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THE UNIVERSITY OF ALBERTA 


RELEASE FORM 

NAME OF AUTHOR Leon Hadsley Grant 

TITLE OF THESIS A MONTE CARLO STUDY OF THE STRENGTH 

VARIABILITY OF RECTANGULAR TIED 
REINFORCED CONCRETE COLUMNS 

DEGREE FOR WHICH THESIS WAS PRESENTED M. Sc. 

YEAR THIS DEGREE GRANTED 1976 

Permission is hereby granted to THE UNIVERSITY OF 
ALBERTA LIBRARY to reproduce single copies of this 
thesis and to lend or sell such copies for private, 
scholarly or scientific research purposes only. 

The author reserves other publication rights, and 
neither the thesis nor extensive extracts from it may 
be printed or otherwise reproduced without the author's 
permission. _ 





THE UNIVERSITY OF ALBERTA 


A MONTE CARLO STUDY OF THE STRENGTH VARIABILITY OF 
RECTANGULAR TIED REINFORCED CONCRETE COLUMNS 

by 

Leon Hadsley Grant 



A THESIS 

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

OF MASTER OF SCIENCE 


DEPARTMENT OF CIVIL ENGINEERING 


EDMONTON, ALBERTA 
SPRING, 1976 
























































THE UNIVERSITY OF ALBERTA 
FACULTY OF GRADUATE STUDIES AND RESEARCH 


The undersigned certify that they have read, ana 
recommend to the Faculty of Graduate Studies and Research, 
for acceptance, a thesis entitled A MONTE CARLO STUDY OF THE 
STRENGTH VARIABILITY OF RECTANGULAR TIED REINFORCED CONCRETE 
COLUMNS submitted. by LEON HADSLEY GRANT in partial 
fulfilment of the requirements for the degree of Master of 


Science 










ABSTRACT 


The safety provisions proposed for use in Canada for 
limit states design involve load factors to account for 
possible overloads and resistance or performance factors to 
account for possible understrength of structural members. 
The purpose of this study was to evaluate the understrength 
or <f> factor applicable to rectangular tied reinforced column 
cross sections based on a probabilistic analysis of the 
results of a Monte Carlo Study. 

Probability models were described for the major 
variables affecting the cross sectional strength. A Monte 
Carlo procedure was used to develop a sample of cross 
section strengths from which the understrength factor was 
calculated. This study showed that the concrete strength 
variability and the steel strength variability were the 
major contributing factors to the variability in cross 
sectional strength. 

The understrength factors calculated from the results 
of this study were found to be in close agreement with the 
understrength factors used in the ACI 318-71 Building Code. 


IV 



















































ACKNOWLEDGEMENTS 


This investigation was made possible by a grant 
provided by the National Research Council of Canada (Grant 
A1673) . 

The author wishes to express his sincere appreciation 
to Dr. J.G. MacGregor under whose direction this study was 
performed. 

The assistance of Dr. S. Ali Mirza in describing the 
variability of steel strength as contained in Appendix A is 
gratefully acknowledged. 


v 


















TABLE OF CONTENTS 


CHAPTER PAGE 

I INTRODUCTION 1 

1 .1 General 1 

1.2 The Monte Carlo Technique 3 

1.3 Development of the Understrength Factor i- 5 

II THEORETICAL BEHAVIOUR OF REINFORCED CONCRETE 

SECTIONS 9 

2.1 The Basic Assumptions for Analysis 9 

2.2 The Stress-Strain Relationship for Concrete 10 

2.3 The Stress-Strain Relationship for Steel 19 

2.4 Numerical Method for Developing the 

Interaction Diagram 19 

III COMPUTER PROGRAM FOR ANALYSIS 27 

3.1 Description of The Monte Carlo Technique 27 

3.2 Description of The Computer Program 28 

3.3 Comparison of Theory With Test Results 43 

IV PROBABILITY MODELS OF VARIABLES AFFECTING SECTION 

STRENGTH 48 

4.1 Concrete Variability 48 

4.1.1 Introduction 48 

4.1.2 Distribution of Concrete Strength 50 

4.1.3 Statistical Description of Concrete 

Strength Variation 53 

4.1.4 Cylinder Strength vs. Design Strength 56 

4.1.5 In-situ Strength of Concrete 58 


vi 


















































TABLE OF CONTENTS CONTINUED 


CHAPTER PAGE 

4.1.6 Probability Model for Concrete 

Strength 61 

4.2 Reinforcing Steel Variability 62 

4.3 Cross Section Dimensional Variability 62 

4.3.1 Introduction 62 

4.3.2 Probability Model for Cross Section 

Dimensions 63 

4.4 Reinforcing Steel Placement Variability 67 

V THE MONTE CARLO STUDY 71 

5.1 Size of Columns and Reinforcement Studied 71 

5.2 Size of Sample Studied 77 

5.3 Results of The Monte Carlo Simulation 87 

5.3.1 General 87 

5.3.2 The Effect of Steel Strength 

Distribution Used 87 

5.3.3 The Effect of the Concrete Strength 

Variation 93 

5.3.4 The Effect of the Variables Studied 97 

5.4 Cross Section Strength 100 

5.5 Calculation of <p Factors 110 

5.5.1 Based on 1 in 100 Understrength 110 

5.5.2 Based on Cornell-Lind Procedure 114 

VI SUMMARY AND CONCLUSIONS 118 

REFERENCES 120 


vii 



























. 


■ 













































TABLF OF CONTENTS CONTINUED 


CHAPTER PAGE 

APPENDIX A VARIABILITY IN REINFORCING STEEL 127 

APPENDIX B COLUMNS STUDIED 145 

APPENDIX C FLOW DIAGRAMS OF THE MONTE CAPLO PROGRAM 150 

APPENDIX D LISTING OF THE MONTE CAPLO PROGRAM 185 

APPENDIX E DESCRIPTION OF INPUT DATA 204 

APPENDIX F NOMENCLATURE 207 


viii 




















































LIST OF TABLES 


Table Description Page 

3.1 Comparison of Ptest/Ptheory With the Value of k^ 45 

3.2 Theory Comparison With Hognestad's Tests II 46 

3.3 Theory Comparison With Hognestad's Tests III 47 

4.1 Concrete Strength Variability 52 

4.2 Concrete Strength in Structures vs. Cylinder 

Strength 60 

5.1 Comparison of the Mean Value of the Patio 
Ptheory/PACI for Sample Sizes of 200, 500 and 

1000 83 

5.2 Comparison of the Coefficient of Variation of the 

Patio Ptheory/PACI for Sample Sizes of 200, 500 

and 1000 84 

5.3 Comparison of the Coefficient of Skewness of the 

Patio Ptheory/PACI for Sample Sizes of 200, 500 

and 1000 85 

5.4 Comparison of the Measure of Kurtosis of the 

Patio Ptheory/PACI for Sample Sizes of 200, 500 

and 1000 86 

5.5 Comparison of the Mean Value of the Patio 

Ptheory/PACI for a Normal and a Modified Log¬ 
normal Steel Strength Distribution 89 

5.6 Comparison of the Coefficient of Variation of the 

Patio Ptheory/PACI for a Normal and a Modified 
Log-normal Steel Strength Distribution 90 


IX 
































































































LIST OF TABLES CONTINUED 


Table Description Page 

5.7 Comparison of the Coefficient of Skewness of the 

Patio Ptheory/PACI for a Normal and a Modified 
Log-normal Steel Strength Distribution 91 

5.8 Comparison of the Measure of Kurtosis of the 

Patio Ptheory/PACI for a Normal and a Modified 
Log-normal Steel Strength Distribution 92 

5.9 Comparison of the Mean Value of the Ratio 

Ptheory/PACI for Concrete Cylinder Strength 
Coefficients of Variation of 10%, 15% and 20% 94 

5.10 Comparison of the Coefficient of Variation of the 

Patio Ptheory/PACI for Concrete Cylinder Strength 
Coefficients of Variation of 10%, 15% and 20% 95 

5.11 Comparison of the coefficient of Skewness of the 

Patio Ptheory/PACI for Concrete Cylinder Strength 
Coefficients of Variation of 10%, 15% and 20% 96 

5.12 Comparison of the Mean Value of the Ratio 

Ptheory/PACI for the 12 in. and 24 in. Columns 109 

5.13 The Understrength Factor for the 12 in. by 12 in. 

Column Based on a Probability of Understrength of 

1 in 100 111 

5.14 The Understrength Factor for the 24 in. by 24 in. 

Column Based on a Probability of Understrength of 

1 in 100 112 

5.15 The Understrength Factor for the 12 in. by 12 in. 


x 






























LIST OF TABLES CONTINUED 


Table Description 

Column Based on <f> = y e -(BaV R 

R 

5.16 The Understrength Factor for the 24 in. by 24 in. 
Column Based on <j> = y D e SaV R 

K 

A-1 Summary of Selected Studies on Steel Strength 
B-1 Properties of the 12 in. Column Assumed in the 
Calculations 

B-2 Properties of the 24 in. Column Assumed in the 
Calculations 


Page 

116 

117 

130 

146 

147 


xi 

















































































LIST OF FIGURES 


Figure Description Page 

2.1 Compression Block Parameters 11 

2.2 Some Suggested Stress-*Strain Curves for Confined 

Concrete 13 

2.3 The Kent and Park Stress-Strain curve for 

Concrete 15 

2.4 The Stress-Strain Curve for Concrete Used in This 

Study 20 

2.5 The Stress-Strain Curve for Steel Used in This 

Study 21 

2.6 Typical Moment Curvature Diagram 23 

2.7 Basic Notation Used in the Flexural Analysis of 

Reinforced Concrete Sections 24 

3*1 The Monte Carlo Technique 29 

3.2 Condensed Flow Diagram of the Monte Carlo Program 31 

3.3 Condensed Flow Diagram of the Subroutines ACI and 

ASTEEL 33 

3.4 The ACI Interaction Diagram 34 

3.5 Condensed Flow Diagram of the Subroutine CURVE 36 

3.6 Condensed Flow Diagram of the Subroutine THMEAN 37 

3.7 Condensed Flow Diagram of the Subroutine THEORY 39 

3.8 Condensed Flow Diagram of the Subroutine AXIAL 40 

3.9 Condensed Flow Diagram of the Subroutine FSTEEL 41 

3.10 Condensed Flow Diagram of the Subroutine STAT 42 

4.1 Relationship Between Standard Deviation and Mean 


Xll 
















































































LIST OF FIGURES CONTINUED 


Figure Description Page 

Strength of Concrete 54 

4.2 Histogram of Cross Section Dimensional Variation 

reported by Tso and Zelman 64 

4.3 Histograms of Cross Section Dimensional Variation 

Reported by Hernandez and Martinez 66 

4.4 Histogram of Variation in Concrete Cover Reported 

by Hernandez and Martinez 70 

5.1 Histogram of the Frequency of Column Sizes vs. 

Column Size 72 

5.2 Histogram of the Percentage of Reinforcing Steel 

in All Columns 73 

5.3 Histogram of the percentage of Reinforcing Steel 

in Columns Less Than 16 in. 74 

5.4 Histogram of the Percentage of Reinforcing Steel 

in Columns 16 in. to 24 in. 75 

5.5 Histogram of the Percentage of Reinforcing Steel 

in Columns 24 in. to 36 in. 76 

5.6 Final Column Cross sections Studied 78 

5.7 Mean Value of the Ratio Ptheory/PACI vs. e/h for 
Sample Sizes of 200, 500 and 1000 for a 12 in. 

Square Column and Modified Log=*normal Steel 
Strength Distribution 80 

5.8 Coefficient of Variation of the Ratio 
Ptheory/PACI vs. e/h for Sample Sizes of 200, 500 


xiii 






























































































































LIST OF FIGURES CONTINUED 


Figure Description Page 

and 1000 for a 12 in. Square Column and Modified 
log-normal Steel Strength Distribution 81 

5.9 Coefficient of Skewness of the Ratio Ptheory/PACI 

vs. e/h for Sample Sizes of 200, 500 and 1000 for 
a 12 in. Square Column and Modified Log-normal 
Steel Strength Distribution 82 

5.10 Standard Deviation Squared of the Ratio 

Ptheory/PACI vs. e/h for the Variables Affecting 
Column Strength for a 12 in. Square Column and 
Modified Log-normal Steel Strength Distribution 99 

5.11 Dispersion of Strengths of an Eccentrically 

Loaded 12 in. Square Column 101 

5.12 Dispersion of Strengths of an Eccentrically 

Loaded 24 in. Square Column 102 

5.13 Normal Cumulative Frequency Plot of the Ratio 

Ptheory/PACI for the 12 in. Column, e/h = 0.10 105 

5.14 Log-normal Cumulative Frequency Plot of the Ratio 

Ptheory/PACI for the 12 in. Column, e/h = 0.10 106 

5.15 Normal Cumulative Frequency Plot of the Ratio 

Ptheory/PACI for the 24 in. Column, Pure Moment 107 

5.16 Log-normal Cumulative Frequency Plot of the Patio 

Ptheory/PACI for the 24 in. Column, Pure Moment 108 

5.17 The Understrength Factor 4> vs. e/h Based on a 
Probability of Understrength of 1 in 100 for the 


xiv 
















LIST OF FIGURES CONTINUED 


Figure Description Page 

12 in. and 24 in. Columns 113 

A”1 Steel Strength Distribution for Grade 40 

Reinforcing Bars 132 

A*»2 Probability Density Function for Grade 40 Bars 134 

A°3 Steel Strength Distribution for Grade 60 

Reinforcing Bars 135 

A-4 Probability Density Function for Grade 60 Bars 136 

A-5 Effect of Bar Diameter on Steel Strength, Grade 

40 140 

A-6 Effect of Bar Diameter on Steel Strength, Grade 

60 141 

B-1 Nominal or Designer's Properties of the 12 in. 

and 24 in. Columns 148 

B-2 Mean Values of the Properties of the 12 in. and 

24 in. Columns 149 


xv 




















































































CHAPTER I 


INTRODUCTION 


lil Ge neral 

It is generally recognized that there is some degree of 
uncertainty in the design equations used to calculate the 
resistance of a reinforced concrete section. The strength of 
a reinforced concrete section is calculated by the designer 
as a constant nominal value but it is recognized that the 
ultimate strength of a reinforced column is affected by 
variations in: 

Concrete strength 

Steel strength 

Cross section dimensions 

Location of steel reinforcement 

Eccentricity of load 

Rate of loading 

Amount of creep and plastic flow 

The three most common approaches that have been used to 
estimate the variability of the ultimate strength of a 
reinforced concrete section are: 

The technique of error statistics or regression 
analysis applied to the results of full scale or 
laboratory tests. 

Direct statistical evaluation of means and 


1 
















































2 


standard deviations from the means and standard 
deviations of the individual parameters involved. 

The Monte Carlo Technique in which the variables 
affecting the cross section strength are treated as 
random variables and are randomly chosen and used to 
calculate a population of ultimate strengths based on 
structural theory. 

The method of error statistics has been applied to sets 
of test results in various fields and has been accepted as a 
method of analyzing test data. This method of using test 
data has the disadvantage of requiring many tests to produce 
reliable results. More important, however, the sample may 
never be representative of the population due to testing 
procedures and systematic errors. Construction tolerances 
may not be adequately modeled, for example. 

When the cross sectional strength can be calculated 
with relatively simple analytical expressions, standard 
statistical techniques can be used to calculate the mean 
error and coefficient of variation of the cross sectional 
strength based on the descriptions of the distributions of 
the individual variables. This procedure becomes awkward if 
the strength expressions become complex. 

The Monte Carlo Technique has been used to model a 
population of values in various fields. This method has the 
disadvantage of requiring a statistical description of each 


























































































3 


individual variable which affects the final variable being 
studied. The Monte Carlo Technique has the advantage of 
being able to generate a large size sample using computer 
simulations rather than actual test data. 

Since the error statistics method of predicting 
strength has been considered insufficient and too costly for 
developing probability models of cross section strength and 
the equations used to calculate the strength of reinforced 
concrete cross sections are relatively complex, the Monte 
Carlo Technique has become popular. 

JL.2 The Mont e Car lo T echnique 

The Monte Carlo Technique is a method of obtaining 
information about the total system performance from the 
individual component characteristics. It consists of 
generating many total systems from the component data and 
analyzing the sample of total systems. 

This procedure has been used by various researchers to 
model the variability of structure strength and loading 
conditions. Housner and Jennings 30 have used this procedure 
to develop "Artificial Earthquakes" from which the various 
effects of earthquakes could be measured. Using the data 
generated with the Monte Carlo Technique close agreement was 
found with actual measured values. 

Warner and Kabalia 72 have described a method of 
















































4 


developing the strength and serviceability of a real 
structure using the Monte Carlo Technigue. The strength of 
an idealized axially loaded reinforced concrete column was 
calculated including the effects of variations in the 
material and geometric properties. 

Allen 5 has presented a probability distribution of the 
ultimate moment and ductility ratio for reinforced concrete 
in bending. The ultimate moment and ductility ratio were 
obtained using prediction equations and probability 
distributions of the parameters. The computations were based 
on the method of using the Monte Carlo Technique described 
by Warner and Kabalia 72 . The results showed that the 
probability distributions of the ultimate moment and 
ductility ratio are affected by material properties, 
duration of loading, steel percentage and geometric 
properties. 

In this study the Monte Carlo Technigue was used to 
develop a probability model of the strength of a rectangular 
tied reinforced concrete column. The actual probability 
distribution developed was that for the ratio of the 
theoretical load capacity to that computed in accordance 
with the ACI design equations, Ptheory/PACI, for specific 
values of e/h or eccentricity of axial load. This study 
shows the effect of variations in the concrete strength, 
steel strength, cross section dimensions, location of 
reinforcing steel and steel percentage on the probability 










































































































5 


distribution of the strength of a reinforced concrete 
section under axial load and bending moment. 

1.3 Developm ent of the Under strength Fac tor 

The ACI 318-71 Building Code Requirements for 
Reinforced Concrete 3 requires that the design equation 
follow the format of: 

^iY D D+T L L (1.1) 

Where <p is an understrength factor, R is the nominal 
calculated resistance or strength, L and D are the live and 
dead loads respectively and Y L and Y D are the load factors 
to account for uncertainties in the loads. 

Generally the procedure used for determining the values 
of ■ 4> , , and y d has been to rely on ’’common sense and 
experience” along with a semi-mathematical approach. These 
factors may also be determined using a logical mathematical 
approach using the probabilistic concepts. 

The first consistent proposal for design based on the 
concept of probability appears to have been made by 
Torroja 68 . This proposal was based on the concept of limit 
states in which the design loads and resistance have a 
specified probability of being exceeded. 

Easier 10 has suggested that the coefficient of 
variation may be used as a probabilistic but distribution 



















































































6 


free safety measure. He has proposed a rational scheme for 
splitting the safety factor in the partial load and 
resistance factors of practical codes. He accounts for the 
most uncertain variables such as workmanship by the use of a 
separate safety factor which may not be defined explicitly. 
Ang and Amin 7 have developed an alternative approach which 
uses judgement factors. 

Cornell* 9 has suggested a probability based code format 
known as the first order or second moment format. The basic 
ACI Code design equation remains unchanged for this format, 
whereas the code specified values for loads and factors may 
be changed in absolute value. Cornell 19 suggests a method of 
calculating the values of the loads and factors along with a 
method of calculating a coefficient of variation of the in 
place structure resistance and load effects. 

The probabilistic design code suggested by Cornell 
implies that the understrength and overload factors are 
dependent on a predetermined reliability factor and the 
uncertaintity in the components which affect the structure 
safety. In view of this, equations must be used to give the 
best estimate of the mean load and strength conditions along 
with their variability rather than the current approach 
based on estimates of the material strengths. 

The second moment theory developed by Cornell reduces 
to the requirement that the mean safety margin be greater 
than or egual to a specified 0 of its standard deviation so 








































































7 


that: 



( 1 . 2 ) 


Lindas has extended Cornell's approach to code formats 
of higher order and demonstrated a method of calibrating a 
partial safety factor format to Cornell's as well as Ang and 
Amin's format. Since it is possible to choose 6 by 
calibrating probabilistic code formats to existing codes, 
the parameters may be adjusted to yield designs comparable 
to existing code designs. This leads to a more acceptable 
implementation of probabilistic code concepts initially. 

Siu et al. 63 have presented a method of code 
calibration which may be used to calibrate probabilistic 
code formats with existing code formats as well as to 
compare various probabilistic code formats. 

In this study understrength factors for rectangular 
tied reinforced concrete columns were calculated directly 
from the distribution of column strength and a probability 
of understrength of 1 in 100 and have been compared with the 
understrength factors calculated on the basis of the first 
order second moment format. 

The form of the second moment format used was that 
developed by Cornell, Lind and ACI Committee 348: 



-BaV, 


R 


(1.3) 



































































The derivation of this equation has been reviewed 
Gregor 40 . 


by 












































CHAPTER II 


THEORETICAL BEHAVIOUR OF REINFORCED CONCRETE SECTIONS 

The Ba sic Assu m ption s for A naly sis 

If an analytical expression is to be used to determine 
the ultimate strength of a reinforced concrete cross section 
a number of assumptions must first be made. The following 
basic assumptions were made for the analysis: 

(a) Plane sections remain plane, that is, the strain in 
the concrete or steel is directly proportional to the 
distance from the fibre to the neutral axis. 

(b) The concrete stress is a function of the strain as 
expressed by the modified Kent and Park stress strain curve 
for concrete for the theoretical calculations. 

(c) The steel stress is a function of the strain as 
expressed by an elastic plastic stress strain curve. 

W There is no slip between the concrete and steel 
reinforcing. 

(e) Bending in one plane is assumed and biaxial bending 
is neglected. 

(f) Stability failure of the member is not included. 

(g) The stiffness in bending of the individual layers 
of steel reinforcement is neglected. 


9 











































































10 


(h) The effect of duration of loading is neglected. 

2. 2 The Stress ^ Str ain Rel ationship for Co ncret e 

The properties of the compressive stress block of a 
concrete flexural member may be defined by the parameters 
k , and k^ as shown in Figure 2.1. These parameters 
depend on the shape of the stress-strain curve for concrete. 

In North America the most widely accepted stress-strain 

curve for concrete is that proposed by Hognestad 29 which 

consists of a second order parabola up to a maximum stress 

f” at a strain e and then a linear falling branch. 

Hognestad 1 s 28 curve was obtained from results of tests on 

eccentrically loaded short columns in which he found that 

f"=0.85f•. 
c c 

There is controversy as to whether the shape of the 
stress-strain curve for concrete is affected by a strain 
gradient. Sturman, Shah and Winter 67 concluded that the peak 
occured at a 2C% higher stress and a 50% higher strain for 
eccentrically loaded prisms compared to concentrically 
loaded prisms. In Hognestad's tests this was not observed. 
There may be no significant effect of the presence of a 
strain gradient but its presence, if anything, will improve 
the properties of the compression block. There is no doubt, 
however, that the presence of a strain gradient delays the 
appearance of longitudinal cracking in the compression zone. 
















' 





















11 



Figure 2.1 


Compression Block Parameters 






















































12 


In columns the concrete is confined by the ties to some 
extent, depending on the type of ties used. The confinement 
due to the ties does not affect the concrete strength until 
there has been some yielding of the concrete to cause a load 
in the ties. At low levels of stress the ties will not be 
stressed and therefore the concrete will act as unconfined 
concrete. Tests have shown that when the stress in the 
concrete approaches the maximum uniaxial strength, 
deterioration of the concrete causes an outward expansion 
perpendicular to the load causing a stress in the ties which 
in turn causes a confining pressure. In this case spiral 
ties are more effective than rectangular ties since the 
spiral is able to exert pressure for its entire length 
whereas the rectangular ties tend to exert pressure at the 
corners and not along their entire length. This is due to 
the relatively flexible bar between the corner points. As a 
result the concrete is confined at the corners and in the 
centroidal core of the member. Even though the rectangular 
ties are not as effective as the spiral ties, they do 
produce a significant increase in ductility of the core as a 
whole. 

Some stress-strain curves proposed for concrete 
confined by rectangular ties are shown in Figure 2.2. In 
Chan*s 17 trilinear curve the range OAB approximates the 
curve for unconfined concrete and the slope BC depends on 
the lateral confinement. Soliman and Yu's 64 curve consists 


































































Figure 2. 


1 3 



CHAN'S CURVE 



SOLIMAN & YU'S CURVE 



ROY &SOZEN'S CURVE 


Some Suggested Stress-Strain Curves for 
Confined Concrete 














































































14 


of a parabola and two straight lines. Values for the 
critical points are related to the properties found from 
tests on eccentrically loaded prisms. Boy and Sozen 57 
conducted tests on axially loaded prisms and suggested that 
the descending branch of the stress-strain curve could be 
replaced by a straight line. The strain at 50% of the 
maximum stress on the falling branch e^ Qc was related to the 
volumetric ratio of the transverse steel. 

Boy and Sozen 57 concluded that rectangular hoops did 
not increase the concrete strength. Other investigators such 
as Chan 17 , Soliman and Yu 64 , Bertero and Felippa 11 , and 
Busch and Stockl 59 have observed an increase in strength due 
to closely spaced rectangular ties. 

Kent and Park 36 , on the basis of experimental evidence 
have proposed the stress-strain curve shown in Figure 2.3 
for confined and unconfined concrete. This curve combines 
many of the features of the previously described curves. The 
ascending region AB is represented by a second order 
parabola in common with the Hognestad 29 curve. The confining 
steel is assumed to have no effect on the stress strain 
relationship before the maximum stress. Kent and Park 35 used 
a maximum stress in bending equal to f^,, that is, k^=1.0 in 
Figure 2.1. Sturraan, Shah and Winter*s 67 work suggests that 
the value of k 3 =1.C is conservative where there is a strain 
gradient. Kent and Park 36 assume the strain, e Q , at maximum 
stress to be 0.002 which is in the range commonly accepted 













































STRESS 


15 



STRAIN 


Figure 2.3 


The Kent and Park Stress-Strain Curve for 
Concrete 






























































16 


for unconfined concrete. Confinement may increase the 
maximum strain but this will occur after the maximum stress 
is reached. Region AB for the Kent and Park 36 curvo is 
expressed using: 


f = 


c 


f' 

c 



( 2 . 1 ) 


In this study the value for k^ was taken as 0.85 based 

on comparison with Hognestad*s 29 test results, (See Section 

3.3). To allow compatibility between the ACI equation for 

modulus of elasticity and the strain at about 0.4^, the 

strain z at a maximum stress of k f', was taken as: 
o 3 c 


e 

o 


1.8 f ? 
c 


E 

c 


( 2 . 2 ) 


The region of the curve after the maximum stress is 

linear from e and f m and is described by the strain in 

the concrete at 50% of the maximum stress as suggested by 

Roy and Sozen. The slope of the falling branch increases 

rapidly with an increase in concrete strength. This suggests 

that is dependent on f'. This can easily be observed by 

5ou c 

the fact that high strength concrete is more brittle than 

low strength concrete. For concrete that is not laterally 

restrained, Kent and Park suggest that the strain e at 50% 

5ou 

of f£ is: 


e = 3.0 + 0.002f' 
5ou c 

f' - 1000 
c 


(2.3) 


For concrete confined by rectangular ties the slope of 



























































































































17 


the falling branch is reduced. This is due mainly to the 
restraint supplied by the ties. Kent and Park 35 expressed 
this in terms of the ratio of the volume of the ties to the 
volume of the concrete core within the ties. Kent and Park 36 
expressed the volumetric ratio as: 

p" = 2.0 (b"+d M ) A" (2.4) 

__ s 

b"d"s 


Corley 18 suggested that the compression steel should be 
included in the volumetric ratio. In this study the 
compression steel was included in the volumetric ratio which 
was expressed as: 


p M = 2.0 (b"+d") A" + A' S 
__s_ s 

b"d"s 


(2.5) 


The descending branch of Kent and Park's 36 curve for 
confined concrete may be described by: 



where: 


z 


0.5 


£ 5oh 


+ 


’5ou 


e 

o 


( 2 . 6 ) 


(2.7) 


and: 

E 50h = 3/4 *"W (2 ’ 8) 

Kent and Park 36 assumed that confined concrete could 
sustain a stress of 0.20^ at an infinite strain as shown by 


the dashed line in Figure 2.3. In this study the descending 























































































































18 


region was assumed to continue to zero. 

