OZ of stress release 


1441 


ae Eee 


Sera eee 


SSS Se Eee 


iI 





Se See 


Ree ean 


NO 
of stress release 


LEGE, 
0% 
10% 
20% 
30% 
40% 
S0% 
60% 
70% 


ra Ee Sa ares | 








=5 -6 “4 -2 0) 2 


Distance to Tunnel Axis (D) 


Figure C.26 Normalized Settlements: K,=1, ¢,/yD=2.5, H/D=6 


1442 





4 
yoo 


LEGENO 
=> of stress release 
10% 


2 
To a 


1 
aS 
a 
aes een) eee Een eee eee Ene 


ete 


-1 
s 
2 


=e 





-4 


-5 


a ee 


Distance from Surface (D) 
-3 


-6 





=10-3 


-0.3 -O.1 Or (55) ORS Ger Gin) Vat 
(Settl./Crown Settl.) 


10 


0. 


-0.05 











-0.20 






10% LEGEND 
OZ of stress release 
10% 


20% 
+—+ 31 } 
KAO 4 


Se 
60% 
70% 


(Surf, SettlJ/(Crown Settl.) 


-0.35 





-0.50 


-10 -8 -6 -\ =2 0 2 y 6 8 10 
Distance to Tunnel Axis (D) 


Figure C.27 Normalized Settlements: K,=1, ¢,/yD=1.25, H/D=6 


LEGEND 
dL of stress release 
10% 
202 
+——+ 301 


owe ree re ere ee 


30% 10% 


SS 











Distance from Surface (0) 
E\0s9e -6m -7.) Ge —-Sm Ue 38-26-10 
i 
a- eo - oe 
sg a ee es ee eh hee s 


=Os Smee Ol Q.1 0.3 0.5 Oe 0.9 tend 
(Settl./Crown Settl.) 











o 
= 4 
| ] 
. Oo 
~~ . | 
- o 
- 1 4 
30% { 
eh z 
je} 
(= (=) 
bg 
= 4 
ei | 
pan a LEGEND 7| 
wu O% of stress release 1 
a 10% 4 
car as { 
yoy zs & 
S 7 
Qo ] 
=a 
’ 
o 
w 
ro) 
i) 
-10 -8 -6 <4 -2 0 2 Q 6 8 10 


Distance to Tunnel Axis /D) 


Figure C.28 Normalized Settlements: K,=1, c,/yD=0.625, H/D=6 


(Surf, Settl.)/(Crown Settl.) 


10 


-0.35 -0.20 -0.05 0. 
a eae (ee yo oe te a ase 


-0.50 





‘ 
- 
oO 


Distance from Surface (D) 


=] 
Taal 


=i) jb) a 


-6 


=10'-9 


1444 


[ LECEND 

35) OL c. stress release 

10% 

li 20% 

+—+ 301 

x—X «403 

S—SD sox 
f 503 10% 


Se a Sa Sa Cae Ses 








ORS 
(Settl./Crown Settl.) 


Q.S$ 0.7 0.9 Lol 





‘ 
@ 
‘ 
n 


ree | 


stress release 


° 
eo 
SS 
QR 
Se ees es ae ee | 


=—" 


ho. 





-\ =2 0 2 4 6 
Distance to Tunnel Axis (D) 


@ 
- 
oO 


Figure C.29 Normalized Settlements: K,=1, ¢=20°, H/D=6 


10 


0. 


man 
a SS 
= 3 
~ 

u 
ep) 

Cc 
2 
j=} 
(KS 
[S| a 
XN oo 
en OF 
= 

~ 
_ 
u 
n 

~ 9 
SS = 
So 
By 8 
So 
w 
3 
i] 
Pagure 


Distance from Surface (D) 


+ 


SC a fee oor em ae (ara as) aaa eas aaa ara 


~ 


r 





A) 26 65 4h 3 
Sea a eo 


-8 


-10-9 





r 


LEGENO 
OX of stress release 


ey as 
a 
a 

) oe!) a oe 


70% 10% 


NW 


) oe) ee Gere) 





ms oe 
a aS ee ae 


ees 


cl (a ae ceil (aaa an ee ia i 








SS SS re a eT re 
=0. 1 0.1 0.3 0.5 On 0.9 to1 
(Settl./Crown Settl.) 


‘ 
oO 
Ww 


= ae eee 


70% 





1445 


ce 
LON ne een release 
B—5) 10% 
— a 
<—x 40 4 
=F. | 
70% 7 
-10 -8 -6 -4 -2 0 2 4 6 8 10 
Distance to Tunnel Axis (D) 
C.30 Normalized Settlements: K,=1, $=30°, H/D=6 


ettl.) 


