Introducing Size Effect into Design Practice and Codes for Concrete Infrastructure

Transcription

1 Introducing Size Effect into Design Practice and Codes for Concrete Infrastructure Part 1: Part 2: Part 3: Designing Against Size Effect on Shear Strength of Reinforced Concrete Beams without Stirrups Size Effect and Design Safety in Concrete Structures Under Shear Probabilistic Size Effect in Fracture Mechanics of Quasi-Brittle Materials Principal Investigator: Professor Zdnek Bazant A final report submitted to the Infrastructure Technology Institute for TEA- 21 funded projects designated A456, A466, A475, A488, A497, A206, and A210 DISCLAIMER: The contents of this report reflect the views of the authors, who are responsible for the facts and accuracy of the information presented herein. This Document is disseminated under the sponsorship of the Department of Transportation University of Transportation Centers Program, in the interest of information exchange. The U.S. Government assumes no liability for the contents or use thereof.

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23 NORTHWESTERN UNIVERSITY Size Effect and Design Safety in Concrete Structures under Shear A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Civil Engineering By Qiang Yu EVANSTON, ILLINOIS June 2007

24 3 ABSTRACT Size Effect and Design Safety in Concrete Structures under Shear Qiang Yu Safe design requires a valid mechanical model and correct probabilistic analysis. Concrete, which is an archetypical quasibrittle material, typically exhibits stable crack propagation in many types of failure. Therefore, a scientific approach requires analyzing concrete failure on the basis of fracture mechanics. One of the simplest ways to incorporate fracture mechanics into design practice is through the size effect. Statistical analysis reveals that an unevenly sampled database cannot be used to extract the size effect formula for shear failure of concrete beams purely empirically. Therefore the basic form of this size effect formula should be selected first. Simple dimensional analysis of the shear capacity of reinforced concrete beams yields the asymptotic properties of size effect, which are characterized by a plastic strength limit for sufficiently small sizes and an asymptotic LEFM (Linear Elastic Fracture Mechanics) trend for very large sizes. Together with the established small- and large-size second-order asymptotic properties of the cohesive crack model, the analysis unambiguously demonstrates that the energetic size effect law is a correct choice as the basic form of the design formula for concrete beams under shear. Then the new proposed shear design formula is verified and calibrated by a large number of experimental test data. Its validity for practical situations that are not fully covered by the current experimental database is supported by computer simulations based on microplane

25 4 model. For correct fracture analysis to be made, the fracture energy values need to be measured. For this purpose an improved version of Guinea et al. s method is presented, to reduce the statistical problem to linear regression by exploiting the systematic trend of size effect. Another obstacle to safe design is the misleading use of understrength factors in the current code provisions. The current design codes for concrete structures contain covert understrength factors. This prevents distinguishing between different combinations of separate risks due to statistical scatter of material properties, the error of the design formula and the degree of brittleness of failure mode, and also makes any prediction of structural reliability impossible. As a remedy, it is shown how the covert understrength factor of the design formula could be made overt. Its coefficient of variation can be based on the supporting test data and on the specified type of probability distribution, which then also implies the failure probability cutoff.

26 5 ACKNOWLEDGMENTS First, I want like to thank professor Zdeněk P. Bažant, my advisor, who provided me with the opportunity to pursue my doctoral degree in Northwestern University. His persistence in solving difficult problems, his creativity in coming up with the most unique ideas, his unbelievable enthusiasm and his quest for the perfection, has inspired and impacted my personal and professional growth. It has been a great honor to work with him. I am also grateful to professor Issac M. Daniel, Professor Edwin C. Rossow, Professor Leon M. Keer of Northwestern University, and Professor Roberta Massabo of Genova University (Italy) for serving as the committee members of my Ph.D. defense and qualifying examination. I thank my colleague and friend Dr. Goangseup Zi for his great contribution in my first paper. And I would like to give my thanks to Dr. Sze-Dai Pang; Dr. Zaoyang Guo; Dr. Peter Grassl; Dr. Gianluca Cusatis and Dr. Ferhun C. Caner. The inspiring discussions with them always refreshed my thoughts. I appreciate the generous help and cheerful spirit of Mr. Craig Neumanm, Mr. Dave Ventre and Ms. Jeanie Uchiyama. I am also thankful to Steve Albertson for his instruction and help during my experiment. The help from Ms. Raissa Ferron is also deeply appreciated here. I would like to thank the wonderful friends that I ve made at Northwestern University: Dr. Hang Cheng, Dr. Zhihui Sun, Dr. Yong Zhu, Dr. Jian Yang, Dr. Ying Xu, Dr. Yujun Xie, Dr. He Yan, and Dr. Xiaohua Wu with whom I had a lot of joyous time.

27 6 I would like to extend my sincere gratitude to my former advisor, Professor Aizhu Ren of Tsinghua University. Her encouragement and trust led me to achieve what I have today. Finally, I am indebted to my wife Ms. Hong Zhang, whose eternal beauty, selfless sacrifice and unending support has carried me through.

