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Shake table test of Oak Street Bridge bent: correlation of analysis and experiment Khoshnevissan, Manzar 1998

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Shake Table Test of Oak Street Bridge Bent: Correlation of Analysis and Experiment by M A N Z A R KHOSHNEVISSAN B.Sc, Sharif University of Technology, Tehran, Iran, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required^standards THE UNIVERSITY OF BRITISH COLUMBIA April 1998 © Manzar Khoshnevissan, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of d j v . L P ^ y V ? C ^ ' I A J The University of British Columbia Vancouver, Canada Date dp. i /Z% / 79 DE-6 (2/88) ABSTRACT Experiments on full- or reduced-scale structures, especially shake table tests, have provided insight into the static and dynamic behavior of structures subjected to earthquakes. Experimental testing is both expensive and time-consuming, and sometimes it is simply not practical to conduct seismic tests. Analytical and numerical methods, such as computer modeling techniques, are good alternatives to experimental testing, providetl that they are reliable and realistic. It is important to know with precision how reliably these computer programs predict the seismic response of real structures. Tliis thesis is a comparative study of the experimental and analytical responses of a 0.27 scale model for a two-column bent of the Oak Street Bridge tested on a shake table. The purpose of this study is to assess two nonlinear structural analysis programs, CANNY-E (Li, 1995) and RUAUMOKO (Carr, 1996), by perforating a correlation analysis between test and analytical results. A detailed study and interpretation of the recorded data from six levels of excitation was perfonned by detennining the dynamic and material characteristics of the specimen through different stages of excitation. Subsequently, an analytical model for the bent was proposed that consisted of 8 beam elements and 2 column elements. The linear and nonlinear responses of the analytical model subjected to the shake table earthquake motion simulations, were investigated using the CANNEY-E and RUAUMOKO computer programs. The sequences of plastic hinge formation were investigated through a push-over analysis using these two computer programs. Finally, the ability of the two computer programs to predict reliably the structural behavior of the reinforced concrete bridge bent was assessed by comparing analytical with experimental results. ii In spite of limitations such as the need to adjust the model's damping ratio and hysteresis parameter, both CANNY-E and RUAUMOKO were able to provide a good representation of the dynamic response of the specimen in tenns of its fundamental frequency, relative displacement time hi story at the cap beam level, base shear time histories, absolute acceleration time histories, and the overall hysteresis responses. The main sources of error in the nonlinear analyses were due to the estimation of damping and stiffness of the structure, the modeling of the elements, the global modeling, and the numerical methods applied to calculate the response. It was found that CANNY-E is generally a more powerful program than RUAUMOKO, as the former produced more reliable and more accurate results than the latter. However, neither of the programs was able to consider moment-shear interaction or shear failure demonstrated by the specimen during the tests. iii T A B L E OF CONTENTS A B S T R A C T ii T A B L E O F C O N T E N T S iv L I S T O F T A B L E S viii L I S T O F F I G U R E S x A C K N O W L E D G M E N T xiv D E D I C A T I O N xv CHAPTER! INTRODUCTION 1 1.1 P R E L I M I N A R Y R E M A R K S 1 1.2 M O T I V A T I O N 2 1.3 O B J E C T I V E 3 1.4 P R O J E C T B A C K G R O U N D 4 1.5 R E L A T E D R E S E A R C H 6 1.6 S C O P E O F T H E T H E S I S 10 1.7 T H E S I S L A Y O U T 11 CHAPTER! TESTING PROCEDURE 13 2.1 G E N E R A L DESCRIPTION O F T H E S P E C I M E N 13 2.2 G R A V I T Y L O A D I N G 15 2.3 D Y N A M I C L O A D I N G 15 2.3.1 Shake Table Testing 15 2.3.2 Hammer Impact 16 2.4 I N S T R U M E N T A T I O N 17 2.4.1 External Instrumentation 17 iv 2.4.2 Internal Instrumentation 18 CHAPTER 3 E X P E R I M E N T A L TEST RESULTS AND INTERPRETATION 19 3.1 O V E R V I E W 19 3.2 D Y N A M I C C H A R A C T E R I S T I C S 19 3.2.1 Evaluation of Natural Frequencies from Experimental Data 19 3.2.2 Evaluation of Damping from Experimental Data 30 3.2.3 Filter Analysis 31 3.2.4 Time Histories 34 3.3 M A T E R I A L C H A R A C T E R I S T I C S 39 3.3.1 Hysteresis curves 39 3.3.2 Strain Gauge Evaluation 49 3.3.3 Evaluation of Forces 56 CHAPTER 4 C O M P U T E R PROGRAMS AND PROPOSED A N A L Y T I C A L M O D E L 66 4.1 O V E R V I E W 66 4.2 T H E C A N N Y - E C O M P U T E R P R O G R A M 66 4.2.1 The Element Library 67 4.2.2 Solution Technique 71 4.2.3 Analysis options 73 4.2.4 Hysteresis Modeling , 73 4.2.5 Damage Index 74 4.3 T H E R U A U M O K O C O M P U T E R P R O G R A M 75 4.3.1 The Element Library 76 4.3.2 Solution Technique 77 4.3.3 Analysis Options 77 V 4.3.4 Hysteresis Modeling 78 4.3.5 Damage Index --78 4.4 G E N E R A L A S P E C T S O F T H E C O M P U T E R M O D E L F O R C A N N Y - E A N D R U A U M O K O 79 4.4.1 Material Properties 82 4.4.2 Beam Modeling in CANNY-E 83 4.4.3 Column Modeling in CANNY-E 89 4.4.4 Link (Truss) Element Modeling in CANNY-E 92 4.4.5 Hysteresis Modeling in CANNY-E 92 CHAPTER 5 PUSH-OVER ANALYSIS 96 5.1 I N T R O D U C T I O N 96 5.2 C A N N Y - E 97 5.2.1 The Tri-Linear Model 98 5.2.2 The Bi-Linear Model 99 5.3 R U A U M O K O 101 5.4 C O M P A R I S O N S T U D Y 104 CHAPTER 6 C O R R E L A T I O N OF A N A L Y T I C A L AND E X P E R I M E N T A L RESULTS OF 0.27 S C A L E M O D E L 106 6.1 I N T R O D U C T I O N 106 6.2 E A R T H Q U A K E R E C O R D 106 6.3 F U N D A M E N T A L F R E Q U E N C I E S 107 6.4 D A M P I N G 108 6.5 A N A L Y T I C A L R E S P O N S E A N D C O M P A R I S O N S T U D Y 110 6.5.1 Linear Analysis I l l 6.5.2 Non-linear Analysis 119 CHAPTER! CONCLUSIONS 167 7.1 I M P O R T A N T F E A T U R E S O F T H E C O M P U T E R P R O G R A M S 168 vi 7.2 V A L I D I T Y O F T H E A N A L Y T I C A L R E S U L T S 169 7.3 C O M M E N T S O N A N A L Y S I S 170 7.4 R E C O M M E N D A T I O N S F O R F U T U R E W O R K 171 BIBLIOGRAPHY 173 APPENDIX A 0.27 Scale Oak Street Bridge Bent Drawing 176 APPENDIX B Sample Input-Output Files for C A N N Y - E and R U A U M O K 0 . . 1 7 8 vii LIST OF TABLES Table 2.1 Test Program .