S 


Figure C.31 Normalized Settlements: K 


(Surf, Settl.J/(Crown 


=O), 39 -0.20 -0.05 0.10 


-0.50 








wo 
4 
=F LECENO 7 
ae =) OF ch stress release 4 
nk 20% 4 
nF — O40 | 
50% ] 
[ 60% 
a 702 al 
ran) b 802 
“a Oo aan 
uv 
oe L 908 108 
ce 1 a he 
A wt 
Ee Pp 
ja) Gal 5 
(es i} 
ee r 
ee 
5 - 
e OPE 
as } 
(sl or, 
r 
iT 
Ce ie 
r 
if 
So 
= OS ee Oleut 0.1 Oe Q.S Ciena 0.9 ESt 
(Settl./Crown Settl.) 
4 
aa a 
r 4 
[ | 
Is 4 
t 903% 1 
: f 1 
} 
[ | 
4 
r | 
[ 108 LEGEND 1 
OX of stress release 1 
r 10% 
++ jn | 
< 
—_ aor 4 
SO 
0% | 
70% 
80% 
90% 
-10 -8 -6 -4 -2 0 2 4 6 8 














Distance to Tunnel Axis (D) 


o- 1, 


¢=40°, H/D=6 


1446 





1447 


LEGEND 
OX of stress release 
10% 


3 
SE En a 
paps) 
oO 
we 
eas ee ee 


0 1 

5 ee aa 
=~ 

4 


10% 40% 


ae 


-4 
qt 
> 
ree ee ee ee ee ere 





Distance from Surface (D) 
-5 =3) 


-6 








SS) ee ee hee 











o 
An’ 
cd r 
= ||! 
I Se eae ee a ee ee ae re er ee er a 
-0.3 -0.1 0.1 0.3 0.5 Jor 0.9 te 
(Settl./Crown Settl.) 
oa 
3 
= 
4 
2 
_ y | 
ee L 5 
~ j=) } 4 
~ 1 
a) ; 
et | 
> «4 
Hee) 
6 ca Van | 
lara! \ r 
rs r LEGEND | 
5 t OZ of stress release 1 
5 . a 
ae 408 Goes po | 
_— 
as —X in ; 
{=} 
wn fou 
— L = 
} { 
2 | 
© 


={0 -8 -6 -4 -2 0 2 my 6 8 10 
Distance to Tunnel Axis (D) 


Figure C.32 Normalized Settlements: K.=0.8, $=20°, H/D=6 


° 


1448 


LEGEND 
QL of stress release 
10% 
— & 
t 
“x O40 


Sergi 


x ee ee | 


| 


1 


10% 60% 


1 
Naat, See I 





me 





=) 


af 


Distance from Surface (D) 


-6 


=) 
Galery laa 


-4 -3 
oe ee eee eee ee ee Ee ee eee et 








Oro 


S0),2) ely st 0.1 0.3 0.5 0.7 0.9 
(Settl./Crown Settl.) 


oo 
- 











So 

Z | 

2 Fr a 4 
= fs [ | 
rome 4 
~ ir 
wu 
(op) - 
ia 
ras} r 
Sete 10 
S sf 
=: 
ae y Sete heme | 
vu of stress release 
ee [ a { 
ee OF 4 
aS x«— = 
* —— 60% 

So 

w 

oS 

1 

-10 -8 -6 <4 -2 0 2 q 6 8 10 


Distance to Tunnel Axis (D) 


Figure C.33 Normalized Settlements: K,=0.8, $¢=30°, H/D=6 


° 


1449 





LEGEND 


ree | 


a 
j 
4) 
Q 


10% 70% 


80% 


PEEP E EEE 


+ 


Distance from Surface (D) 
3 


So | 


=) 


-8 


-4 
rat ae a ea a og a ae a eo ee ee oe oe ee Ue ae, 








-10-9 


1 
oO 


“E) nse yt 0.1 0.3 0.5 Oe? 0.9 teat 
(Settl./Crown Settl.) 














oO 
ro] 
r 
P 
aw aet 
ee Ae 
4 . 
By YS) 
- ] 
Ee t } 
a r 10% 4 
a Go 
cue | | 
Bgl 
rs r ae LEGENO 4 
vi 708% OZ of stress release j 
en e 
Cee +—+ jon ; 
& amb << «nt 
si eS a 
Sp) i 
ma =5 
70% 
sot 
oOo - 
w 
o f 
‘ 
-10 -8 -6 <4 -2 0 2 q 6 8 10 


Distance to Tunnel Axis (D) 


Figure C.34 Normalized Settlements: K,=0.8, 9=40°, H/D=6 


(Surf. Settl.)/(Crown Settl.) 