28 7 DEDICATION To my father and mother

29 8 Table of Contents List of Tables 12 List of Figures 13 Chapter 1. Introduction 22 Chapter 2. Formulation of Size Effect Formula for Concrete Beams under Shear Introduction Experimental Database Used Method to Interpret Existing Database Variables in Problem of Shear Failure and Size Effect Small- and Large-Size Asymptotes Dictated by Dimensional Analysis Second-Order Asymptotic Properties Size Effect Transition Between Asymptotes Difference Between Notched Specimens and Unnotched Structures with Large Cracks Question of Uniqueness of Size Effect Law Conclusions 57 Chapter 3. Verification and Calibration of Proposed Size Effect Formula Introduction 66

30 Verification by Comparisons with Individual Size Effect Data Expressions for Size Effect Law Parameters and Their Justification Statistical Calibration by Least-Square Fitting of Database Other Formulas for Beam Shear Conclusions 91 Chapter 4. Statistical Analysis and Computational Simulation of Size Effect Formula for Beams under Shear Introduction Minimizing Statistical Bias in Evaluating Current Database Failure Probability for Large Size Structure Design Situations Not Fully Covered by Current Database Shear Strength of Beams with Stirrups Conclusions 123 Chapter 5. Choice of Standard Fracture Test for Concrete and Its Statistical Evaluation Introduction Difference in Physical Meanings of Fracture Energies G f and G F Statistical Scatter of G f and G F, and Mean Trends Basic Testing Methods Level I and Level II Testing Level I Test: Measure G f or G F? Review of Relationship Between Size Effect Law and Fracture Parameters 147

31 Notched-Unnotched Method(NUM) Exact Size Effect Curve of Cohesive Crack Model Improved Direct and Inverse Formulae for the Cohesive Size Effect Curve Statistical Analysis of Notched and Unnotched Test Data Numerical comparison of statistical methods Approximate Size Effect Regression Using Tensile Strength and Notched Specimens of One Size Ambiguity of TPFM Due to Unloading Conclusions 174 Chapter 6. Covert Understrength Factors and Brittleness in Concrete Design Codes Introduction Examples of Existing Fringe Formulas in Concrete Design Codes How to Eliminate Deficiencies of Fringe Design Formulas Design Criterion Implied by Codes: Apparent and Actual Reliability Analysis with Several Understrength Factors Approximating Brittleness Effect by Reducing Dynamic Strength Concepts of Static and Dynamic Ductility of Structures True Safety Factor and Its Obscuring Effect on Forensic Evidence How to Judge Whether an Experiment Supports the Code Incorrect Use of f c in Estimating Scatter in Shear or Tension Conclusions 219 Chapter 7. Conclusion 231

32 11 References 242 Vita 255

33 12 List of Tables 2.1 ESDB database (ACI-445, 2002) Statistical evaluation and comparison of results Statistical comparison for small samples (m = 24, ω W eibull = 4.8%). 194

34 13 List of Figures 1.1 Catastrophic failures of concrete structures due to shear. (a) Air Force warehouse; (b) Sleipner oil platform; (c),(d) Palau bridge Histogram of beam depths (number of test data in each beam depth interval of 5 in. (130 mm) versus the depth in inches) Example of contamination of database due to variation of uncertain factors other than size, showing some typical test data included in the ACI-445F database, compared to the size effect curve and the broad range data from the Toronto tests and the Northwestern tests (the latter shifted to compensate for their much higher brittleness) Example of the effect of shifts in the size range of a highly scattered database on the slope of the regression line ACI-445F database for beam shear and plots of various size effect formulas Example of fallacious statistical analysis: (a),(c) Hypothetical perfect data generated so as to match exactly the size effect law for four different concretes; (b),(d) incorrect inference made by regression of the combined data set. Note that shifting of the chosen size range of data can yield any desired slope of regression line. 63

35 (a) Softening stress-separation curve of cohesive (or fictitious) crack model; (b) Geometry of reinforced concrete beam under shear Two test series of geometrically similar beams with a significant size range, fitted by size effect law (Eq. 2.18). Top: optimum fit of data of Bažant and Kazemi (1991, Northwestern); middle: optimum fit of data of Podgorniak-Stanik (1998, Toronto) and Yoshida (20000, Toronto); bottom: constrained optimization of combined Northwestern and Toronto data. Left: Plots used in nonlinear optimization; Right: Optimization results shown in linear regression plots (note the disagreement with power laws of exponents 1/3 and 1/4) Size effect in two classical series of shear tests of very large beams conducted in Japan (CoV = coefficient of variation) Optimum fits by size effect law (Eq. 2.18) of remaining existing data that have a non-negligible size range but have gross deviations from geometrical similarity (all coordinates are log d, d in mm) (a) Kani s (1967) data showing the effect of shear span ratio d/a on v c ; (b) Kani s data replotted in variables that give linear regression according to proposed Eq. (3.6) for v c ; (c) Küng s 1985 data (see Reineck et al. 2003) showing the dependence of shear strength v c on steel ratio ρ w ; (d) data of Shioya and Akiyama showing the effect of changing maximum aggregate size d a. 97