- : 16 Table 3.1 Comparison of Natural Frequencies and Natural Periods 29 Table 3.2 Measured Damping 31 Table 3.3 Maximum Values of Acceleration, Displacement, and Base Shear 35 Table 3.4 Progressing of Local Yielding 5 1 Table 3.5 Maximum Shear From the Elastic Approach - 10% Run 59 Table 3.6 Maximum Moment From the Elastic Approach - 10% Run 59 Table 3.7 Maximum Shear From the Elastic Approach - 40% Run 60 Table 3.8 Maximum Moment From the Elastic Approach - 40% Run 60 Table 3.9 Maximum Shear From the Elastic Approach - 60% Run 61 Table 3.10 Maximum Moment From the Elastic Approach - 60% Run 61 Table 3.11 Maximum Shear From the Elastic Approach - 80% Run 62 Table 3.12 Maximum Moment From the Elastic Approach - 80% Run 62 Table 3.13 Maximum Shear From the Elastic Approach - 120% Run 63 Table 3.14 Maximum Moment From the Elastic Approach - 120% Run 63 Table 3.15 Maximum Shear From the Elastic Approach - 150% Run 64 Table 3.16 Maximum Moment From the Elastic Approach - 150% Run 64 Table 3.17 Comparison of Forces 65 Table 4.1 Computer Model Node Coordinates 82 Table 4.2 Material Properties of Steel Bars (Unit: MPa) 82 Table 4.3 Shear Strength of Beam Sections (kN) 88 Table 4.4 Column Flexural and Shear Strength (kN-m) 91 Table 4.5 Column Axial Load (kN) 91 Table 5.1 Beams properties : E=27400 MPa, 1=0.0008389 m 4 97 Table 5.2 C A N N Y - E : Hinge Sequences Using aTri-linear Backbone Curve 101 Table 5.3 C A N N Y - E : Hinge sequences Using a Bi-linear Backbone Curve 101 viii Table 5.4 R U A U M O K O : Hinge Sequences Using a Bi-linear Backbone Curve 102 Table 6.1 Fundamental periods Obtained from Test, C A N N Y , and R U A U M O K O 108 Table 6.2 Damping Ratios 109 Table 6.3 Comparison of Base Shear Maximum Values 113 Table 6.4 Comparison of Displacement Maximum Values 113 Table 6.5 10% Test - C A N N Y Results from Different Runs 121 Table 6.6 10% Test - C A N N Y - E Results from Different Runs 126 Table 6.7 Maximum Values from 10% Test, C A N N Y - E , and R U A U M O K O 128 Table 6.8 40% Test - C A N N Y - E Results from Different Runs 131 Table 6.9 40% Test - C A N N Y Results from Different Runs 132 Table 6.10 Maximum Values from 40% Test, C A N N Y - E , and R U A U M O K O 137 Table 6.11 60% Run - C A N N Y Results from Different Runs compared to the Experiment 138 Table 6.12 Maximum Values from 60% Test, C A N N Y - E , and R U A U M O K O 140 Table 6.13 80% Run - C A N N Y - E Results from Different Runs 146 Table 6.14 80% Run - R U A U M O K O Results from Different Runs. 148 Table 6.15 Maximum Values from 80% Test, C A N N Y - E , and R U A U M O K O 148 Table 6.16 120% Run - C A N N Y - E Results from Different Runs 155 Table 6.17 120% Test - R U A U M O K O Results from Different Runs 156 Table 6.18 Maximum Values from 120% Test, C A N N Y - E , and R U A U M O K O 156 Table 6.19 Maximum Values from 150% Test, C A N N Y - E , and R U A U M O K O 162 ix LIST OF FIGURES Fig. 2.1 The Test Specimen Layout, Davey (1996) 14 Fig. 2.2 Hammer Sensor Locations, Davey (1996) 16 Fig. 2.3 External Instrumentation, Davey (1996) 18 Fig. 2.4 Internal Instrumentation, Davey (1996) 18 Fig. 3.1 Block Diagrams of A Structure in Time and Frequency Domain 20 Fig. 3.2 Dynamic Response of the 0.27 Scale Specimen - 10% Test 23 Fig. 3.3 Dynamic Response of the 0.27 Scale Specimen - 40% Test 24 Fig. 3.4 Dynamic Response of the 0.27 Scale Specimen - 60% Test 25 Fig. 3.5 Dynamic Response of the 0.27 Scale Specimen - 80% Test 26 Fig. 3.6 Dynamic Response of the 0.27 Scale Specimen- 120% Test 27 Fig. 3.7 Dynamic Response of the 0.27 Scale Specimen - 150% Test 28 Fig. 3.8 Natural Frequencies Vs. Run Amplitude from Hammer and Shake Table Tests 29 Fig. 3.9 5% Run - Power Spectrum of T5 Sensor, Strain at the top of the Beam 32 Fig. 3.10 150% Run - Power Spectrum of T5 Sensor, Strain at the top of the Beam 32 Fig. 3.11 150% Run - Power Spectrum of Bent Acceleration 33 Fig. 3.12 Absolute Acceleration Time Histories at Shake Table & Bent 36 Fig. 3.13 Relative Displacement Time Histories at the Top of Beam & Mass Block 37 Fig. 3.14 Base Shear Time Histories 38 Fig. 3.15 Hysteresis Curves - 10%, 40%, 60%, 80%, 120%, and 150% Tests 41 Fig. 3.16 Hysteresis Curves During 5 Sec. Intervals - 10% Test 42 Fig. 3.17 Hysteresis Curves During 5 Sec. Intervals - 40% Test 43 Fig. 3.18 Hysteresis Curves During 5 Sec. Intervals - 60% Test 44 Fig. 3.19 Hysteresis Curves During 5 Sec. Intervals - 80% Test 45 Fig. 3.20 Hysteresis Curves During 5 Sec. Intervals - 120% Test 46 Fig. 3.21 Hysteresis Curves During 5 Sec. Intervals - 150% Test 47 Fig. 3.22 Comparison of Hysteresis Curves for Critical Duration 48 Fig. 3.23 Strain Gauge Locations 51 ; Fig. 3.24 Experimental Curvatures (milli strain/mm) - 10% Test 52 x Fig. 3.25 Experimental Curvatures (mili strain/mm) - 40% Test 53 Fig. 3.26 Experimental Curvatures ( milli strain/mm) - 60% Test 54 Fig. 3.27 Experimental Curvatures (milli strain/mm) - 80% Test. 55 Fig. 3.28 Experimental curvature of the column 56 Fig. 3.29 Elastic Analysis Approach 57 Fig. 4.1 Beam Modeling 68 Fig. 4.2 Moment-Rotation Skeleton Curve for Beam Elements (Khoshnevissan and Li, 1996). 70 Fig. 4.3 Curvature Distribution Assumption (Khoshnevissan and Li, 1996) 70 Fig. 4.4 The Theoretical Model 80 Fig. 4.5 Details of Member Sections 81 Fig. 4.6 Material Stress-Strain Curves 83 Fig. 4.7 Node Position 83 Fig. 4.8 Moment-Rotation Skeleton Curve for Beam Element 85 Fig. 4.9 Column Modeling 89 Fig. 4.10 Effect of Variation of Axial Load on Flexural Capacity of Columns 91 Fig. 4.11 CANNY Hysteresis Model #15 95 Fig. 5.1 Model Outline 97 Fig. 5.2 CANNY-E: Static Push-Over Analysis Using a Trilinear Backbone Curve 100 Fig. 5.3 Period of the Specimen at Different Stages of Loading 100 Fig. 5.4 CANNY-E: Static Push-Over Analysis Results from Two Different Models 101 Fig. 5.5 CANNY & RUAUMOKO: Static Push-Over Analysis Results 103 Fig. 5.6 RUAUMOKO: Static Push-Over Analysis for Different Plastic-Hinge Lengths 103 Fig. 5.7 Damage Distribution and Mechanism 105 Fig. 6.1 ComparisonBetween Magnitudes of Inertia, Viscous Damping, and Base Shear Force obtained from RUAUMOKO 112 Fig. 6.2 Elastic Analysis - Results from CANNY and RUAUMOKO using Shake Table Recorded Acceleration from 10% Test. Relative Displacement Time Histories at the Cap Beam Level and Base Shear Time Histories 114 Fig. 6.3 Elastic Analysis - Relative Displacement Time Histories at the Cap Beam Level: 10% Test, XI CANNY-E, and RUAUMOKO 115 Fig. 6.4 Elastic Analysis - Base Shear Time Histories: 10% Test, CANNY-E, and RUAUMOKO 116 Fig. 6.5 Elastic Analysis - Relative Displacement Time Histories at the Cap Beam Level: 40% Test, CANNY-E, and RUAUMOKO 117 Fig. 6.6 Elastic Analysis - Base Shear Time Histories: 40% Test, CANNY-E, and RUAUMOKO 118 Fig. 6.7 10% Run - Analytical Response Time Histories: CANNY-E: 122 Fig. 6.8 Relative Displacement Time Histories at the Cap Beam Level: 10% Test, CANNY-E, and RUAUMOKO 123 Fig. 6.9 Base Shear Time Histories: 10% Test, CANNY-E, and RUAUMOKO 124 Fig. 6.10 Absolute Acceleration Time histories at the Cap Beam Level: 10% Test, CANNY-E, and RUAUMOKO 125 Fig. 6.11 Base Shear vs. Relative Displacement at the Cap Beam Level: 129 Fig. 6.12 Relative Displacement Time Histories at the Cap Beam Level: 40% Test, CANNY-E, and RUAUMOKO 133 Fig. 6.13 Base Shear Time Histories: 40% Test, CANNY-E, and RUAUMOKO. 