0. 
a 


-0.05 


=05:20 


mati. 1 


-0.35 


-0.50 


Distance from Surface (D) 


T 


~ 


a 





-10 





1450 

















i) Speen reer aie Sse ee Se 
z>+- LEGEND 5| 
O% of stress release j 
10% 
(od 20% | 
4 | 
ait | 
af 
=a 10% 208 
‘ 
o 4 
‘ 
nm 
' 
=> 
' 
cS 
1! 
wo 
' 
= 
’ 
o 
1 
fez) 
i) 
° 
Th 
-0. -0.1 GO. 1 5) 0.5 oar! 0.9 ig 
(Settl./Crown Settl.) 
10% } 
aX 4 
LEGENO 
OX of stress release 
y 10% 4 
208 <3 4 
4 
-8 -6 -4 -2 0 2 q 6 8 10 


Distance to Tunnel Axis (D) 


Figure C.35 Normalized Settlements: K,=0.6, ¢=20°, H/D=6 


1451 


=A LEGEND 4 

OZ of stress release | 

= fe | 
a 

“ Xx tn | 

o— sn | 


oh. 





-4 


-2 
Sy et te a 


Distance from Surface (D) 
-5 ~3 


=6 





=? 


—= +. 
ee Eee Ee Se eee 





-10-9 


“o)<) See 0.1 0.3 0.5 OMT 0.9 ire 
(Settl./Crown Settl.) 


10 





-0.05 





SSS EES EE See 


-0.20 





LEGEND 


i: 
[ 7] OX of stress release 
ie 
- 


Settl.) (Ccown Settl.) 


(Surf. 
-0.35 


uw 
° 
- 
w 

sO rane) (eae) Wee eae eer) ae | 








=O. 50 


=10 -8 -6 -4 -2 0 2 my 6 8 10 
Distance to Tunnel Axis (D) 


Figure C.36 Normalized Settlements: K,=0.6, ¢=30°, H/D=6 


1452 


S 





a a a ee 


LEGEND 


O% of stress release 


10% 


RRSRBR 


LN 


70% 








se ——?s- ———— 222 am 
Se eR een es ee ee Ee eer eee 
h E 2 I O Fit ec - C= a-Si e 8 CSO 


(Q) aoejung wosy 3a9ue1STG 


0.9 


0.7 


Q.S 


1 0.3 
(Settl./Crown Settl.) 


-0.1 Q. 


3 


-0. 


ot 








ok ek dF Ee ee Ee 


“0 





SO°O- Oc 0- SE70- 


("1229S UROLD) 7 TILIS °J4NS) 


OSs0> 


10 


Distance to Tunnel Axis (D) 


- <4 


oO 
= 
i) 


Figure C.37 Normalized Settlements: K,=0.6, ¢=40°, H/D=6 


APPENDIX D - TWICE NORMALIZED GROUND REACTION CURVES 


1453 


1454 


KO=0.6 H/D=6.0 Crown 
0 4 0.8 8 12 1.6 2 2.4) eceaem E5:2 ns. 4 








0.8 Legend 
PHI220 
0.6- PHI=30 
0.4 PHIz40 
’ Regression 
ie} 
3 ORZ 
€ 0 
ie} 
a -0.2 
-0.4 
-0.6 
-0.8 
= . 
0 0.4 0.8 672 1.6 2 2.4 2.8 S22 3.6 4 
U/Uref 
Figure D.1 NNGRC for Crown, K,= 0.6, H/D=6 


1455 


KO=0.6 H/D=3.0 Crown 
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 





Legend 
° Pria20 
4 PHIie5S0 
+ 
° 


PHie4d 


Regression 


Lambda 


Oe 
-0.4 
-0.6 


Figure D.2 NNGRC for Crown, K,= 0.6, H/D=3 


1456 


KO=0:6 FY S="S" Crown 
0 0rd 0-82 1.6 2 OEE OE RP 4 EE 4 


Legend 
PHi2a276 
PHI250 


PHiz40 


Regression 


Lambda 





Figure D:3 NNGRC fOr Crowny Ke=00. Sten D- no 


1457 


KO=0.6 H/D=6.0 Springline 
0 pe ges be abe 4 428 BSG KBed 752 8 





0.8 Legend 
Oo PHIz20 | 
0.6 o PHIiz30 | 
0.4 + PHIi240 | 
; ° Regression 
2) : 
8 0.2 
€ 0 
12) 
a -0.2 
-0.4 
-0.6 
=ORS 
—| 
0 0.8 1.6 2.4 Sine 4 4.8 5.6 6.4 1 2 8 
U/Uref 