The tensile strength of concrete is usually neglected 
in most flexural theories as well as codes of practice. It 
is reasoned that it may be unsafe to take into account the 
tensile strength of the concrete since the concrete may be 
cracked due to shrinkage or other reasons even before any 
load is applied. While the tensile strength of concrete is 
small compared to its compressive strength it has a sizeable 
effect on the resistance and deformation of the uncracked 
section. After the appearance of the first cracks this 
influence becomes smaller and smaller as the load increases. 
This is due to the fact that with the advancement of 
cracking the tensile block becomes closer to the neutral 
axis resulting in a smaller lever arm and a negligible 
addition to the moment capacity. 

In view of the above it was assumed that for the 
purposes of this study an elastic brittle stress”strain 
relationship can represent fairly well the behaviour of 
concrete in tension. An elastic brittle stress-strain 
relationship can be expressed as follows: 



for e < £ 

t - tr 


(2.9) 


and 


a 


t 


0 


for e > £ 

t tr 


( 2 . 10 ) 


The modulus of elasticity of concrete in tension was 

































































' 



































19 


taken as the accepted value in compression: 

E = 57000 /'T'”' 
c c 


( 2 . 11 ) 


The modulus of rupture was taken as the accepted value: 


a = 7.5 /f r 
tr c 


( 2 . 12 ) 


The complete stress-strain curve for concrete used in 
this study is shown in Figure 2.4. 


2.*3 The Stress-Strain Rel ations hip for Steel 

In this study an elastic purely plastic stress-strain 
relationship was assumed for steel as shown in Figure 2.5. 
The modulus of elasticity of steel was taken as 29,000 ksi. 
in tension as well as in compression. The steel stress was 
assumed to increase to the yield point and remain at the 
yield stress for any further strain. This is a conservative 
representation of the steel strength since the effect of 
strain hardening is neglected. 

2 .4 Num erical Method for Developing th e Inte ract ion Diagram 

The inter-relationship between the effects of the axial 
load and applied moment on a reinforced concrete section are 
best shown by an interaction diagram. These diagrams are a 
graphical representation of the envelope of the maximum 
capacities of a reinforced concrete section under various 


























































































































20 



Figure 2.4 


The Stress-Strain Curve for Concrete Used in 
This Study 
























































21 



Figure 2.5 The Stress-Strain Curve for Steel Osed in This 

Study 



















































22 


axial load and moment combinations. 

Using strain compatibility, the moment curvature 
relationships were derived for the section for a number of 
axial load levels using the procedure described in the next 
few paragraphs. The moment curvature relationship developed 
is similar to that shown in Figure 2.6. The maximum moment 
in the moment curvature diagram was taken as the ultimate 
moment for that given load. The various values of load and 
ultimate moment were plotted as an interaction diagram. 

The calculation of the moment curvature diagram was 

started by assuming a strain distribution across the cross 

section and determining the location of the neutral axis and 

the point at which the tensile strains exceeded £ . The 

compression region was then divided into sections with equal 

widths measured perpendicular to the neutral axis, (See 

Figure 2.7). Using the concept of linear strains in the 

cross section the strain at the centroid of each section may 

be determined. Fy assuming the strain is constant over each 

section the resulting stress and total force over the area 

was determined with the aid of the stress~strain curve for 

concrete. The total compressive force supplied by the 

concrete may be expressed as: 

ns 

F c= E f ci bdx (2a3 > 

i=i 

Assuming the maximum tensile stress in concrete occurs 

at a strain of e and assuming a linear stress*"strain curve 

tr 







' 




































































































MOMENT (k-in. 


23 



CURVATURE <£h 


Figure 2.6 


Typical Moment curvature Diagram 
























































































Figure 2.7 Basic Notation Used in the Flexural Analysis of 

Peinforced Concrete Sections 


CROSS SECTION STRAINS STRESSES 















































































































25 


for concrete in tension, the total tensile force may be 
calculated using a triangular stress block. The total 
concrete tensile force may be expressed as: 


F = c b dt 
t t — 


(2.14) 


From the strain distribution the strain in each steel 

bar may be determined. Using the stress-strain curve for 

steel the stress in each bar may be calculated. The total 

steel force may be expressed as: 

nb 


F = y" f . A. 

st Z-j si 1 

i=i 


(2.15) 


The total axial force resisted by the cross section is 
the algebraic sum of the concrete compressive force, the 
concrete tensile force and the steel force. The total moment 
that the section is subjected to may be determined by 
summing the moments of the above forces about the centroidal 
axis. The moment may be expressed as: 


m 


ns 

nb 


y f . x. + 

A-* ci l 

i=i 

y F x . + F x -Pc 

Z-j st si t t 

i=i 

(2.16) 


where c = the distance from the tension steel to the 
centroid of the cross section. 


The first three terms are the moments of the internal 
forces about the tension steel and the last term. Pc, is to 
convert the moment to a moment about the centroid of the 


cross section 















































































































































T he required points on the axial load” moment 
interaction curve were developed by selecting specific axial 
load levels at which the ultimate moment was calculated. At 
each axial load level an initial strain at the extreme 
compression fibre and initial curvature was assumed. For the 
initial curvature the edge strain was incremented until the 
sum of the internal forces and the external specified load 
were balanced within a specified tolerance. After balancing 
the axial loads the moment required to provide equilibrium 
was calculated. The curvature was then incremented and the 
axial load again balanced and the moment calculated. This 
procedure was repeated until the maximum moment on the 
moment curvature curve was calculated. 


The 
gives a 
accuracy 


subroutine THEORY and flow diagram in Section 
further description of the above nrocedure. 
of this procedure is discussed in Section 3.3. 


3.2 

The 
















































































































CHAPTER III 


COMPUTER PROGRAM FOR ANALYSIS 

3.1 Desc ripti on of The Monte Car lo Tec hniqu e 

The Monte Carlo Technique is a method for obtaining 
information about system performance from the performance 
data of the individual components. This method may be called 
a synthetic or empirical method of sampling. It consists of 
simulating many systems by computer calculation and then 
evaluating the performance of the overall system by 
evaluating the performance of the population of synthesized 
systems. 

If a system consists of many components each with a 
number of values, a number of systems could be built to 
measure the performance of the system using each component 
value. Although this would give an indication of the 
variability of the system, it would generally be impractical 
or uneconomical. If there is a relationship between the 
total system performance and each component variable, a 
measurement of the total system performance may be 
calculated without actually building the system. By knowing 
the statistical properties of the distribution of each 
variable and drawing a value from this distribution rather 
than using measured values, it is possible to calculate the 
performance of a specified number of synthetic systems to 
get the variability of the system. 


27 
























. 


























' 




























































28 


This procedure is called the Monte Carlo Technique and 
is shown graphically in the form of a flow diagram in Figure 
3.1. The availability of high speed computers has led to the 
popularity of this technique. 

In this study the Monte Carlo Technique was used to 
generate a family of theoretical axial load-moment 
interaction curves for rectangular column cross sections 
using random values of the variables affecting the cross 
section strength. The random value of each variable was 
based on the statistical properties of each individual 
variable. Each theoretical curve was then compared to the 
ACI axial load-moment interaction curve to obtain a sample 
of ratios of the random theoretical capacity to that based 
on the ACI Code, Ptheory/PACI. These ratios were eventually 
used to calculate <+> or understrength factors for rectangular 
tied column cross sections. 

3^2 Description of The Computer P rogram 

The computer program used in this study is capable of 
developing the axial load^moment interaction diagram for 
rectangular tied column cross sections with the longitudinal 
steel at any location in the cross section. The program is 
capable of developing the interaction diagram using the ACI 
method and assumptions as well as the theoretical 
interaction diagram using a theoretical calculation of 
strength based on material and cross section properties. 



















































29 



Figure 3.1 


The Monte Carlo Technique 










































































































































30 


Figure 3.2 is a condensed flow diagram of the Monte 
Carlo program. The main program consists of the subroutines 
PROP, ACI, CURVE, THMEAN, RANDOM, THEORY and STAT. A 
complete listing of the program with its subroutines may be 
found in Appendix D. Detailed flow diagrams of the 
subroutines are given in Appendix C. 

The subroutine PROP is used to read and write the 
nominal cross section properties. The statistical properties 
of the variables are read and written in the main program. A 
complete description and format of input data is given in 
Appendix E. 

The subroutine ACI is used to calculate the ACI axial 
load^moment interaction diagram using the nominal or 
designer's values of section and material properties. The 
subroutine ACI uses the subroutine ASTEEL to calculate the 
forces in the steel reinforcement in the cross section. The 
capacity under pure axial load, balanced conditions and pure 
moment are first calculated. Using the concept of linear 
strain across the cross section the axial load and 
associated moment are calculated for various strain 
distributions using equations based on sections 10.2.1 to 
10.2.5 and 10.2.7 of ACI 318-71 3 . Tension or compression 
failures are classified by comparing the axial load with the 
axial load at balanced conditions. The value of e/h for each 
load level considered is calculated for use in fitting a 
curve to the interaction diagram. Finally the ACI axial 

















































































31 




Figure 3.2 Condensed Flow Diagram of the Monte Carlo 

Progra m 





































■ 

















































































































































































32 


load-moment interaction diagram is written. A condensed flow 
diagram of the subroutines ACI and ASTEEL is shown in Figure 

3.3. 

The subroutine CDPVE is called a number of times to fit 
a polynomial curve to the interaction diagram developed. The 
interaction diagram is transformed into a curve of axial 
load vs. e/h for axial loads above the balance point or 
compression failures and a curve of moment vs. h/e for axial 
loads below the balance point or tension failures. The two 
part curve fit was used to achieve greater accuracy in 
fitting the curve near the balance point. The use of moment 
rather than the axial load was used for the tension region 
to achieve greater accuracy since e/h approaches infinity as 
P approaches zero. There was no attempt made to force the 
two curves to coincide at the balance point but the last 
point used for fitting the curve above the balance point was 
used as the first point for fitting the curve below the 
balance point. By using the same point in both curve fits a 
close agreement was achieved at the balance point. When the 
subroutine CDPVE is used to fit a polynomial to the 
interaction diagram the calculated points with an associated 
value of e/h greater than 3.0 are eliminated from the curve 
fit since these points may cause large errors. Figure 3.4 is 
a plot of the ACI interaction diagram plotted from the ACI 
calculated values and the ACI interaction diagram plotted 
from values from the curve fit. Figure 3.4 is the 
transformed diagram with axial load vs. e/h and moment vs. 




























































33 



ACI 


Figure 3.3 


Condensed Flow Diagram of the Subroutines ACI 
and ASTEEL 





















































































































































































































































































MOMENT KIP-IN AXIAL LOAD KIPS 


34 



e/h 



Figure 3.4 


The ACI Interaction Diagram 





































































35 


h/e plotted. 

The curve fit used for the interaction diagram required 
a minimum of two points above the balance point and a 
minimum of two points between the balance point and an e/h 
value of 3.0. This resulted in a curve fit for three points 
above and below the balance point. It was determined that a 

curve fit using a minimum of six points resulted in a curve 

\ 

fit with virtually no error above the balance point and a 
maximum error of about 2.5% below the balance point with the 
general error below the balance point in the range of 1% or 
less. On this basis twenty points on the interaction diagram 
were considered sufficient to achieve a satisfactory curve 
fit. 

The subroutine CURVE uses the IBM subroutines GRATA, 
ORDER, MlNV and the modified IBM subroutine MULTR to TMULTR. 
These subroutines are described in Reference 31. A condensed 
flow diagram of the subroutine CURVE is shown in Figure 3.5. 

The subroutine THMEAN uses the subroutine THEORY to 
calculate the theoretical axial load-moment interaction 
diagram using the mean value of the individual variables. 
This subroutine also writes the interaction diagram 
calculated. A condensed flow diagram of the subroutine 
THMEAN is shown in Figure 3.6. 

The subroutine RANDOM is a subroutine which combines 
the IBM subroutine GAUSS and RANDU to calculate random 


















. 

































































36 



Figure 3.5 


Condensed Flow Diagram of the Subroutine CUFVE 



































































































37 



Figure 3.6 


Condensed Flow Diagram 


of the Subroutine THMEAN 


































































38 


values of each variable based on the statistical properties 
of each variable. These IBM subroutines are also described 
in Reference 31. 

The subroutine THEORY was developed to calculate the 
theoretical axial load~moment interaction diagram using the 
subroutines AXIAL and FSTEEL. A specific axial load level is 
chosen in THEORY which in turn calls AXIAL. Using the axial 
load level selected, a strain distribution is determined at 
a given curvature for which the external load and internal 
forces balance. For this curvature the moment required to 
develop the curvature is determined. The above procedure is 
repeated with increasing curvature until the moment capacity 
is determined at each load level. This method produces a 
moment curvature diagram similar to the one shown in Figure 
2.6. The subroutine FSTEEL is used by AXIAL to calculate the 
forces in the reinforcing steel. Figures 3.7 through 3.10 
are condensed flow diagrams of the subroutines THEORY, AXIAL 
and FSTEEL. The theoretical interaction diagram was obtained 
as the locus of the values of M u for each value of P for 
which a moment curvature diagram had been computed as 
explained in Section 2.4. All comparisons of the theoretical 
strength with the ACI strength or Hognestad*s tests were 
done using values of the theoretical strength after the 
interaction diagram was subjected to a curve fit. 

The subroutine STAT is a subroutine used to perform a 
statistical analysis on the ratio Ptheory/PACI for the 
























































39 



Figure 3.7 


Condensed Flow Diagram of the Subroutine THEORY 
















































































































40 



Figure 3.8 


Condensed Flow Diagram of the Subroutine AXIAL 





























- 

























































































































Condensed Flow Diagram of the Subroutine FSTEEL 



































































































42 



Figure 3.10 Condensed Flow Diagram of the Subroutine STAT 











































































































43 


various values of e/h specified. The output from the 
subroutine STAT includes the mean, standard deviation, 
coefficient of variation, coefficient of skewness, kurtosis, 
minimum and maximum value, median and cumulative frequency 
table of the ratio Ptheory/PACI. A condensed flow diagram of 
the subroutine STAT is shown in Figure 3.10. 

3^3 Com parison of Theory With Test Resul ts 

The subroutine used to calculate the theoretical axial 
load-moment interaction diagram was compared with the 
results of tests on rectangular tied columns reported by 
Hognestad. Using Hognestad’s column properties and the total 
eccentricity of the reported failure loads the mean ratio of 
Ptest/Ptheory was calculated to be 1.0068 with a standard 
deviation of 0.064 when k^ used to define the maximum 
compressive stress, k f', was taken equal to 0.85. Table 3.1 
is a summary of the values of the ratio Ptest/Ptheory and 
standard deviation for various values of k^. Tables 3.2 and 
3.3 are the results of a comparison of the theoretical 
calculations with Hognestad*s 29 test results using a value 
of k 3 =0.85. Although the lowest standard deviation was 
obtained for k 3 =C.87, any increased accuracy did not warrant 
abandoning the traditional value of k^=0.85. 

In this study the compression block was divided into 
ten equal segments between the extreme compressive fibre and 
the neutral axis with the strain averaged over the segment 





















































































44 


depth. A comparison of the analysis with an analysis using 
twenty segments showed no significant difference in the 
ratio of Ptest/Ptheory. The mean value of Ptest/Ptheory for 
ten segments was 1.0068 compared with 1.005 for calculations 
using twenty segments. 

In view of the above the calculations in the subroutine 
THEORY were based on ten segments with a value of k^=0.85 
resulting in a mean value of Ptest/Ptheory of 1.0068 with a 
standard deviation of 0.064. Any inaccuracies due to the use 
of the curve fitting subroutine CURVE are included in these 


statistics. 













































































45 


TABLE 3.1 


Comparison of Ptest/Ptheory with the value of 


r 


r 





Ptest/Ptheory 


Std. Deviation 



- r 

1 


-,---- 

1 

- 1 


0.70 | 

1 

1 . C 8 10 

[ 0.080 



1 

0.80 | 

■ 

1.0300 

1 

| 0.066 

fl 



1 

0.85 | 

co 

o 

o 

• 

T— 

| 0.064 

1 



1 

0.86 | 

I 

1.0020 

1 

| 0.062 

i 



o 

• 

00 

0.9980 

1 

| 0.060 

§ 



S 

0.90 | 

fl 

0.9850 

| C .062 

| 



1 

1 . 0C s 

I 

C .9470 

5 0.067 


1 _ 

1 

--- L 


1 

i 

i 















































































































































































46 


Table 3.2 


Theory Comparison With Hognestad's Tests II 


1 - 

1 

Concrete 
Strength 
Psi. 

— i — 

1 

I 

1 

i 

e/h 

i — 

1 

1 

1 

j 

Ptest 

Kips 

— i 

1 

! 

1 

i 

Ptheory 

Kips 

—r 

1 

1 

1 

i 

Ptest/Ptheory 

1 

r 

5810 

1 

0. 276 

1 

1 

284.0 

1 

1 

290.1 

1 

1 

0.978 

1 


5810 

1 

0.540 

1 

152.0 

1 

167.7 

1 

0.906 



5520 

1 

0.534 

I 

162.0 

1 

166.6 

1 

0. 972 



5240 

[ 

0.344 

1 

274.0 

1 

233.0 

1 

1.176 



5170 

1 

0.789 

1 

91.2 

1 

93.7 

1 

0.973 



5170 

1 

1.275 

1 

44.0 

1 

45.9 

1 

C.959 



5100 

1 

1.278 

1 

46.1 

1 

45.9 

1 

1.004 



5100 

1 

0.787 

! 

89.0 

1 

93.7 

\ 

0.950 



4700 

f 

0.785 

1 

94.0 

1 

91.7 

1 

1.025 



4700 

1 

0.535 

1 

156.0 


159.8 

1 

0.977 



4370 

1 

1.279 

1 

44.0 

1 

45.0 

1 

0.977 



4370 

1 

0.782 

1 

89.5 

S 

90.9 

1 

0.985 



4260 

1 

0.532 

1 

146.0 

1 

150.6 

1 

0.970 



4260 

1 

1.278 

1 

43.5 

1 

44.8 

1 

0.971 



4080 

1 

0.007 

! 

456.0 

1 

427.6 

1 

1.066 



4080 

1 

0.27 5 

1 

256.0 

1 

227.8 

1 

1.124 



4040 

1 

0.006 

1 

420.0 

1 

429.8 

! 

0.977 



4 04 0 

1 

0.274 

1 

248.0 

1 

227.9 

1 

1.088 



2300 

1 

1.285 

1 

44.5 

1 

42.9 

1 

1.038 



2020 

l 

0.C10 


225.0 

1 

263.8 

1 

0.853 



1970 

l 

0.278 

I 

141.0 

1 

143.0 

1 

0.981 



1880 

1 

0.788 

( 

73.0 

1 

73.2 

i 

0.998 



1820 

1 

0.532 

1 

99.0 

\ 

99.7 

i 

0.993 



1820 

1 

0.539 

! 

99.0 

I 

98.9 

1 

1.001 



1770 

1 

1.288 

1 

45.0 

1 

42.0 

i 

1.071 



1730 

1 

0.785 

« 

65.5 

1 

71.7 

! 

0.914 



1520 

! 

0.018 

1 

202.0 

1 

221.5 

1 

0.912 



1520 

1 

0.277 

1 

126.8 

i 

130.5 

\ 

0.971 


i 


x 


X- 


L- 


i 


i 












































































































































































































































































































































• 











\ 



































































4 7 


Table 3.3 


Theory Comparison with Kognestad's Tests III 


r 

1 . 

Concrete 

Strength 

Psi. 

~i-r 

I e/h | 

! 1 

1 1 

Ptest 

Kips 

r~ 

1 

1 

1 

f 

Ptheory 

Kips 

r~ 

1 

1 

1 

i 

Ptest/Ptheory 


I 

5350 

S 0.536 | 

220.0 

—T 

1 

218.1 

1 

1 

1.009 



5350 

1 0.787 | 

142.0 

s 

151.0 

I 

0.940 



5100 

I 1.292 | 

88.0 

1 

79.7 

1 

1. 105 



5100 

| 0.793 1 

153.0 

1 

147.2 

1 

1.040 



5050 

| 0.272 | 

326.0 

1 

325.6 

! 

1.001 



4850 

| C.534 | 

210.0 

1 

208.2 

8 

1.008 



4850 

| 1.285 | 

79.0 

1 

79.8 

J 

0.991 



4630 

? 1.292 | 

84.5 

1 

78.0 

1 

1.083 



4300 

| 0.272 | 

303.0 

\ 

293.8 

1 

1.031 



4290 

j 0.534 j 

206.0 

1 

194.3 

1 

1.060 



4150 

| 0.270 | 

315.0 

1 

287.7 

1 

1.095 



4070 

| 0.010 | 

485.0 

1 

514.5 

1 

0.943 



4010 

| 0.276 | 

284.0 

s 

279.8 

! 

1.015 



3870 

j 0.008 | 

500.0 

1 

501.7 

! 

0. 997 



3800 

| 1.291 | 

74.0 

1 

77.9 

1 

0. 950 



3580 

| 0.535 | 

180.0 

1 

179.8 

1 

1.001 



3580 

I 0.789 | 

1 38.8 

1 

135.0 

1 

1.028 



2300 

j 0.276 \ 

252.0 

i 

215.3 

! 

1.171 



2300 

1 0.533 | 

151.0 

i 

145.5 

1 

1.038 



2200 

J 0.272 | 

230.0 

1 

217.9 

i 

1.055 



2070 

| 0.000 j 

353.0 

1 

376.8 

1 

0.937 



2070 

| 0.528 | 

137.0 

I 

141.4 

I 

0.969 



2070 

| 0.787 | 

104.0 

i 

112.0 

( 

0.928 



2070 

\ 1.291 1 

74.5 

I 

72.0 

1 

1.035 



1950 

| 1.289 f 

72.5 

s 

69.0 

1 

1.051 



1950 

j C.784 | 

115.5 

t 

107.1 

1 

1.078 


i _ 


_1-L 


_x_ 


-X- 



















































































































































































CHAPTER IV 


PROBABILITY MODELS OF VARIABLES AFFECTING SECTION STRENGTH 

Concrete V aria b ility 

iL2.JL.-l Introduc t io n 

Concrete, like all other construction materials, is 
variable. This variability is influenced by design, 
production and testing procedures. Research data shows that 
under current design and construction techniques concrete 
which differs from the specified strength is placed in 
structures. These structures have performed satisfactorily 
due to redistribution of stresses, mixing of the under 
strength concrete with over strength concrete within the 
forms, and the fact that the concrete strength increases 
with age after the time at which tests are made. In some 
cases experience has lead to design equations which result 
in conservative designs even though the assumptions used are 
not entirely correct. 

The two broad causes of variations in concrete strength 
are variations in material properties and variations in the 
testing procedures. Since concrete is a heterogeneous 
mixture of cement, water, coarse and fine aggregate, 
entrained air, and in some cases admixtures, variations in 
the final concrete strength are inevitable. Variations in 
any one of the ingredients or a combination of variations in 


48 
















. 













































49 


more than one ingredient will result in a variation in the 
final concrete strength. Variation in the water-cement ratio 
will cause significant strength variation. The water-cement 
ratio may be altered due to poor control of water content, 
variation in moisture content or nonuniformity of the 
aggregate. Variations in the properties or proportioning of 
any of the materials will cause strength variation. The 
methods of transporting, placing and curing will also affect 
the final concrete strength. 

Variations in the testing methods will lead to apparent 
variations in the concrete strength. Variations in testing 
may be due to inconsistent sampling, nonuniform fabrication 
of test samples or poor handling and care of fresh samples 
and variations in temperature and moisture conditions. Also 
the preparation of the samples for testing and the procedure 
used in testing may cause variations in the test strength. 

The control strength is affected by material properties 
and test procedures whereas the structure concrete strength 
will be affected by the material properties and placing 
procedures. This results in different concrete strengths in 
the test specimen and in the structure. The concrete 
strength will differ from place to place in the structure 
due to different placing procedures, curing conditions, and 
the location in the structure. 















































' 
















































50 


4. 1. 2 D istrib u tion of Concrete Stre n gt h 

Generally the distribution of concrete strength has 
been assumed to be a Gaussian or normal distribution. ACI 
Committee 214 2 found that for practical concrete control the 
normal distribution adequately describes the variation in 
concrete strength. Rusch and Rackwitz have presented data 
from an international study of cube and cylinder tests which 
also follows a normal distribution in most cases. 

In establishing understrength factors for members to 
reflect the probability of the material strength being lower 
than the specified strength, the low strength tail ends of 
the curve are important. Eecause little data is available 
for these tail areas, the tail of the curve must be 
extrapolated from the central area of the curve. The normal 
distribution fits the data very well for the majority of the 
data in the central portion of the curve. Some researchers 
have shown however, that the normal curve does not always 
give the best fit in the tail areas. 

Freudenthal 24 , Julian 33 , and Shalon and Reintz 61 have 
shown that the log-normal distribution gives a better fit 
for concrete strength in which the control is poorer than 
average and should be used where extreme values are 
important. Shalon and Reintz 61 have shown that the normal 
curve as a general assumption is valid but in almost every 
case a skew towards the higher strengths was observed, 
especially for cases of high coefficient of variation. Using 
















I 














































51 


the x 2 test as a measure of discrepancy, a discrepancy was 
observed between the actual distribution and the normal 
distribution at the 5% level of significance for a 
coefficient of variation of 23% whereas for a coefficient of 
variation of 14.2% practically no skew was observed. 
Freudenthal 24 suggests the use of the log-normal or the 
extreme distribution to better describe the tail area but 
the extreme distribution has the disadvantage of 
mathematical complexity. 

Table 4.1 is a collection of data from a number of 
statistical studies of concrete strength. The majority of 
researchers have used a normal distribution due to its 
simplicity and the fact that in concrete control it is the 
central area cf the curve that is important. Due to this, 
studies in concrete control are generally not concerned with 
the tail areas of the distribution. 

For concrete strengths with a coefficient of variation 
of 15% or lower the normal curve describes the variation in 
the concrete strength as well as any other distribution. For 
cases where the coefficient of variation is greater than 15% 
a skewed distribution is observed for which a log-normal 
transformation becomes valid to increase the accuracy in the 
tail areas of the curve. 










' 


































52 


Table 4.1 

Concrete Strength Variability 


1- 

-r— 

i 



T" 

1 


*i- 

i 

-1 

| Source 

1 

1 

i 

Test 

i 

l 

Type of 

1 

( Coefficient 

i 

of | 


1 

+- 

■ 

-- r - 

i 


i 

-t 

i 


1 

1 

i 



1 

1 

£ 

l 

Type | 

fl 

No. 

i 

s 

i 

Distribution 

1 

| Variation % 



I 

1 

1 

. j 


1 

1 


l 

i 


l 

1 

a 

1 

1 


1 

1 


i 


j Julian 

1950| 

II 

cyl. 1 

a 

861 

1 

1 

Normal 

l 

| 10.4 

l 


|Cummings 

1 

1953J 

i 

cyl. | 

| 

208 

1 

1 

Normal 

I 9.3 

1 


|Shalon 

1955| 

I 

1 

cube | 

i 


9 

1 

81 

Normal 

J 14.2 

■ 


1 n 

i 

ii i 

i 

1 

cube | 

8 


1 

1 

i 

Log-normal 

| 23.6 

i 


|Bloem 

1 

1955| 

l 

1 

cyl. 1 

1429 

9 

1 

Normal 

l 

| 11.4 

i 


1 " 

1 

ii i 

| 

cyl. | 

354 

1 

1 

8 

Normal 

( 16.4 

1 


|Wagner 

1955| 

i 

| 

cyl. 1 

i 

613 

S 

J 

i 

Normal 

| 11.8 
i 


j Erntroy 

19601 

l 

cube J 

■ 

4000 

1 

1 

i 

-.< —— 

o 

• 

o 

CM 


|Malhotra 

1962| 

cyl. | 

1 

68 

I 

1 

1 


f 13.5 

I 


1 Wagner 

1 

19631 

i 

1 

cyl. i 

9 

688 

i 

! 

i 

Normal 

| 12.4 

» 


1 « 

8 

ii i 

cyl. 1 

a 

688 

i 

I 

fl 

Normal 

| 15.2 

j 


| BPF 

! 