36 (a) Histogram of existing data and its smoothing as a function of beam depth d, used to assign data weights in inverse proportion; (b-d) Comparison of proposed Eq. (3.6) to weighted data, for (b) only those data for which d a is known, (c) complete ACI-445F database, and (d) extended ACI database; (e) comparison of proposed Eq. (3.11)-(3.12) to complete ACI-445F database; (f) comparison of Eq. (3.6) to complete ACI-445F database in a linear regression plot in terms of steel ratio ρ w Left: Shear strength compared to the portion of ACI-445F database for d < 6 in. (150 mm); right: Sensitivity of standard deviation of regression errors to various parameters influencing s L (a) (d) Beam shear failure pattern measured at University of Toronto and its interpretation; (e) load-deflection diagram of a beam with growing diagonal shear crack, and dimensionless load-deflection diagrams with the peak controlled by shear-compression failure (CCM = cohesive crack model, LEFM = linear elastic fracture mechanics) (a) Distribution of tensile cohesive stress σ 1 along diagonal shear crack at peak load in FEM simulation; (b) small and large beams with same mesh size at failure; (c) distribution of compressive stress over the ligament above shear crack tip in small beam; (d) distribution and evolution of compressive stress over the ligament in large beam ACI-ESDB database and statistical regression of centroids of test data with intervals of equal ratio in terms of d. (a): data points and size intervals; (b):

37 16 centroids of the first 4 intervals under the restricted ranges; (c): centroids of the first 4 intervals with a higher ρ w ; (d): centroids of all the intervals ACI-ESDB database and statistical regression of centroids of test data with intervals of equal ratio in terms of ρ w. (a): data points and steel ratio intervals; (b): centroids of the first 4 intervals under the restricted ranges; (c): centroids of the first 4 intervals under another restrict ranges (a) University of Toronto tests (Collins and Kuchma 1999; Angelakos et al. 2001) of shear strength of beams of various sizes; (b) ACI-445F database of 398 data points; (c) Portion of the database for beams from 10 to 30 cm deep (a) Probability distribution of shear strength of beams from 10 to 30 cm deep, based on the ACI-445F database; (b) Distribution for beams 1 m deep inferred from the database Tests and simulations compared to proposed formula. a) ESDB data with ρ w < 1%; b) simulations of beams with different ρ w ; c) Japanese tests of simply supported beams under distributed load and their simulations; d) crack patterns for Japanese beams at maximum load finite element simulations for Bentz s slab question: a) 14 in. deep slab with steel bar at bottom face across whole span; b) 14 in. deep slab with steel bar at bottom face terminated 1.5 d away from support; c)simulations of fixed-end wide beam of different thickness, of sizes within and outside

38 17 the range of proposed formula; d) 42 in. deep slab showing deep beam behavior Size effect of slender beams with stirrups. Left: tests conducted in Stuttgart; right: tests conducted in Perth Crack band finite element simulations for shear failure of beams with stirrups Top: bilinear softening stress-separation curve of cohesive crack model; bottom: typical stress profile throughout the fracture process zone in a notched test specimen of positive geometry Plots of measured versus predicted values of fracture energy G f or G F, obtained by (a) peak-region methods (SEM, TPFM, ECM, 77 data); (b) work-of-fracture method (161 data) (note: s y x = standard deviation of vertical deviations of data points from line of slope 1; s y 1 = standard deviation of the differences of G test f /G pred f from 1) Left: the idea of direct measurement of the softening stress-separation curve; middle: specimen needed to achieve nearly simultaneous separation; right: field surrounding the FPZ in a large structure Bottom: dependence of σ 2 N of singly and doubly notched tensile specimens on size D, calculated from bilinear cohesive crack model for various specimen geometries (the straight dash line in these plots is the size effect asymptote); top: dependence of notch-tip stress σ tip on D. 180

39 Continuation of Fig. 5.4 for the three-point bend and wedge splitting specimens Plots from Fig. 5.4 and 5.5 in expanded horizontal scale emphasizing the small-size behavior Size effect curves of the cohesive crack model and the classical size effect law (SEL, Bažant 1984) plotted in different coordinates illuminating the small-size asymptotics Size effect curves of the cohesive crack model and the classical size effect law (SEL, Bažant 1984) plotted in different coordinates illuminating the large-size asymptotics (a,b) Difference between size effect law fitted to test data and asymptotic size effect law of cohesive crack model, for direct tension specimens with two symmetric notches (a) and for three-point bend specimen (b); (c) size effect lines connecting the points representing the mean ± standard deviation (a) Illustration of notched-unnnotched method; (b) free-body diagram of one half of three-point bend specimen at D 0; (c) extreme cases of unloading-reloading lines used in simulations of Jenq-Shah method (TPFM) for softening stress-separation curve; and (d) for load CMOD diagram Comparison of the approximate formulae (5.22) (direct, on top) and (5.24) (inverse, at bottom) with the dimensionless size effect curves of

40 19 cohesive crack model (solid curves) for 3PB beam (dashed lines: size effect asymptote) Left: Illustration of statistical method A from the random nominal strength data shown on the vertical axis one obtains a set of points of dimensionless sizes on the horizontal axis, each of which implies one value of l 1 and of G f ; right: illustration of Monte Carlo simulation(method A ) from the nominal strength data calculated from the random pair of f ti and σ Nj one obtains a set of points of dimensionless sizes on the horizontal axis, each of which implies one value of l 1 and of G f Identification of G f and f t reduced to linear regression (100 randomly generated input data points for strength and 100 for nominal stress are shown on the vertical lines for D = 0 and D = 155 mm and yield the regression lines drawn and the coefficients of variation listed. Left: large scatter of input; right: small scatter; top: simple regression; bottom: iterated regression Left: illustration of statistical method B extended to multiple notched sizes; right: illustration of statistical method B extended to multiple notched shapes Fitting of cumulative probability curve of G f assuming (a) normal, (b) lognormal, and (c) Weibull distributions size effect lines obtained (after cohesive model correction) for all the pairs of small size random input of unnotched strength values and small size