Figure D.4 NNGRC for Springline, K,= 0.6, H/D=6 


KO=0.6 H/D=3.0 


0 0.8 1.6 2.4 Bird 4 





Lambda 
[@) 


0 0.8 1.6 2.4 3.2 7 


Figure D.5 NNGRC for Springline, K 


° 


Springline 


4.8 


= 0.6, 


5.6 


H/D=3 


6.4 U2 


Legend 


PHIz20 


° 

a PHI2z30 
+ PHIz40 
° 


Regression 


1458 


Lambda 


KO=0.6 H/D=1.5 Springline 


0 0.8 1.6 2.4 Siz 4 4.8 5.6 6.4 











U/Uref 


Figure D.6 NNGRC for Springline, K,= 0.6, H/D=1.5 


(ee 


1459 





Lambda 


(3 2.4 2.8 
U/Uref 


Figure D.7 NNGRC for Floor, K,= 0.6, H/D=6 


Legend 


23574 


PHiz20 
PHIs3S0 
PHIiz40 


Regression 





3.6 


1460 


Lambda 


Figuse D.SaNNGRGafor. Floor, 


K = 


° 


O66; 


H/D=3 


Legend 
PHIz20 
PHi2z30 
PHIz40 


Regression 





1461 


Lambda 


Figure D.9 NNGRC ‘for "Fltcon, Mie 0056, #H/Dsl. > 


Legend 
PHi220 
PHIz3SO 
PRI2Z40 


Regression 





1462 


KO=0.8 H/D=6.0 Crown 
0 Gt eo Aas) 12 1.6 2 EE: 





Lambda 
oO 


Figure D.10 NNGRC for Crown, K,= 0.8, H/D=6 


857 


3:2 


3.6 


Legend 
PHie20 
PHIz30 
PHin40 


Regression 


arG 


1463 


KO=0:8 H7D=3.04 Crown 
0 0 bag Os mmm ree 2 TET ae eee 





0.8 Legend 
PHia20 
0.6 PHIaSO 
0.4 > PHIz40 
Regression 
fe) 
3 OnZ 
iS 0 
lo) 
+ -0.2 


0.8, H/D=3 


Figure D.11°NNGRC*®for*€rown ek 


° 


1464 





KO=0.84/H/D=1.55-Crown 
0 0.4 GNOgaigG.2 1.6 2 


2.4 2.8 32 S50 


Legend 
PHI220 
PHiasO 
PHia40 


Regression 


Lambda 


a 
" 


Figure D.12 NNGRC. for Crown, K,= 0.8, H/D=1.5 





1465 


KO=0.8 H/D=6.0 Springline 


0 0.4 0.8 72 1.6 2 





Lambda 
oO 


-0.2 
-0.4 
-0.6 


-0.8 


U/Uref 


Figure D.13-NNGRC for Springline, K 





° 


2.4 


2.4 


2.8 


2.8 


0286 


Se74 


52 


H/D=6 


3.6 


Legend 
PrHiz20 
PHIzSO 
Priz40 


Regression 


3.6 


1466 





1467 


KO=0.8 H/D=3.0 Springline 


0 0.4 0.8 eee 1.6 Z 2.4 2.8 32 3.6 4 








Legend 
0.6 Pei220 


PrHie3sO 
0.4 PHIi240 
Regression 


Lambda 


2.4 2.8 3.2 até 4 


Figure D.14 NNGRC for Springline, K,= 0.8, H/D=3 


KO=0.8 H/D=1.5 Springline 
0 Oy eee eee eee 2 5, tommy i: mene 1 rae 


a8 SS, Legend 

PrHi220 
0.6 PHI230 
PHiz40 


0.4- 


Regression 


0.2 


Lambda 


1(0},,72 
-0.4 
=—On6 


-0.8 


Figure D.15 NNGRC for Springline, K,= 0.8, H/D=1.5 





1468 


KO=0.8 H/D=6.0 Floor 


Lambda 
e 


-0.4 
-0.6 


—O.o 


Figure D.16 NNGRC for Floor, K, 


2 


2.4 


On 85 


2.8 


H/D=6 


SZ 


S101} 


Legend 


° 
4 
+ 
° 


PHIs20 
PHIiz3S0 
PHie40 


Regression 





1469 


Lambda 


KO=0:34/R/D=5.OFloee 
0 ry een): meee 1.6 2 32 wena 





2.4 Zi8 


2 
U/Uref 


Figure D.17 NNGRC for Floor, K,= 0.8, H/D=3 


Legend 


ear 


PHI220 
PHIasO 
PHia40 


Regression 


a6 


1470 


1471 


KO=0.8 H/D=1.5 Floor 
0 0.4 0.8 1.2 1.6 2, 2.4 2.8 3.2 5.6 4 





a Legend 
PHI220 
oF PHI=30 
0.4 PHie4d 
Regression 
ie) 0.2 
2 
€ 0 
Ol. 
SS -0.2 