19631 

i 

cyl. | 

a 

975 

1 

1 

i 

Normal 

| 12.4 

1 


i h 

1 

1964| 

s 

cyl . | 

| 

200 

s 

1 

i 

Normal 

| 10.9 

■ 


| Virginia 

s 

19651 

i 

l 

cyl. | 
a 

210 

i 

1 

l 

Normal 

| 7.2 

« 


l 

! Hwy . 

8 

■ 

1 

1 

a 


1 

i 


8 

1 


| Soroka 

1968 | 

cyl. 1 

68 

I 

1 

i 

Normal 

! 15.2 

i 


J Eiley 

1 

1971 S 

i 

I 

cyl. 1 

i 

50,000 

i 

1 

i 

Normal 

| 13.6 

1 


«-— 

1 

L 

i 

L 


1 


1 

-J -- 

i 







































































' 








































































































































































































































































































































53 


4.1.3 S tati s ti cal Desc ript ion of Concrete St ren gth 
Variation 

The average strength and variation in strength of 
concrete cylinder tests may be described by the mean, 
standard deviation and coefficient of variation. The 
coefficient of variation has become the accepted measure of 
concrete strength variation. 

Depending on the control of the concrete operations the 
coefficient of variation may range from 5% for laboratory 
conditions to as high as 3C% for uncontrolled conditions. 
The 30% value is unacceptable under present construction 
techniques and the 5% value is not practical for field 
conditions. On the Skylon Tower 37 at Niagara Falls, Ontario 
coefficients of variation ranging from 6.8% to 9.8% were 
achieved using exceptional control methods. This suggests a 
minimum value for site conditions. The Bureau of 
Reclamation 2 consistently achieves a coefficient of 
variation of about 15% which suggests a value for better 
than average control or good control. Table 4.1 indicates 
that the coefficient of variation in many cases is between 
15% and 20% which suggests that 20% is a reasonable maximum 
value. 

An ASTK 2 task force working on the question of concrete 
strength suggested a coefficient of variation of 20% when no 
control data is available for the average job. Figure 4.1 
illustrates that the coefficient of variation varies but, on 




































































1000 


54 



(iSd) NOI1VIA3Q aaVQNViS 


Relationship Between Standard Deviation and 
Mean Strength of Concrete 


Figure 4.1 


1000 2000 3000 4000 5000 6000 7000 8000 9000 

MEAN STRENGTH (PSI) 





















































































55 


the average, is less than 20%. In this study the levels of 
control were divided into three classes with corresponding 
coefficients of variation as follows: 


Excellent Control 

10% 

Average Control 

15% 

Poor Control 

20% 


The total variation in concrete strength must include 
the variation in concrete strength within a single batch. 
This in batch test variation may be considered as a 
variation in testing procedures or a variation in the actual 
concrete strength. The variation in concrete strength in a 
single batch will include the effects of mixer 
inefficiencies. Comparison of samples taken from different 
locations in the mixer may be used to evaluate the variation 
within a single batch. In this study the levels of control 
for within batch tests were divided into three classes with 
corresponding coefficients of variation as follows: 


Excellent Control 

4% 

Average control 

5% 

Poor Control 

6% 


Figure 4.1 illustrates that the standard deviation and 
the coefficient of variation are not a constant for 
different strength levels. Due to this the mean strength 
along with the coefficient of variation is reguired to 
adequately describe the strength variation. The relationship 
between the mean strength and the standard deviation shown 





































56 


in Figure 4.1 was developed using the data from several 
continents. The relationships shown by Murdock, Erntroy and 
Rusch 47 indicate that above a certain value of mean strength 
the standard deviation remains constant while below this 
value the coefficient of variation remains constant. The ACI 
data indicates a constant standard deviation for all 
strength levels. 

The differences in values reported by the different 
sources may be partially explained by the type of data used. 
The specimens of Erntroy and Murdock 47 were 6 in. cubes 
while the ACI specimens were standard cylinders. The data 
reported by Rusch 47 contains test specimens of both types. 
Neville 46 notes that cube tests tend to be more variable 
than cylinder tests. The relationship reported by Erntroy 
and Murdock were based on individual test values whereas the 
ACI values were based on two specimens per test. The Rusch 
data again contains both types of data. 

On the basis of test data available and reported it 
appears that the standard deviation remains constant for 
concrete strengths above a value of 3500 to 4000 psi. and 
the coefficient of variation is constant for strength levels 
below 3500 psi. 

4.1.4 Cylinder Stre ngth vs. Design S tr e ngth 


The average concrete strength required by the ACI 
































































57 


Building Code 3 must exceed the value of the design strength, 
by at least: 

400 psi. if the standard deviation is «300 psi. 

550 psi. if the standard deviation is 300 to 400 psi. 

700 psi. if the standard deviation is 400 to 500 psi. 

900 psi. if the standard deviation is 500 to 600 psi. 

If the standard deviation of the test cylinders exceeds 
600 psi. or if a suitable record of test results is not 
available proportions shall be used which provide an average 
strength 1200 psi. greater than the design strength. After 
test data becomes available the amount by which the average 
must exceed the design strength may be reduced such that the 
probability of a test being 500 psi. below the design 
strength is 1 in 100 and the probability of the average of 
three consecutive tests being below the design strength is 1 
in 100. 

The amounts by which the average strength must exceed 
the design strength in the ACI Code are based on the 
following criteria: 

1. The probability of less than 1 in 10 that a random 
individual strength test will be below f^. 

f = f ? + 1.282a 
cr c 


(4.1) 
















. 









































58 


2. The probability of 1 in 100 that an average of three 
consecutive strength tests will be below f* 


f = f' + 1.343a 
cr c 


(4.2) 


3. The probability of 1 in 100 that an individual 

strength test will be more than 500 psi. below f*. 

c 

f cr = f c + 2 - 326a -500 (4.3) 


where: 

f^ = the design concrete strength 

f c = the average cylinder strength 

a = the standard deviation individual tests 

4. 1,5 I n-situ S tren gth of Con c ret e 

The concrete strength in a structure is not clearly 
defined as some specific multiple of the standard cured 
cylinder strength. Most researchers agree that the strength 
of the concrete in the structure is somewhat lower than the 
standard test cylinder strength. 

Tests by Petersons 48 on columns under well controlled 
laboratory conditions suggest that the strength of the 
concrete in the structure ranges from 90% to 70% of the 
standard cylinder strength. Bloem 12 suggests the strength of 
the concrete in columns is 80% of the standard cylinder 



















































































5 9 


strength for all but the top 10 in. of the column. Allen's 5 
study of beams failing in flexure suggests the strength of 
concrete in the cases of compression failure to be 90% of 
the cylinder strength. Table 4.2 gives the average ratios of 
core strengths to cylinder strengths from various 
researchers. 

Petersons 46 reviewed the data available on core 
strength as compared to standard cylinder strength and 
concluded that the three most important factors affecting 
the strength of the concrete in the structure are: 

1. The strength level of the concrete” The ratio 
between the strength of the concrete in the structure and 
the standard cylinder strength decreases as the strength 
level increases. 

2. The curing of the concrete” The difference between 
the minimum curing acceptable and good curing can be 
approximated by a factor of 0.9. 

3. The location of the concrete in the structure” Tests 
by several researchers have indicated that the concrete in 
the top foot of columns is weaker than the concrete in the 
remainder of the column. This may be explained by the 
increased water cement ratio at the top due to water 
migration after the concrete is placed. The reduction in 
strength is of the order of 15% of the strength of the 
remainder of the column. 
















































Table 4.2 


Concrete Strength in Structure vs. Cylinder Strength 




Researcher 


Core Strength 

Ratio -- 

Cylinder Strength 


| Kaplan 

—i- 

1 

1 

0.74 



8 

S 

1 

0.96 



1 

\ 

a 

0.90 


1 Petersons 

1 

1 

i 

0.90 



1 

1 

| 

0.88 


| Bloem 

1 

1 

a 

0.83 


| Campbell and Tobin 

1 

! 

1 

0.87 


i 

I 

(1 


1 































































































































61 


The reduction in the concrete strength in the structure 
is partially offset by the requirement that the average 
cylinder strength must be from 700 to 900 psi. greater than 
the design strength to meet existing design codes. Based on 
this observation and on the equations from Allen and Bloem 40 
the mean strength for minimum acceptable curing may be 
expressed as: 

f , „ , s = (0.675f' + 1.1) ksi (a a ) 
c(structure) c 


h.1.6 P robability Model for Concre te S treng th 


The variation in concrete strength was described with a 


normal distribution and a mean value of: 



with a coefficient of variation: 


V 



2 2 
V 2 + V 3 


(4.5) 


(4.6) 


where: 

V = the variation in test cylinder strengths 

1 

V = the variation between real strength of cylinders 

2 

and measured cylinder strengths, "in°test 
variation" 


V = the variation of in*»situ strength relative to 
3 ' 

cylinder strength 












' 























































































62 


The basic cylinder strength variation was taken as 0.15 
with a basic in-test variation of 0.04 and a variation of 
0.10 for differences between in-situ and test cylinder 
strengths. Checks were also made with test cylinder 
variation of 0.10 and 0.20. 

4 .2 Reinforcing Steel V ar iability 

The variability of the strength of the reinforcing 
steel was described with a normal distribution as well as a 
modified log-normal distribution. The complete discription 
of the reinforcing steel strength distribution used is given 
in Appendix A. 

4^3 Cross Sectio n Dimensional Variability 

Introduction 

Geometric imperfections are the variations in the 
dimensions, shape, lines, grades and surfaces of as-built 
structures compared to the specified values. Variations in 
cross section dimensions, verticality of columns, 
misalignment and intial curvature of columns are inevitable 
in structures. Geometric imperfections arrise during each 
phase of the construction process. Variations in the size 
and shape are particularly dependent on the size, shape and 
quality of the forms used for manufacture. Setting out and 
assembly affect the geometry of the overall structure and 































































































63 


are dependent on construction techniques and construction 
and inspection personnel. 

Data from field measurements of imperfections is needed 
for various purposes such as for the evaluation of specified 
tolerances, construction performance and theoretical 
probability models. It is important that data be collected 
which is complete and uniform. Unfortunately at present 
there is not a uniform method of collecting and reporting 
this data. Without some degree of standardization it is 
difficult to compare the results of measurements made by 
various investigators with any degree of reliability. 

4 .3.2 P robabil ity Model for Cross Section Dime nsio ns 

The variation in column cross section dimensions has 
been reported by Tso and Zelman 69 . Their results are 
summarized in the histogram in Figure 4.2. The dimensional 
measurements were made to the nearest 1/4 in. in conjunction 
with a study of the strength variation in concrete. The data 

is based on 299 columns from 8 buildings. The nominal 

« 

dimensions ranged from 12 in. to 30 in.. Usually two 
measurements were made at each of five levels over the 
storey height of the column. The mean variation was found to 
be + 0.06 in., that is, the width or thickness averaged 0.06 
in. larger than the specified value with a standard 
deviation of C.28 in.. Tso and Zelman*s 69 measurements 
indicate the distribution of dimensional variations to be 






















' 































64 



- 1.0 - 0.5 0 0.5 1.0 1.5 


VARIATION (in) 


Histogram of Cross Section Dimensional 
Variation Reported by Tso and Zelman 


Figure 4.2 



























































































































































65 


normal. 

The variation in the dimensions of one size of column 
has been reported by Hernandez and Martinez 28 . Their results 
are summarized in the histogram in Figure 4.3. The 
measurements were made at five levels over the storey height 
of the column. At each level the width and thickness were 
measured at each face and at the centre line of the column, 
seventeen columns were studied with a nominal cross section 
of 11.811 in. (30 cm.) by 19.685 in. (50 cm.). A mean 
variation was found to be + 0.15 in., that is, the width or 
thickness averaged 0.15 in. larger than the specified value 
with a standard deviation of 0.157 in.. A normal 
distribution also describes the variation in column 
dimensions reported by Hernandez and Martinez 28 . 

Fiorato 23 has reported a mean deviation of 0.0118 in. 
(0.3 mm.) to 0.276 in. (7.0 mm.) with a standard deviation 
ranging from 0.063 in. (1.6 mm.) to 0.154 in. (3.9 mm.) for 
precast beams and columns ranging in size from 7.87 in. (200 
ram.) to 23.62 in. (600 mm.). These values are based on a 
collection and comparison of published data from field 
measurements, primarily from Sweden. These values may not be 
considered comprehensive but do give an indication of the 
variations which may occur in prefabricated structures. 

As stated earlier, due to inconsistencies in measuring 
and reporting techniques, comparison of data on column 
dimensions from various researchers is difficult. The 










































































FREQUENCY % FREQUENCY 


66 




SIZE (cm) 


o 

o 

CM 

OO 

■cj 

o 

o 

CM 

OO 


o 

«■> 

tv 

O 

CM 

in 

OO 

o 

CO 

CO 

OO 

r— 

CS 

<> 

o 

o 

d 

o 



i-C 

r-^ 

CM 

CM 

CM 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 


Figure 4.3 


Histograms of Cross Section Dimensional 
Variation Reported by Hernandez and Martinez 


































































































67 


majority of researchers are interested in actual 
construction tolerances in which a maximum and minimum 
tolerance is reported rather than a mean value and its 
standard deviation. 

In this study a normal distribution was used to 
describe the variation in column dimensions with a mean 
value of + 0.06 in. and a standard deviation of 0.28 in.. 
Tso and Zelman's 69 results were used since they are based on 
North American data and are based on a larger sample size 
than that obtained by Hernandez and Martinez 28 . 

4.4 Reinforcin g Steel Placement V ariabil ity 

Redkop 53 has developed models describing the error in 
placing reinforcing steel in rectangular tied columns based 
on test data from measurements on several columns in several 
buildings. He describes the variation in steel placement 
with respect to the specified cover for the steel in the 
exterior layers and the specified position for the interior 
steel. The error in steel placement may be described by the 
normal probability distribution. Redkop 53 observed that the 
placement error was a function of the column size as well as 
construction practices. Since statistical data available 
does not suggest a complicated relationship, a linear 
relationship between column size and placement error was 
assumed with a normal distribution of scatter. 


Based on Redkop 1 s 53 data the error in placement of the 






.*■ 











■ 

















68 


interior steel may be described with: 

e = 0.04 in. 
n 

a = 0.2035 + 0.0329 h (4.7) 

The placement of the steel in the exterior layers may 
be described with: 

C = C + 0.250 + 0.0039 h (4.8) 

a sp 

a = 0.166 

where: 

e R = placement error of interior steel in inches. 

° = standard deviation in inches. 

h = column dimension perpendicular to the neutral 
axis. 

c_ = actual cover of exterior steel in inches. 

CL 

c = specified cover of exterior steel in inches. 

S p 

Based on Bedkop's 53 data the mean variation in concrete 
cover of the exterior steel is * 0.315 in., that is, the 
actual cover on the average is 0.315 in. larger than the 
specified cover, with a standard deviation of 0.166 in.. 
Hernandez and Martinez report a mean variaton of + 0.473 in. 
with a standard deviation of 0.13 in.. The smaller standard 
deviation of the Mexican data is due to measurements being 
taken only from one size of column whereas the measurements 



































































* 





































69 


reported by Redkop S3 were taken from various sizes of 
columns. Figure 4.4 is a histogram summarizing the results 
of concrete cover reported by Hernandez and Martinez 17 . The 
normal distribution may be used to describe the variation in 
the concrete cover for both sets of measurements. 

In this study the error in steel placement was 
described by Equations 4.7 and 4.8 with a normal 
distribution. Negative cover is not a problem since with 1 
1/2 in. nominal cover negative cover does not occur before 
the value of cover is the mean value minus 10.54 standard 
deviations. The probability of the value of cover being the 
mean value minus eight standard deviations is approximately 
6.22x10*6. 





































































70 



COVER (cm) 


Figure 4.4 


Histogram of Variation in Concrete Cover 
Reported by Hernandez and Martinez 





















































































































































CHAPTER V 


THE MONTE CARLO STUDY 

5. 1 Siz e of Columns and R einforcement Stu died 

For this study the size of columns and reinforcement 
selected was based on a limited study of columns in existing 
structures. A column take-off was done on five buildings 
including a high rise office building, a parking garage, a 
university building, a hospital and an industrial type 
building. 

Figure 5.1 is a histogram of the frequency of column 
size vs. column size. This histogram indicates that the 
majority of columns are 24 in. or smaller. The high 
percentage of columns in the 52 in. to 56 in. range is due 
to the small number of buildings studied in which one was a 
high rise with large columns throughout. From the histogram 
of column sizes the 12 in. column and the 24 in. column were 
taken as representative of the smaller and larger sizes of 
columns. 

Representative reinforcing steel percentages were 
chosen in the same manner as the column sizes. Figure 5.2 is 
a histogram of the reinforcing steel percentage used in all 
columns. Figures 5.3 through 5.5 are histograms of steel 
percentages used in the various sizes of columns. From these 
histograms it can be seen that the most commonly used steel 
percentage ranges from 1% to 3%» Based on these histograms a 


71 






































50 



SIZE OF COLUMN (in) 


Figure 5.1 


Histogram of the Frequency of Column Sizes 
Column Size 


vs. 


























































































73 



% STEEL 


Figure 5.2 


Histogram of the Percentage of Feinforcing 
Steel in All Columns 





















































































































































































7 4 



Figure 5.3 


Pistoqram of the Percentage of Reinforcing 

Steel in Columns Less Than 16 in. 



























































75 


0 



1.5 


3.0 4.5 

% STEEL 


6.0 


7.5 


Figure 5.4 Histogram of the Percentage of Feinforcing 

Steel in Columns 16 in. to 24 in. 






















































































































76 



% STEEL 


_L_ 

6.0 


Figure 5.5 


Histogram of the 
Steel in Columns 


Percentage of Reinforcing 
24 in. to 36 in. 


75 















































































77 


steel percentage of 1% was chosen for a lower limit and a 
steel percentage of 3% was chosen for an upper limit. 

The final column cross sections chosen are shown in 
Figure 5.6. The basic column was 12 in. by 12 in. with a 
nominal steel percentage of 1%. A 24 in. by 24 in. column 
was chosen to have a low variability of strength with a 
nominal steel percentage of 3%. The nominal or designer's 
concrete and steel strengths were 3000 psi. and 40000 psi. 
respectively. These strengths and properties were chosen to 
get an estimate of the upper and lower bounds of the 
variabilities. 

Interaction diagrams for the two sections are presented 
in Figures 5.11 and 5.12 and will be dicussed more fully in 
Section 5.4. The balanced eccentricity, e^/h, was 0.4 for 
the 12 in. column and 0.5 for the 24 in. column. The columns 
are fully described in Appendix B with their nominal 
properties and the mean values and standard deviations of 
the variables affecting column strength. 

5^2 Size of Sample Stud ied 

For this study a sample size was required which would 
give reasonable results compared to a large sample size 
without using an excessive amount of computer time. Sample 
sizes of 1000, 50C and 200 were used to determine the 

smallest practical sample size. 












































78 



4# 6 bars 

p = 1.22% 


#3 @ 12 in. 


■< 


24 in. 





12# 11 bars 
p = 3.25% 


#4 @ 12 in. 


Figure 5.6 


Final Column Cross Sections Studied 























































































79 


Figure 5.7 is a plot of the mean value of Ptheory/PACI 
vs. e/h for each sample size. The mean value is practically 
independent of the sample size so that any of the sample 
sizes could be used to determine the mean value of 
Ptheory/PACI. 

Figure 5.8 is a plot of coefficient of variation of the 
ratio Ptheory/PACI vs. e/h for the three sample sizes. The 
coefficient of variation for the sample size of 500 is 
practically the same as the coefficient of variation for a 
sample size of 1C00 over the range of e/h less than 1.0. 
Since a good correlation was found between the sample size 
of 500 and the sample size of 1000 below an e/h value of 1.0 
the sample size of 500 was acceptable when the mean and 
coefficient of variation were needed as output. 

Figure 5.9 is a plot of the coefficient of skewness of 
the ratio of Ptheory/PACI vs. e/h for the three sample 
sizes. The coefficient of skewness for a sample size of 500 
is not significantly different from the coefficient of 
skewness for a sample size of 1000. 

Tables 5.1 through 5.4 are tables of comparison of the 
mean values, coefficients of variation, coefficients of 
skewness and kurtosis of the ratio of Ptheory/PACI for 
sample sizes of 200, 500 and 1000. 

All the tables and graphs of comparison indicate no 
meaningful increase in accuracy in using a sample size of 





















































































MEAN VALUE OF 


8 C 


1.35 


1.30 

0 

CL< 


£ 1.25 
O 



Mean Value of the Ratio Ptheory/PACI vs. e/h 
for Sample Sizes of 200, 500 and 1000 for a 12 
in. Sguare Column and Modified Log-normal Steel 
Strength Distribution 


Figure 5.7 

















































































































































































COEFFICIENT OF VARIATION 


81 


16 



e/h 


Figure 5.8 Coefficient of Variation of the Patio 

Ptheory/PACI vs. e/h for Sample Sizes of 200, 
500 and 1000 for a 12 in. Square column and 
Modified Log-normal Steel Strength Distribution 


























































































































































COEFFICIENT OF SKEWNESS 


82 


1.3 


1.1 



Coefficient of Skewness of the Patio 
Ptheory/PACI vs. e/h for Sample Sizes of 2C0, 
500 and 1000 for a 12 in. Square Column and 
Modified Log-normal Steel Strength Distribution 


Figure 5.9 













































































83 


Table 5.1 


Comparison of the Mean Value of the Ratio Ptheory/PACI 
Sample Sizes of 290, 500 and 1000 




Sample 


\ Size 
\ 


e/h \ 


200 




---- 

| 

-1- 

| 


— r 

i 


1 


0.0 

| 1.22535 

| 

1 

1 

1.22497 

i 

i 

a 

1.22010 



0.05 

I 

| 1.17705 

1 

1 

s 

1.17698 

i 

i 

i 

1.17245 



0.10 

1 

! 1.15581 

a 

1 

1 

| 

1.15575 

i 

i 

i 

1.15143 



0.15 

i 

| 1.13981 

1 

i 

I 

1.13985 

i 

i 

1.13559 



0.20 

1 

J 1.12921 

1 

! 

i 

1.12937 

i 

i 

a 

1.12505 



0.30 

| 1.12255 

I 

1 

1 

1.12239 

i 

i 

a 

1.11790 



0.40 

1 

| 1.12228 

1 

1 

1 

1.12039 

i 

i 

1.11645 



0.50 

1 

| 1.06353 

1 

i 

i 

1.06030 

i 

i 

i 

1.05756 



0.60 

\ 

1 1.04519 

1 

1 

1.04139 

I 

s 

i 

1.03926 



0.70 

1 

| 1.03560 

s 

1 

i 

1.03100 

i 

i 

1.02921 



0.80 

1 

| 1.03162 

l 

1 

a 

1.02598 

1 

! 

g 

1.02434 



0.90 

f 

| 1.03116 

1 

1 

1 

1.02439 

I 

1 

■ 

1.02278 



1.00 

1 

$ 1.03285 

l 

1 

1 . 02498 

i 

1 

■ 

1.02334 



1.50 

I 

| 1.05147 

! 

1 

g 

1.03976 

• 

1 

g 

1.03766 



OxD 

1 

| 1.02563 

1 

1 

i 

1.01598 

i 

1 

i 

1.01551 


1 


1 

_j-—- 

i 

(— 


1 

—i— 


i 


500 


1000 


for 































































































84 


Table 5.2 


Comparison of the Coefficient of Variation of the Patio 
Ptheory/PACI for Sample Sizes of 200 r 500 and 1000 


r 


T 


T 







Sample 


\ Size 
\ 

% 

e/h' \ 


200 


500 


1000 


I—-— 


0.0 

l 

| 0.15521 

1 

0.05 

1 

| C.15230 

| 

0. 1C 

| 0.14802 

0. 15 

1 

| 0.14625 

0.20 

i 

| 0.14394 

g 

0.30 

1 

| 0.13696 

0.40 

I 

J 0.11955 

j 

0.50 

9 

[ C. 10186 

0.60 

I 

| 0.09741 

0.70 

I 

5 0.0.840 

0.80 

1 

1 0.10085 

0.90 

I 

j 0.10313 

I 

1.00 

1 

\ 0.10470 

a 

1.50 

1 

j 0.10497 

CO 

I 

J 0.10744 

II 


i——l 


0. 15491 

1 

1 

0.15549 

0.15159 

1 

1 

e 

0.15234 

0. 14802 

1 

S 

t 

0.14871 

0. 14549 

1 

1 

1 

0.14619 

0.14352 

1 

1 

0.14428 

0. 13693 

s 

! 

0.13798 

0.11908 

1 

1 

1 

0.12018 

0.10190 

1 

1 

9 

0.10117 

0.09705 

S 

1 

i 

0.09620 

0.0.733 

I 

I 

| 

0.09661 

0.09929 

I 

1 

1 

0.09889 

0.10127 

1 

1 

a 

0. 10124 

0.10266 

1 

1 

I 

0.10295 

0.10241 

1 

! 

1 

0.10396 

0.10590 

1 

I 

0.10441 


J______I 























































































































































































































' 










































































































































































85 


Table 5.3 


Comparison of the Coefficient of Skewness of the Patio 
Ptheory/PACI for Sample Sizes of 200, 500 and 1000 


r 


T 


n 


Sample 


Size 

s 

e/V x 

\ 

i 

l 

l 

\ 1 
si 

200 

0.0 

1 

1 

1 

1 

C.24511 

0.05 

1 

1 

1 

0.22741 

0.10 

1 

\ 

0.20932 

0.15 

1 

1 

1 

0.19112 

0.20 

1 

1 

1 

0.17406 

0.30 

! 

\ 

l 

0.13074 

0.40 

1 

1 

I 

0.14910 

0.50 

i 

1 

a 

0.12192 

0.60 

1 

1 

0.33447 

0.70 

1 

1 

1 

0.53488 

0.80 

1 

1 

2 

0.65065 

0.90 

1 

! 

0.70518 

1.00 

1 

S 

0.72474 

1.50 

1 

1 

0.65006 

oo 

1 

1 

0.82210 


I____L 


500 

1000 

j 

0.10136 

0.04325 


0.08661 

0.02044 


0.06424 

-0.00657 


0.C4905 

-0.01670 


0.03912 

-0.01357 


-0.00716 

-0.03286 


-0.02379 

-0.06445 


-0.00978 

0.05817 


0.18334 

0.30627 


0.36708 

0.51101 


0.49013 

0.64057 


0.55422 

0.71160 


0.57855 

0.74623 


0.50968 

0.7278^ 


0.68282 

0.81786 


-- - 

_— 

__i 


Normal and log-normal distributions have coefficients of 
skewness of 0.0 and 0.5 to 1.5 respectively. 













































































































































































% 







































































■ 































































































































































86 


Table 5.4 


Comparison of 

the 

Measure 

of 

Kurtosis 


of the 

Fat 

Ptheory/PACI for Sample Sizes 

of 

200, 500 

and 1000 


i 

■ 


i 

i 


r— 

i 


fl 


-1 

1 

i 

1 

i 

Sample 

1 

! 


i 

i 

■ 


1 

i 

■ 


I 

\ 

i 

k 

1 

1 

1 

L 

\ Size 

\ 

e/h\ 

\ 

\ 

1 

1 

1 

1 

1 

200 

8 . 

1 

1 

1 

1 

i . 

500 

1 

1 

I 

1 

1 

i 

1000 

1 

1 

1 

! 

1 

j 

r 

i 


I 

a 


1 

1 


J 

1 


1 

i 

i 

i 

0.0 

8 

1 

1 

3. 43756 

1 

1 

| 

3.09968 

! 