41 20 random input of notched nominal strength values (Left: high scatter; right: low scatter; top: 3 data sizes; bottom: 6 data size) ACI (1962) small beam database used to justify the current ACI code formula for shear force capacity V c due to concrete in reinforced concrete beams with and without stirrups, and reductions specified or implied by ACI standard (2002) that was justified by this database ACI-445 database for shear force capacity V = V c ( shear strength in ACI terminology) of beams without stirrups, and optimum fits by size effect law (left) and by an approximation with straight asymptotes (right); after Bažant and Yu 2003, 2005a,b Example of scatter of 12 compression strength tests, mean strength f c (in ACI notation, f cr), and specified strength f c used in design Probability density and cumulative distributions of load and resistance of structure, entering the reliability integral Comparison of imperfection effects in (a) thin shell buckling and (b) quasibrittle fracture Dependence of maximum load and dynamic ductility on peak morphology and on kinetic energy K imparted to the structure Dependence of static ductility AB on peak morphology, postpeak and structural stiffness C 229

42 Two shapes of load-deflection peaks and load drop based on imparted kinetic energy K. 230

43 22 CHAPTER 1 Introduction The first and the most important fundamental in structural engineering is the structural safety. The definition of structural safety dictates the prevention of harm to the users and occupants of the structures we design. This absolute requirement has been one of the greatest challenges for the structural engineers since the first building was designed, and it will still be in the future. Currently, the condition of safe structural design is commonly written in the general form Structural Resistance Safety Factor Load Effect (1.1) where the structural resistance is determined by the design formula based on applied structural analysis; the safety factor provides the safety margin for the design due to the uncertainties associated with the material properties and applied loads, as well as the simplification or incompleteness of the applied structural analysis. It directly shows that a safe design can be achieved only after a valid mechanical model is established and correct probabilistic analysis applied. Concrete, invented in 1824, is now the most widely used engineering material in terms of volume, simply because most of the structures on this planet have been being built with it. Therefore the safety of concrete structures is always a serious concern in the structural field. For a long time, the structural analysis models for concrete structures have largely

44 23 been based on the classical allowable stress design, as well as the theory of limit states (or plastic limit analysis), which underlies the current design code. In these traditional structural analysis models, the material failure at one point is completely determined by the stress and strain tensors at that point. However, more and more catastrophic failures and accumulating experimental evidence reveal that the traditional mechanical models cannot sufficiently capture some constitutive behaviors of concrete, especially when the failure is associated with stable crack propagation. Concrete, known as a quasibrittle material, displays stable crack growth before the maximum load is approached in shear, torsion, punching, compression crush, and many other types of failures. This means that in all these types of failures, analysis models based on the strength limit or the theory of plasticity could not provide an accurate prediction of structural resistance to the structural engineers who are conducting the design. Usually, the traditional structural approaches give an overestimation of the structural resistance, which then leads to an unsafe design no matter how sophisticated the mechanical models are. Therefore, a new structural analysis approach, which must be able to accurately describe the constitutive behaviors of concrete during crack propagation, should be introduced into our design codes to overcome the inadequacy of current design methods. Fortunately, after decades of study, the picture is clear now that the new structural analysis model should be based on fracture mechanics, in which the material failure is mainly determined by energy criteria. Some effective mechanical models based on fracture mechanics, for example, the cohesive crack model, crack band model and nonlocal damage model, now are widely used to analyze the concrete performance during crack propagation

45 24 and lead to good agreement with the experimental results. A salient property of all these models is that a size effect is inevitably exhibited when concrete or other quasibrittle material is studied. Actually, one of the simplest ways to incorporate fracture mechanics into the design code is through the size effect, which means the structural strength changes with the size of structure. Like most of the code provisions, the shear design formula of ACI code, which was formulated on the basis of a database collected in 1962, gives no size effect on the shear strength (American Concrete Institute Code 318, Building, 2005). However, investigations in the catastrophic failures of Air force warehouse (failed in 1956, Fig. 1.1a), Sleipner oil platform (sank in 1991, Fig. 1.1b), and Palau bridge (collapsed in 1996, Fig. 1.1c,d) indicate that the exclusion of size effect in the current design formula is one of the major causes of these catastrophes. More compelling recent experimental evidence shows that the design formula for reinforced concrete beams under shear should include the size effect. Therefore, the main objective of this thesis is to determine how to incorporate the size effect into the current design code, and then provide a safe resistance prediction for the structural design. Besides the design formula, a correct probabilistic analysis is also essential to a safe design, given all the uncertainties associated with the structural design. However, the intuitive use of a single safety factor to cover all the uncertainties related to structural design is not a scientific way to study the reliability and failure probability of a structure. Thus, to study the separate risks due to the randomness of material properties, the error of the design formula and the degree of brittleness of failure mode is also an important objective of this work.