134 Fig. 6.14 Absolute Acceleration Time Histories at the Cap Beam Level: 40% Test, CANNY-E, and RUAUMOKO 135 Fig. 6.15 Base Shear vs. Relative Displacement at the Cap Beam Level: 40% Test, CANNY-E, and RUAUMOKO 136 Fig. 6.16 Relative Displacement Time Histories at the Cap Beam Level: 141 Fig. 6.17 Base Shear Time Histories: 60% Test, CANNY-E, and RUAUMOKO 142 Fig. 6.18 Absolute Acceleration Time Histories at the Cap Beam Level: 143 Fig. 6.19 Base Shear vs. Relative Displacement at the Cap Beam Level: 144 Fig. 6.20 Hysteresis Loops at Different Time Intervals: 60% Test and CANNY-E 145 Fig. 6.21 Relative Displacement Time Histories at the Cap Beam Level: 149 Fig. 6.22 Base Shear Time Histories, 80% Test, CANNY-E, and RUAUMOKO 150 Fig. 6.23 Absolute Acceleration Time Histories at the Cap Beam Level: 151 Fig. 6.24 Base Shear vs. Relative Displacement at the Cap Beam Level: 152 Fig. 6.25 Parametric Study of CANNY-E - 80% Run 153 Fig. 6.26 Relative Displacement Time Histories at the Cap Beam Level: 157 Fig. 6.27 Base Shear Time Histories, 120% Test, CANNY-E, and RUAUMOKO 158 xii Fig. 6.28 Absolute Acceleration Time Histories at the Cap Beam Level: 159 Fig. 6.29 Base Shear vs. Relative Displacement at the Cap Beam Level: 160 Fig. 6.30 Relative Displacement Time Histories at the Cap Beam Level: 163 Fig. 6.31 Base Shear Time Histories, 150% Test, CANNY-E ; and RUAUMOKO 164 Fig. 6.32 Absolute Acceleration Time Histories at the Cap Beam Level: 165 Fig. 6.33 Base Shear vs. Relative Displacement at the Cap Beam Level: 166 X l l l A C K N O W L E D G M E N T I would like to express my first, and foremost gratitude to my supervisor, Dr. Robert G. Sexsmith, for his guidance, inestimable assistance, and constant encouragement during the course of this research. His knowledge, modesty, understanding, and support are sincerely appreciated. I would also like to express my special gratitude to Dr. Carlos Ventura for his excellent guidance throughout this research and during the reading of this thesis and for his very useful comments. The advice and assistance of Dr. Kang Wing Li during his stay at the University of British Columbia is also appreciated. I wish to thank my colleagues and friends, Caroline, Jennifer , Hong, David, Ye, and Thomas for sharing knowledge and for providing me a lovely and enjoyable environment in "Timber Lab". Special thanks to my colleague and friend, Ahmed Ebrahim, for studying together and for being a generous source of help. Many thanks to my dear little son, Mahon, for being patient with a student mom. Finally, my very special and heartfelt gratitude goes to my husband, Masoud, for his love, patience, unconditional support, and endless encouragement during the whole course of this study. xiv DEDICATION To my wonderful parents XV C H A P T E R 1 INTRODUCTION 1.1 Preliminary Remarks Recent earthquakes around the world have demonstrated that the seismic vulnerability of structures leads to catastrophic events that often result in substantial losses of life and capital. Structures designed before modem specifications for earthquake resistant design are more likely to suffer dramatic damage or collapse in tire event of an earthquake. Design, construction, and retrofit of structures are major concerns in potential earthquake regions. One of the first approaches to improving the design of structures and especially to improving the seismic resistance of older structures is to arrange research programs which use both experimental and analytical techniques. Experiments on full- or reduced-scale structures can provide insight into the static and dynamic behavior of real structures. The reliability of analytical techniques can also be validated by test results. There are several different methods for seismic testing, including static (push-over) tests, slow cyclic tests, shake table tests, and pseudo-dynamic tests. Among these techniques shake table testing is particularly effective because it uses a real earthquake time history and evaluates the inelastic response of a structure in a realistic way. Despite its effectiveness, experimental testing is both expensive and time-consuming, and sometimes it is simply not practical to conduct seismic tests. Analytical and numerical tools, especially computer modeling 1 Chapter 1 2 techniques, are good alternatives to experimental testing in these situations, provided that they are reliable and realistic. It is, therefore, important to know with precision how reliably these computer programs predict the seismic response of real structures. 1.2 Motivation There are numerous commercial programs available, such as SAP90 (Wilson and Habibullah, 1992) and ETABS (Habibullah, 1992), that accurately predict the linear static and dynamic behavior of structures. However, most structures respond non-linearly and defomi inelastically when subjected to moderate or severe earthquakes. In these cases it is necessary to use non-linear computer programs which can simulate such non-linear behavior. With the advent of powerful new computers, dynamic analysis of nonlinear structures is finding its way into engineering practice. Although the vast majority of analysis in engineering offices follows the traditional pattern of elastic or linear analysis, when refinements in design or detailing of more accurate response is necessary (particularly for such severe loading effects as those due to earthquakes, blast, etc.), nonlinear analysis techniques become a necessity. In response to this demand there are new computer programs, such as ANSR-III (1982), ID ARC (1992), DRAIN-2DX (1993), CANNY-E (1995), and RUAUMOKO (1996), which can perioral structural non-linear dynamic analyses. Most of these programs fonn the element tangent stiffness and solve the equations of motion for the structure by transforming the dynamic problem into a static equivalent of the fonn K . Ax = AF, where K represents the tangent stiffness matrix, Ax the deformation increment and AF the incremental forces (Inaudi and Llera, 1992). As long as the structure remains within elastic limits, these computer programs can predict the response of the structure very accurately. Their performance and Chapter J 3 accuracy in nonlinear analysis of structures, however, might not be as accurate due to the use of different algorithms in modeling nonlinear static and dynamic behaviors. Among the nonlinear computer programs (some of which were mentioned earlier), CANNY-E (Li, 1995) and RUAUMOKO (Carr, 1996) are rather new programs that take advantage of the most recent advances in the theory of nonlinear structures and numerical computations. CANNY-E (or simply CANNY) is a three-dimensional non-linear dynamic structural analysis computer program, while RUAUMOKO is a two-dimensional non-linear dynamic analysis computer program. Recently developed, these programs have yet to be extensively evaluated by the academy or the industry. A new shake table experiment carried out on a 0. 27 scale model of the Oak Street Bridge bent in the earthquake laboratory at UBC (Davey, 1996) has provided us an excellent opportunity to compare experimental results with analytical results obtained from the foregoing programs. 1.3 Objective To understand the seismic response of a two-column bridge bent of the Oak Street Bridge, a combined experimental and analytical study was performed on a 0.27 scale model of the bridge bent. A series of shake table tests (Davey, 1996) provided the data necessary for this study. It is anticipated that the degree of correlation between the experimental and analytical results would determine the reliability of the analytical approaches. As such, the main objectives of this thesis are: 1. to evaluate the experimental results obtained from a 0.27 scale model of the Oak Street Bridge that was tested on a shake table; Chapter 1 4 2. to examine the dynamic properties of the specimen at different levels of excitation and to assess tire effect of damage on dynamic characteristics of the specimen from the experimental test results; 3. to determine the reliability and applicability of two non-linear computer programs, CANNY-E and RUAUMOKO, in predicting the seismic response of the specimen when compared with the shake table test results; 4. to compare analytical responses from two computer programs, CANNY-E and RUAUMOKO. 