1 

1 

3.05566 

1 

1 

■ 

i 

i 

i 

0.05 

1 

1 

1 

3.49080 

1 

1 

1 

3.11334 

1 

i 

3.07339 

1 

1 

i 

i 

i 

0. 10 

1 

1 

3.49196 

8 

i 

1 

3.11702 

1 

1 

1 

3.08846 

i 

1 

1 

s 

i 

0.15 

1 

1 

3.46624 

I 

1 

1 

3.10238 

S 

1 

fl 

3.08172 

1 

I 

1 

i 

i 

0.20 

1 

1 

3.42402 

1 

s 

s 

3.07604 

1 

s 

fj 

3.05890 

1 

I 

1 

i 

i 

0.30 

I 

1 

3.41098 

s 

1 

a 

3.03726 

s 

1 

1 

3.03096 

1 

1 

i 

i 

i 

0.40 

s 

I 

1 

3.39169 

i 

1 

g 

3.13717 

1 

1 

1 

3.16894 

1 

1 

i 

! 

1 

0.50 

1 

1 

3.45578 

I 

1 

3.18934 

1 

1 

3.30727 

1 

1 

1 

1 

1 

0.60 

5 

S 

3.28053 

1 

j 

I 

3.10547 

1 

1 

3.50337 

1 

! 

i 

1 

I 

0.70 

s 

i 

3.15174 

l 

1 

3.11376 

1 

s 

1 

3.72044 

1 

\ 

■ 

1 

! 

0.80 

1 

1 

3.10245 

l 

1 

1 

3.15105 

1 

1 

1 

3.88468 

! 

i 

■ 

S 

\ 

0.90 

1 

1 

3.09217 

s 

I 

1 

3.16791 

1 

I 

1 

3.99297 

1 

1 

8 

I 

S 

1.00 

1 

( 

3.09422 

8 

I 

fl 

3.15747 

1 

! 

i 

4.06460 

j 

l 

I 

s 

1.50 

\ 

1 

3.07980 

i 

1 

2.96226 

! 

1 

■ 

4.01951 

1 

1 

1 

I 

1 

I 

L_ 

DO 

1 

1 

1 

_ a- 

3.19293 

1 

1 

_j—_ 

3.09842 

1 

! 

1 

_i_ 

4.08547 

1 

1 

J 


A normal distribution has a Kurtosis of 3.0 













































































































‘ 


























- 

















































































































87 


1000 over a sample size of 500. On this basis sample size of 
500 was used for all subsequent calculations. 

5. 3 Results of The Monte Carlo S im ul atio n 


5.3.1 G eneral 

In this Monte Carlo study the relationship between the 
theoretical axial load-moment interaction diagram and the 
ACI axial load-moment interaction diagram was determined. 
This relationship was calculated using the Monte Carlo 
Technique to give the mean ratio of the theoretical strength 
divided by the PCI strength along with its standard 
deviation at various e/h values. From the mean ratio, the 
standard deviation and the type of distribution an 
understrength or <{> factor was calculated. 

To aid in the development of an understrength factor 
the effect of the variation in concrete strength, steel 
strength, cross section dimensions and location of the 
reinforcing steel was studied. In addition the effect of the 
type of distribution of steel strength used was studied 
using a normal distribution and a modified log-normal 
distribution. 


5.3.2. The E ffect of Steel S trength Distrib u tio n Use d 


The effect on the strength of the column cross section 






















































































































88 


of the type of distribution of steel strength was studied 
using a normal and a log-normal distribution of steel 
strength. Both types of distribution can be fitted to the 
data on steel strength as shown in Appendix A. Tables 5.5 
through 5.8 are tables of comparison of the mean values, 
coefficients of variation, coefficients of skewness and 
measure of kurtosis of the ratio Ptheory/PACI for 
calculations based on steel strength normally distributed 
and steel strength which follows a modified log-normal 
distribution. 

The distribution assumed for the variation in the steel 
strength did not significantly affect the mean ratio of the 
theoretical strength to the ACI strength as shown in Table 
5.5 but did affect the distribution of the ratio in the 
tension failure region of the interaction diagram. When the 
modified log-normal steel strength distribution was used, 
the distribution of the ratio of theoretical strength to the 
ACI strength approached a log-normal distribution for values 
of axial loads below the balance point. If a normal 
distribution of the steel strength were used, this ratio was 
normally distributed. This is shown by the coefficient of 
skewness given in Table 5.7. For normally distributed steel 
yield strengths the coefficient of skewness remained close 
to zero throughout corresponding to a normal distribution. 
With the log-normal assumption the coefficient of skewness 
approaches 1.0 for tension failures corresponding to a log¬ 
normal distribution, ( See also Figure 5.9 ). The use of the 





























































































89 


Table 5.5 


Comparison of the Mean Value of the Patio Ptheory/PACI for a 
Normal and a Modified Log-normal Steel Strength Distribution 


1— 

1 

1 

1 

1 

1 

1 

-1- 

1 

Distribution| 

\ 1 

x x Type | 

\ i 

e/hv | 

xj 

- —---r~ 

1 

1 

! 

Normal | 

! 

I 

1 

i 

Mod. log-normal 

—i 

i 

1 

II 


-r 

i 

1 

2 



s 

1 

1 

0.0 

i 

i 

1.22495 1 

1.22497 


i 

1 

0.05 

s 

i 

1 

1.17712 | 

| 

1.17698 


1 

1 

0.10 

i 

i 

1 

1.15605 | 

■ 

1.15575 


1 

1 

0. 15 

i 

5 

1 

S 

1.14012 f 

1 

1.13985 


1 

1 

0.2C 

! 

1 

I 

1.12946 | 

• 

1. 129 37 


1 

1 

0.30 

1 

3 

1 

1.12212 | 

1.12239 


1 

1 

0.40 

! 

! 

1 

1.12012 | 

1 

1.12C39 


1 

1 

0.50 

! 

1 

i 

1.05996 | 

a 

1.06030 


1 

1 

0.60 

f 

1 

! 

1.04163 f 
« 

1.04139 


1 

1 

0.70 

1 

i 

i 

1.03157 | 

* 

1.03100 


1 

S 

0.80 

3 

1 

1 

1.02663 | 

1.02598 


1 

1 

0.90 

i 

\ 

s 

1.02495 S 

1.02439 


1 

l 

1.00 

s 

! 

1 

1.02535 | 

i 

1.02498 


i 

s 

1.50 

1 

1 

1 

1.03889 | 

■ 

1.03976 


3 

1 

oo 

1 

i 

l 

1.01677 | 

| 

1.01598 


1 


1 

_—L-- 

1 

1- 


_i 



































































































































































































































































































G0 


Table 5.6 


Comparison of the Coefficient of Variation of the Patio 
Ptheory/PACI for a Normal and a Modified Loq-normal Steel 
Strength Distribution 

i—-1--—-1—---1 

lli I 

I Distribution! I I 


\ 


1 

x x Type 

e/h v ^ 

\ 

! 

1 

1 

1 

\i 

Normal j 

1 

( 

1 

• 

Mod. Log~-normal 

I 

1— 

0.0 

T 

! 

1 

A 

! 

0.15464 \ 

0.15441 

I 


0.05 

1 

1 

I 

I 

0.15189 S 

1 

0.15159 



0.10 

1 

I 

1 

0.14831 1 

0.14802 



C. 15 

1 

1 

1 

1 

0.14576 S 

0.14549 



0.20 

! 

1 

1 

0.14378 | 

6 

0.14352 



0.30 

! 

1 

1 

0.13729 S 

| 

0.13693 



0.40 

1 

\ 

1 

0.12078 | 

0.11908 



0.50 

1 

1 

1 

0.10391 | 

0.10190 



0.60 

\ 

1 

! 

0.10051 \ 

■ 

0.09705 

I 


0.70 

! 

1 

! 

0.10183 S 

SI 

0.09733 



0.80 

1 

! 

1 

0.10437 I 

0.09929 


I 

0.90 

! 

1 

1 

0.10658 | 

0.10127 



1.00 

1 

1 

5 

0.10798 | 

i 

0.10266 



1.50 

1 

i 

« 

0.10689 | 

9 

0. 10241 



oo 

1 

i 

1 

0.10807 | 

I 

0. 10590 


!_ 


1 

i — 

! 

L 


i 








































































































































































































































































































91 


Table 5.7 


Comparison of the Coefficient of Skewness of the Fatio 
Ptheory/PACI for a Normal and a Modified Log-normal Steel 
Strength Distribution 


I 


k 


Distribution 


\ 



X \ Type | 

1 

e/h \ | 

\ I 

Normal 

1 

1 

! 

1 

< 

Mod. Log-normal 

i 



a 


1 

1 


1 


0.0 

i 

i 

■ 

0.09843 

1 

i 

0.10136 



0.05 

I 

i 

i 

0.08254 

I 

0.08661 


1 

0.10 

i 

i 

a 

0.05519 

1 

i 

0.06424 



0.15 

i 

1 

■ 

0.03722 

i 

1 

| 

0.04905 



0.20 

i 

1 

0.02797 

1 

1 

1 

0.03912 



0.30 

! 

1 

-0.00223 

s 

1 

l 

-0.00716 

! 


0.40 

1 

1 

-0.04916 

1 

! 

a 

-0.C2379 



0.50 

! 

1 

-0.10565 

! 

! 

1 

0.00978 

1 

a 


0.60 

i 

1 

—0 . 08048 

1 

S 

< 

0.18334 

1 


0.70 

I 

1 

-0.03075 

1 

i 

0.36708 

i 


0.80 

s 

i 

0.00709 

1 

1 

| 

0.49013 

1 


0.90 

i 

i 

0.02321 

1 

s 

1 

0.55422 



1.00 

i 

i 

0.02142 

1 

1 

1 

0.57855 



1.50 

s 

i 

-0.03508 

1 

1 

1 

0.50968 



oo 

1 

! 

0.04325 

1 

1 

1 

C.68282 


1_ 


1 

j— 


! 

1- 


i 

























































































































































































92 


Table 5.8 

Comparison of the Measure of Kurtosis of the Patio 
Ptheory/PACI for a Normal and a Modified Log-normal 
Distribution of Steel Strength 


I 




Distribution 


Type 

\ 

e/h \ ^ 

\ 

1 

1 

1 

1 

Normal 1 

1 

1 

1 

i 

Mod. Log 

0.0 

8 

! 

l 

1 

3.10221 | 

fl 

3.09968 

0.05 

1 

1 

0 

1 

3.11561 | 

fl 

3.11334 

0.10 

1 

1 

S 

3.12425 1 

fl 

3.11702 

0.15 

8 

! 

1 

1 

3.10326 I 

3.10238 

0.20 

1 

! 

1 

8 

3.06481 $ 

i 

3.07604 

0.30 

1 

1 

1 

3.04098 J 

3.03726 

0.40 

1 

1 

1 

3.13422 | 

1 

3.18301 j 

fl 

2.13717 

0.50 

1 

I 

3.18934 

0.60 

1 

i 

i 

2.95718 j 

3.10547 

0.70 

8 

8 

i 

2.81651 f 

6 

3.11376 

0.80 

! 

i 

S 

2.76677 j 
! 

2.74429 | 

fl 

3.15105 

0.90 

1 

1 

3.16791 

1.00 

1 

8 

1 

2.72148 8 

i 

3. 15747 

1.50 

1 

8 

! 

2.65068 | 

i 

2.96226 

oo 

1 

1 

1 

2.69439 | 

3.09842 


































































93 


modified log-normal steel strength distribution resulted in 
a larger 4> factor at the 1% level of probability of failure 
than that for the normal distribution of steel strength in 
the tension region of the interaction diagram. The 
calculation of this term is discussed in Section 5.5. 

The type of steel strength distribution used did not 
significantly affect the calculated value of the <j> factor at 
the 1% level of probability of failure in the compression 
failure region of the interaction diagram. The distribution 
of the ratio of the theoretical strength to the ACI strength 
in the compression failure region followed a normal 
distribution for both types of steel strength distribution. 
This may be explained by the failure in the compression 
region being dependent on the concrete strength rather than 
the steel strength. 


5 .3. 3 Th e Effec t of The Conc re te Str en gth Va ri ation 

The effect of the coefficient of variation of the 
concrete cylinder strength was studied by keeping all other 
variables at their mean values and using a cylinder strength 
variation of 10%, 15% and 20%. The overall coefficients of 
variation of in=situ strength were 13.6%, 17.6% and 22% as 
computed per Egn. 4.6 in Section 4.1.o. Tables 5.9 through 
5.11 are tables of comparison the mean values, coefficients 
of variation and skewness of the ratio Ptheory/PACI for 
various values of e/h and cylinder strength coefficients of 
































































































94 


Table 5.9 


Comparison of the Mean Value of the Patio Ptheory/PACI for 
Concrete Cylinder Strength Coefficients of Variation of 10%, 
15% and 20% 


! 

(^Coefficient of 
\ 

\ Variation 

\ 

e/h ^ ^ 


10 % 



o 

• 

o 

1 

1 

1.18146 

—l— 

1 

1 

| 

1.22622 

“T 

1 

1 

1 

— 

1.28386 


0.05 

! 

1 

1.13652 

1 

1 

I 

1.17774 

1 

s 

a 

1.23088 


0.10 

1 

1 

1.11642 

I 

! 

a 

1.15546 

! 

I 

A 

1.20598 


0.15 

i 

I 

a 

1.10207 

1 

l 

| 

1 . 14 005 

i 

1 

I 

1.18897 


0.20 

i 

! 

1.09345 

i 

1 

1.13111 

1 

s 

1 

1.17910 


0.30 

S 

1 

1.09010 

! 

1 

1 

1.12616 

I 

l 

1.17157 


C. 40 

! 

s 

1.08946 

§ 

1 

■ 

1.11784 

s 

! 

a 

1.15384 


0.50 

1 

\ 

1.04264 

1 

1 

l 

1.06022 

s 

i 

■ 

1.08408 


0.60 

! 

1 

1.03276 

1 

! 

1.04554 

i 

! 

1.06304 


0.70 

! 

1 

1.02775 

I 

1 

1.03794 

1 

I 

s 

s 

I 

1.05140 

I 

0.80 

\ 

1 

1.02602 

1 

I 

1 . 03474 

1.04549 


0.90 

i 

i 

1.02646 

! 

1 

1 . 03426 

1 

f 

1 

1.04316 


1.00 

! 

I 

1.02829 

1 

I 

i 

1.03544 

1 

I 

1 

1.04310 


1.50 

! 

1 

1.04443 

1 

1 

B 

1 . 04915 

1 

I 

1 

1.05515 

s 

oo 

I 

I 

1.01756 

\ 

! 

1 . 02138 

1 

! 

1.02887 

! 

t _ 


I 

L- 


i 

i— 


1 

8— 

j 


15 % 


20 % 




























































































































































































































95 


Table 5.10 


Comparison of the Coefficient of Variation of the Ratio of 
Ptheory/PACI for Concrete Cylinder Strength Coefficients of 
Variation of 10%, 15% and 20% 


I 


k c 


oefficient of 


x Variation 
\ 

\ 

e/h x 


10 % 


I— 

0.0 

-.~ 

1 

1 

a 

0.11363 

—i— 

J 

S 

0.14790 

—r~ 

\ 

1 

I 

0 . 18607 


0.05 

I 

i 

a 

0.11066 

{ 

1 

g 

0.14377 

1 

1 

0.18100 

1 

0 . 10 

i 

i 

0.10804 

I 

1 

| 

0.14022 

! 

I 

9 

0.17669 


0 . 15 

i 

i 

0 . 1 C 695 

1 

1 

a 

0 . 13842 

1 

I 

1 

0.17369 


0.20 

i 

i 

0.10667 

i 

j 

i 

0 . 13762 

f 

! 

• 

0.17215 


0.30 

i 

i 

0.10229 

j 

i 

a 

0.13189 

i 

! 

1 

0.16487 


0.40 

! 

f 

0.08303 

S 

1 

0.10936 

I 

1 

0.13953 


0.50 

1 

l 

0.05665 

1 

I 

0.07589 

1 

S 

0.10000 

0.07803 


0.60 

1 

1 

0.04427 

1 

S 

l 

0.05873 

1 

1 

I 

1 


0.70 

1 

! 

0.03672 

1 

1 

i 

0.04840 

0.06355 


0.80 

1 

S 

0.03191 

II 

1 

0.04191 

1 

s 

1 

0.05381 


0.90 

1 

1 

0.02881 

1 

1 

a 

0.03789 

I 

1 

g 

0.04742 


1.00 

i 

1 

0.02679 

i 

i 

a 

0.03551 

1 

i 

0.04337 


1.50 

1 

1 

0.02844 

i 

s 

• 

0.03400 

1 

1 

s 

0.03872 


oo 

! 

! 

0.02689 

i 

1 

2 

0.03184 

I 

s 

1 

0.03738 



! 

j 


1 

L 


1 

1 

j 


15 % 


20 % 

















































































































































































































































































































96 


Table 5.11 


Comparison of the Coefficient of Skewness of the Ratio of 
Ptheory/PACI for Concrete Cylinder Strength Coefficients of 
Variation of 10%, 15% and 20% 


r 


i 


Coefficient of 

s. 

\ 

v ^Variation 

e/h ^ \ 

\ 

i-— 

1 

1 

1 

| 10% 

1 

1 

1 

si 

—r~ 

1 

1 

1 

1 

! 

1 

1 

i. 

15% 

r~-—— 

20% 

0.0 

! 

1 

I 0.02523 

l 

1 

1 

0.02478 

0.02786 

0.05 

1 

1-0.01578 

| 

1 

! 

| 

-0.01428 

0.00366 

0. 10 

1 

f-0.04927 

i 

1 

i 

-0.03626 

-0.00627 

0. 15 

S 

1-0.05004 

I 

5 

1 

-0.04489 

-0.02282 

0.20 

I 

1-0.02722 

1 

1 

1 

-0.04231 

-0.04249 

0.3C 

1 

1-0.01351 

1 

1 

-0.05370 

-0.07949 

0.40 

1-0.17206 

1 

1 

8 

-0.15911 

-0.17109 

0.50 

I 

1-0.21277 

I 

1 

i 

-0.22505 

-0.23879 

0.60 

i 

1-0.38463 

1 

S 

-0.30449 

-0.35791 

0. 70 

1 

j-C. 50581 

1 

1 

i 

-0.38219 

-0.50237 

0.80 

1 

1-0.56889 

I 

1 

fl 

-0.43285 

-0.62403 

0.90 

1 

|-0.55617 

1 

1 

8 

-0.41673 

-0.67631 

1.00 

1 

J-0.47411 

1 

i 

1 

-0.32853 

-0.65730 

1.50 

1 

| 0.09033 

1 

1 

■ 

-0.05914 

-0.51679 

OxO 

1 

j 0.26793 

s 

1 

1 

0.41395 

0.39448 


1 

.j-- 

I 

L. 


j- 


-i 





























































































































































































' 




























I 






























































97 


variation. 

The mean value of the ratio Ptheory/PACI increased in 
the compression failure region of the interaction diagram 
with increasing cylinder strength coefficient of variation. 
This increase may be explained by the increased mean 
concrete strength required by ACI 318-71 Section 4.2 to 
account for the increased coefficient of variation. There 
was no significant increase in the theoretical strength in 
the tension region due to the increased mean concrete 
strength. Again this may be explained by the compression 
failures depending on the concrete strength and the tension 
failures depending on the steel strength. 

As the coefficient of variation of the concrete 
cylinder strength was increased the overall coefficient of 
variation of the ratio Ptheory/PACI increased. Again the 
increase in overall coefficient of variation was greater in 
the compression region of the interaction diagram where the 
concrete strength has more effect on the cross section 
strength than in the tension failure region. 


5^3_j_4 The Effe ct of Th e Variables Stud ied 

The effect of the variation in concrete strength, steel 
strength, cross section dimensions and location of the 
reinforcing steel was determined for the 12 in. by 12 in. 
cross section with a nominal steel percentage of 1%. A 


















































































































98 


coefficient of variation of concrete cylinder strength of 
15% was used for this study. Each variables effect was 
studied with all other variables at their mean value. 

Figure 5.1C gives a graphical representation of the 
overall variation in cross section strength for various e/h 
values for each of the variables and for all the variables 
combined. The plot of standard deviation squared vs. e/h 
indicates the major component causing variation in cross 
section strength in the compression region of the 
interaction diagram is the variability in the concrete 
strength. The effect of the variability in the concrete 
strength becomes minimal in the tension failure region. This 
may be explained by the fact that the full strength of the 
concrete is not utilized in the tension failure region such 
that the high concrete strengths have no effect on the 
variability of the cross section strength. 

The effect of the variability in the steel strength on 
the overall cross section strength variability is greater in 
the tension region where the steel strength controls the 
cross section capacity. The effect of the steel strength 
variability in the compression failure region is again 
minimal due to the concrete strength being the controling 
factor. 


The effect of the variability of the concrete strength 
and the steel strength are about the same at the balance 
point. This is to be expected since the failures in a 




























































STANDARD DEVIATION SQUARED 


99 


0.035 


0.030 


0.025 - 


0.020 - 


0.015 - 


0.010 


0.005 - 


0 



0 0.3 0.6 0.9 1.2 1.5 


e / h 


Ptgure 5 . IQ Standard Deviation Squared of the Patio 

Ptheory/PACI vs. e/h for the Variables 
Affecting Column Strength for a 12 in. Square 
Column and Modified Log-normal Steel Strength 
Distribution 









































































IOC 


randomly selected sample would depend on the concrete and 
steel strength equally at the balance point. 

The effect of the variability in the cross section 
dimensions and the location of the steel was very small for 
both compression and tension failures. The most significant 
effect occured for the cases of pure axial load and pure 
moment. 

The total variability in the cross section strength may 
be closely approximated by the expression: 

V = V Z + V + V / (5.1) 

t tc ts td 

where may be the total standard deviation or coefficient 

of variation and V , V to and V td are the standard deviation 
or coefficient of variation of the cross sectional strength 
if only the concrete strength, steel strength or the 
dimensions are varied separately. 

5.4 Cross Sect ion Stren gth 

Figures 5.11 and 5.12 are plots of the interaction 
curves for the 12 in. and 24 in. columns based on a modified 
log-normal distribution of steel strength. The mean strength 
indicated is the mean strength calculated from a sample size 
of 500 using the Monte Carlo Technique and the theoretical 
calculation of cross section strength. The maximum and 
minimum strength curves are also calculated from the Monte 
Carlo calculations. The ACI ultimate strength is the cross 



































































































1000 


101 



(sd !>|) avoi 1VIXV 


Figure 5.11 Dispersion of Strengths of an Eccentrically 

Loaded 12 in. Square Column 


MOMENT (ft. - kips) 




































































4000 


102 



(sd!>i) avoi 1VIXV 


Figure 5.12 


Dispersion of 
Loaded 24 in. 


Strengths of an Eccentrically 
Sguare Column 


MOMENT (ft.-kips) 





































































































































































103 


section capacity calculated using the ACI 318-71 Building 
Code. The ACI factored strength is the ACI ultimate strength 
divided by 1.4/0.7 corresponding to the lowest possible load 
factor. The ACI factored strength corresponds to the normal 
service load conditions. The discrepancy in the mean 
strength of the 24 in. column immediately above the balance 
point is due to the reinforcing steel placement in the 
column cross section. As a result of the reinforcing steel 
at the centre of the cross section shifting from compression 
to tension steel the capacity of the section appears to 
increase to a second balance point but the first downturn of 
the curve is not a true representation of the capacity of 
the cross section. 

The dispersion of cross section strength is plotted at 
selected values of e/h. In each case the dispersion of the 
cross section strength is a normal distribution for the 
compression failure region and a log-normal distribution for 
the tension failure region. At the balance point the 
dispersion of cross section strength may be represented 
equally well with either a normal or log=normal 
distribution. 

Table 5.12 is a comparison of the mean value and 
coefficient of variation of Ptheory/PACI for a sample size 
of 5C0 for the cross sections of 12 in. by 12 in. with 1% 
steel and 24 in. by 24 in. with 3% steel. 


In the compression failure region the variation in the 































































. 














































































104 


strength of the 12 in. and the 24 in. columns are similar 
but the mean value of Ptheory/PACI is larger for the 12 in. 
column. This is due to the higher dependence on the concrete 
strength. Since the ratio of the mean concrete strength to 
the nominal concrete strength is higher than the ratio of 
the mean steel strength to the nominal steel strength the 
capacity in the compression region increases for decreasing 
steel percentages relative to the ACI capacity. 

The variability of the theoretical strength of the 24 
in. by 24 in. column is greater than that of the 12 in. by 
12 in. column in the tension region. This may be due to the 
increase in steel percentage from 1% to 3%. Also the mean 
value of the ratio Ptheory/PACI is larger in the 24 in. 
column in the tension region due to the increased steel 
percentage. 

Figures 5.13 through 5.16 are cumulative frequency 
plots of the ratio Ptheory/PACI for the 12 in. and 24 in. 
columns at selected e/h values. A comparison of these and 
similar cumulative frequency plots for a normal and a log¬ 
normal dispersion of the ratio Ptheory/PACI and the data in 
Tables 5.1 to 5.4 and 5.9 to 5.11 suggest that for 
compression failures the dispersion may be represented by 
the normal distribution and for tension failures the 
dispersion may be represented with a log-normal 


distribution. 




























































































































CUMULATIVE FREQUENCY 


105 





0.0 0.4 0.8 1.2 1.6 2.0 


^THEORY / ^CI 


Figure 5.13 Normal Cumulative Frequency Plot of the Fatio 

Ptheory/PACI for the 12 in. Column, e/h = C.10 



































































CUMULATIVE FREQUENCY 


106 



p theory/ ^ci 


Figure 5.14 Log-normal Cumulative Frequency Plot of the 

Fatio Ptheory/PACI for the 12 in. Column, 

e/h = 0.10 



















































































































CUMULATIVE FREQUENCY 


107 





THEORY 


/ 


ACX 


Figure 5.15 Normal Cumulative Frequency Plot of the Ratio 

Ptheory/PACI for the 24 in. Column, Pure Moment 













































































99 . 

99 

98 

95 

90 

80 

70 

60 

50 

40 

30 

20 

10 

5 

2 

1 


1 08 



1.0 10.0 

^THEORY / P ACI 


Log-normal Cumulative Frequency Plot of the 
Batio Ptheory/PACI for the 24 in. Column, Pure 
Moment 









































































109 


Table 5.12 


Comparison of the Mean Value of the Ratio Ptheory/PACI for 
the 12 in. and 24 in. Columns 


r 


T 


T 


12 in. x 12 in. Column 


24 in. x 24 in. Column 




I 

I 



i 




a 





1 


i 

i 


1 


■ 

i 


e/h 

1 

i 

1 

Mean Value 

1 

1 

1 

c.o.v. 

1 

s 

1 

Mean Value 

1 

| C.O.V. 

1 



1 

i 


1 

i 


1 

1 


J 

e 



1 

1 


1 

1 


! 

s 


1 

1 


0.0 

1 

1 

I 

1.22497 

1 

1 

■ 

0.15441 

i 

i 

a 

1.17160 

| 0.13031 


0.05 

1 

i 

8 

1.17698 

1 

1 

I 

0.15159 

i 

i 

i 

1.11903 

1 

| 0.13367 


0.1C 

1 

1 

1 

1.15575 

1 

1 

| 

0.14802 

s 

s 

1 

1.08382 

1 

| 0.14170 

fi 


0.15 

1 

1 

1.13985 

? 

1 

C.14549 

I 

\ 

1 

1.06790 

i 

s 0.14099 

• 


0.2C 

I 

1 

1.12937 

1 

1 

0.14352 

1 

s 

1 

1.07079 

s 

| 0.13601 

a 


0.30 

1 

1 

1.12239 

i 

1 

0.13693 

1 

1 

1 

1.12003 

i 

i 0.11802 

8 


0.40 

s 

i 

1.12039 

I 

1 

l 

0.11908 

1 

1 

1 

1 . 17984 

| 0.11500 

i 


0.50 

1 

f 

1.06030 

! 

1 

s 

0.10190 

1 

1 

1 

1 . 19744 

| 0.10429 

I 


0.6C 

! 