46 25 This thesis is organized as follows. In Chapter 2, we begin with determining the correct approach to formulating the design formula for reinforced concrete beams without stirrups under shear loading. The shear design provisions in current building code of ACI give no size effect on the shear strength. In view of the enormous financial input for large scale size effect testing, both the 1962 ACI database and the recent ACI 445F database, serving as the experimental foundation of design formula, consist of beams most of which are too small. Thus a correct size effect formula cannot be extracted purely empirically from the database. Therefore, the correct approach to determine the size effect formula is first to select the basic form of size effect formula based on a sound theory which must be able to capture similar size effects in other types of failure with the same physical source, occurring in concrete as well as other quasibrittle materials. This selected size effect formula should be verified by comparison with individual test series in which geometrical similarity is adhered to. Then the size effect formula for shear design is calibrated by the entire database. The general approximate mathematical form of the size effect formula to be calibrated by experimental data can be deduced from two facts: (1) the failure is caused by cohesive (or quasibrittle) fracture propagation; and (2) the maximum load is attained only after large fracture growth (rather than at fracture initiation). Simple dimensional analysis yields the asymptotic properties of size effect, which are characterized by (1) a constant beam shear strength v c (i.e., absence of size effect) for sufficiently small beam depths, and (2) the LEFM size effect v c d 1/2 for very large beam depths d. Together with the recently established small- and large-size second-order asymptotic properties of the

47 26 cohesive (or fictitious) crack model, this suffices to unambiguously support a size effect formula of the general approximate form v c = v 0 (1+d/d 0 ) 1/2 (where v 0, d 0 are constants), which has been applied to many types of failure for concrete structures. In Chapter 3, experimental verification by least-square fitting of those existing individual data sets that have a broad size range is presented. The size effect formula is compared to the reduced-scale tests conducted at Northwestern University, and the normal-scale tests conducted at University of Toronto. The salient property of the asymptotic slope of -1/2 for fracture mechanics is verified. The comparison with Japanese tests and other individual test series with a significant size range also does not contradict this size effect trend. Subsequently, empirical prediction formulas for the size effect parameters, consisting of the asymptotic small-size strength v 0 and the transitional size d 0, are calibrated by leastsquare regression of (1) a recent ACI database with 398 data points, and (2) a combination of this database with large-scale Japanese tests and Northwestern reduced-scale model tests. The appropriate approach to conduct this nonlinear statistical regression is studied here to prevent the misleading use of model safety factor γ based on elementary population statistics. Also, some alternative proposals for dealing with the size effect in beam shear are discussed. From theoretical analysis and computational simulation, it transpires that neither the Japan Society of Civil Engineers (JSCE) code based on Weibull statistical theory nor the crack-spacing hypothesis developed on the modified compression field theory can capture the size effect trend correctly.

48 27 In Chapter 4, supplementary statistical analysis and computational simulation are conducted to verify the proposed design formula. The equal-interval-method, used to minimize the statistical bias, is adopted in the statistical regression. The least-square regression on the centroids of intervals in terms of size strengthens the finding that the size effect trend in beam shear has a slope of -1/2 in logarithmic plot. Then statistical analysis of strength distribution and investigation on failure probability for large size beams are carried out on using the individual data set tested at University of Toronto. It reveals that ignoring the size effect would lead to intolerable probability of failure for beams 1 m deep. Next, finite element simulations are performed to explore the practical situations which are not fully covered in the recent ACI database. The simulation results conducted on beams with low longitudinal steel ratios, simply supported beams under distributed load, and fixed-end beams under distributed load, agree with the proposed design formula very well. The computational simulation is also undertaken to investigate the size effect of reinforced concrete beams with shear reinforcement. Combined with the comparison against the individual test series extracted form literature, it is found that the stirrups can only mitigate the size effect significantly, but cannot suppress the size effect. Next, in Chapter 5, the choice of standard test for fracture energy of concrete is studied, and its statistical evaluation is analyzed. In view of the importance of fracture energy for the size effect in quasibrittle materials, shown in preceding chapters, it is necessary to introduce a proper fracture test standard for concrete and other quasibrittle materials.

49 28 Although fracture energies G F and G f, corresponding to the areas under the complete softening stress-separation curve and under the initial tangent of this curve, are two independent fracture characteristics, it is demonstrated that only one can be measured is G f in basic (level I) standard test. Among various approaches to measure G f, the method proposed by Guinea et al. is the most robust and optimal. However, the uncertain statistical correlation between σ N and f t in this method makes it impossible to identify G f and f t by statistical regression. Therefore an improved version of Guinea et al. s method is presented to reduce the statistical problem to linear regression by exploiting the systematic trend of size effect. This is made possible by noting the fact that, according to the cohesive (or fictitious) crack model, the zero-size limit σ N0 of a notched specimen is independent of F f and thus can be easily calculated from the measured f t. Then, the values of σ N0 obtained, together with the measured σ N values of notched specimens, are used in statistical regression based on the exact size effect curve calculated in advance from the cohesive crack model (generally accepted as the best simple fracture model for concrete) for the chosen specimen geometry. In Chapter 6, we analyze the reliability consequences of the fact that the current design codes for concrete structures contain covert (or hidden) understrength (or capacity reduction) factors. This prevents distinguishing between different combinations of separate risks due to the statistical scatter of material properties, the error of the design formula and the degree of brittleness of failure mode, and also makes any prediction of structural reliability (or survival probability) impossible. The covert formula error factor is implied by the fact that the design formula was calibrated to pass not through the mean but through the fringe (or periphery, margin) of the supporting experimental data. The