1.4 Project Background Owing to serious existing deficiencies in the seismic behavior of many major bridges in the Lower Mainland (Crippen International Ltd., 1993), the British Columbia Ministry of Transportation and Highways decided to organize some laboratory test programs at the University of British Columbia (Anderson et al., 1995). The Oak Street Bridge, constructed in the 1950's and designed for a peak horizontal acceleration of only 0.04g (Williams, 1994), was recognized as the most vulnerable bridge to an earthquake (English, 1993). In 1995, several specimens at 0.45 scale of a typical two-column bent of die Oak Street Bridge underwent slow cyclic tests in the Structural Engineering Laboratory at UBC (Seefhaler, 1995). The as-built specimen failed in shear, and showed very low ductility. The peak lateral load was 265 kN at the lateral displacement of 9 mm (Anderson et al., 1995). When the test was repeated for four retrofitted bridge bents, both seismic capacity and ductility increased and overall strength improved considerably. Williams (1994) investigated the correlation between the test results and the analytical approach. The non-linear computer program ID ARC (Kunnath, 1992) was used for the analysis, and hysteresis response and damage indices were the two subjects of comparison. The correlation between hysteresis responses from the Chapter 1 5 experimental and analytical approaches was good. The modified Park and Ang (1985) damage index, which is included in IDARC, provided a good estimate of the damage state of the bents in a cyclic displacement-controlled laboratory test. The slow cyclic test also provided a good source of data for the comparison of analytical and experimental curvature distribution in bridge bents under slow cyclic test (English, 1996). In this case, the computer program DRAIN-2DX (Powell and Campbell, 1993) was used for simulation. Generally, there was agreement between the shapes and the magnitudes of the curvatures obtained from both test and computer analysis. The need for more strain gauges to obtain more detailed curvatures, the sensitivity of the experimental results to the strain gauge location, and the proper installation of gauges were also discussed. The limitations of the DRAIN-2DX in reflecting the beam-column joint responses was also stressed. In 1996, a 0.27 scale model of the Oak Street bridge bent was built and tested on the shake table in die Earthquake Laboratory at UBC (Davey, 1996). Based on the scaling theory, the capacity of the shake table, and the available size of reinforcing steel, a 0.27 scale was selected for a typical series of the Oak Street Bridge bents. A concrete mass block with several steel plates on top simulated the structure's dead load. The gravity load was transferred to the bent through five rubber bearing locations, while the lateral load was transferred to the specimen through two connections provided by using Dywidag™ post-tensioning bars. The lateral support system was provided via four wire rope tie-downs. The columns were hinged at the bottom. The specimen was instrumented internally and externally and testing was monitored through strain gauges, accelerometers and displacement transducers, as well as two video cameras. The specimen was subjected to two types of excitation: 1) a hammer impact test to obtain the natural frequencies of the bent, and 2) a shake table test to simulate an earthquake motion. The Joshua Tree Fire Chapter 1 6 Station E-W component from the 1992 California Landers Earthquake (Naeim and Anderson, 1993) was used in the shake table test. Both the peak acceleration and time increment of the excitation time liistory were scaled. The shake table tests were run mainly at 7 different levels of excitations: 10%. 20%, 40%, 60%, 80%, 120%, and 150% of the scaled PGA of the reference earthquake. For the 0.27 scale test, the maximum base shear was 111.1 kN during the 120% run. During the 150% run, the table peak acceleration was 1.13g and the cap beam relative displacement was 35.3 mm. The test results were compared with those from the slow cyclic test, including structural behavior, failure mode, and hysteresis loops. The modes of the failure from both slow cyclic and shake table tests were due to shear failure of the cap beam. The hysteresis curves were also compared. The bent demonstrated less degradation of strength from shake table tests than from the cyclic test. The recorded data during the shake table test provided a unique opportunity for a comparison study. Later, this 0.27 scale model was retrofitted and tested again on the shake table The results of this experiment are reported by Chen and Sexsmith (1998) and Chen (1998). 1.5 Related Research In early 1980's, a 1/5 scale model of a 7-story reinforced concrete frame-wall structure was tested on a shake table at the University of California (Aktan, 1983). This experiment was part of the comprehensive US-Japan cooperative-research program on the behavior of structures subjected to earthquake. In addition to experimental work, analytical studies were performed at the University of California, Berkeley. Several reports and papers were published based on this experimental and analytical study (Charney and Bertero, 1982), (Aktan et al., 1983), (Bertero et al., 1984), (Charney, 1991). Chapter 1 1 Charney (1991) studied the correlation of the analytical and experimental inelastic responses of the scaled model and addressed the difficulties and uncertainties involved in using inelastic analysis. The physical model was subjected to a total of 63 tests, including free vibration, harmonic shaking, and simulated earthquakes. Two different signals from the Japanese Miyyagi-Okki earthquake and the California Taft earthquake were used in these tests. The maximum responses were investigated and it was observed that failure was caused by fracturing of longitudinal reinforced concrete at the base of the main shear wall. For comparison, a two dimensional analytical model was created and the computer program DRAIN-2D (Kaanan and Powell, 1973) and two other auxiliary computer programs were used in the analysis. One component model (Giberson, 1967) was used for modeling the girders in the analysis by DRAIN-2D. A one-component model consisting of a linear elastic beam element, with a nonlinear plastic hinge was located at each end. A two-component beam-column element of DRAIN (Clough et al., 1965) was used for column modeling. The end walls were modeled as truss elements, while for transverse beams a simple extension spring was used. The analytical model was then subjected to a set of three different monotonically-increasing static lateral local patterns and the global analytical response of the model was computed. Also the sequence of plastic hinge formation was studied and the analytical and experimental global response were compared. The general correlation was found to be satisfactory. The experimental response formed an upper bound for the analytical responses. For a given value of roof displacement, the analytical base shear was on the average 22% different from the experimental values, while die difference between the fundamental frequency of the analytical model and the test structure