0 0.4 0.8 HazZ4 1.6 


Figure D.18 NNGRC for Floor, K,= 0.8, H/D=1.5 


° 


KO=1.0 H/D=6.0 Crown 
0 os as}, Pane so Pome EL 2 2h neeel8 


Lambda 


Figure D.19 NNGRC for Crown, K,= 1.0, H/D=6 


Swe 


356 


Legend 
PHIz20 
PRIz30 


Priz4ad 


Regression 





1472 


0.8 
0.6 
0.4 
0.2 


Lambda 
oO 


= (0) 572 
-0.4 
-0.6 


= O58 


Figure D.20 NNGRC for Crown, 


° 


H/D=3.0 
2 





1208 


2.4 


Crown 
2.8 


H/D=3 


S22 


Legend 


° 
4 
+ 
° 


PHI=20 
PHI230 
PHIz40 


Regression 
| 


3.6 


1473 





KO=1.0 
0 04 08 1.2 1.6 


Lambda 
(eo) 


Figure D.21 NNGRC for Crown, 





H/D=1.5 Crown 
2 Pisewme Pei 


2 pee 7 yhss 
U/Uref 


Ke= 150). W7De5..5 


° 


Be 


Bie4 


als 


Legend 
PHie20 
PHIeSO 
PHIz40 


Regression 


se 


1474 





1475 


KO=1.0 H/D=6.0 Springline 
2 


0 0.4 0.8 EZ 1.6 2.4 28 ome 3.6 4 
1 . 
0.8 Legend 
Oo PHI=20 
0.6 a PHIieSO 
0.4 + PHia40 
. ° Regression 
ie) : 
3 0.2 
€ 0 
eo} 
+ -0.2 


-0.4 
—Or6 


—O55 





2 
U/Uref 


Figure D.22 NNGRC for Springline, K,= 1.0, H/D=6 


KO=1.0 H/D=3.0 Springline 
0 GrdmmemeGrgnmmnnd 1.6 2 i anil A meee 


i 








Lambda 
oO 


2.4 2.8 332 


2 
U/Uref 


Figure D.23 NNGRC for Springline, K,= 1.0, H/D=3 


3.6 


Legend 
PrRI=20 
PHi2s0 
PHieed 


Regresition 


3.6 


1476 





KO=1.0 H/D=1.5 Springline 
1.2 2 


1.6 2.4 2.8 B574 S15 


fe) 0.4 0.8 


Legend 
° PHIe20 
PHIz3O 
PrHia40 


Regression 


Lambda 


Figure D.24 NNGRC for Springline, K,= 1.0, H/D=1.5 





1477 


KO=1.0 
0 04 amaetiOis 1.2 1.6 


Lambda 


Figure D.25 NNGRC for Floor, 


H/D=6.0 Floor 
2 


2a4 


2 2.4 
U/Uref 


° 


2.8 


2.8 


H/D=6 


Lye 


3)-(5] 


Legend 


3.2 


PRIe20 
PRIeZSO 
PHIe40 


Risigcolisiog 





3.6 


1478 


1479 


KO=1.0 H/D=3.0 Floor 
0 0.4 0.8 1.2 1.6 2 7 Agee 2 Soeee ) aneS 4 


1 | 


Legend 
PHI220 
PHIeSO 
PHim40 


Regression 


Lambda 





Figure D.26 NNGRC for Floor, K,= 1.0, H/D=3 


1480 


KO=1.0 H/D=15 Floor 
0 0.4 0.8 ther 1.6 2 2.4 2.8 Sows 3.6 4 





0.8 Legend 
PHIs20 
0.6 PHIesO 
0.4 PHin40 
Rogression 
ie) 2 
4 
€ 0 
ie) 
SS -0.2 


-0.4 


-0.6 


2.4 2.8 3, 


Lee) 


2 3.6 4 
U/Uref 


Figure D.27 NNGRC for Floor, K.= 1.0, H/O=i5 


°