50 29 covert material randomness factor is the ratio of the reduced concrete strength required for design to the mean of the strength tests. As a remedy, the covert understrength factor of design formula should be made overt, its coefficient of variation (based on the supporting test data) should be specified, and the type of probability distribution (e.g., Gaussian or Weibull) indicated (which then also implies the probability cutoff). Furthermore, it is proposed that the currently used empirical understrength factor, which accounts mainly for the risks of structural brittleness (or lack of ductility), should be based on the expected maximum kinetic energy that could be imparted to the structure. The reliability integral taking into account the randomness of both the load and structural resistance is generalized for the case of multiple (statistically independent) understrength factors. Finally, the study is concluded in Chapter 7 by highlighting the important results obtained in this work. Parts of each of Chapters 2, 3 and 6 were published in Journal of Structural Engineering ASCE. One other paper, focusing on fracture energy which is studied in Chapter 5, was published in International Journal of Fracture. And another paper based on Chapter 4 is accepted by ACI Structural Journal.

51 30 Figure 1.1. Catastrophic failures of concrete structures due to shear. (a) Air Force warehouse; (b) Sleipner oil platform; (c),(d) Palau bridge.

52 31 CHAPTER 2 Formulation of Size Effect Formula for Concrete Beams under Shear 2.1. Introduction Although a provision for size effect in shear failure of reinforced concrete beams was incorporated into some design codes more than a dozen years ago, the current ACI Building Code (ACI , Equation 11-3) specifies the contribution of concrete to the crosssection shear strength of reinforced concrete members as: V c = 2 f c b w d (2.1) where f c = required compression strength of concrete (psi), d =beam depth from top face to longitudinal reinforcement centroid (inch), b w =web width (inch). This code formula, giving a size independent shear strength of reinforced concrete beams, was justified on the basis of ACI-ASCE database (1962), which involved only small beams, of average beam depth 13.4 in. (340 mm). During the last few years, a compelling experimental evidence obtained by properly scaled tests of large-size beams has become available. It has now become clear the design formula for reinforced concrete beams without shear stirrups must include the size effect. The problem is how to best interpret the test results, and how to best describe the size effect mathematically in a sufficiently simple

53 32 and practical manner, without violating certain restrictions that have crystallized from theoretical researches during the last two decades. The size effect is measured in terms of the nominal strength, generally defined as σ N = P/bd where P is the maximum (or ultimate) load (or load parameter), b is the structure width and d is the characteristic dimension (or size) of the structure. The size effect is characterized by comparing σ N for geometrically similar structures of different sizes d. According to the classical allowable stress design, as well as the theory of limit states (or plastic limit analysis) which underlies the current design codes for reinforced concrete structures, the nominal strength σ N is independent of the structure size. We say that in this case there is no size effect. It has been generally proven that the size effect is absent from all structural analysis methods in which the material failure at a point of the structure is decided exclusively by the stress and strain tensors at that point, and that a size effect inevitably arises if the material failure criterion involves energy. This is the case of fracture mechanics, provided that, at maximum load, the crack, or the fracture process zone (i.e., microcracking zone), or both, is not negligible compared to structure dimensions. A size effect is exhibited by all the theories of failure which involve some material characteristic length, l 0. This is always the case when the failure criterion involves both the stress and fracture energy because the ratio of energy per unit area to stress has the dimension (Nm/m 2 )/(N/m 2 ) = m. An example is the cohesive crack model, as well as the crack band model (which is almost equivalent, Bažant and Planas 1998), and also the nonlocal damage model. The cohesive (or fictitious) crack model, originated by Barenblatt (1959), developed by Rice (1968) and pioneered for concrete by Hillerborg et al. (1976),

54 33 is an approximation now generally regarded as the best compromise between simplicity and accuracy. The salient property of these models is that attainment of the strength limit at a material point means only that fracture can begin but not that it must proceed to create a crack. To proceed, a sufficient energy must be supplied Experimental Database Used Thousands of experiments have been conducted around the world to assess the shear capacity of concrete members, although only a small fraction of them were specifically aimed at the effect of size. To provide a foundation for new shear design provisions, ACI subcommittee 445F extracted, from a collection of more than 1000 data, a new database of 398 data, called ESDB (Evaluation Shear Database, Reineck et al. 2003). Only beams with no shear reinforcement, subjected to three-point or four-point loading, are included in this database. All the beams have rectangular cross section except that 24 are T-beams. The beam depth ranged from 4.33 in. to in., the shear span ratio a/d from 2.41 to 8.03, the compression strength f c of concrete from 1,828 psi to 16,080 psi, longitudinal steel ratio from 0.14% to 6.64%, and the maximum aggregate size, known for only for 341 data points, from 0.25 in. to 1.5 in..; see table 2.1. The ESDB database has been adopted for the present studies in ACI 446, even though the rationality and impartiality of the criteria used to select the data have been questioned (Bažant 2004a, Bažant and Yu 2004a, Yu 2004, Bažant et al. 2006, Yu and Bažant 2006, and Kazemi 2006): for instance, the largest beams ever tested, up to 3 m deep (Iguro et al. 1985, Shioya et al. 1989, and Shioya et al. 1994) were excluded from the ESDB