1 

1.04139 

1 

1 

0.09705 

I 

i 

1 

1.13196 

| 0.10118 

8 


0.70 

1 

1 

1.03100 

1 

1 

0.09733 

i 

1 

fl 

1.07567 

\ 0.10021 
| 


0.80 

\ 

\ 

1.02598 

1 

5 

0.09929 

1 

I 

s 

1.07338 

| 0.09812 

1 


0.90 

! 

1 

1.02439 

1 

1 

1 

0.10127 

i 

i 

i 

1.06804 

i 0.09731 

1 


1 . oc 

S 

I 

1.02498 

1 

0.10266 

3 

i 

s 

1.06246 

| 0.09703 

1 


1.50 

i 

i 

1.03976 

I 

! 

0. 10241 

1 

1 

1 

1.04738 

| 0.09961 

s 


oo 

\ 

\ 

1.01598 

l 

\ 

0.10590 

1 

1 

a 

1.04568 

S 0.10978 

I 



1 

_L- 


1 

l 


! 

u 


i 

i 

i 















































































































































































































































































































































































lie 


5. 5 Ca l cula t ion of Factor s 

5*5 . 1 Based on 1_ in J.00 Under st rengt h 

Tables 5.13 and 5.14 are tables of the calculated <}> 
factor based on a probability of understrength of 1 in 100 
and a normal dispersion of cross section strength in the 
compression failure region and a log-normal dispersion of 
cross section strength in the tension failure region. Figure 
5.17 is a plot of the $ factor for the 12 in. and the 24 in. 
columns vs. e/h based on a probability of understrength of 1 
in 100. The $ factors in the ACI Code are related to a 
probability of understrength of 1 in 100 40 . 

















































Ill 


Table 5.13 


The Understrength Factor for the 12 in. by 12 in. 
Based on a Probability of Understrength of 1 in 100 


r 


T 


l 


e/h 


Mean Value 


Std. Dev. 


$ Factor 






0.0 

\ 

1 

« 

1.22497 

0.05 

1 

1 

1 

1.17698 

0.10 

1 

1 

I 

1.15575 

0. 15 

I 

1 

« 

1.13985 

0.20 

1 

1 

I 

1.12937 

0.30 

I 

1 

a 

1.12239 

0.40 

1 

1 

1.12039 

0.50 

! 

1 

1 

1.06030 

0.60 

I 

1 

i 

1.04139 

0.70 

1 

1 

1 

1.03100 

0.80 

i 

1 

a 

1.02598 

0.90 

s 

I 

1.02439 

1.00 

! 

8 

i 

! 

1 

1.02498 

1.50 

1.C3976 

0^3 

i 

f 

■ 

1.01598 


i_—«- 


0.18915 

\ 0.78 

0.17842 

| 0.76 


I 

0.17108 

1 0.76 

0.16583 

\ 0.75 

0.16208 

8 0.75 

0.15369 

| 0.76 

0.13342 

| 0.81 

0.10804 

1 0.83 

0.10107 

f 0.83 

0.10035 

J 0.8 2 

0.10187 

| 0.81 

0.10374 

I 0.81 

0.10522 

o 

00 

• 

o 

0.10649 

| 0.82 

l 

1 0.79 

0.10759 


j__—_ 


Column 















































































































































































































































































































































































112 


Table 5.14 


The Understrength Factor for the 24 in. by 24 in. Column 
Based on a Probability of Understrength of 1 in 100 


r 




e/h 


Mean Value 


Std. Dev. 


<t> Factor 


1 — - — 

i 

i 


-r~ 

I 


T 

1 


I 0.0 

1 

1 

1 

1.17160 

8 

1 

a 

0.15267 

1 

I 

e 

0.82 

| 0.05 

1 

1 

1.11903 

i 

l 

0.14958 

i 

1 

1 

0.77 

| 0.10 

1 

1 

1.08382 

i 

1 

0.15358 

i 

i 

a 

C .73 

1 0.15 

1 

S 

1.06790 

i 

1 

i 

0.15057 

I 

i 

i 

C . 7 2 

| 0.20 

1 

1 

1 

1.07079 

i 

l 

0.14564 

1 

1 

0.73 

| 0.30 

1 

1 

1. 12003 

i 

1 

I 

0.13218 

I 

1 

0.81 

S 0.40 

1 

\ 

I 

1 . 17984 

i 

l 

i 

0.13568 

J 

i 

0.86 

| 0.50 

i 

1 

1 . 19744 

1 

i 

0.12489 

1 

1 

a 

0.91 

| 0.60 

1 

1 

1.13196 

1 

1 

I 

0.11453 

1 

! 

fl 

0.87 

| 0.70 

( 

1 

1.07567 

1 

1 

0.10779 

1 

! 

1 

0.85 

| 0.80 

1 

1 

1.07338 

1 

0. 10532 

1 

1 

i 

0.85 

S 0.90 

1 

I 

1.06804 

1 

1 

l 

0.10393 

1 

1 

0.85 

I 1.00 

f 

5 

1.06246 

\ 

1 

i 

0.10309 

1 

I 

1 

0.84 

f 1.50 

l 

I 

1.04738 

1 

■ 

0.10433 

1 

J 

1 

0.83 

J Cx^> 

1 

J 

1.04568 

i 

1 

I 

0.11479 

l 

1 

0.82 

i 

! 

L 


1 

-- i~ 


i 

— *- 



j 

















































































































































































































































UNDERSTRENGTH FACTOR 


113 



e/h 


Fiaure 5 17 The understrength Factor 4> vs. e/h Based on a 

Probability of Understrength of 1 in 100 for 
the 12 in. and 24 in. Columns 



































































114 


5.5.2 Based on Cornell-Lind Procedure 


It has been proposed that future code revisions have 


factors based on the equation: 


-8aV RaV 

R y D e R > Uy e P u 
R — u 


(5.2) 


or: 


<j> R > XU 


where: 


a “3aV 

$ = Y R e R 


= the understrength factor 


> _ e«v 

A = Y u e u 


- the overload factor 


y R = Ptheory/PACI 


8 = safety index 


= 3.5 for probability of failure of 1.1 x 10 ~ 4 


= 4.0 for probability of failure of 3.2 x 10 


-s 


V = the variability of the strength or resistance 

B 


V u = the variability of the loads 


a = 0.75 




































































































































































115 


The a value is used to allow the separation of the 
effects of the variability of the member strength and the 
variability of the member loading. 

Tables 5.15 and 5.16 are tables of the <p factor for the 
12 in. and 24 in. columns based on the above equation. In 
these tables the $ factors are based on values of 6 of 4.0 
for compression failures and 3.5 for tension failures. The 
lower probability was used for the compression failures due 
to the sudden brittle mode of failure. 

The 12 in. square column cross section was chosen to 
display a large variability. For this column, <t> was 
G.78±0.03 throughout the entire range of e/h studied. The 24 
in. column was chosen to have a low variability. For this 
column, was 0.79±0.09. The large variability in the 24 in. 
column was due to the discrepancy in the theoretical 
strength discussed in Section 5.4. 























































































116 


Table 5.15 


The Understrength Factor for the 12 in. by 12 in. Column 

„ , , -3aV„ 

Based on cf) = ye R 

K 


r 


L 


e/h 

T— 

1 

\ 

I 

J 

Mean 

—!— 

1 

1 

1 

C.O.V. 

—i— 

1 

I 

1 

i 

$ Factor 

0.0 

1 

J 

1 

I 

1.22497 

1 

1 

1 

1 

0.15441 

i 

! 

1 

fl 

0.77 

0.05 

1 

1 

1 

1. 17698 

! 

1 

■ 

0.15159 

1 

i 

9 

0.75 

0. 10 

1 

1 

fl 

1.15575 

1 

1 

1 

0.14802 

! 

I 

1 

0.74 

0. 15 

1 

I 

1 

1.13985 

J 

1 

fl 

0.14549 

\ 

\ 

fl 

0.74 

0.20 

I 

! 

1.12937 

s 

! 

0. 14352 

1 

s 

fl 

0.78 

0.30 

1 

f 

1.12239 

? 

1 

fl 

0.13693 

1 

1 

1 

0.78 

0.40 

I 

1 

1.12039 

1 

1 

0.11908 

§ 

1 

0.78 

C. 50 

I 

1 

1.06030 

f 

1 

0.10190 

1 

1 

0.81 

0.60 

1 

1 

1.04139 

1 

I 

0.09705 

I 

1 

i 

0.81 

0.70 

S 

1 

1.03100 

I 

1 

0.09733 

s 

l 

i 

1 

1 

1 

1 

0.80 

0.80 

! 

1 

1.02598 

1 

1 

0.09929 

0.79 

0.90 

i 

1 

1.02439 

s 

I 

fl 

0.10127 

0.79 

1.00 

! 

1 

1.02498 

f 

5 

1 

0.10266 

1 

1 

s 

0.78 

1.50 

i 

1 

1.03976 

1 

1 

| 

0.10241 

1 

s 

■ 

0.80 


1 

1 

s 

1.01598 

1 

1 

1 

0.10590 

i 

i 

i 

0.77 


1 




Avg = 0.779 































































































































































































































































































117 


Table 5.16 


The Understrength Factor for the 24 in. by 24 in. Column 
Based on <j> = y e -|3otV ’ 


R 


R 


e/h 


Mean 


C.O.V. 




a 

— 


—i— 

g 


-1 


0.0 

i 

i 

i 

1.17160 

0.13031 

1 

1 

a 

0.79 



0.05 

1 

i 

i 

1 . 11903 

0.13367 

i 

t 

i 

0.75 



0.10 

i 

i 

1.08382 

0.14170 

i 

i 

a 

0.71 



0.15 

s 

i 

i 

1.06790 

0.14099 

1 

I 

a 

0.70 



0.20 

i 

! 

1.07079 

0.13601 

1 

! 

fi 

0.71 



0.30 

1 

i 

1.12003 

0.11802 

1 

\ 

g 

0.79 



0.40 

1 

1 

1 . 17984 

0.11500 

1 

1 

1 

0.84 



0.50 

! 

1 

1. 19744 

0.10429 

1 

s 

1 

0.88 



0.60 

1 

1 

1.13196 

0.10118 

1 

f 

III 

0.84 



0.70 

1 

\ 

1.07567 

0.10021 

s 

s 

1 

0.83 



0.80 

1 

1 

1.07338 

0.09812 

1 

1 

0.83 



0.90 

1 

I 

1.06804 

0.09731 

1 

s 

1 

0.83 



1.00 

l 

I 

1.06246 

0.09703 

8 

8 

i 

0.82 

a 

1 

1.50 

1 

1 

1.04738 

1.04568 

_ 

0.09961 

0.10978 

-- 

3 

1 

I 

0.81 

i 

1 

O <D 

1 

1 

s 

J — 

8 

i 

5 

j — 

0.78 

i 


<j> Factor 


Avg = 0.794 





















































































































































































CHAPTER VI 


SUMMARY AND CONCLUSIONS 

In this study probability models were developed to 
describe the variability of the major variables affecting 
the strength of a reinforced concrete section. Based on data 
from a literature search the concrete strength, cross 
sectional dimensions and location of reinforcing steel were 
described with a normal distribution as described in Chapter 
IV. The steel was described with a normal and modified log~ 
normal distribution of yield strength as discussed in 
Appendix A. 

A Monte Carlo study was performed using the probability 
models developed to determine the variability in the cross 
sectional strength of a 12 in. square and a 24 in. square 
tied reinforced concrete column. The results of this study 
show that the variability of the concrete strength is the 
major contributing factor to cross sectional strength 
variability in the compression failure region and the 
variability in the steel strength is the major contributing 
factor to cross sectional strength variability in the 
tension failure region. The effect on the overall strength 
variability of the dimensional variability and the 
variability in the location of the steel strength were found 
to be minor compared to the effects of the concrete and 
steel strength variability. The type of distribution assumed 


118 


















. 






















































119 


for the steel strength variability was found to 
significantly affect the overall strength variability in the 
tension failure region only. 

The $ or understrength factors were calculated based on 
a probability of understrength of 1 in 100 and based on the 
first order second moment procedure developed by Cornell and 
Lind. The calculated values of <j> were in close agreement 
with those used in the ACI 318~71 Code for column cross 
sections but significantly different for the case of pure 
bending. This suggests that the <p factors used in the ACI 
Code are adequate and may be conservative for rectangular 
tied column cross sections but seem to be unconservative for 
bending tension failures. 
















































REFERENCES 


1. ACI Committee 318, ” Commentary on Building Code 

Requirements for Reinforced Concrete ( ACI 318-71 ) 
", American Concrete Institute, Detroit, 1971. 

2. ACI Committee 214, ” Realism in the Application of 

Standard 214-65 ”, American Concrete Institute, 

Publication SP-37, 1973. 

3. ACI Standard 318-71, ” Building Code Requirements for 

Reinforced Concrete ( ACI 318-71 ) ”, American 

Concrete Institute, Detroit, 1971. 

4. Allen, D.E., ” Statistical Study of the Mechanical 

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National Research Council, Ottawa, April 1972. 

5. Allen, D.E., " Probabilistic Study of Reinforced 

Concrete in Bending ”, Technical Paper NRC 11139, 
National Research Council of Canada, Ottawa, 1970. 

6. Ang, A.H-S. and Cornell C.S., " Realibility Bases of 

Structural Safety and Design ”, American Society of 
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1755-1769. 

7. ASCE Committee on Structural Safety, " Structural Safety 

A Literature Review ”, American Society of Civil 
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8. Baker, M.J., ” The Evaluation of Safety Factors in 

Structures: Variation in the Strength of Structural 
Materials and Their effect on Structural Safety ”, 
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College, London, July 1970. 

9. Bannister, J.L., ” Steel Reinforcement and Tendons for 

Structural Concrete, Part 1: Steel for Reinforced 
Concrete ”, Concrete, Vol. 2, No. 7, July 1968, pp. 
29 5-306. 

10. Basler, E., " Analysis of Structural Safety ”, ASCE 

Annual Convention, Boston, Massachusetts, 1960. 

11. Bertero, V.V. and Felippa, C., ” Discussion of Ductility 

of Concrete ", Proceedings of the International 
Symposium on Flexural Mechanics of Reinforced 
Concrete, ASCE-ACI, Miami, November 1964, pp. 227- 

234. 


120 
















































































121 


12 . 


Bloem, D.L., " Concrete Strength in Structures ", 

Journal of the American Concrete Institute, 
Proceedings, Vol 65, No. 3, March 1968, pp. 176- 
187. 


13. Bloem, D.L., " Concrete Strength Measurement-Cores 

Versus Cylinders ", American Society for Testing 
and Materials, Proceedings, Vol. 65, 1965, pp. 668- 
686 . 

14. Bloem, D.L., ” Studies of Uniformity of Compression 

Strength Tests of Ready Mixed Concrete ", American 
Society for Testing and Materials, Bulletin No. 
206, May 1955, pp. 65-70. 

15. Cady, P.D., ” Statistical Evaluation of Concrete Tests 

", American Society of Civil Engineers, Journal of 
the Construction Division, Vol. 89, No. C01, March 
1963, pp. 19-31. 


16. Campbell, R.fi. and Tobin, R.E., " Core and Cylinder 

Strengths of Natural and Lightweight Concrete ", 
Journal of the American Concrete Institute, 
Proceedings, Vol. 64, No. 4, April 1967, pp. 190- 
195. 


17. Chan, W.L., ” The Ultimate Strength and Deformation of 
Plastic Hinges in Reinforced Concrete Frameworks ", 
Magazine of Concrete Research, Vol. 7, No. 21, 
November 1955, pp. 121-132. 


18 


19 


20 


21 


22 


Corley, W.G. , 
Concrete 


II 


Beams 


Rotational Capacity of Reinforced 
American Society of Civil 


n 


Engineers, Journal of the Structural Division, Vol. 
92, No. ST5, October 1966, pp. 121-146. 

Cornell, C.A., " A Probability Based Structural Code ", 
Journal of the American Concrete Institute, Vol. 
66, No. 12, December 1969, pp. 974-985. 

Costello, J.F. and Chu, K., " Failure Probabilities of 
Reinforced Concrete Beams ", American Society of 
Civil Engineers, Journal of the Structural 
Division, Vol. 95, No. ST10, October 1969, pp. 
2281-2304. 


Cummings, A.E., " Strength Variations in Ready Mixed 
Concrete ", Journal of the American Concrete 
Institute, Vol. 51, No. 4, April 1955, pp. 765-772. 


Drysdale, R.G., " Variation of Concrete strength in 
Existing Buildings ", Magazine of Concrete 
Research, Vol. 25, No. 85, December 1973, pp. 201- 
207. 


































■ 






































































122 


23. Fiorato, A.E., " Geometric Imperfections in Concrete 

Structures ", Chalmers University of Technology, 
Goethenberg, Sweden, National Swedish Building 
Research Document Number 5, 1973. 

24. Freudenthal, A.M., M Safety and the Probability of 

Structural Failure ", American Society of Civil 
Engineers, Transactions, Vol. 121, 1956, pp. 1337- 
1375. 

25. Gamble, W.L., " Some Observations of the Strength of 

Large Reinforcing Bars ", Journal of the American 
Concrete Institute, Vol. 70, No. 1, Januray 1973, 
pp. 31-35. 

26. Hahn, G.J. and Shapiro, S.S., " Statistical Models in 

Engineering ", John Wiley and Sons, Inc, New York, 
1967. 

27. Haris, A., " Minimum Eccentricity Requirements in the 

Design of Reinforced Concrete Columns ", The 
University of Texas at Austin, Ph. D. Disertation, 
1972, Engineering, Civil. 

28. Hernandez, A.A. and Martinez, F.V., " Variaciones de las 

Demensiones y de la Posicion del Acero de Refuerzo 
en las Estructuras de Concreto ", Universidad 
Nacional Autonoma de Mexico, Facultad de 
Ingenieria, 1974. 

29. Hognestad, E., "A Study of Combined Bending and Axial 

Load in Reinforced Concrete Members ", Bulletin No. 
399, Engineering Experiment Station, University of 
Illinois, Urbana, November 1951. 

30. Housner, G.W. and Jennings, P.C., " Generation of 

Artificial Earthquakes ", American Society of Civil 
Engineers, Journal of the Engineering Mechanics 
Division, Vol. 90, No. EMI, February 1964, pp. 113- 
150. 

31. IBM Application Program GH20-0205-4, " Systera/360 

Scientific Subroutine Package, Version III 
Programers Manual ", Fifth Edition (1970), 
International Business Machines Corporation 1968. 

32. Johnson, A.I., " The Determination of the Design Factor 

for Reinforced Concrete Structures ", Symposium on 
the Strength of Concrete Structures ", Institution 
of Civil Engineers - Cement and Concrete 
Association, London, May 1956. 

33. Julian, O.G., " Discussion of Reference 21 ", Journal of 

the American Concrete Institute, Vol. 51, No. 12, 



















































. 












































123 


December 1955, pp. 772-4, 772-8. 

34. Julian, O.G., " Synopsis of First Progress Report of 

Committee on Factors of Safety ", American Society 
of Civil Engineers, Journal of the Structural 
Division, Vol. 83, No. ST4, July 1957, pp. 1316-1, 
1316-22. 

35. Keenan, W.A. and Feldman, A., " The Yield Strength of 

Intermediate Reinforcing Bars Under Rapid Loading 
", Report, University of Illinois, Civil 
Engineering Studies Structural Research Series, No. 
197, March 1960. 

36. Kent, D.C. and Park, R., " Flexural Members With 

Confined Concrete ", American Society of Civil 
Engineers, Journal of the Structural Division, Vol. 
97, No. ST7, July 1971, pp. 1969-1990. 


37. Lampert, P., Wegmuller, A. and Thurlman, B., " Einfluss 
der Dehngeschwindigkeit auf Festigkeitswerte von 
Armierungsstahlen ", Institut fur Banstatik ETH 
Zurich, Bericht Nr. 10, April 1967. 


38. Lauer, L.R. and Rigby, R.J., " Concrete Proportioning 

and Control for the "Skylon" ", Journal of the 
American Concrete Institute, Vol. 63, No. 9, 
September 1966, pp. 897-909. 

39. Lind, Niels C., " Consistent Partial Safety Factors ", 

American Society of Civil Engineers, Journal of the 
Structural Division, Vol. 97, No. ST6, June 1971, 
pp. 1651°1669. 


40. Mac Gregor J.G., " Safety and Limit States Design for 

Reinforced Concrete ", The 1975-76 Visiting 
Lectureship, Canadian Society of Civil Engineers, 
University of Alberta. 


41. Malhotra, V.M., " Maturity of Concrete and Its Effect on 
Standard Deviation and Coefficient of Variation ", 
Journal of the American Concrete Institute, Vol. 
59, No. 5, May 1962, pp. 729-730. 


42. 


Mather, B. and Tynes, W.O., " Investigation 

Compressive Strength of Molded Cylinders 
Drilled Cores of Concrete ", Journal of 
American Concrete Institute, Vol. 57, No. 
January 1961, pp. 767-778. 


of 

and 

the 

1 , 


43. Maynard, D.P. and Davis, S.C., " The Strength of In-situ 

Concrete ", The Structural Engineer, Vol. 52, No. 
10, October 1974, pp. 369-374. 































124 


44. Narayanaswamy, V.P. and Gadh f A.D., ” Characteristic 

Strength of Reinforcing Steel ”, Journal of the 
Institute of Engineers (India), Civil Engineering 
Division, Vol. 53, Part CI2, November 1972, pp. 85- 
88 . 

45. National Academy of Sciences-National Research Council, 

” The AASHO Road Test, Report 2, Materials and 
Construction ”, Highway Research Board of the NAS- 
NRC division of Engineering and Industrial 
Research, Special Report 61B, Publication No. 951, 
Washington, D.C., 1962. 

46. Neville, A.M., ” The Relation Between Standard Deviation 

and Mean Strength of Concrete Test Cubes ”, 
Magazine of Concrete Research, Vol. 11, No. 32, 
July 1959, pp. 75-84. 

47. Newlon, H.H., ” Variability of Portland Cement Concrete 

”, National Conference on Statistical Quality 
Control Methodology in Highway and Airfield 
Construction ”, Proceedings, University of 
Virginia, Charlottesville, Virginia, 1966, pp. 259- 
289. 

48. Petersons, N., " Should Standard Cube Test Specimens Be 

Replaced by Test Specimens Taken From Structures ”, 
Materiaux et Constructions, RILEM, Paris, Vol. 1, 
No. 5, 1968, pp. 425-435. 

49. Plowman, J.M., Smith, W.F., and Sheriff, T., ” Cores, 

Cubes and the Specified Strength of Concrete ”, The 
Structural Engineer, Vol. 52, No. 11, November 
1974, pp. 421-426. 

50. Plum, N.M., " Quality Control of Concrete Its Rational 

Bases and Economic Aspects ”, Institution of Civil 
Engineers, Proceedings, Vol. 2, Part 1, London, 
1953, pp. 311-333. 

51. Rackwitz, R., ” Statistical Control in Concrete 

Structures ”, C-4, CEB International Course on 

Structural Concrete, Laboratorio Nacional de 
Engenharia Civil, Lisbon, 1973. 

52. Rao, N.R.N., Lohrmann, M. and Tall, L., ” The Effect of 

Strain on the Yield Stress of Structural Steels ”, 
American Society for Testing and Materials, Journal 
of Materials, Vol. 1, No. 1, March 1966, pp. 241- 
262. 

53. Redekop, D., ” A Study of Reinforced Concrete Columns in 

Existing Buildings ”, Master of Engineering Thesis, 
Me Master University, August 1971. 































i 






































































125 


54. Riley, o. and Cooper, S.B., " concrete Control on a 

Major Project ", Journal of the American Concrete 
Institute Proceedings, Vol. 68, No. 2, February 
1971, pp. 107-114. 

55. Roberts, N.P., " The Characteristic Strength of Steel 

for Reinforcing and Prestressing Concrete ", 
Concrete, Vol. 1, No. 8 August 1967, pp. 273-275. 

56. Robles, F., " Strength Factors: Material and Geometrical 

Aspects ", ASCE-IABSE International Conference on 
Tall Buildings, Lehigh University, Proceedings, 
Vol. Ill, 1972, pp. 907-921. 

57. Roy, H.E.H. and Sozen, M.A., " Ductility of Concrete ", 

Proceedings of the International Symposium on 
Flexural Mechanics of Reinforced Concrete, ASCE- 
ACI, Miami, November 1964, pp. 213-224. 

58. Rusch, H., " Die Streung der Eigenschaften von 

Schwerbeton ", Symposium on Concepts of Safety of 
Structures and Methods of Design, International 
Association for Bridge and Structural Engineering, 
London, 1969, pp. 63-74. 

59. Rusch, H. and Stokl, S., " Der Einfluss von Bugelin and 

Druckstaben auf das Verhalten der Biegedruckzone 
von Stahlebetonbalken ", Bulletin No. 148, 
Deutscher Ausschuss Fur Stahlbetonbau, Berlin, 
1963, p. 75. 

60. Sargin, M., " Stress-Strain Relationships for Concrete 

and the Analysis of Structural Concrete Sections ", 
SM Study No. 4, Solid Mechanics Division, 
University of Waterloo, 1971. 

61. Shalon, R. and Reintz, R.C., " Interpretation of 

Strengths Distribution As a Factor in Quality 

Control of Concrete ", RILEM Symposium on the 

Observation of Structures, Vol. 2, Lisbon 
Laboratorio Naciano de Engenharia Civil, 1955, pp. 
100-116. 

62. Sexsmith, R.G. and Nelson, M.F., " Limitations in 

Applications of Probabilistic Concepts ", Journal 
of the American Concrete Institute, Proceedings, 
Vol. 66, No. 10, October 1969, pp. 823-828. 

63. Siu, W.W.C., Parimi, S.R. and Lind, N.C., " Practical 

Approach to Code Calibration ", American Society of 
Civil Engineers, Journal of the Structural 
Division, Vol. 101, No. ST7, July 1975, pp. 1469- 

1480. 


























































126 


64. Soliraan, M.T.M. and Yu, C.W., " The Flexural Stress- 

Strain Relationship of Concrete Confined by 
Rectangular Transverse Reinforcement ", Magazine of 
Concrete Research, Vol. 19, No. 61, December 1967, 
pp. 223-238. 

65. Soroka, I., " An Application of Statistical Procedures 

to Quality Control of Concrete ", Materiaux et 
Constructions, RILEM, Paris, Vol. 1, No. 5, 1968, 

pp. 437-441. 

66. Soroka, I., " On Compressive Strength Variation in 

Concrete ", Materiaux et Constructions, RILEM, Vol. 
4, No. 21, 1971, pp. 155-161. 

67. Sturman, G.M., Shah, S.P. and Winter, G., " Effect of 

Flexural Strain Gradients on Microcracking and 
Stress-Strain Behaviour of Concrete ", Journal of 
the American Concrete Institute, Proceedings, Vol. 
62, No. 7, July 1965, pp. 805-822. 

68. Torroja, E., " Philosophy of Structures ", University of 

California Press, Berkley, California, 1958. 

69. Tso, W.K. and Zelraan, I.M., " Concrete Strength 

Variation in Actual Structures ", Journal of the 
American Concrete Institute, Proceedings, Vol. 67, 
No. 12, December 1970, pp. 981-988. 

70. Wagner, W.K., " Discussion of Reference 21 ", Journal of 

the American Concrete Institute, Proceedings, Vol. 
51, No. 12, December 1955, pp. 772-14, 772-16. 

71. Wagner, W.K., " Effect of Sampling and Job Curing 

Proceedures on Compressive Strength of Concrete ", 
Materials and Research Standards, August 1973. 

72. Warner, R.F. and Kabaila, " Monte Carlo Study of 

Structural Safety ", American Society of Civil 
Engineers, Journal of the Structural Division, Vol. 
94, No. ST 12, December 1968, pp. 2847-2859. 





























V 




















' 









































APPENDIX A 


VAPIABILITY IN REINFORCING STEEL 


Introduction 

The three main sources of variation in steel strength 

are: 

(1) variation in the strength of material, 

(2) variation in area of the cross-section 
of the bar, and 

(3) variation in the rate of loading. 