was only about 2%. Chapter 1 8 The next phase of the analysis was to subject the mathematical model to dynamic excitation, and to compare the computed response to the experimentally measured response. In the mathematical model, die structure was considered to have significant cracks at the beginning of the analysis. The analysis was perfonned using the damping values previously obtained through the shake table test, and an integration time step of 0.005 seconds. The best possible correlation with the existing mathematical model yielded peak base shears and overturning moments of approximately 85 percent of the corresponding experimental values. The main source of error in computing the global response was due to ignoring damping forces. Modeling errors were associated with the modeling of tire individual elements of the structure, and since recalculation of the static response with a smaller time step yielded no significant difference in response, the numerical error was not the cause of the discrepancy. Neglect of the strain rate for steel and concrete in the formulation was also discussed. Aside from experimental errors, other differences between computed and measured response can be associated with differences between the initial frequency in the experimental and analytical models, differences between effective damping in the experimental and analytical models, and errors in modeling the hysteretic response of the individual elements of the structure. The variation of computed response with input signal was also studied and the variation of computed response with the initial frequency of the model was discussed. It was observed that as the initial frequency of the structure decreased, the amplitude of roof displacements increased while the magnitude of peak base shear remained relatively constant. The time histories of peak roof displacement and base shear at 2.75 Hz indicated that the analysis was almost in phase with the experimental response. Even though this model provided better correlation, the analytical peak roof displacement and base shear were still considerably low, and hence the degree of correlation obtained was still described as poor. Chapter 1 9 The author concluded that it is not easy to obtain a good correlation between the test, and the analytical results for dynamic inelastic response. There are many parameters which make this task difficult. For example, uncertainty in modeling, sensitivity to the relationship between the initial frequency of the analytical model and the frequency characteristics of the ground motion, and uncertainties in the determination of the viscous damping are some of the reasons that make nonlinear dynamic analysis difficult and unreliable, unless several analyses, accounting for these uncertainties, are carried out. Another correlation study between a shake table test result and an analytical response was conducted by Filiatrault (1990) at the University of British Columbia. In order to examine die ultimate lateral load carrying capacity and to study the dynamic behavior of timber shear walls, a non-linear computer program SWAP (Shear Wall Analysis Program), was developed. The program was able to perfonn both static and dynamic analysis. The analytical model was validated by comparing it to the full-scale racking and shaking test results, which were also performed at die University of British Columbia (Dolan 1989). The shear walls were 2.4 m x 2.4 m. The sheathing panels were 1.2 m x 2.4 m of 9 mm plywood which were connected to the frame with 63.5 mm long galvanized nails. In SWAP, the dynamic analysis was carried out by solving the differential equations of motion in each time step. The mass matrix was diagonal, and for computational simplicity, the damping matrix was proportional only to the mass matrix. Newmark's average-constant acceleration algorithm (Clougli and Penzien, 1993) was applied for integration. The results from static and free-vibration tests were compared with those from analytical approaches. A general agreement was observed. The ultimate loads predicted by SWAP were within 9% of the average test peak loads, and the maximum discrepancy of the initial fundamental frequencies was 15%. The Chapter 1 10 computed displacement at the top of the wall from dynamic analysis was compared to the measured one, and the correlation for both amplitude and phase were found reasonable. One of the main advantages of the SWAP program is that the shear walls can have arbitrary geometry, so that walls with openings can be analyzed. Although the computer program provided reasonable results, it was restricted to two-dimensional shear walls. Another limitation is that, in dynamic analysis, the equilibrium equations are linearized within a time-step, At. Therefore, the accuracy of the solution depends upon the size of the time step. In dynamic analysis, the comparison was made only for the top displacement of the walls. Sensitivity of results to other parameters, such as acceleration, forces, hysteresis loops, and damage, was not considered. 1.6 Scope of the Thesis This thesis is a comparative study of the experimental and analytical responses of a 0.27 scale model for a two-column bent of the Oak Street Bridge under shake table tests. The experimental work was carried out and reported by Davey (1996). The detailed test results from six levels of excitation are studied and interpreted. An analytical model is presented and time history analyses are performed by using two computer programs: CANNY-E (Li, 1995), and RUAUMOKO (Carr, 1996). Comparisons are drawn for six levels of excitation. The sequences of plastic liinge formation are investigated through a push-over analysis using these two computer programs. The advantages and disadvantages of the programs are discussed accordingly. Chapter 1 11 1.7 Thesis Layout The thesis is divided into seven chapters. In Chapter 2, an overall description of the specimen, gravity loading, dynamic testing, and instrumentation used during testing is presented. Assessment of test results from six consecutive tests is provided in Chapter 3. The natural frequencies of the specimen from both shake table and hammer tests are obtained and compared. The damping coefficients are computed from the recorded data. Filtering of the experimental data for purposes of noise removal is explained. Absolute acceleration, relative displacement, and base shear time histories for three different runs, 10%, 60%, and 150%, are plotted and compared. Hysteresis loops for six levels of excitation are studied and the non-linear behavior is compared in successive tests. Strain gauge records are evaluated and the curvature distribution over the cap beam and columns are presented. Finally, the computed forces are validated. Chapter 4 describes two computer programs, CANNY-E and RUAUMOKO, in detail. The advantages, disadvantages and limitations of each program are discussed. Details of mathematical models including model layout, geometric data, material property, and hysteresis models are described. Chapter 5 examines a push-over analysis on a 0.27 scale model using CANNY-E and RUAUMOKO. The distribution of plastic hinges is investigated and a comparative study of results obtained from each computer program is perforated. In Chapter 6, a time history analysis for the 0.27 scale model is perfonned on analytical