55 34 database, based on the fact that they were subjected to distributed load, a combination of which with point loads in the same database was thought to complicate interpretation. But this position disregards the fact that the code provision must apply to both. The reduced-scale beam tests at Northwestern University (Bažant and Kazemi 1991), with aggregate size 4.8 mm and beam width b w = 48 mm, were excluded with the explanation that, inexplicably, only beams with b w > 50 mm were admissible; these tests, however, exhibited the most systematic size effect trend, had an exceptionally broad size range (1 : 16), and achieved the highest brittleness number among all the available tests, thus simulating the brittleness of very large beams (b w equaled 10 maximum aggregate sizes, which is not only adequate but also, after a width increase by mere 4%, would have technically qualified these data for inclusion in the ESDB database and would not have distorted interpretation since it is generally accepted that the beam width has no effect on v c ; see Kani 1967) Method to Interpret Existing Database Despite the questionable criteria to select data points, the ESDB database is adopted here to study the failure behavior of concrete beams under shear. The objective now is to determine how to extract a correct size effect formula from the database which includes various parameters other than size and shows enormous scatter as described in table Obstacles for Determining Shear Design Formula Empirically Could a size effect formula be determined purely empirically from the ESDB database? We need to address this question to place the problem in proper perspective. Many

56 35 formulas in concrete design codes can of course be developed purely empirically because it is possible to obtain adequate test data for the entire range of practical interest, and to conduct experiments that sample this range statistically uniformly, without bias. An example is the expression for the elastic modulus in terms of the compressive strength, or the effect of reinforcement ratio in various code specifications. Unfortunately, the size effect is not a problem of that kind Unevenly Sampling of Database. Fig. 2.1 shows the histogram of the number of test data versus beam depth d according to the database. The size effect is of practical concern mainly for beam depths ranging from 1 m to 15 m (the depth of the failed Koror Bridge girder in Palau was 14.2 m). Unfortunately, 86% of all the available test data pertain to beam depths less than 0.5 m, 99% to depths less than 1.1 m, and 100% to depths less than 2.0 m. The coefficient of variation ω of the deviations of a size effect formula from the points of the database will therefore be totally dominated by small beam depths for which the size effect is unimportant. Thus it can easily happen that some formula that gives the smallest ω for such data could be completely wrong for very large sizes while another formula that might give a higher ω could be much more realistic for large sizes. So, data fitting alone is not the way to develop a size effect formula for the ACI code. For an unbiased, purely empirical, validation of a formula, the test data would have to be distributed uniformly over the entire range of interest. In view of the costs of large scale tests, we cannot even hope to acquire such a database. Therefore, we must extrapolate. But the extrapolation, visualized by Fig. 2.1, cannot be accomplished empirically. In keeping with the motto of former ACI president s inaugural

57 36 message (Izquierdo-Encarnación 2003), ars sine scientia nihil est, a sound scientific support is required. The scientific theory, in turn, should be verified by properly scaled size effect tests on one and the same concrete, and especially by reduced-scale model tests in which the dimensionless size (characterized in a shape-independent manner by the brittleness numbers β, explained in Bažant and Planas, 1998, Bažant 2002a, Bažant 2004b) can attain the highest possible values Uncertain Influences from Parameters other than Depth. The most serious obstacle to extracting size effect information from the ACI-445F database is the fact that the vast majority (more than 97%) of its 398 data points come from tests motivated by different objectives (such as the effect of concrete type, reinforcement, shear span, etc.), in which the beam depth was not varied at all. To document the problem, see Fig. 2.2, which shows some such test data (marked by an oval) in comparison with the size effect law (Bažant 1984) and with the data points from two test series of the broadest available size range. These data contaminate the database by irrelevant scatter, caused by influences that cannot be eliminated because they are poorly understood. Such contamination widens the scatter band of the database and masks the size effect trends of the individual data series with a significant size range. When this database is fitted with a power law (a straight line in Fig. 2.3), the best-fit exponent (slope of the straight line) will depend on the beam size distribution in the database; see Fig. 2.3 which illustrates how the shifting of a hypothetical data cloud to smaller or larger sizes can yield any power law with exponent between 0 and 1/2. Obviously, such statistical inferences are not objective; they depend on the frequency of test data in various size intervals, which is a subjective choice of the experimenters, influenced by the funds available.