The variability of yield strength depends on the source 
and the nature of the population. The variation in strength 
within a single bar is relatively small, while the in-batch 
variations are slightly larger. However, variability Of 
samples derived from different batches and sources may be 
high. This is expected since rolling practices and quality 
measures vary for different countries, different 
manufacturers and different bar sizes. Furthermore, the 
cross-sectional areas vary due to differences in the setting 
of the rolls, and this adds to the variation. Mill tests are 
generally carried out at a rapid rate of loading (ASTM 
corresponds to 1040 micro-in/in/sec) and have the tendency 
of reporting the unstable high yield point rather than the 
stable low yield point. Since the strains in the structure 
are induced at a much lower rate than the mill tests, mill 
tests tend to overestimate the strength of reinforcement. 


127 












































































128 


hence another source of variation. 

An examination of the test data revealed that the bars 
of large diameter tended to develop less strength (4, 24, 
55) than #3 to #11 bars. Thus, for the purpose of 
statistical evaluation, the #14 and #18 bars were studied 
separately from the other sizes. Also the #2 bars were not 
included in this study because of their rare use for 
structural concrete. 

In this study the terms Grade 40, Grade 50 and Grade 60 
refer to reinforcing bars with minimum specified yield 
strength of 40, 50 and 60 ksi, respectively, even though the 
bars in question may not have been produced according to 
ASTM or CSA specifications. Only data for deformed bars has 
been included. In some cases data for cold-worked bars has 
been considered but most of the data is for hot-rolled bars. 

Va riation in S teel Strength 

Different values for the yield strength of steel may be 
obtained depending on how it is defined. The static yield 
strength based on nominal area seems to be desireable 
because the strain rate is similar to what is expected in a 
structure and designers use the nominal areas in their 
calculations. Most mill tests, however, are conducted with a 
rapid rate of loading, and the strength is generally 
referred to actual areas* For these reasons the yield 
strength corresponding to rapid strain rate and measured 






























129 


area is discussed in this section, the effects on this 
strength of variations in cross-sectional area and rate of 
loading are dealt with in the succeeding sections. 

A review of literature on steel strength showed that 
the coefficient of variation was in general in the order of 
1% to 4% for individual bar sizes and 4% to 7% overall for 
data derived from any one source. When data was taken from 
many sources the coefficient of variation increased to 5% to 
8% for individual sizes and 10% to 12% overall. A summary of 
selected studies from literature (4,8,9,33,43) is shown in 
Table A-1. 

The data reported by Allen 4 and Julian 33 on Grade 40 
and 60 steel bars showed close agreement with a normal 
distribution (with respective mean and standard deviation) 
in the range approximately 5 to 95 percentile but differ 
from the normal distribution outside this range. Some 
authors have suggested other types of distributions such as 
skewed distribution (9,54), truncated normal (31) and Beta 
distribution (20). These suggestions were, however, based on 
a particular set of data and only approximated the 
distribution of the population from which the data was 
drawn. Nonetheless, they suggest that the yield strength is 
a phenomenon that can be described by a particular 
theortical distribution with certain limitations. The normal 
distribution seems to correlate very well in the vicinity of 
the mean values for different populations of yield strength/ 












. 
































Summary of Selected Studies on Steel Strength 


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131 


but it is a crude approximation at low and high levels of 
probability where the steel strength distribution curves 
tend to have certain minimum and maximum values rather than 
following the theoretical tails. This is expected since 
there are always some quality controls that are used to 
attain a certain minimum yield strength with the result that 
the manufacturing of steel is not truly a random process. 
Furthermore, certain data indicated a positive skewness, 
particularly when derived from different sources and mixed 
together. Theoretically a log-normal distribution should be 
a better fit for this case than a normal distribution since 
it takes into account the skew nature of the data. However 
the logarithmically distributed values of yield strength at 
low and high levels of probability did not show a 
significant improvement over normally distributed values of 
available data. Therefore, it was decided to empirically 
establish a "modified" log-normal distribution that would 
yield correlate with the North American data on yield 
strength. 


Values of (f y »34ksi) are plotted on log-normal 
probability paper in Figure A-1 for the data from Julian 33 
and Allen 4 for Grade 40 reinforcing bars grouped together. 
The values were found to be in good agreement with a log“ 
normal distribution in the range from the 0.01 percentile to 
the 99th percentile. The modification constant of 34 ksi was 
established by trial and error. The corresponding frequency 


curve. 


of the grouped data and the 


the histogram 

































































































CUMULATIVE FREQUENCY 


132 



(fy-34) ( ksi) 


Figure A~1 


Steel strength Distribution for Grade 40 
Reinforcing Bars 











































































133 


corresponding normal frequency distribution curve are shown 
in Figure A-2 for the purpose of comparison. The mean value 
of the data was found to be 48.8 ksi with a maximum value of 
66 ksi and a coefficient of variation of 10.7%. 

Similarly, values of (£ y »55ksi) for Grade 60 
reinforcing bars from mill tests reported by Allen 4 were 
found to be log-normally distributed in the range from the 
0.01 percentile to the 99th percentile as shown in Figure A- 
3. The frequency curves and histogram for Grade 60 steel are 
shown in Figure A-4. The mean value for the data was 71.5 
ksi with a maximum value of 90 ksi and a coefficient of 
variation of 7.7%. 

In both cases the modified log-normal curve is a better 
approximation at the lower end of the curve while the normal 
curve is better at the high end of the curve. 

Variation in Steel Cross - Sectional Area 

The actual areas of reinforcing bars tends to deviate 
from the nominal areas due to the rolling process. The 
designers do not have this information readily available to 
them, and hence use the nominal areas in their calculations. 
For this reason this variation should be incorporated in the 
strength of steel. 

variation in the ratio of measured to nominal 
areas (A /A R ) was studied as a measure of the variation in 


















_ 















































































FREQUENCY 


134 



34 38 42 46 50 54 58 62 66 70 74 


STEEL STRESS (ksi) 


MODIFIED LOG-NORMAL DISTRIBUTION _ 

C * r 1 fX-X 

PDF = -t= exp [- 2 


ya x /2TT 


) ] 


X 


X = 1.19456, a = 0.14112 

X 

c = 0.43429 

y = (fy - 34 ksi) , x = LOG^y 


NORMAL DISTRIBUTION 

X = 48.8 ksi, a = 5.506 ksi 

X = fy ksi 


Figure A-2 


Probability Density Function for Grade 40 Bars 















































































































































CUMULATIVE FREQUENCY 


135 



(f y -55) (ksi) 


Figure A~3 


Steel Strength Distribution for Grade 60 
Feinforcing Bars 























































































FREQUENCY 


136 



STEEL STRESS (ksi) 

MODIFIED LOG-NORMAL DISTRIBUTION 

PDF “ yo^TUT* I 1 

2 X X 


1.19456, a = 0.14112 
x 


X = 

c = 0.43429 

y = (fy - 55 ksi), x = 
NORMAL DISTRIBUTION 


LOG 10 y 


X = 71.8 ksi, a = 5.506 ksi 

X = fy ksi 


Figure A-4 Probability Density Function for Grade 60 Ears 








































































































































































































the cross-sectional area of reinforcing bars. The values of 
^ e /A n are reproduced from available literature (4, 8, 43) in 
Table A°1. Table A-1 indicates that the data reported by 
Baker 8 for Grade 60 steel demonstrates high mean value and 
coefficient of variation. Such values cannot be explained in 
definite terms. It is possible that the collected data 
contained a good percentage of values from mills with old 
rolls that increased the mean and coefficient of variation. 
Furthermore, British rolling practice may differ from 
Canadian practice. For these reasons, these values were not 
included in the analysis. 

The ratios of A e /A n from tests on Grade 40 and 60 
reinforcing bars, manufactured in Canada (Study No. 1 and 3 
in Table A-1), were plotted on normal probability paper. 
These values exhibited close agreement in the range from the 
5th to the 95th percentile for Grade 40 steel and from the 
2nd to the 98th percentile for Grade 60 bars with a normal 
distribution. When the values for both studies were combined 
they resulted in a normal distribution in the range between 
the 4th and 99th percentile with a mean value of 0.988 and a 
coefficient of variation of 2.4%. The effect of such a small 
coefficient of variation is not large enough to have any 
significant effect on the coefficient of variation of A s f y . 
For this reason a single value for A e /A n seems to be more 
appropriate. Allen 4 has suggested a value of 0.97 for A 0 /A n . 
This seems to be a conservative estimate of the average 
values of A e /A n shown in Table A*-1 r 


and close to ASTM 




.. 






























138 

rolling tolerances that allow an average ratio as low as 
0.965 and a minimum single value up to 0.940. 

Effect of Rate of Loa din g 

The apparent yield strength of a test specimen 
increases as the strain rate or the rate of loading 
increases. Since mill tests on steel specimens are generally 
carried out at much greater strain rates (approximately 1040 
micro-in/in/sec) than encountered in a structure, they tend 
to overestimate the yield strength. A strain rate of 1 
in/in/sec may increase the yield strength of Grade 40 steel 
as much as 50% over the static yield strength (34). 

Tests conducted on steel coupons of A36, A441 and A514 
steel (51) demonstrated a yield strength reduction more or 
less of the same value for all types of steel with decrease 
in the rate of strain. The equation developed by Fao 5 * on 
the basis of these tests gives values of static yield 

strength that are 4.8 ksi and 3.4 ksi less than the yield 

strengths obtained at cross-head speed of 1000 and 200 
micro~in/in/sec respectively. NRC tests on Grade 40 bars (4) 
showed a reduction of approximately 3 ksi in the mean yield 

strength when speed of the testing machine was dropped from 

208 micro—in/in/sec to static. This value correlates well 
with the one obtained from Bao's equation. Similarity, for 
Grade 40 bars, it has been shown at the University of 
Illinois (34) that the difference between the yield strength 




















































inr 






























139 


at a strain rate of 1040 micro-in/in/sec and the strength at 
a strain rate of 20 micro-in/in/sec is about 9% or 4 ksi. 
ETH tests (36) for high strength reinforcement demonstrated 
a reduction of 3 ksi for static conditions. 

For evaluation of the static yield strength from mill 
tests, Allen 4 has suggested a decrease of 4 ksi. This value 
seems to be a reasonable estimate for the available test 
data. 

Effect of Bar Diameter 

The strength of steel tends to vary across the cross- 
section of a reinforcing bar with the highest strength near 
the outside of the bar. This is probably due to the cold¬ 
working of circumferential sections of bars during the 
rolling process. Thus the mean yield strength is expected to 
decrease with increasing diameter. The variation of the mean 
yield strength with size is plotted in Figures A-5 and A-6. 
The data shown in the figures were taken from several test 
series for Grade 40 and Grade 60 reinforcement 
(4,44,8,9,24). For bars with relatively small diameter the 
effect of this variation is small and not clearly 
established. For large diameter bars such as #14 and #18 
this effect becomes more significant. In addition, the ASTM 
specifications allow the use of small specimens machined 
from samples of large diameter bars for testing purposes. A 
specimen machined to a smaller diameter from a guarter-piece 



















































































































140 


100 


90 


80 


«/> 

dS 70 


o 

z 

lu 60 

f— 

oo 


50 


40 


30 


0 

8 4 

• A 


O 

□ 


A 

A 


A MILL TESTS (AASHO) 

• DATA SYSTEM(AASHO) 
□ NRC TESTS (ALLEN) 

O MILL TESTS (ALLEN) 


□ 

O 


O 


O 

□ 


3 4 5 


8 9 11 

BAR SIZE 


14 


Figure A-5 Effect of Bar Diameter on Steel Strength, 

40 


Grade 




































































































































































































141 


100 


90 


80 


«/> 

— 70 


o 

Z 

111 

c£ 60 
H— 

CO 


50 


40 


30 


□ REPORTED BY BANNISTER 
A MANUFACTURER 1 (BANNISTER) 
▲ MANUFACTURER 2 (BANNISTER) 
• REPORTED BY BAKER 
O REPORTED BY GAMBLE 


^ g • 


^ 8 8 • t 


□ 


3456789 10 

BAR SIZE 


14 


18 


Effect of Bar Diameter on Steel Strength, Grade 
60 


Figure A-6 







































































































































































142 


of a full size bar tends to show higher yield strength than 
the bar itself (24). Since some manufacturers may use these 
tests as a measure of quality control, the #14 and #18 bars 
tend towards a higher probability of passing through quality 
controls without developing the required strength. 

An extremely limited amount of data is available for 
#14 and #18 bars. Tests on Grade 40, #14 bars carried out by 
Allen 4 showed that the mean yield strength of #14 bars was 
44 ksi, a 15% decrease from the strength of #3 to #11 bars 
produced by the same manufacturer. Some data has been 
reported by Gamble 24 for #14 and #18 bars of Grade 60 steel. 
The mean yield strengths were 60 ksi for #14 and 55 ksi for 
#18 bars. These strengths were referred to the nominal 
areas. Using the mean yield strength of Grade 60, #3 to #11 
bars as 71.5 ksi (as per Study No. 3 in Table A~1) and a 3% 
adjustment for the deviation from the nominal area, the 
reduction in strength is approximately 13% for #14 bars and 
21% for #18 bars. This comparison is, however, not truly 
justified since the data for both studies was not drawn from 
the same source. Nonetheless, it strongly indicates the 
understrength of #14 and #18 bars. Until more data is 
available, it seems reasonable that the yield strength of 
#14 and #18 bars should be reduced at least 15% below the 
yigj_d strength of reinforcing bars with smaller diameter. 


































































































143 


Summary 

The modified log-normal distribution curves shown in 
Figures A-1 through A-4 seem to correlate well, particularly 
near the lower tails of the curves, with the available North 
American data for Grade 40 and Grade 60 reinforcing bars. 
The Probability Density Function for these curves can be 
calculated using the following equation: 

PDF = c 

ya »/ 2n 

x 

where: 



c = 0.43429 


y 

y 



34 ksi for Grade 40 bars 

55 ksi for Grade 60 bars 


x = 


Log 

10 


y 


x = 1.14482 for Grade 40 bars 


x = 1.19456 for Grade 60 bars 

a = 0.14866 for Grade 40 bars 
x 

a = 0.14112 for Grade 60 bars 

x 

The mean yield strength of the selected data was found 
to be 48.8 ksi (c.o.v. = 10.7%) for Grade 40 bars and 71.5 
ksi (c.o.v. = 7.7%) for Grade 60 bars. The modification 

constants were empirically established and found to be 34 











































































* 














































144 


ksi and 55 ksi for Grade 4C and Grade 60 steel respectively. 

A value of 0.97 for the ratio A e /A R seems to be 
reasonable to account for deviations from the nominal areas. 
Similarly, for the evaluation of the static yield strength, 
at least 4 ksi should be deducted from the yield strength- 
obtained in mill tests or at high strain rates allowed by 
ASTM specifications. 

When calculating the yield strength of #14 and #18 
reinforcing bars from the strength of bars of smaller sizes 
at least a 15% reduction should be used to account for the 
effect of the large diameter. 






















































































































APPENDIX B 


COLUMNS STUDIED 

This appendix contains the details of the two major 
columns studied. Tables B-1 and B-2 are tables of the 
properties of the 12 in. and 24 in. columns respectively. 
Figures B-1 and B-2 are diagrams of each column showing the 
designer's properties and the mean values of the column 
properties used in the Monte Carlo calculations. 


145 

























































146 


Table B- 1 


Properties of the 12 in. Column Assumed in the Calculations 

Specified Mean In- o C.C.V. 

situ 


Material Strength s 


Concrete Strength 

3000 

psi. 

3712 

psi. 


0.17 

Steel Yield Strength 

40 

ksi. 

48.8 

ksi. 

1.41 

ksi. 

Dimensions 

b, h 

12.00 

in. 

12.06 

in. 

0.280 

in. 

d 

9.75 

in. 

9.51 

in. 

0.166 

in. *=-- 

d' 

2.25 

in. 

2.55 

in. 

C.166 

in. --- 

*s 

1.76 

sg.in. 

on cm 




A s 

0.88 

sg.in. 




0C~j ar 1 c 

s 

12.00 

in. 

12.00 

in. 

— - 

<c cs e 

b", d" 

9.00 

in. 

8.47 

in. 

0.166 

in. —- 

A" 

0.11 

sg .in. 

0.11 

sg. in 

• 



Indivi dual Longitud i nal S teel Bars 


ASB (1) 

to ASB (4) 

0.44 

sg .in. 

0.44 

sg.in. --- 

DS (1) , 

DS (2) 

2.25 

in. 

2.55 

in. 

0.166 in. 

DS(3) » 

DS (4) 

9.75 

in. 

9.51 

in. 

0.166 in. 


ta lt . 












































































































147 


Table B-2 

Properties of the 24 in. Column Assumed in the Calculations 

Specified Mean In- a C.O.V. 

situ 

Ma terial Strenq hts 


Concrete Strength 


3000 psi. 

3712 

psi. 



C. 176 

Steel Yield Strength 


4 0 k si. 

4 

CO 

• 

00 

ksi 

1, 

.41 

ksi. 

«T3 c 

Dimensions 











b, h 

24 

o 

o 

• 

in. 

24 

.06 

in . 

0. 

,280 

in. 


d 

21 

.30 

in. 

21 

.Cl 

in. 

0 . 

.166 

in. 


d* 

2 

.70 

in. 

3 

.05 

in. 

0 . 

,166 

in. 

mo mo 

A s 

18 

.72 

sq.in. 

- 

— 



3 CEE- ru 


a,a6m 


7 

o 

CO 

• 

sg .in. 

- 

— 



— - 



S 

12 

o 

o 

• 

in. 

12 

o 

o 

in. 




b", d" 

20 

.50 

in. 

19 

.87 

in. 

0. 

166 

in. 

«o 


A” 0.20 sq.in. 0.20 sq.in. --«* —- 

Indi vi dual L ongitudinal S teel Bars 
ASB(1) to ASB(12) 1.56 sq.in. 1.56 sq.in. — 

DS(1) to DS(5) 2.70 in. 3.05 in. 0.166 in. --- 

DS(6) , DS(7) 12.00 in. 12.07 in. 0.993 in. 

21.30 in. 21.01 in. 0.166 in. —- 


DS(8) to DS(12) 








































































































































































































































148 



A 

c 

ID 

v 


c 

C\l 


f = 40,000 psi 

f = 3,000 psi 

A $ — 2# 6 bars 
A; - 2# 6 bars 
A" - #3 @12 in 


12 in. x 12 in. COLUMN 



f = 40,000 

f y = 40,000 psi 

f' = 3,000 psi 

A — 7# 11 bars 

S 

A' — 5# 11 bars 

S 

A'' — #4 @12 in. 
in sets of 3 


24 in. x 24 in. COLUMN 


Figure B-1 


Nominal or Designer's Properties of the 12 in. 
and 24 in Columns 

































































- 









































































































































149 



f y = 48,800 psi 

f = 3712 psi 
A — 2# 6 bars 

S 

A' — 2# 6 bars 

S 

A"- #3 @12 in 

S 


12 in. x 12 in. COLUMN 



f y = 48,800 psi 

f c = 3712 psi 

A — 7# 11 bars 

S 

A; - 5# 11 bars 

A"- #4 @12 in. 
in sets of 3 


24 in. x 24 in. COLUMN 


Figure B— 2 M©an Valu©s of fh© Piropoirtiss of fh© 12 in. 

24 in Columns 


and 













































































































































































































































APPENDIX C 


FLOW DIAGRAMS OF THE MONTE CARLO PROGRAM 

This appendix contains detailed flow diagrams of the 
complete Monte Carlo Program including: 

The Main Program 
Subroutine PROP 
Subroutine ACI 
Subroutine ASTEEL 
Subroutine CURVE 
Subroutine THMEAN 
Subroutine THEORY 
Subroutine AXIAL 
Subroutine FSTEEL 
Subroutine RANDOM 
Subroutine STAT 






150 




























































MONTE CARLO PROGRAM 

























, 









































































































152 
















































































































£ 



II 1 MQ 

V 

JJ 1, IMo 


I = 1, NV 

I 


4 - 


SD 

= STDV (1) 

CONST = FCONST (1) 

RM 

= RMEAN (1) 

ITP = ITYPE (1) 


CALL RANDOM 



FY = 0.97 (X (2) -4000) 

5 

BB = X (3) Dll = X (6) 

H = = X (4) DC = X (7) 

B11 = X{5) DD = X (8) 

3Z 

I = 1, NB 


Nl = I + 8 


DS (1) = 

= X (Nl) 




(!) 


























































































































































































































































































1 55 












































































































156 


SUBROUTINE PROP 

























































































































157 


SUBROUTINE ACI 



A 

































































' 



































































158 



I 


































































































































































































1 59 



£ 


P (J) = FCCONC + FST 


COMPM = FCCONC*(DD - B1 ■ C/2 


SM = 0.0 


I = 1, NB 


SM = SM + SBM (I) 


BM (J) = COMPM + SM - P (J)*(DD - H/2) 


EOH (J) =BM (J)/(P(J)*H) 4- 


YES 



YES 


WRITE PO, PB, BMO 


© 


SBM (I) = (FSS (I) - FCS (l)*ASB (l))*DD - DS (I)) 


WRITE ACI INTERACTION 
DIAGRAM 































































































































































160 


SUBROUTINE ASTEEL 































































161 


SUBROUTINE CURVE 


DEFINE EOH1 (1) 

1 = 1, NN 

< 

r 

i = 

1, N 



YES 


CONTINUE 


r 

•i 

NP 

= 1 

1 


EOHB = 
Ml = M 

EOH (1) 



YES ^ 

M = 

NP—2 












































































































































































































164 


§ 



© 


I = 1, JJ 

m 


P (II) = P(ll) + COE (I)*[EOH1 (ll)**(l - 1)] 


CONTINUE 


NNN = II 


I = NP. N 


DIFE = 3.0 — EOH (I) 



CONTINUE 

n 

► 

1 

NM = N 

- NP + 1 



M = 

Ml 



N = 1 - 1 





M = NM — 2 
































































































































































































165 






























































































© ® 



YES 


SUMIP = ANS (4) - SUM 



SUMIP > 0.0 
YES 


SUM = ANS (4) 
COE (1) = ANS (1) 



J = 

U 




t_ 


COE (J + 1) = B (J) 


_ 

[ 


JJ = I + 1 


T 


CONTINUE 


I 


II = NNN, NN 


BM (II) = 0.0 

~T~ 


I = 1, JJ 



































































































167 
















































































































168 


SUBROUTINE THMEAN 

















































































































































































































169 


SUBROUTINE THEORY 





































































































































































































































1 70 






































































































































172 


SUBROUTINE AXIAL 



A 


B 






















































































































































































173 



© 


P11 = (2*(B11 + D11)*AS11 + ASC*S)/(B11*D11*S) 


E50H = 0.75*P11 VB11/S 

E50U = (3 + 0.002*FC)/ (FC - 1000) 



z = 
zz 

EU 

0.5/(E50H + E50U - EO 
= 0.5/(E50U - EO) 

= 2*(E50H + E50U) - EO 





DX = C/10 






1 = 1,10 



% . 





i A,=l 



* 


X (1) = C - AI*Dx + Dx/2 

E (1) = PHI*X(I) + ECO 

B (1) = BB 


-® 



B (I) = B11 





















































































































































































































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177 


SUBROUTINE F STEEL 
























































































































































































































178 


SUBROUTINE RANDOM 































































1 79 


SUBROUTINE ST AT 




















































































18C 



C 






























































































































































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184 























































APPENDIX D 


LISTING OF THE MONTE CAPLO PPOGPAM 

This appendix contains a complete listing of the Monte 
Carlo Program. The modified IBM Subroutine MULT to MULTR is 
also listed. The listing includes: 

The Main Program 
Subroutine THMEAN 


Subroutine 

A Cl 

Subroutine 

ASTEEL 

Subroutine 

PROP 

Subroutine 

CURVE 

Subroutine 

RANDOM 

Subroutine 

THEORY 


Subroutine AXIAL 
Subroutine FSTEEL 
Subroutine STAT 
Subroutine TMULTR 


185 




























































186 


$ LIST F2 
1 
2 

3 

4 

5 

6 

7 

8 
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55 


ON * PEI NT* 

£******************************«********************** 

COMMON N.EOH1 (13) ,PO,BMO,DCS,DTS 

COMMON FC,FY,ES,BB,H,DC,DD,AS, AS 11,B1 1 ,D1 1 , S , C , 27 , NED 
COMMON PH I,EO,J,2, ECC, EY , FCONCC (20) , ASC, EOH (40) , FCONST (25) 
COMMON X (16 000) , EC(20) , B (2 0) , P (4 0) ,BMM (1000) , BM(40) , FCS (20) 
COMMON FST, E (20) ,NB,DS (20) ,ASB(2 0) ,FS5 (2 0) ,SBM (20) 

DIMENSION PACI (15),PTH (15,2000),RP (15, 200 0) 

DIMENSION FMFAN (25) ,STDV (2 5) ,ITYPE (25) 

DIMENSION CLASS (3 1) ,CFREQ(31) ,NGBOSS (31) 

C BEAD QUANTITIES NEEDED FOE MONTE CARLO SIMULATIONS 
READ (5,500) NV,NS,FY1 
500 FORMAT (215,F9.2) 

BEAD (5,511) RMEAN1,RMEAN2,IY,NRU 

511 FORMAT (2F15.5,2110) 

C READ NOMINAL PROPERTIES 

CALL PROP (NS) 

C BEAD STATISTICAL PROPERTIES OF VARIABLES 
DO 5 1=1,NV 

READ (5,510) RMEAN(I) ,STDV(I) , FCONST (I) ,ITYPE(I) 

510 FOFMAT(3F15.5,I5) 

5 CONTINUE 

C WRITE STATISTICAL PROPERTIES OF VARIABLES 
WRITE (6,512) 

512 FORMAT (’1*,//////27X,•DISTRIBUTION PROPERTIES OF VARIABLES') 
WRITE (6,519) NS,NRU 

519 FORMAT (31X,• (• , 14 , •SIM•,15,•) •//) 

WRITE (6,513) 


513 

1000 

514 

515 

516 
522 
520 


MEAN-VALUE STD-DEVI ATION 


FORMAT (16X,' 

DO 1000 1=1,NV 

WRITE (6,514) RMEAN (I) ,STDV (I) ,FCONST(I) ,ITYPE (I) 

FORMAT (16X,3F15.5,15) 

WRITE (6,515) 

FORMAT (////2GX,' FC (MEAN-VALUE) FY (MEAN-VALUE) '/) 

WRITE (6,516) RMEAN1,RMEAN2 
FORMAT (20X,2F15.5) 

WRITE (6,522) FY1 
FORMAT (///21X,'FY LIMIT=*,F9.2) 

WRITE (6,520) IY 
FORMAT (///2IX, 1 ISEED=',110) 

CALCULATE THE ACI INTERACTION DIAGRAM 
CALL ACI (NS) 

FIT A POLYNOMINAL TO THE ACI INTERACTION DIAGRAM 
CALL CURVE 

WRITE THE ACI INTERACTION DIAGRAM AFTER THE CURVE FIT 

( 6 , 100 ) 

('1’,//////23X,'****ACI INTERACTION DIAGRAM****') 
(6,519) NS,NRU 
(6,517) 

(30X,* (AFTER CURVE FIT)'//) 

(6,103) 

(19X,'P(J) LBS' ,6X,'M(J) LB-IN' ,7X,'E0H(J) •) 

DO 6 0=1,13 

WRITE (6,104) P (J) ,BM (J) ,EOHl (J) 

FORMAT (/16X, 3E1 5.7) 

PACI (1)=PO 


FCONSTANT TYPE'/) 


100 

517 

103 
6 

104 


WRITE 

FORMAT 

WRITE 

WRITE 

FORMAT 

WRITE 

FORMAT 





















































































56 

57 

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115 


187 


PACI (15)-8R0 
DO 1 1=1,13 
1 PACI (1 + 1) =P (I) 

C CALCULATE MEAN THEORY INTERACTION DIAGRAM 
CALL THMEAN (F. MEAN , P M E A N 1 , RME A N 2 , NS) 

C FIT A POI.YNCMJNAL TO THE THEORY INTERACTION DIAGRAM 
CALL CURVE 

C WRITE MEAN THEORY INTERACTION DIAGRAM AFTER CURVE FIT 
WRITE (6,101) 