models of the bent using the CANNY-E and RUAUMOKO computer programs. The improvement of die model's frequency is explained. The earthquake record and damping value used in the analysis are studied. Similarities and Chapter 1 12 differences between the two computer models, the first from CANNY-E and the second from RUAUMOKO, are investigated by a linear analysis. Non-linear analyses for the two models are carried out for six different levels of excitation. The time histories for absolute acceleration, relative displacement, and base shear are plotted and compared to the ones from experimental tests. A strong emphasis is placed on assessing the correlation between experimental results and analytical results obtained from the two programs. In Chapter 7, conclusions related to tire work done in this thesis are explained. The chapter concludes with recommendations for future research. C H A P T E R 2 TESTING PROCEDURE 2.1 General Description of the Specimen The 0.27 scale model of the Oak Street Bridge was designed and tested on the shake table of the Earthquake Engineering Research Laboratory at U B C (Davey, 1996). The geometric parameters of the 0.27 scale reinforced concrete model were designed to comply with the requirements of a reduced scale model of the prototype. The same prototype material was used in the specimen. The average concrete strength for the specimen was assumed to be 40.5 MPa, while the initial Youngs modulus of elasticity was set to 30351 MPa. An average yield stress of 461 MPa was used for #3 reinforcing steel. Fig. 2.1 shows the test specimen layout. The two-column bent was hinged at the base. The overall dimensions of the specimen were 1573 mm in height by 4304 mm in width. The beam was cantilevered at both ends by 703 mm. The dimensions of the columns and the beam were 327 mm x 327 mm and 290 mm x 411 mm, respectively. The figure in Appendix A illustrates the specimen with more details concerning the reinforcement arrangement. More detailed information can be found in Davey (1996). The weight of the superstructure was simulated by a concrete mass block with several steel plates on top. The center of gravity of the mass was 411 mm above the top of the cap beam in order to comply with the scaling of the center of gravity for the bridge deck of the prototype. The concrete mass block was 4360 mm x 500 x 235 mm, with a total weight of 12.3 k_N, and to prevent damage during testing process, it was reinforced longitudinally. The steel plates provided 76.7 kN of weight. The size of each plate was 1500 mm 13 Chapter 2 14 x 600 mm with three different thicknesses: 9 mm, 31.75 mm, and 63.5 mm. These plates were connected to the concrete mass block by hooked steel ready-rods Davey (1996). The mass block rested on five rubber pad bearings located on the beam, which corresponded to the prototype girder locations. The two interior bearings were pinned after the mass block was placed, and provided the connection between the bent and the mass block. The lateral load was therefore transferred through these connections. A total of four Dywidag post tensioning bars were used to provide these pinned connections. Four wire ropes perpendicular to the main axis of the bent were used as lateral bracing to restrain out-of-plane motions. Fig. 2.1 The Test Specimen Layout, Davey (1996). Chapter 2 15 2.2 Gravity Loading A scaled dead-load weighing 89 kN, was provided by a one-beam spanning concrete mass block with steel plates on top. The dead load was transferred to the specimen through five rubber pad bearings to simulate five superstructure bearing locations. The specimen had a total weight of 18.2 kN. 2.3 Dynamic Loading The specimen was excited only in the longitudinal direction. Two types of dynamic testing were performed : a shake table test and a hammer impact test. These dynamic loads are described in the following sections. 2.3./ Shake Table Testing In search of an appropriate earthquake record that could follow the design spectrum and could cause damage to the specimen, the scaled Joshua Tree Fire Station record in E-W direction from the Landers Earthquake was used as the base excitation to represent an earthquake motion (Davey, 1996). This earthquake, which occured in 1992, was characterized by a peak ground acceleration of 0.28g and a magnitude of 7.5 on the Richter scale (Naeim and Anderson, 1993). The earthquake record was modified and then scaled both in duration and acceleration. After scaling, the peak ground acceleration was 0.7g with a duration of about 10 seconds. The test was carried out in incremental steps, starting from a very low-amplitude excitation (to check the shake table performance and the data acquisition system) up to 1.5 times of the full-scale earthquake (at Chapter 2 16 which point the structure failed). Table 2.1 lists different levels of the excitation input used in the experiments. The "Run %" in the table is percentage of the scaled Joshua Tree record with peak ground acceleration of 0.7g. The data sampling frequency in each run was 200 Hz Table 2.1 Test Program. Run# Run % PGA (g) Input 1 5 0.035 Joshua Tree Record 2 10 0.07 Joshua Tree Record 3 20 0.14 Joshua Tree Record 4 40 0.28 Joshua Tree Record 5 60 0.42 Joshua Tree Record 6 80 0.56 Joshua Tree Record 7 120 0.84 Joshua Tree Record 8 150 1.05 Joshua Tree Record 2.3.2 Hammer Impact An instrumented hammer was used to excite the specimen. This method of testing was an easy way to obtain the natural frequencies of the specimen for comparison with frequencies obtained from shake table tests. The hammer consisted of an integral piezoelectric force sensor which measured the input signals. Hammer impact was applied to the end point of the cap beam in the longitudinal direction. As shown in Fig. 2.2, four accelerometers were installed on the specimen to measure its response. Output Output Output r Outpuc Hammer Input w Fig. 2.2 Hammer Sensor Locations, Davey (1996). Chapter 2 17 2.4 Instrumentation Both internal and external sensors were used to record the global and local responses of the specimen. A total of forty-eight channels of data were used for each test, consisting of two sets of data collection banks with 16 filtered channels, and an extra 16 unfiltered channels. The instrumentation incorporated strain gauges, accelerometers, and displacement transducers. The sampling frequency was 200 Hz, i.e., the signal from each instrument was recorded every 0.005 seconds. A detailed description of the data acquisition system can be found in Davey (1996). The testing was also monitored through two video cameras, one recording at a normal speed and the other recording at a high-speed. The normal-speed video camera captured the overall motion of the specimen while the high-speed one recorded the development of cracks in the beam-column joint. The speed of the latter was 1000 frames per second. 2.4./ External Instrumentation Fig. 2.3 shows a schematic layout of external instrumentation. Six accelerometers were employed to record the absolute acceleration of the specimen and the shake table. The arrows in the figure show the direction of the recorded data. The accelerometers at both ends of the cap beam were capable of measuring the acceleration in longitudinal and transverse directions. While die recorded data in the transverse direction was used to analyze the amount of torsion induced in the specimen, the normal and high-speed video cameras were used to capture the behavior of the specimen. The absolute displacements of the shake table and the specimen were monitored by Linear Variable Differential Transducers (LVDT) which were installed at three different locations on both the shake table and the specimen. Chapter 2 18 I (§)—•displacement transducer (§) strain gauge @—•accelerometer Fig. 2.3 External Instrumentation, Davey (1996). 