58 futility to Select a Size Effect Formula Empirically. A size effect formula for shear strength, motivated by fracture mechanics, was proposed by Bažant and Kim in Since that time a number of other formulas have appeared; see Fig. 2.4 which includes: (1) The size effect law based on fracture mechanics and energy release arguments (Bažant 1984; Bažant and Kazemi 1991; Bažant and Planas 1998; Bažant 2002a; Bažant 2004b); (2) An extended form of that law for fractures in which the cohesive stresses are never reduced to zero but exhibit a finite residual strength (Bažant 1987); (3) A formula resulting from an enhancement of the modified compression field theory (MCFT) based on a hypothesis that crack spacing causes size effect (Vecchio and Collins 1986; Collins et al. 1996; Collins and Kuchma 1999; Angelakos et al. 2001); (4) The CEB-FIP formula, introduced empirically (Comité Euro-international du Béton 1991); (5) Similar Carpinteri s (1994) multifractal scaling law (MFSL); (6) The formula of Japan Concrete Institute (Japan Society of Civil Engineers 1991), motivated by Weibull statistical theory a theory that assumes the failure to occur right at the initiation of a macroscopic crack, and the size effect to be caused by randomness of local material strength; (7) A power law of exponent 1/2 corresponding to linear elastic fracture mechanics (LEFM) and supplemented with an upper bound (small-size cut-off); and

59 38 (8) A purely empirical power law of exponent 1/3 proposed in 2003 by an ACI subcommittee. For comparison, Fig. 2.4 shows all the data points of the existing ACI-445F database. What is striking in this figure is that the very different curves of the aforementioned formulas look almost equally good (or equally bad). The reason is that the size range covered by the data is not broad enough and the scatter is enormous. The size range cannot be significantly extended without very large financial outlays. The main cause of the enormous scatter is that test data for different concretes, different shear spans, different reinforcement ratios, etc., are mixed in one and the same database. This manmade scatter cannot be filtered out to a significant extent because its causes are poorly understood. The problem is compounded by the fact that most of the data sets included in the database involve only a single beam size (depth) or a negligible size range because their original purpose was to clarify influences other than the size effect. Some efforts are being made to choose among various formulas by comparing the coefficients of variation of the errors of each formula, calculated for the existing ACI-445F database. But such efforts are futile. The coefficients of variation of the deviations of the formula from the data points are almost the same for all the formulas, good or bad. The arbitrariness of such a comparison then inevitably leads a committee to a political choice Correct Approach to Determine Shear Design Formula So, is the existing ACI-445F database useless? Not at all. But it should be used only for calibrating the size effect formula after the basic form of that formula has already been

60 39 selected, and nothing else. The selection of the best form of the formula must be based on a sound theory. The theory must be such that it could also capture similar size effects in other types of failure with the same physical source, occurring in concrete as well as other quasibrittle materials. The theory should of course be experimentally validated. This can be done only by comparisons with individual test series in which, ideally, no parameter but the beam size is varied and the size range is broad enough in relation to the inevitable random scatter. Because of the high random scatter in beam shear tests, the size range should be at least 1:8. Especially, the concrete must be one and the same, and geometrical similarity of the beams of various sizes should be maintained as closely as possible, so as to prevent polluting the data set by uncertain influences other than those of the size. Currently there exist only about 11 data series, from 8 investigator teams (Leonhardt and Walther 1962; Kani 1967; Bhal 1968; Iguro et al. 1985; Bažant and Kazemi 1991; Shioya and Akiyama 1994; Collins and Kuchma 1999; Angelakos et al. 2001), satisfying these criteria at least approximately (a few more have a significant size range but grossly depart from geometrical similarity). Only two of them, namely the 1991 Northwestern tests (Bažant and Kazemi 1991) and the recent Toronto tests( Collins and Kuchma 1999; Angelakos et al. 2001), satisfy these criteria quite closely. Of these two, the Northwestern ones are reduced-scale model tests, which have the advantage that they achieve (thanks to the reduced scale) the largest dimensionless size so far, as measured by the brittleness number β. Recently, while limiting consideration to test series in which the beam depth was significantly varied, the aggregate of such test series was evaluated jointly, without paying

61 40 attention to the trends of the individual test series. However, such an approach is again misleading. To illustrate it, consider Fig. 2.5 showing bi-logarithmic plots of log v c versus log d (on the left) for two sets of four hypothetical data series with the same range of beam depth d (in log-scale), generated so as to match perfectly the curves of the theoretical size effect law v c = v 0 (1 + d/d 0 ) 1/2 (discussed later), in which v 0 and d 0 are empirical parameters depending on the type of concrete. The set on top is obtained by a frugal investigator, who has modest funding and must therefore test smaller (less expensive) beams, and the set at the bottom is obtained by a wealthy investigator, who has greater funding and can thus afford to test larger beams. Each investigator conducts the size effect tests for four different concretes each of which is the same for both investigators (and all the other influencing parameters, including the steel ratio ρ w and shear span a/d, are also the same for both). The curve of the size effect law for each concrete is different, characterized by different values v 01, v 02, d 01, d 02 of the size effect law parameters v 0 and d 0. Assuming that both of these investigators do not know the theoretical size effect law and regard these perfect data as one combined database, they see only the data pictures on the right of Fig Because of the high scatter of the combined database on the right, each investigator, looking at his combined database, can at best infer a straight line trend in the bi-logarithmic plot, which corresponds to a power-law size effect. By statistical regression, the frugal investigator thus finds the mean size effect v c d 1/4 (which happens to coincide with the JSCE code specification in 1991), while the wealthy investigator finds the mean size effect v c d 1/3 (which happens to coincide with a recent recommendation by one code-preparing subcommittee). Thus, because of not having checked the trends of the individual data series, both investigators are led to erroneous conclusions. Their