101 FORMAT ('1',//////1 Q X,•****MEAN THEORY INTERACTION DIAGRAM****') 
WFITE (6,519) NS,NPU 
WRITE (6,517) 

WRITE (6,103) 

DO 3 J=1,13 

3 WRITE (6,104) P (J) ,BK(J) ,EOH1 (J) 

PTK (1 , 1) = PO 
PTH (15,1) =EMO 
DO 15 1=1,13 
15 PTH (1+1,1) =P (I) 

WFITE (6,105) 

105 FORMAT (//18X,•MEANTH/ACI' ,2X , * EOH’) 

C CALCULATE AND WRITE RATIO MEAN THEORY/ACI 
DO 17 1=1,15 
IF (I.EQ.1) EOH2=0.0 
IF (I.GT.1) EOH2=EOHl (1-1) 

IF (I.EQ.15) EOH2=99.9 9 
RP (I , 1) =FTH (I, 1) /PACI (I) 

17 WRITE (6,106) RP(I,1),EOH2 

106 FORMAT (16X,2F10.5) 

C MONTE CARLO CALCULATION OF THEORETICAL STRENGTH 
DO 4 JJ=1,NS 
DO 10 1=1,MV 
SD = STDV (I) 

RM=RMEAN(I) 

CONST-FCONST (I) 

ITP=ITYFE (I) 

CALL RANDOM (IY,SD,RM,CONST,ITP,V) 

X(I)=Y 
10 CONTINUE 

IF (X(1) .IE. (RMEAN(1)-3.3*STDV (1))) X (1) =RMEAN (1) - 3.3* STD V (1) 
FC=X(1) 

IF (X (2) .GT.FY1) X (2) =FY 1 
FY= (X(2)-4000.0)*0.97 
BB=X (3) 

H=X (4) 

B11 = X (5) 

Dll =X (6) 

DC=X (7) 

DD = X (8) 

DO 2 1=1,NB 
Nl=l+8 

2 DS(I)=X (NI) 

C CALCULATE THEORETICAL INTERACTION DIAGRAM 
CALL THEORY 

C FIT A POLYNOMIAL TO THE THEORY INTERACTION DIAGRAM 
CALL CURVE 
PTH (1, JJ) =PO 
PTI! (15,JJ) =BMO 
DO 9 1=1,13 
9 FTH (1+1,OJ)-P(I) 












































































































































116 

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188 


C CALCULATE RATIO PTHEOPY/PACI AT SPECIFIED E/H VALUES 
DO 44 1=1 ,15 

R P (1 , J J) =PTH (I,JJ)/PACI (I) 

44 CONTINUE 
4 CONTINUE 

C STATISTICAL ANALYSIS OF PTHEOSY/PACI FOR EACH SPECIFIED E/H VALUE 
DO 45 1=1,15 
DO 8 JJ=1,NS 
L=J J 

8 X (I,) =RP (I, JJ) 

N = NS 

CALL STAT (RK,SD,COV,COS,COK,RMIN,RHAX,UN2,UH3,UM4,IBAX,IHIN,RMED, 
1CLASS,CFREQ,NGROSS) 

IF (I.EQ.1) ECH2=0.0 
IF (I.GT.1) EOH2=EOH1(1-1) 

IF (I.EQ.15) EOH2 = 99.9 9 
WRITE (6,518) 

518 FORMAT (* 1•,//////31X,*******+THEORY/ACl***♦***') 

WRITE (6,22) EOH2 

22 FORMAT (,/3 1 X ,********* EOH= ', F5.2 ,' ********) 

WRITE (6,519) NS,NRU 
WRITE (6,16) 

16 FORMAT (31X, •<*> STATISTICAL EVALUATION <*>') 

WRITE (6,20) 

20 FORMAT (/I6X,'MEAN-VALUE SD-DEVIATION CO-VARIATION CO-SKEWNESS K 
1URTOSIS') 

WRITE (6,25) EM,SD,COV,COS,COK 
25 FORMAT (13X,4F13.5,FI 0.5) 

WRITE (6,30) 

30 FORMAT (/17X,'MIN-VALUE (SIMULN NO.) MAX-VALUE (SIMULN NO.) 

1 MEDIAN') 

WRITE (6,35) RMIN,IMIN,RMAX,IMAX,RMED 

35 FORMAT (1 3X,F13.5,113,F13.5, 113, F10.5) 

WRITE (6,36) 

36 FORMAT (/25X,'MOMENTS ABOUT THE MEAN') 

WRITE (6,37) 

37 FORMAT (18X,'2ND-MOMENT 3RD-MOMENT 4TH-MOMENT') 

WRITE (6,38) UM2,UM3,UM4 

38 FORMAT (16X,3E15.7) 

WRITE (6,39) 

39 FORMAT (23X,•CUMULATIVE FREQUENCY TABLE') 

WRITE (6,40) 

40 FORMAT (19X,* CLASS-NO. UPPER-LIMIT %CUM-FREQ. GROSS-NUMBER•) 

DO 45 111=1,31 

45 WRITE (6,50) III,CLASS (III),CFREQ(III) ,NGROSS (III) 

50 FORMAT (15X,I13,2F13.5,1I13) 

WRITE (6,1505) 

1505 FORMAT ('1',//////26X,'TOTAL POPULATION: PHI FACTORS') 

WRITE (6,519) NS,NRU 

C STATISTICAL ANALYSIS CF PTHEORY/PACI FOR ALL E/H VALUES COMBINED 
L=0 

DO 7 1=1,15 

IF (I.EQ.1) EOH2=0.0 

IF (I.GT.1) EOH2=EOH1(1-1) 

IF (I.EQ.15) EOH2=99.99 
WRITE (6,1515) EOH2 

1515 FORMAT (/26X,'***EOH=',F5.2,'***'/) 

WRITE (6, 1500) (RP(I,JJ), JJ = 1,NS) 

1500 FORMAT (21X,5F10.5) 

DO 7 JJ- 1,NS 













































■ 


































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231 

232 

233 

234 

235 


1 89 


L=L + 1 

7 X(L)= RP (I,JJ) 

N = L 

CALL STAT (RM,SD,COV,COS,COK,RMIN,RMAX,0M2,UM3,UM4,IMAX,IMIN,RMED, 
1CI,ASS,CFREQ,NGROSS) 

DC 11 1=1,15 
IHAX=IMAX-NS 
IF (IMAX.LE.O) GO TO 12 

11 CONTINUE 

12 IMAX=IMAX+NS 
DO 13 1=1,15 
IMIN=IMIN-NS 

IF (IMIN.LE.O) GO TO 14 

13 CONTINUE 

14 IMIN=IMIN+NS 
WRITE (6,21) 

21 FORMAT ('1',//////26X,• <*> 

(6,519) NS,NRU 

( 6 , 20 ) 


FORMAT 

WRITE 

WRITE 

WRITE 

WRITE 

WRITE 

WRITE 

WRITE 

WRITE 

WRITE 

WRITE 

DO 55 

WRITE 


TOTAL STATISTICAL EVALUATION <*>') 


(6,25) 

(6,30) 

(6.35) 

(6.36) 

(6.37) 

(6.38) 

(6.39) 

(6.40) 

1=1,31 

55 WRITE (6,50) 

1600 CONTINUE 

WRITE (6,1900) JJ 

1900 FORMAT {'1 • ,/20X,'***',15,'***') 

STOP 

END 

C***************************************************** 


RM,SD,COV,COS,COK 

RMIN,IMIN,RMAX,IMAX,RMED 

UM2,UM3,0M4 

I,CLASS (I) ,CFREQ (I) ,NGROSS(I) 


C 

C***************************************************** 

SUBROUTINE TKMEAN (RMEAN,RMEAN1,RMEAN2,NS) 

COMMON N,EOH1 (13) ,PO,BMO,DCS,DTS 

COMMON FC,FY,ES,BB,H,DC,DD,AS,AS11,B11,D11,S,C,ZZ,NRU 
COMMON PHI,EO,J,Z,ECC,EY,FCONCC(20) ,ASC,EOH (40) ,FCONST(25) 
COMMON X (16 000) , EC (20) ,B (20) ,P (4 0) ,BMM (1000) ,BM (40) ,FCS (20) 
COMMON FST,E (20) ,NB,DS (20) ,ASB (20) ,FSS (20) ,SBM (20) 

DIMENSION RMEAN(25) 

C SET EACH VARIABLE EQUAL TO ITS MEAN VALUE 
FC=R MEAN 1 

FY=(RMEAN2-4000.0)*0.97 
BB = RMEAN (3) 

H = RMEAN (4) 

B11 = RMEAN (5) 

Dll = RMEAN (6) 

DC=RMEAN (7) 

DD = RMEAN (8) 

DO 2 1=1,NB 
NI=I+8 

2 DS (I) = RMEAN (NI) 

C CALCULATE THEORETICAL INTERACTION DIAGRAM 
CALL THEORY 

C WRITE MEAN THEORY INTERACTION DIAGRAM 
WRITE (6,100) 














































































236 

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242 

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261 

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285 

286 

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288 

289 

290 

291 

292 

293 

294 

295 


19C 


100 FORMAT (' 1 ' ,//////19X, • ****ME\N THEORY INTERACTION DIAGRAM****') 
WRITE (6,514) NS,NRU 

519 FORMAT (31X,• (•,14, 'SIM',15,•)•//) 

WRITE (6,101) 

101 FORMAT (//I 9 X, ' P (J) LBS ' , 6 X, • M (J) LB-IN',7X,•EOH (J) ') 

DO 4 J = 1,N 

WRITE (6,102) P (J) ,BM (J) ,EOH (J) 

102 FORMAT (/16 X , 3E 1 5.7) 

4 CONTINUE 

WRITE (6,103) 

103 FORMAT (//20X,'PO LBS',7X,'BMO LB-IN') 

WRITE (6,104) PO , B MO 

104 FORMAT (16X,2E15.7) 

RETURN 

END 

C***************************************************** 

c 

c***************************************************** 

SUBROUTINE ACI (NS) 

C 

C THIS SUBROUTINE CALCULATES THE ACI INTERACTION DIAGRAM 

C 

COMMON N,EOH1(13),PO,BMO,DCS,DTS 

COMMON FC,EY,ES,B3,H,DC,DD,AS,AS 11,B11,D11,S,C,ZZ,NRU 
COMMON PHI,EO,J,Z,ECC,EY,FCONCC (20) ,ASC,FOH(40) ,FCONST (25) 

COMMON X (16000) ,EC (20) ,E (20),P (40),BMM(1000),BM(40) ,FCS (20) 

COMMON FST,E (20) ,NB,DS (20) ,ASB(20) ,FSS(2C) ,SBM (20) 

IF (FC.LE.400C.0) GO TO 1 
B1=0.85-0.05*(FC-4CGC.0) / I 000.0 
IF (B1.LE.0.65) B 1 = 0.6 5 
GO TO 4 

1 B1=0.85 
4 E4=0.003 

C CALCULATE PURE AXIAL LOAD CAPACITY 
PO=0.85*FC*(BB*H-AS)+AS*FY 

C CALCULATE AXIAL LOAD CAPACITY AT BALANCED CONDITIONS 
PB=0.85*E1*FC*EB*DTS* (0.00 3/(FY/ES + 0.003)) 

C CALCULATE PURE MOMENT CAPACITY 
AST=AS-ASC 
AA=BB*FC*B1*0.85 
AB=0.003*ASC*ES“AST*FY 
AC=-0.003*ASC*ES*DCS 
RA=SQPT(AB**2-4.0*AA*AC)/(2.0*AA) 

C1=(-AB/(2.0*AA))-RA 

IF (Cl . I.E .0.0) C 1 = R A- (AB/ (2. 0*AA) ) 

ES2 = 0.003* (Cl-DCS)/Cl 
IF (ES2.GE.(FY/ES)) GO TO 9 

BMO= ASC*ES2 *F,S* (DTS-DCS) + ( AST*FY-ASC*ES2 *ES) * (DTS-B1 *C 1/2.0) 

GO TO 8 

9 BMC= ASC*FY*(DTS-DCS) + ( (AST-ASC)*FY)*(DTS-(AST-ASC) *FY/(1.7*FC*BE)) 
C INITIALIZE STRAIN IN TENSION STEEL 
8 E1=0.0019 
J =0 

2 J=J+1 

IF (J.EQ.1) GO TO 5 
C MODIFY TENSION STEEL STRAIN 

IF (El.GT.-0.001) E1=E1-0.0005 
IF (El .LF.-0.001) E1= E1-0.001 

C CALCULATE NEUTRAL AXIS DEPTH 
C = F4*DD/ (E4-E1) 






















































































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297 

298 

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355 


non on 


191 


PHI=E9/C 

C CALCULATE FORCES IN STEEL BARS 
CAIL ASTEEL (E4) 

C CALCULATE CONCFETE COMPRESSIVE BLOCK FORCE 
IF (C.GE.(H/B1)) C=H/B1 
FCCONC=0.85*FC*B1*BB*C 
C CALCUI.ATE AXIAL LOAD LEVEL 
P(0)=FCCONC+FST 

C CALCULATE BENDING MOMENT DUE TO CONCFETE COMPRESSIVE FORCE 
COMPM=FCCONC*(DD-B1*C/2.0) 

SM = 0.0 

C CALCULATE BENDING MOMENT DUE TO STEEL FORCES 
DO 3 1=1,NP 

SBM (I) = (FSS (I) -FCS (I) * ASB (I) ) * (DD-DS (I) ) 

3 SM=SM+SBM(I) 

C CALCUI.ATE TOTAL BENDING MOMENT CAPACITY 
BK(0)=C0MPM*SM-P(J)*(DD-H/2.0) 

GO TO 6 

5 P(J)= 0.85*FC* ( (BB*H)-AS) ♦ AS*FY 
BM (J)=0.0 

C CALCULATE ECCENTRICITY E/H 

6 EOK (J) =BM (J) / (P (J) *H) 

IF (J.GE.20) GO TO 7 

IF (EOH (J) .LT.2.0) GO TO 2 

7 N=J 

C WRITE THE ACI INTERACTION DIAGRAM 
WRITE (6,100) 

100 FORMAT ('1',//////23X,'****ACI INTERACTION DIAGRAM****') 
WRITE (6,519) NS,NRU 

519 FORMAT (3 1X, • (' , 14 , * SIM • ,15, •) • //) 

WRITE (6,101) 

101 FORMAT (//19X, ' P (0) LBS ' , 7 X, ' P (B) LBS • , 6 X , • M (0) LB-IN') 
WRITE (6,102) PO,PB,3 MO 

102 FORMAT (/I6X,3E15.7) 

WRITE (6,103) 

103 FORMAT (//19X,'P(J) LBS ' ,6X,' M' (J) LB-IN • ,7X, • EOH (J) •) 

DO 20 J=1, N 

WRITE (6,104) P (J) ,BM (J) ,EOH (J) 

104 FORMAT (/16 X , 3E 1 5.7) 

20 CONTINUE 

RETURN 

END 

£*********$******************************************* 

SUBROUTINE ASTEEL (E4) 

THIS SUBROUTINE CALCULATES THE ACI FORCES IN THE STEEL 
COMMON N,EOH1 (13) , PO,BMO,DCS,DTS 

COMMON FC,FY,ES,BB,H,DC,DD,AS,AS 11,B11,D11,S,C,ZZ,NFU 
COMMON PHI,EO,J,Z,ECC,EY,FCONCC(20) ,ASC,EOH (4 0) ,FCONST (2 5) 
COMMON X (16000) , EC (2 0) , B (20) , P (40) , BMM (1 000) , BM (90) ,FCS (20) 
COMMON FST,E (20) ,NB,DS (20) ,ASB (2 0),FSS (2 0) ,SBM (2 0) 

EY=FY/ES 

FST = 0.0 

DO 4 1=1,NP 

E (I) =E4-PHI*DS (I) 

IF (DS (I) .GE.C) GO TO 5 
FCS(I)=0.85*FC 












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192 


FSC=-FCS (I) * AS B (I) 

GO TO 8 
5 FSC=C.0 
FCS (I)=0.0 

8 IF (E (I) .GE.EY) GO TO 1 
IF (E (I) . LE.-EY) GO TO 2 
GO TO 3 

1 FSS (I) =FY*ASB (I) 

GO TO 4 

2 FSS (I) =-FY* ASB (I) 

GO TO 4 

3 FSS (I) =E (I) *ES*ASB (I) 

4 FST=FST+FSS (I) +FSC 
RETURN 

END 

C******** ********************************************* 

c 

C** *************************************:t<*** ********** 

SUBROUTINE PROP (NS) 

THIS SUBROUTINE READS AND WRITES THE COLUMN PROPERTIES 
COMMON N.EOH1 (13) ,PO,BMO,DCS,DTS 

COMMON FC,FY,ES,BB,H,DC,DD,AS, AS 11, B1 1, D11 , S , C , ZZ , NF. U 
COMMON PHI,EC,J,Z,ECC,EY,FCCNCC (20) ,ASC,EOH(4 0) ,FCONST(25) 
COMMON X (1 6000) ,EC (20) , B (2 0) , P (40) ,BMM (1000) ,BM (40) ,FCS (20) 
COMMON FST,E (20) ,NB , DS(20) ,ASB(20) , FSS(20) , SBM(20) 

C FC=CONCRETE STRENGTH 

C FY=STEEL STRENGTH (PSI) 

C ES=STEEL MODULUS OF ELASTICITY (PSI) 

C BB=CROSS SECTION WIDTH (IN) 

C H=CROSS SECTION DEPTH (IN) 

C DC=DISTANCE TO THE COMPRESSION STEEL (IN) 

C DD=DISTANCE TO THE TENSION STEEL (IN) 

C ASC=AREA OF COMPRESSION STEEL (SQ IN) 

C AST=AREA OF TENSION STEEL (SQ IN) 

C AS=TOTAL AREA OF STEEL (SQ IN) 

C AS11=AREA OF STIRRUP (SQ IN) 

C B11 =VIIDTH OF CORE (IN) 

C D11=DEPTH OF CORE (IN) 

C S=SPACING OF STIRRUPS (IN) 

READ (5,100) BB,H,DD,DC,AS,ASC 

100 FORMAT (6F5.2) 

READ (5,101) FC,FY,ES 

101 FORMAT (3F10.0) 

READ (5,102) DCS,DTS,S,B11,D11,AS11 

102 FORMAT (6F5.2) 

READ (5,110) NB 

110 FORMAT (113) 

DO 1 1=1,NB 

READ (5,111) ASB (I) , DS (I) 

111 FORMAT (2F5.2) 

1 CONTINUE 

WRITE (6,103) 

103 FORMAT ('1 1 ,//////3CX,'COLUMN CROSS SECTION PROPERTIES’) 
WRITE (6,519) NS,NRU 

519 FORMAT (31X,'(',14,'SIM’,I5,')'//) 

WRITE (6, 104) • 

104 FORMAT (16 X,’FC (PSI) • ,3X,•FY (PSI) ',5X,•ES (PSI)’) 

WRITE (6,105) FC,FY,ES 












































































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105 FOFKAT (/16X,1F6.1,5X,1F7.1,4X,1F10.1) 

WRITE (6,106) 

106 FORMAT (//16 X , • B (IN) • , 5X , • H (I N) • , SX , ' D (IN) • , 4X , • DC (I N) • 

1 ,4X,•AS (SCIN) ',3X , *ASC (SQIN) ') 

WRITE (6,107) RB,H,DD,DC,AS,ASC 

107 FORMAT (/1 2 X, 6F1 0.2) 

WRITE (6,108) 

108 FORMAT (//16X,* DCS(IN) ',2X,'DT3(IN) ',4X,'S (IK) *,4X,'B 1 1 (IN) •, 
14X,•Dll (IN) •,2X,'AS 11 (SQIN)•) 

WRITE (6,109) PCS,DTS,S,B11,D11,AS11 

109 FORMAT (/I2X,6F10.2) 

WRITE (6,112) 

112 FORMAT (//16X, • NB« ,4X, • ASB (I) • ,5X,+ DS (I) •) 

DO 114 1=1,NF 

WRITE (6,113) KB , ASB (I) , DS (I) 

113 FORMAT (15X, 113,2F10.2) 

114 CONTINUE 
RETURN 
END 



SUBROUTINE CURVE 


THIS SUBROUTINE FITS A POLYNOMIAL TO THE INTERACTION 
DIAGRAM 

COMMON N,EOH1 (13) ,PO,BMO,DCS,DTS 

COMMON FC,FY,ES,BB,H,DC,PD,AS,AS11,B11,D11,S,C,ZZ,NRU 
COMMON PHI,EO,J,Z,ECC,EY,FCONCC(2 0) ,ASC,EOH(40) ,FCONST (25) 
COMMON X (16000) ,EC(20) ,B(20) ,P(40) ,BMM (1 000) , BM (40) ,FCS (20) 
COMMON FST,E (20) ,NB,DS (20) ,ASB (20) ,FSS (20) ,SEM (20) 

DIMENSION DI (4 00) ,D(7C) , SB (10) ,T (10) ,COE (11) 

DIMENSION XBAR (11) ,STD (11) ,SUMSQ (11),ISAVE(11) ,ANS (10) 
DIMENSION XX (500) ,BBB (10),EE (10) 

M = 1 0 

EOH1 (1)=0.05 
EOH1(2)=0. 10 
EOH1 (3)=0.15 
EOH1 (4)=0.20 
EOH1 (5)=0.30 
EOH1 (6)=0.40 
F.OK1 (7) =0.50 
EOH1 (8)=0.60 
EOH1 (9)=0.70 
EOH 1 (10)=0.80 
EOH1 (11)=0.90 
EOH1 (12)=1.00 
EOH 1 (13) =1.50 
N = N- 1 

DO 2 J = 1 , N 
P(J) =P (J+1) 

BM (J) =BM (J+1) 

2 EOH (J) =EOH (J+1) 

DO 10C 1=1,N 
DIFil = BM (I + 1) -BM (I) 

IF(DIFM) 105,100,100 
100 CONTINUE 
105 N P=I 

EOHE = ECH (I) 




















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194 


mi = m 


IF (M.GE. (NP-1)) M = NP-2 
L=NP*n 

DO 110 1=1,NP 
J = L ♦ I 

XX (I) = EOH (I) 

110 XX(J)=P(I) 

LL=L+NP 

CALL GDATA (NP,H,XX,XBAR,STD,D,SUMSQ) 

MM = M -*■ 1 


SUM = 0.0 
DO 200 1=1,n 
ISAVE (I) =1 

CALL ORDER ( HP!, D , MM, I , IS AV E, DI, EE) 

CALL H-INV (DI, I, DET,BBB,T) 

CALL TMULTR (NP,I,XBAR,STD,SUMSQ,DI,EE,ISAVE,BBB,SB,T,ANS) 
IF (ANS (7)) 220,130,130 
130 SUHIP = ANS (4)-SUM 

IF (SOMIP) 220,220,150 
150 SUM = ANS (4) 

COE (1) =ANS (1) 

DO 160 J = 1,I 
160 COE (J+1) =BBB (J) 

11=14-1 

JJ=I41 

200 CONTINUE 
220 NN=13 

DO 240 11=1,NN 
ECH2=EOHl (II) 

IF (EOH2.GT.EOHB) GO TO 250 
P (II)=0.0 
DO 245 1=1,JJ 

245 P (II) =P (II) +COE (I) * (EOH1 (II) ** (1-1) ) 

240 CONTINUE 
250 NNN=II 

DO 101 I=NP,N 
DIFE=3.0-EOH (I) 

IF (DIFE) 102,101,101 

101 CONTINUE 
GO TO 103 

102 N=I—1 

103 NM=N-NP4l 
M = M 1 

IF (M.GE. (NM-1)) M=NM-2 
L=NH*M 

DO 310 1=1,NM 
J = L + I 

XX(I)=1.0/EOH (I + NP-1) 

310 XX(J)=BM(I+NP-1) 

CALL GDATA (NM,M,XX,XBAR,STD,D,SUMSQ) 

M M = M♦ 1 


330 


SUM=0.0 
DO 300 1=1,M 
ISAVE (I) =1 

CALL ORDER (MM,D,MM,I,ISAVE,DI,EE) 
pa t T MTNV ET/ B E B /T ) 

CALL ?SSlTR (N-M.I,XBAR, S TD, S UM S Q, D I,EE,ISAVE, B BB,S B ,T,ANS) 

IF (ANS (7)) 320,330, 330 

SUMIP= ANS(4)-SUM 
IF (SUMIP) 320,320,350 























































































































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195 


350 SUM=ANS(4) 

COE (1)= ANS (1) 

DO 360 J = 1,I 
360 COE (J+ 1) =BBB (J) 

JJ=I+1 

300 CONTINUE 
320 DO 340 II=NNN,NN 
BM (II)=0.0 
DO 345 1=1,JO 

345 BH (II) =BM (II) + COE (I) * ( (1.0/EOH1 (II))** (I-1)) 

P (II) = BM (II) / (H* EOH1 (II)) 

340 CONTINUE 
N4=NNN—1 
DO 370 1=1,N4 
370 BM(I)=P(I) *H*EOH1 (I) 

RETURN 

END 

C***************************************************** 

C 

C**************************»************************** 

SUBROUTINE RANDOM (IY,SD,SB,CONST,ITP, V) 

C 

C THIS SUBROUTINE GENERATES VALUE OF THE VARIABLES KITH 
C THE MEAN, STANDARD DEVIATION, AND DISTRIBUTION GIVEN 

C 

COMMON N,EOHl (13) ,PO,BMO,DCS,DTS 

COMMON FC,FY,ES,BB,H,DC,DD,AS,AS 11,B11,D11,S,C,ZZ,NRU 
COMMON PHI,EO,J,Z,ECC,EY,FCONCC(20),ASC,EOH(40),FCONST(25) 
COMMON X (16C00) , EC (20) ,B (20) ,P (4 0) ,BMM (1000) ,BM (40) ,FCS (20) 
COMMON FST,E (20) ,NB,DS (20) ,ASB (20),FSS(20) ,SBM(20) 

A=0.0 

DO 50 1=1,12 
IY=IY*65539 

IF (IY.LT.0) IY=IY+2147483647+ 1 

Y=IY*0.4656613E-9 
50 A= A ♦ Y 
A=A - 6.0 
V=A*SD+RM 

IF (ITP.EQ.1) V= 10.0** V 

IF (ITP.EQ.2) V=10.0**V+CONST 

RETURN 

END 

C ****************************************** *********** 

C 

£***************************************************** 

SUBROUTINE THEORY 


C 

c 

c 


c 


THIS SUBROUTINE CALCULATES THE THEORETICAL P-M DIAGRAM 


N,EOH 1(13) ,PO,BMO,DCS,DTS 

FC,FY,ES,BB,H,DC,DD,AS,AS11,B11,D11,S,C,ZZ,NRU 

PHI,EO,J,Z,ECC,EY,FCONCC (20) ,ASC,EOH(4 0) ,FCONST (2 5) 

X(16000) ,EC (20) ,B (20),P(40),BMM(1000),BM(40) ,FCS (20) 

FST, E (20) , NB, DS (2 0) , ASB (20) , FSS (20) ,SBP1 (20) 

FC*0.85 

(FC.EQ.1000.0) FC=1000.1 


COMMON 
COMMON 
COMMON 
COMMON 
COMMON 
FC 
IF 

ECC=5700C.0*SQRT (FC) 


EO=1.8 *FC/ECC 
EY=FY/ES 

CALCULATE PUPE AXIAL LOAD CAPACITY 







































































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n n n no 


196 


PO=FC*BE*H+AS*FY-AS*FC 

J=0 

1 J=J*1 

C SET AXIAL LOAD LEVEL 

IF (J. EQ. 1) GO TO 5 
IF (J.EQ.21) P (J) =0.0 

IF (P (J-1) .LE. (0.6*PO) ) P(J)=P(J-1)-0.034*PO 
IF ( P (J-1) .LE. (C. 1 *PO) ) P(J) = P (J -1) -0 . 04 * PC 
IF (P (J- 1) ,GT. (0.6 + PO) ) P (J) =P (J- 1)-0. 16*PO 
IF (P (J) .LE.0. C) P (J) =0.0 
IF (J.EQ.2) P(J) =P(J-1)-0.08*PO 
C CALCULATE MOMENT CAPACITY AT SPECIFIED AXIAL LOAD 
CALL AXIAL 