2.4.2 Internal Instrumentation A total of 24 strain gauges were attached to the reinforcing steel throughout the specimen. They were installed in locations which were expected to experience the most severe damage. While 20 of the strain gauges were mounted on longitudinal reinforcing bars in cap beams and columns, the remaining four strain gauges were attached to two stirrups to monitor the shear failure. The locations of all strain gauges are shown in Fig. 2.4. Fig. 2.4 Internal Instrumentation, Davey (1996). C H A P T E R 3 E X P E R I M E N T A L TEST RESULTS AND INTERPRETATION 3.1 Overview Dynamic behavior and material characteristics of the specimen are the two main subjects in this chapter. After calibrating the data, collected by strain gauges, accelerometers, and potentiometers, the dynamic properties of the specimen were studied, and its natural frequency and damping rado were calculated for each individual test. The data was low-pass filtered and response time histories were plotted. Then, using the filtered data and some other results from dynamic studies of the specimen, hysteresis loops, forces, and other material characteristics were studied. 3.2 Dynamic Characteristics A particularly important aspect of this quantitative study is the determination of the dynamic characteristics of the specimen. In this section, evaluation of natural frequencies, damping, and time histories are studied, and the data filtering process is discussed. 3.2.7 Evaluation of Natural Frequencies from Experimental Data Theoretical Background - Vibration signals from experimental testing are usually measured as a time history record. The Fourier Transform converts signals from time domain to frequency domain. Frequency 19 Chapter 3 20 response analysis is a very powerful tool for obtaining basic properties of a linear or nonlinear structure that has undergone displacements under periodic or nonperiodic loads. From the block diagram in Fig. 3.1, using Discrete Fourier Transform (DFT) of the input acceleration, Z(t), and the output displacement of the structure, X(t), we can determine which frequencies the input force and the output displacement have been rich in (resonance frequency), and, accordingly, what the maximum amplitudes have been. Also, from the input-output data, we can compute the Frequency Response Function (FRF) of the structure if it is assumed to have a linear behavior. The FRF may then be used to obtain the natural frequencies and the damping ratios of the structure. structure structure Z(t) * jh(t)dt X(t) Z(co> H(co) X ( G » (a) time domain (b) frequency domain Fig. 3.1 Block Diagrams of A Structure in Time and Frequency Domain. Assuming that the records correspond to the response of a linear elastic system, the DFTs of the shake table acceleration and of the relative displacement at the cap beam level of the specimen can be computed using such programs as DFT123 (Ventura, 1992) or MATLAB (Little and Shure, 1990). The FRF of an equivalent single-degree-of-freedom (SDOF) system may be obtained from Fig . 3.1b, as: FRF (a) = i^- = Re(co)+/ Im(co) Z(co) Chapter 3 21 where "Re" and "Im" indicate the real part and the imaginary part of the FRF, respectively. Since the DFTs of the relative displacement X(co) and the shake table acceleration Z(co) are generally complex numbers, tire computed FRF of the structure is also in a complex form. Its amplitude and phase angle can be computed from: I FRF (co) I = A/Re(co)2 + Im(to)2 < / W = t a n - ' ( ^ ) Re(co) where "I... I" and "< ..." mean absolute value (magnitude) and phase angle of the FRF, respectively. Determination of the Natural Frequency - Two types of vibration were performed on the specimen:: impact test using an instrumented hammer and shake table test. The applied load in the impact tests and the direction of the excitation in the shake table tests were in the longitudinal direction, and the fundamental frequencies were found for this direction. For the hammer tests, the outputs from three accelerometers alongside the cap beam and one at the base of the west column were recorded. Since the readings from all four accelerometers were supposed to be the same, any of them could be taken as an output sensor. For the hammer tests, calculations were performed using the FRF computer program (Horyna and Ventura, 1995). The MATLAB computer program was employed to obtain the natural frequencies from the shake table acceleration records (Figs. 3.2(a) to 3.7(a)) and tire relative displacement records at the cap beam level (Figs. 3.2(c) to 3.7(c)). The "spectrum" function in MATLAB's Signal Processing Toolbox (Little and Shure, 1990) was used to Fourier-transfonn the input-output signals and to estimate the FRF of the system (the solid curves in Figs. 3.2(e),(f) to 3.7(e),(f)). The "spectrum" function utilized the Welch method of power spectrum estimation and a Harming window to smooth the frequency analysis results (Oppenheim, Chapter 3 22 1975). Since the FRF analysis yields the best results for a linear system, only the high intensity regions in the test data (between 7 to 14 seconds in Figs. 3.2(a),(c) to 3.7(a),(c)) were windowed for the analysis. Next, in order to determine the natural frequency of the structure for each test, a linear second order system was fitted (the dotted lines in Figs. 3.2(e),(f) to 3.7(e),(f)) to the experimental FRF data using the MATLAB's "invfreqs" function (see also Sec. 3.2.2). Since the FRF of a second order system was quite smooth, for each test the FRF's peak amplitude was located at a frequency where the phase angle was 90 degrees. This frequency was taken as the natural frequency of the system. The results from botii hammer tests and dynamic tests are summarized in Table 3.1. While there was close agreement between the frequencies obtained through hammer tests and shake table tests at low-amplitude excitations, the gaps between the measured frequencies from both methods increased as the amplitude of excitation increased. The ratio of the fundamental frequency measured during the shake table test to the one measured using the hammer test changed from 0.84 for the 10% test to 0.71 for the 150% test. Figs. 3.2 to 3.7 show the dynamic response of the 0.27 scale specimen during the 10% to 150% shake table tests. Fig. 3.8 compares the frequencies obtained through the shake table tests to the frequencies computed from the hammer tests. The computed fundamental frequency differed to some extent depending on the technique used in its determination. One of the major causes for the discrepancy was due to the nonlinear behavior of the structure, e.g., cracking and steel debonding, wliich were more pronounced during strong earthquake excitations. The displacement amplitude during the hammer test was not big enough to allow the cracks to open completely; hence, the section and global stiffness corresponded to a minor crack state. The shake table