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@X MBBJS aMSKSHWIS Digitized by the Internet Archive in 2019 with funding from University of Alberta Libraries https://archive.org/details/Grant1976 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. 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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 1 3C r ~ “ 1 ~rr II “Tf 1 > ~ r 3 - I cn 1 in vO 0 1 n > i 1 1 1 * t 1 t 1 t • i l t 1 >s 1 it 'rH O « 1 r~ 1 \— I o «" 1 1 e | 1 B B It ,rH| • | 3 3 ii ' 3 , O 1 | » * II i u l- " " ___ __ __ I 01 U) u Icp, >1 3 3 3 CP II H i i— r- o iO CM o CT>I 1 ^ 1 Chi 3 4J lO, 3 01 1 x> CO o i CO cr « o 10, X 3 3 3 0) ii 1 1 cu r*n j < T- C 7 > o 1 cn 1 o r~ 1 I M | 3 3 3 3 >i •H S-J 3 II i SC • • • 1 • • l • * o CQ a; Q) CP •H 3 3 c# 0J CP 1 1 1 O o o o r~ r-* j J Cu rH rH r*H rH U u 3 r* 11 I 0 ) rH *H rH 3 3 3 3 3 3 1 f — i 1 cc 1 *c << *“3 S 5 523 pa CQ 00 II 1 i_n r- o r h 1| s_ . V . | • • m O'! CT li ^1 1 i « i 1 i 1 C 1 ll ' CD. O StU o 1 i 1 i 1 t 0 o| "H II H Nl • *• H W | 4 -> 1 l 1 i i 1 x> X 1 M | n 33 33 Ii 1 3 3 3 3 3 3 3 ir | CM o ac 1 1 +•> I T 5 re) TO 3 3 3 3 3 * • • i . 0 53 3 3 3 •H •H rH rH rH ll «c j hJ 1 O r— 3 ’ 1 3 M 1 3 3 3 1 33 33 Cr> On cr> 3 1 H X 3 3 3 1 3 3 3 3 3 II l»l | l 1 3 a 1 U U U ( M M CW PO W 1 in D CO CM a 3 I li |rH| CO r— o r~ | 1 3 0 1 3 | cn o o CM SC u 1 ll | 3 | « » • • I h - -1- ( 33 3 ■H | o r- r~ II ,-H| 3 m 1 i 1 I 8 ? 1 1 cn I l> a; M o o i i i i 1 0 o s II •hI s: W | X> X l 1 l S 1 X x I ,-H a) | rH rH 'll 1 X 01 Xi rH rH 1 Xi Xi B Xi X 5 1 CO CM CO m| . w 3 1 3 -H •H i 3 3 i ■ 3 3 II Im 1 in D CO r— 1 a> 0 S~i JB 1 tH rH i ►H H . 1 (T» cn r—« fH I II IL 1 1 * • t • i 1 1 o O o r~ i Us , , r u CU 1 It ' i • 1 1 I a> U | SI 1 > r | r- 3 CM in t 1 » X) 4 -. n a a) a' 1 <D a; t i i II .'Hi o t I • • • • 8 1 3 1 1 e O 3 1 3 c 3 i 3 3 i i i l 1 r- CM T~ CM | 1 1 1 3 o o o O I O 0 5 » i IrH . r — | ss m j ll 3 L 0 . aJt - 3 - . J I— -c - — — — ■ Tl CM zt in 00 3 O r-* r~ 1 4 ~J II I>. 3 *H j O I H O 1 3 pi i CM r~ r» o ro r* VO vO | 03 , i | CU M in in r- 3 <o r- uo vO vO 1 r-H M a> CP 1 rH | CQ f CM r— in zt co vO T II HO M- W 1 J 3 jC Cu 1 d" S CO ro CO r- in co I I x B E , 1 r~ r- vo ro II X> 1 1 1 o in O'! r-| 1 o 3 r— On • • • • ' H 523 in i II 3 I | • 1 co 3 r* 1 1 ■i d 1 > U . C/ 31 • h'o 1 0 r ~ 1 1 &s?. 1 0 t s t i 1 o o c| 1 r~ r-~ ll 10 1 N i • W •H 1 1 I 1 S X ■P X 1 r*~ r— II •-hI u i I 1 1 1 1 1 1 ** ** 33 iCOI in CO vO r~ I v * rr o o o o 11 rH | • • t • 1 N 1 CO CO r"~ r- «r* r r- r— a; 3| o CM in in| 4 r *t 4 * 4 * 4 * ii 'H * 3 L_ - H - 1 c/i | UN * t| >H 103 , . in in o 0 o 0 o ll l o zt CM T— CM* | X> X X H |rH ( 1 • • • • • . 1 ^ % II >3 zt CO O cn ocl 1 CO 4 >fc in CO 00 co m 3 on | 3 | —■* | m in vo vO, j 4 * =*t ?w= 41 44 - 4 *r 4 * II lr c 1 3 *H 1 1- - rH i 3 w, c 0 ii 1 1 5 0 0 c. ' “f- II > CP 1 X X I II 1 li x X XI j 0 ) |’H | E 3 S —* I f 1 P l CD 1 II 03 1 1 o ct 03 1- °l 1 o o o o o o o o o | 3 | 1 9 t • • • r H 3 zT VC in x> vO in ‘nC II ll M CT> CO 3 CM 3 | c 1 - - 1- 1 1 1 1 1 zt -3 VO vO vo J D i >i cu 1 1 II ll 3 CP TP X 3 1 1 'VJ 3 1 X \y>. r*-H e 3 ss 1 CM i CO in 10 r~ 00 0 ^ ll J- 3 e 4-> 3 CO 3 ". 1 L CM in vo r-' CC CM _J 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 174 175 ■ 176 177 SUBROUTINE F STEEL 178 SUBROUTINE RANDOM 1 79 SUBROUTINE ST AT 18C C 181 183 184 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 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 11C 111 112 113 114 115 187 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) 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 188 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 ■ 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 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) 236 237 238 239 240 24 1 242 243 244 24 5 246 247 246 249 250 251 252 253 254 255 256 2 57 258 259 2oG 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 262 283 284 285 286 287 288 289 290 291 292 293 294 295 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) 296 297 298 299 300 3C1 302 303 309 305 306 307 308 309 310 311 312 313 319 315 316 317 318 319 320 321 322 323 329 325 3 26 327 328 329 330 331 332 333 339 335 336 337 338 339 390 391 392 393 399 395 396 397 398 399 350 351 352 353 359 355 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 . 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 4 14 415 o n n 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 • 416 4 17 4 18 4 19 420 4 21 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 4 39 440 44 1 442 443 444 445 446 447 448 4 49 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 noon 193 105 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) 476 477 478 479 480 481 482 483 484 485 486 467 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 194 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) 320,320,350 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 564 585 586 587 588 569 590 591 592 593 594 595 195 350 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 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 6 '4 8 649 650 651 652 653 654 655 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 656 657 656 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 68C 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 7 13 714 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 . 716 717 718 719 720 721 722 723 7 29 725 726 727 728 7 29 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 198 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) =0.0 776 777 778 779 780 781 782 783 764 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 e26 827 828 829 830 831 832 833 834 835 199 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