IF (P (J) .EQ.0.0) BMO=BM (J) 

IF (P(J) . EQ. 0.0) GO TO 7 
GO TO 6 

5 P(J)=PO 
BM (J)=0.0 

6 EOH (O) =BM (J)/(P (J) *H) 

GO TO 1 

7 N=J-1 

C ELIMINATE EREATIC POINTS ON THE INTERACTION CURVE 
M = N 

HJ = N- 1 

DO 8 I J=3,NJ 

IF (BM (IJ) .GE.BM (IJ-1) ) GO TO 8 
IF (BM (IJ+ 1) .LE.BM (IJ- 1) ) GO TO 8 
M = M-1 

DO 9 JJJ=IJ, N.I 
P(JJJ)=P (JJJ + 1) 

BM (JJJ) =BM (JJJ+ 1) 

9 EOH (JJJ) =EOH (JJJ* 1) 

8 CONTINUE 
N = M 

RETURN 

END 

C********$ A***************************************** 

***************************************************** 

SUBROUTINE AXIAL 

THIS SUBROUTINE CALCULATES THE MOMENT AFTER BALANCING P 
COMMON N,EOH1(13) ,PO,BMO,DCS,DTS 

COMMON FC,FY,ES,BB,H,DC,DD,AS,AS 11,311,D11,S,C,ZZ, NRU 
COMMON PHI,EO,J,Z,ECC,EY.FCONCC(20),ASC,EOH(40),FCONST(25) 
COMMON X (16000) ,EC (2 0) ,B (2 0) ,P (4 0) ,3MM(1000) ,BM (4 0) ,FCS (20) 
COMMON FST,E (2 0) ,NB,DS (20) ,ASB (2 0) ,FSS (20) ,SBM (20) 

PK1=0.0000001 
PHIH=PHI*H 
11= 1 

14 E4=0.002 
EINCR=0.002 
33 E4 = E4-El NCR 

ElNCR=EIMCP/2.0 
32 E4 = E4 + EINCF( 

,FCCOffC = 0.0 
C=F4/PHI 
ECO= (C-H)* PHI 
IF (C.GE.H) C=H 























































































































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7 13 

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7 15 


197 


IF (C.LT.H) ECO=0.0 
ASC=0.0 
DO 34 1=1,NB 

IF (DS(I).LE.C) ASC=ASC*ASB(I) 

34 CONTINUE 

C CALCULATE PARAMETERS OF THE CONCRETE STRESS STRAIN CURVE 
P11=(2.0*(B11«-D11) * AS 1 1 ♦ ASC*S) /(B11*D11*S) 
E5CH=0.75*P11*SQRT(B11/S) 

E50U=(3.0+0.002*FC)/(FC-1000.0) 

IF (E50U.LE.0.0) E50U=0.06 

Z=0.5/(E50H«-E50U-EO) 

ZZ=0.5/(E50U-EO) 

EU = 2.0* (E50H+E50U)-EO 
DX=C/10 

C CALCULATE THE CONCRETE COMPRESSION BLOCK FORCE 
DO 23 1=1,10 
AI = I 

X (I) =C-AI* DX + DX/2 
EC (I) = PHI* X (I) «• ECO 
B (I) =BB 

C MAXIMUM STRAIN FOR UNCONFINED COMPRESSION 0.004 
IF (EC (I) .LE.EO) GO TO 3 
IF (EC (I) .GE.0.004) GO TO 21 
IF (EC (I) .GT.EO) GO TO 4 

3 FCC= FC*(2.0*EC(I)/EO-(EC (I)/EO)**2) 

GO TO 22 

4 FCC=FC* (1. 0- Z* (EC (I)-EO) ) 

ECU = FC* (1.0-ZZ* (EC (I)-EO)) 

IF (FCC.LE.0.0) FCC=0.0 
IF (FCU.LE.0.0) FCU=0.0 

FCONCC (I) =FCC*DX*B11*FCU*DX* (B (I) -B 11) 

GO TO 23 

21 B (I) =B 11 

FCC=FC* (1.0-Z* (EC (I) -EO) ) 

IF (FCC.LE.0.0) FCC=0.0 
IF (X (I) .GE. (C-DC) ) ?CC=0.0 

22 FCONCC (I)=FCC*DX*B(I) 

23 FCCONC=FCCCNC+FCONCC(I) 

C CALCULATE THE CONCRETE TENSION BLOCK FORCE 
IF (C.GE.H) GO TO 25 
SFC = SQRT (FC) 

ET=7.5*SFC/ECC 
TC=ET/PHI 
TCA=H-C 
RTC = TC A/TC 

IF (TC.GT.TCA) TC=TCA 
IF (TC.LE.TCA) RTC=1.0 
FCONCT=-RTC*7.5*SFC*TC/2.0*BB 
GO TO 18 
25 FCONCT=0.0 
18 CALL FSTEEL (E4) 

C CHECK FORCE COMPATIBILITY 

PAXIAL=FCCONC*FCONCT+FST 

TOLA = P (J)*0.02 

IF (P(J) .EQ.0.0) TOLA= 0.001*PO 
TOL=P (.7) -PAXI AL 
IF (TOL.LT.-TOLA) GO TO 33 
IF (TOL.GT.TOLA) GO TO 35 
GO TO 36 

35 IF (F4.GE.EU) GO TO 44 






























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198 


IF(EINCR.GE.0 .0C00001) GO TO 32 

36 compm=o.o 

C CALCULATE THE MOMENT DUE TO THE CONCRETE COMPRESSION FORCE 
DO 24 1=1,10 

24 COMP M = COMPM♦FCONCC(I)* (DD-C+X(I)) 

SM = 0.0 

C CALCULATE THE MOMENT DUE TO THE STEEL FORCES 
DO 13 I=1,NB 

SBM(I) = (FSS(I)-FCS (I) ♦A SB (I)) *(DD-DS(I)) 

13 SM=SM+ SBM (I) 

IF (FCONCT.EQ.0.0) TC=0.0 
C SUM THE MOMENTS ABOUT THE TENSION STEEL 

BMM(II)=COMPM+SM+FCONCT* (DD- (C+2.0+TC/3.0))-P (J)*(DD-H/2.0) 
IF (II.EQ.1) GO TO 17 
TOLBMA = ABS (BMM (II- 1)*0.01) 

BMTOL=BMM (II)-BMM (II-1) 

IF (BMM(II).LE.0.0) GO TO 42 
IF (BMTOL.GE. (0.5*TOLBMA)) GO TO 41 
IF (BMTOL.LE.-TOLBMA) GO TO 42 
GO TO 16 
17 PHINCR=0.001 
GO TO 41 
44 E4=0.001 

PHIH=PHIH—PHINCR 
PHINCR=PHINCR/5.0 
PHIH=PHIH+PHINCR 
PHI=PHIH/H 
ElNCR=EINCR/2.0 
GO TO 32 

42 PHIH=PHIH-PHINCR 
PHI NCR = PHINCR/5.0 
41 PHIH=Ph'IH + PHINCR 
PHI=PHIH/H 
11=11+1 
GO TO 14 

16 BM (J) =BMM (II-1) 

CONTINUE 

RETURN 

END 

Q& &$#*# +$*******#******** *********************** £#*■$*# 

c 

SUBROUTINE FSTEEL (E4) 

C 

C THIS SUBROUTINE CALCULATES THE THEORY FORCES IN THE STEEL 

C 

COMMON N,EOHl (13) ,PO,BMO,DCS,DTS 

COMMON FC,FY,ES,BB,H,DC,DD,AS,AS11,B11,D11,S,C,ZZ,NRU 
COMMON PHI,EO,J,Z,ECC,EY,FCONCC(20),ASC,EOH(40),FCONST(25) 
COMMON X(16000) ,EC (20) ,B (20) ,P (40),BMM (1000) ,BM (4 0) ,FCS (20) 
COMMON FST,E (20) ,NB,DS (20) ,ASB (2 0) ,FSS (2 0) ,SEM (20) 

FST=0.0 

DO 4 1=1,NB 

E(I)=E4-PHI*DS(I) 

IF (DS (I) .GE.C) GO TO 5 
IF (E (I) .GT. EO) GO TO 6 
FCS(I)= FC* (2.0*E (I)/EO-(E(I)/EO) **2) 

GO TO 7 

6 FCS(I)=FC*(1.0-Z*(E (I)-EO)) 

IF (FCS (I) . LE. 0.0) FCS (I) =0.0 




































































































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835 


199 


7 FSC=-FCS (I) *ASB (I) 

GO TO 3 

5 FSC=0.0 
FCS(I)=0.0 

8 E (I) =- F, (I) 

IF (E(I).GE.EY) GO TO 2 
IF (E (I) . LF.-EY) GO TO 1 
GO TO 3 

1 FSS (I)=FY*ASB(I) 

GO TO 4 

2 FSS (I)=-FY*ASB(I) 

GO TO 4 

3 FSS (I) =-E (I) *ES*ASB (I) 

4 FST=FST + FSS (I) +FSC 
RETURN 

END 

C+******* : *‘*****’** , $'*<'******** ; < c *************’$ l ’t t >!'<‘***** ; fc , l‘ I ! < 

c 

C********** ; * i **'S'*******’<‘******************** ; <'******#*** 

SUBROUTINE STAT (Fa,STDV,COV,COS,COK,RMIN,EMAX,UM2,UM3,UH4,IMAX,IM 
1IN,BMED,CLASS,CFFEQ,NG ROSS) 

C 

C THIS SUBROUTINE CALCULATES THE MEAN, COEFFICIENT OF 
C VARIATION, COEFFICIENT CF SKENNESS, COEFFICIENT OF 
C KUPTOSTS,AND CUMULATIVE FREQUENCY TABLE 
C 

COMMON N,EOH 1 (13) ,PO,BMO,DCS,DTS 

COMMON FC,FY,ES,BB,H,DC,DD,AS,AS11,B11,D11,S,C,ZZ,NFU 
COMMON PHI,EO,J,Z,ECC,EY,FCONCC(20) ,ASC,EOH(40) ,FCONST (25) 

COMMON X (16000) ,EC (20) ,B (2 0) ,P (4C),EMM(1000),BM(40) ,FCS (20) 

COMMON FST,E (2 0) ,NB,DS (20) ,ASB (2 0) ,FSS (2 0) ,SBM (20) 

DIMENSION CLASS (31) ,CFREQ (31) 

DIMENSION Y (800 2) ,DIF(1600 0),NGFOSS(31) 

SUM=0.0 
DO 10 1=1,N 
10 SUM= SUM + X(I) 

RM=SUM/N 
S UM = 0.0 
DO 20 1=1,N 

20 SUK=SUM+(X(I)-RM )**2 
U M2 = SUM/N 

STDV = SQPT (SUM/ (N- 1) ) 

COV= STDV/R M 
SUM=C.0 
DO 30 1=1,N 

30 SUM=SUM+ (X (I)-RM) **3 
UM3=SUM/N 

COS = SUM/(N* (ST DV ** 3) ) 

S0M=0.0 
DO 40 1=1,N 

40 SUM=SUM+ (X (I)-RM) **4 
UM4 = SUM/N 

COK = SUM/ (N* (STDVv + U)) 

RMAX = X (1) 

IMA X=1 
DO 50 1=2,N 

IF (RMAX. T.T. X (I) ) IMAX = I 
50 RMAX = X (IMAX) 

RMIN = X (1) 

IMIN =1 










































































































836 

837 

838 

839 

840 

841 

842 

843 

844 

845 

846 

847 

848 

849 

850 

851 

852 

853 

854 

8 55 

856 

857 

858 

8 59 

860 

861 

862 

863 

864 

865 

866 

867 

868 

869 

870 

871 

87 2 

873 

874 

875 

877 

878 

879 

880 

881 

882 

883 

884 

885 

886 

887 

888 

889 

890 

891 

892 

893 

894 

895 

896 


200 


DO 60 1 = 2,N 

IF (RMIN.GT. X (I) ) IMIN=I 
60 R MIN = X(IMIN) 

ClNT=0.05 
CLASS (1)=0.50 
DO 70 1=2,30 

70 CLASS (I) =CLASS (I- 1) ♦CINT 
CLASS(31)=2.0 
DO 90 11=1,31 
NPOINT = 0 
DO 80 1=1,N 

80 IF (CLASS (II) .GT. X (I) ) NPOINT= NPOINT-*-1 
NGROSS (II)=NPOINT 
90 CFEEQ (II) = (100.0*NPOINT)/N 
DO 100 1=1,31 

100 CLASS (I)=CLASS(I)-0.00001 

Y (1) =PBIN 
N1=N/2+2 

DO 66 11=2,N1 
NPOINT=1 

DIF (1) =X (1) -Y (II- 1) 

DO 65 1=2,N 

IF (X (I) .EQ.0.0) GO TO 65 
DIF (I) =X (I)-Y(II-I) 

IF (DIF (I) .LT.DIF(NPOINT)) NPOINT=I 

65 CONTINUE 

Y (II)=X (NPOINT) 

X(NPOINT)=0.0 

66 IF (NPOINT.EQ.1) X (NPOINT)=RMAX*1.0 
N 1 = N 1-1 

DO 67 1=1,N1 

67 Y (I) = Y (I* 1) 

RMED = (Y (N1— 1)+Y(N1))/2.0 
RN = (N/2.0)-(N/2) 

IF (RN. GT.0.1) RMED=Y(N1) 

RETURN 

END 

C SUBROUTINE MULTR 

C* ********************************************** ****** 

c 

C PURPOSE 

C PERFORM A MULTIPLE LINEAR REGRESSION ANALYSIS FOR A 

C DEPENDENT VARIABLE AND A SET OF INDEPENDENT VARIABLES. THIS 

C SUBROUTINE IS NORMALLY USED IN THE PERFORMANCE CF MULTIPLE 

C AND POLYNOMIAL REGRESSION ANALYSES. 

C 

C USAGE 

C CALL MULTR (N,K,XBAR,STD,D,RX,RY,ISAVE,B,SB,T,ANS) 

C 

C DESCRIPTION OF PARAMETERS 

C N 

C K 

C XBAR 

C 

C STD 

C 

C D 

C 
C 


- NUMBER OF OBSERVATIONS. 

- NUMBER OF INDEPENDENT VARIABLES IN THIS REGRESSION. 

- INPUT VECTOR OF LENGTH M CONTAINING MEANS OF ALL 

VARIABLES. H IS NUMBER OF VARIABLES IN OBSERVATIONS. 

- INPUT VECTOR OF LENGTH M CONTAINING STANDARD DEVI¬ 

ATIONS OF ALL VARIABLES. 

- INPUT VECTOR OF LENGTH M CONTAINING THE DIAGONAL OF 
THE MATRIX OF SUMS OF CROSS-PRODUCTS OF DEVIATIONS 
FROM MEANS FOR ALL VARIABLES. 
























































































897 

898 

899 
90C 

901 

902 

903 
909 

905 

906 

907 

908 

909 

910 

911 

912 

913 
919 

915 

916 

917 

918 

919 

920 

921 

922 

923 
929 
926 
9 27 

928 

929 

930 

931 

932 

933 
939 

935 

936 

937 

938 

939 

990 

991 
991.1 

992 

993 
999 

995 

996 

997 

998 

950 

951 

952 

953 
959 

955 

956 

957 


C 

C 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 


c 

c 

c 

c 

c 

c 

c 

c 


RX - INPUT MATRIX (K X K) CONTAINING THE I 
INTERCORRELATIONS AMONG INDEPENDENT V 
RT - INPUT VECTOR OF LENGTH K CONTAINING I 
TIONS OF INDEPENDENT VARIABLES WITH D 
VARIABLE. 

ISAVE - INPUT VECTOR OF LENGTH K+1 CONTAINING 
INDEPENDENT VARIABLES IN ASCENDING OP 
SUBSCRIPT OF THE DEPENDENT VARIABLE I 
THE LAST, K «■ 1, POSITION. 

B - OUTPUT VECTOR OF LENGTH K CONTAINING 

COEFFICIENTS. 

SB - OUTPUT VECTOR OF LENGTH K CONTAINING 
DEVIATIONS OF REGRESSION COEFFICIENTS 
T - OUTPUT VECTOR OF LENGTH K CONTAINING 

ANS - OUTPUT VECTOR OF LENGTH 10 CONTAINING 

INFORMATION.. 

ANS (1) INTERCEPT 

ANS (2) MULTIPLE CORRELATION CCEFFICI 

ANS (3) STANDARD ERROR OF ESTIMATE 

ANS (9) SUM OF SQUARES ATTRIBUTABLE T 
SION (SSAR) 

ANS (5) DEGREES OF FREEDOM ASSOCIATED 
ANS (6) MEAN SQUARE OF SSAR 
ANS (7) SUM OF SQUARES OF DEVIATIONS 
SION (SSDR) 

ANS (8) DEGREES OF FREEDOM ASSOCIATED 
ANS (9) MEAN SQUARE OF SSDR 
ANS (10) F-VALUE 

REMARKS 

N MUST BE GREATER THAN K+1. 


NVFKSF OF 
APIABLES. 
NTERCORKELA- 
EPENDENT 

SUBSCRIPTS OF 
DER. THE 
S STORED IN 

REGRESSION 

STANDARD 

T-VALUES. 

THE FOLLOWING 


ENT 

O REGRES- 
WITH SSAR 
FROM REGRES- 
WITH SSDR 


SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED 
NONE 


METHOD 

THE GAUSS-JORDAN METHOD IS USED IN THE SOLUTION OF THE 
NORMAL EQUATIONS. REFER TO W. W. COOLEY AND P. R. LOHNES, 
'MULTIVARIATE PROCEDURES FOR THE BEHAVIORAL SCIENCES', 

JOHN WILEY AND SONS, 1962, CHAPTER 3, AND B. OSTLE, 
•STATISTICS IN RESEARCH', THE IOWA STATE COLLEGE PRESS, 
1959, CHAPTER 8. 


SUBROUTINE TMULTR {NPN,K,XBAR,STD,D,PX,RY,ISAVE,BBB,SB,T,A NS) 
COMMON N,EOHl(13),PO,BMO,DCS,DTS 

COMMON FC,FY,ES,BB,H,DC,DD,AS,AS11,B11,D11,S,C,ZZ,NFU 
COMMON PHI,EO,J,Z,ECC,EY,FCONCC(20),ASC,EOH(40),FCONST(25) 
COMMON X (16000) ,EC (20) ,B (20) ,P (40) ,BMM (1000) ,BM(U0) ,FCS(20) 
COMMON FST, E (20) ,N3,DS(20) ,ASB(20) ,FSS(2C) ,SBM(2C) 

DIMENSION XBAE (11) ,STD (11) ,D(11) , RX (4C0) , R Y (1C) 

DIMENSION ISAVE(II) ,EBB(10),SB(1C) , T (1 0) , ANS (10) 


IF A DOUBLE PRECISION VERSION OF THIS ROUTINE IS DESIRED, THE 
C IN COLUMN 1 SHOULD BE REMOVED FROM THE DOUBLE PRECISION 
STATEMENT WHICH FOLLOWS. 


DOUBLE PRECISION XEAP,STD,D,RX,RY,B,SB,T, 
FN,FK,SSARH,SSDRM,F 


ANS,RM,BO,SSAR,SSDR,SY, 













































































































956 

959 

960 

961 

962 

963 

964 

965 

967 

968 

969 

971 

972 

973 

979 

975 

976 

977 

978 

979 

980 

961 

982 

983 

984 

985 

986 

987 

988 

989 

990 

991 

992 

993 

994 

995 

996 

997 

998 

999 

1000 

1001 

1002 

1003 

1004 

1005 

1006 

1007 

1008 

1009 

1010 

1011 

1012 

1013 

1014 

1015 

1016 

1017 

1018 

1019 


20 2 


c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 


1 00 


110 


c 

c 

c 


c 

c 

c 


c 

c 

c 

120 

c 

c 

c 


THE C HOST ALSO BE REMOVED FF OK DOUBLE PRECISION STATEMENTS 
APPFARING IN OTHER ROUTINES USED IN CONJUNCTION WITH THIS 
ROUTINE. 

THE DOUBLE PRECISION VERSION OF THIS SUBROUTINE MUST ALSO 
CONTAIN DOUBLE PRECISION FORTRAN FUNCTIONS. SORT AND ABS IN 
STATEMENTS 122, 125, AND 135 MUST BE CHANGED TO DSQRT AND DABS. 


KM = K♦ 1 

BETA WEIGHTS 

DO 100 J= 1 , K 
BBB (J)=0.0 
DO 110 J= 1,K 
L1 = K* (J- 1) 

DO 110 1=1,K 
L=I. 1+1 

BBE (J) =B3B (J) +RY (I) *FX (L) 

RM = 0.0 
BO=0.0 
L1=ISAVE (MM) 

COEFFICIENT OF DETERMINATION 

CO 120 1=1,K 

RM = RM + BBB (I) *RY (I) 

REGRESSION COEFFICIENTS 

L = ISAVE (I) 

BBB (I) =BBB (I) * (STD (LI) /STD (L) ) 

INTERCEPT 

BO=EO+BBB(I)*XBAR(L) 

BO=XBAR(Li)-BO 

SUM OF SQUARES ATTRIBUTABLE TO REGRESSION 
SSAR = RM*D (LI) 


C 

c 

c 

122 


MULTIPLE CORRELATION COEFFICIENT 
RM= SQRT ( ARS(RK)) 


SUM OF SQUARES OF DEVIATIONS FROM REGRESSION 

SSDR = D (LI) -SSAR 
IF (SSDR.EQ.0.0) SSBR=0.1 

VARIANCE OF ESTIMATE 

FN = N PN-K-1 
SY = S SDR/FN 


C STANDARD DEVIATIONS OF REGRESSION COEFFICIENTS 

C 

DO 130 J=1» K 



























































































































203 


1020 



L1 = K * (J~ 1) ♦ J 

1 021 



L=ISAVE (J) 

1022 


125 

SB (J) = SQRT ( A 

1023 

C 



1024 

c 


COMPUTED T- 

1025 

c 



10 26 


130 

T (J) =BBB (J) /SB 

1028 

c 


STANDARD ER 

1030 


135 

S Y = SQRT( ABS( 

1032 

c 


F VALUE 

1034 



FK=K 

1035 



SSARM=SSAR/FK 

1036 



SSDRM=SSDR/FN 

1037 



F=SSARM/SSDRM 

1039 



ANS (1) =BO 

1040 



ANS (2)=RM 

1041 



ANS (3) =SY 

1042 



ANS (4)=SSAR 

1043 



ANS (5)=FK 

1044 



ANS (6)=SSARM 

1045 



ANS (7)=SSDR 

1046 



ANS(8)=FN 

1047 



ANS (9)-SSDRM 

1048 



ANS (10)=F 

1 049 



RETURN 

1050 



END 


OF ESTIMATE 


END OF FILE 
$SIGNOFF 
































































appendix e 



DATA 

INPUT FOR THE MONTE CARLO PROGRAM 


Note: 

All units are in inches and pounds. 


Card 

Columns 

Data Description 

Format 

1 

1- 5 

Number of Variables (NV) 

15 


6-10 

Number of Simulations (NS) 

15 


11-19 

Limiting Steel Strength (FY1) 

F9.2 

Note: 

The limiting steel strength is a maximum 

i value of 


steel strength which could reasonably be expected. 


This is 

required so that extremely high 

values of 


steel strength are not used for the 

theoretical 


calculations. 


2 

1-15 

Mean Concrete Strength (RMEAN1) 

F15.5 


15-30 

Mean Steel Strength (RMEAN2) 

F15.5 


31-40 

Initial Seed (IY) 

110 


41-50 

Number of Fun (NRU) 

110 

Note: 

The initial seed is any integer. This 

number is 


required 

to initiate the random number 

generating 


subroutine. The number of run is any 

identifying 


number for the specific run. 


3 

1- 5 

Width of Column (BB) 

F5.2 


6-1 0 

Depth of Column (H) 

F5.2 


11-15 

Distance From the Compression Face 

to 



Longitudinal Steel Closest to the 




Tension Face (DD) 

F5. 2 


16-20 

Distance From Compression Face to 



Nearest Longitudinal Steel (DC) 


F5.2 


204 


















































































































205 


DATA INPUT CONTINUED 


Card 


6 

7 


Note 


8 


Columns 
21-25 
26-30 

1-10 
11-20 
21-30 
1- 5 

6-10 


Data Description 

Total Longitudinal Steel Area (AS) 

Longitudinal Compression Steel Area 
(ASC) 

Concrete Design Strength (FC) 

Steel Yield Strength (FY) 

Steel Modulus of Elasticity (ES) 

Depth From Compression Face to the 
Centroid of Compression Steel (DCS) 
Depth From Compression Face to the 
Centroid of Tension Steel (DTS) 

11-15 Spacing of Steel Ties (S) 

16-20 Width of Ties (B11) 

21-25 Depth of Ties (Dll) 

26-30 Area of Steel Tie (AS11) 

Number of Longitudinal Bars (NB) 

Area of Individual Steel Bars (ASB(I)) 
6-11 Distance From Compression Face to the 
Individual Steel Bars (DS (I)) 

This card is repeated for each longitudinal bar 
1-15 Variable Mean Value (F.MEAN (I) ) 


1 * 

1 - 


Format 
F5.2 

F 5.2 
F1 0.0 
F 1 0.0 
F 10.0 

F5.2 


T4 Variable Standard Deviation (STDV (I)) 

31-45 Variable Constant (FCONST(I)) 


F5.2 
F 5.2 
F5.2 
F5.2 
F5.2 
13 

F5.2 

F5.2 

t 

F15.5 
F 15.5 
F15. 5 


46-50 Variable Distribution Type (ITYPE(I)) 15 
Note: This card is repeated for each variable. In this 

















































































































206 


Card 


DATA INPUT CONTINUED 

Columns Data Description Format 

program the order of variables is as follows: 

Concrete Strength 
Steel Strength 
Cross Section Width 
Cross Section Depth 
Core Width 
Core Depth 

Distance From Compression Face to Nearest 
Longitudinal Steel 

Distance From Compression Face to the 
Longitudinal Steel Furthest From the 
Compression Face 

Distance From the Compression Face to 
Each Longitudinal Bar 









































































. 






























APPENDIX F 


A " 

A S 


b" 


^sp 

d” 

D 

e/h 


e 


n 


“ct 


f • 

c 


'cr 


h 

L 


NOMENCLATURE 

Cross sectional area of tie steel, one side of 
column 

Width of column core 

Actual cover of exterior steel layers 
Specified cover of exterior steel layers 
Depth of column core 
Dead load 

Eccentricity of axial load divided by the column 

dimension perpendicular to the neutral axis 

Error in placement of interior steel layers 

Modulus of elasticity of concrete in compression 

Modulus of elasticity of concrete in tension 

Modulus of elasticity of steel 

Concrete stress 

Concrete design strength 

Mean in-situ concrete strength 

Average concrete cylinder strength 

Depth of cross section 

Live load 


M Mean value of (B-S) 

(F“ S) 

PACI ACI calculated axial load 

Ptheory Axial load calculated from subroutine theory 
Ptest Axial load from Hognestad’s tests 

K Nominal resistance or strength 


Spacing of ties 


2 07 














































































































208 


NOMENCLATURE CONTINUED 



£ 

C 


w 5oh 


£ 

O 


£ 

t 


£ 

tr 


£ 

u 


5ou 


P 


P 


11 


a 


°(R-S) 


4 > 


Coefficient of variation of 
Separation function, 0.75 
Safety index 
Dead load factor 
Live load factor 
Concrete strain 

Increase in strain at 50% of maximum stress due to 

confinement of concrete by tie steel 

Concrete strain at maximum stress 

Concrete tensile strain 

Concrete strain at rupture in tension 

Crushing strain of unconfined concrete 

Concrete strain at 50% of maximum stress of 

unconfined concrete 

Steel percentage 

Tie steel volumetric ratio 

Standard deviation 

Standard deviation of (R-S) 

Stress in tension 

Rupture strength of concrete 

Understrength factor