excited the specimen more strongly than the hammer did, and, as a result, the damage state was more clearly observed in the shake table test. Chapter 3 23 (a) Shake Table Acceleration Time History (b) Shake Table Acceleration Power Spectrum 0.50 2 0.00 -0.50 0.10 10 20 Time (sec) 30 c 0> Q 1 0.05 <u a. </> a 3 o o. 0.00 10 15 Frequency (Hz) 20 25 (c) Relative Displacement Time History at Cap Beam Level Time (sec) (d) Relative Displacement Power Spectrum at Cap Beam Level 200 Z 100 a. in 10 15 Frequency (Hz) 20 25 ( e) F R F of Bent Relative Displacement (f) Phase Angle of Bent Relative Displacement 0.02 0.01 .a < 0.00 200 2 3 4 5 6 7 8 9 10 Frequency (Hz) -200 2 3 4 5 6 7 8 Frequency (Hz) 9 10 Fig. 3.2 Dynamic Response of the 0.27 Scale Specimen -10% Test. Chapter 3 24 (a) Shake Table Acceleration Time History (b) Shake Table Acceleration Power Spectrum 10 20 Time (sec) 30 0.40 r c Q I 0.20 5 o Q. 0.00 10 15 Frequency (Hz) 20 25 (c) Relative Displacement Time History at Cap Beam Level 12 r : (mm) 8 -4 -c i a) 0 -lisplac -4 --8 --12 -10 20 30 Time (sec) (d) Relative Displacement Power Spectrum at Cap Beam Level 400 c a> a oi Z 200 a> a. co o a. 10 15 Frequency (Hz) 20 25 Fig. 3.3 Dynamic Response of the 0.27 Scale Specimen - 40% Test. Chapter 3 25 (a) Shake Table Acceleration Time History (b) Shake Table Acceleration Power Spectrum 0.50 c o E 0.00 -0.50 2.00 r 10 20 Time (sec) 30 a. in 0.00 L 10 15 Frequency (Hz) 20 25 (c) Relative Displacement Time History at Cap Beam Level 12 r 10 20 Time (sec) (d) Relative Displacement Power Spectrum at Cap Beam Level 800 r 400 3 o a. 10 15 Frequency (Hz) 20 25 (e) F R F of Bent Relative Displacement (f) Phase Angle of Bent Relative Displacement 0.03 u. 0.02 .Q < 0.01 0.00 200 U) c < 100 2 -100 -200 5 6 7 8 Frequency (Hz) 10 Fig. 3.4 Dynamic Response of the 0.27 Scale Specimen - 60% Test. Chapter 3 26 (a) Shake Table Acceleration Time History 1.00 r -= 0.50 D) 0.00 -0.50 -1.00 10 20 Time (sec) 30 (c) Relative Displacement Time History at Cap Beam Level 16 12 8 4 0 -4 -8 -12 -16 — 0 10 20 30 Time (sec) (b) Shake Table Acceleration Power Spectrum 0.40 r Z 0.20 0) Q. CO 0.00 10 15 Frequency (Hz) 20 25 (d) Relative Displacement Power Spectrum at Cap Beam Level 1000 r o 500 h a> a. co 10 15 Frequency (Hz) 20 25 (e) F R F of Bent Relative Displacement 0.02 r Frequency (Hz) (f) Phase Angle of Bent Relative Displacement 200 r -200 2 3 4 5 6 7 8 9 10 Frequency (Hz) Fig. 3.5 Dynamic Response of the 0.27 Scale Specimen - 80% Test. Chapter 3 27 (a) Shake Table Acceleration Time History (b) Shake Table Acceleration Power Spectrum 1.00 0.80 -1.00 10 20 Time (sec) 30 in c • I 0.40 a. in 0.00 5 10 15 20 Frequency (Hz) 25 (c) Relative Displacement Time History at Cap Beam Level 10 20 Time (sec) (d) Relative Displacement Power Spectrum at Cap Beam Level 4000 o 2000 a a. in 3 o a. 10 15 Frequency (Hz) 20 25 (e) F R F of Bent Relative Displacement (f) Phase Angle of Bent Relative Displacement 0.02 200 a m 2 -100 -200 4 5 6 7 8 Frequency (Hz) 10 Fig. 3.6 Dynamic Response of the 0.27 Scale Specimen-120% Test. Chapter 3 28 (a) Shake Table Acceleration Time History 1.00 r -= 0.50 2 0.00 -0.50 -1.00 10 20 Time (sec) 30 (b) Shake Table Acceleration Power Spectrum 2.00 0.00 10 15 Frequency (Hz) 20 25 (c) Relative Displacement Time History at Cap Beam Level 10 20 Time (sec) (d) Relative Displacement Power Spectrum at Cap Beam Level 20000 (3 10000 a. co 10 15 Frequency (Hz) 20 25 (e) F R F of Bent Relative Displacement 0.04 r . 0.03 - I 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) (f) Phase Angle of Bent Relative Displacement 200 r _200 ' 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) Fig. 3.7 Dynamic Response of the 0.27 Scale Specimen -150% Test. Chapter 3 29 Table 3.1 Comparison of Natural Frequencies and Natural Periods from Hammer and Shake Table Tests. Tests Frequency (Hz) Hammer Test Frequency (Hz) Shake Table Test Period (sec) Hammer Test Period (sec) Shake Table Test 7.77 - 0.129 -Test 5% 7.77 7.77 0.129 0.129 Test 10% 6.67 5.61 0.150 0.178 • Test 40% 5.76 4.82 0.174 0.207 Test 60% 5.52 4.37 0.181 0.229 Test 80% 5.27 4.02 0.190 0.249 Test 120% 4.47 3.65 0.224 0.274 Test 150% 3.57 2.55 0.280 0.392 Test Fig. 3.8 Natural Frequencies Vs. Run Amplitude from Hammer and Shake Table Tests. During severe earthquake excitations, the fundamental frequency can decrease substantially. Table 3.1 and Fig. 3.8 show that the original frequency of the specimen reduced from 7.77 Hz to 2.55 Hz at the final level. On the other hand, the reduction of the fundamental frequency from the 5% test to the 150% test was about 67%. Most of the collected data corresponded to the nonlinear behavior of the specimen as it experienced severe excitations. However, obtaining the fundamental frequency and damping ratio in such a Chapter 3 30 system purely from linear theory and the collected data might still provide us some insight into the general behavior of the specimen. 3.2.2 Evaluation of Damping from Experimental Data Theoretical Background - A damping force may result from air resistance (viscous damping), external friction (coulomb friction damping), or internal friction (hysteretic damping). Although it is difficult to relate damping resistance to the physical characteristics of a structure, an estimate can be obtained from experimental measurement of the response of a structure under a given excitation. If a structure behaves only elastically, viscous damping is the only means by which energy dissipation is effected. When the velocity is small, viscous damping force is proportional to the velocity. Generally, for reinforced concrete structures a typical value of 5 percent of critical damping is assumed. As stated earlier, hysteretic damping reflects internal friction within the body. It can be measured from a load-defonnation curve or the hysteretic loop. The area under a complete loop identifies the loss of energy per cycle. Since the 0.27 scale bent was subjected to some earthquake excitations, the hystersis loops from the test did not have a regular shape. Although it was still possible to compute, with difficulty, the hysteretic damping from those loops, the resulting values were very approximate. For this reason, it was decided to compute only the "viscous damping" from the recorded data. Determination of Damping Ratio - Several methods exist to obtain the damping ratio of a linear system, such as the half power method and resonance amplification method (Clough and Penzien, 1993). These techniques are designed for situations of pure harmonic excitation. For an input-output data set obtained from a laboratory earthquake test, there is a technique which is based on finding a transfer function model Chapter 3 31 that represents the dynamics of the structure and, most nearly, reproduces the measured input-output relationship in the neighborhood of the structure's fundamental frequency band. This method is sometimes referred to as "parametric modeling", since it provides an optimal estimate of the parameters of the input-output model in a least-squares sense (Little and Shure, 1990). To use the technique, the structure's behavior is approximated by a second order system in a narrow region of its first resonance frequency, as: XU(o)= b A(j(o) (yco) 2+2c;co„(yco) + (to„) 2 where A is the input acceleration, X is the output displacement, j = , and b is a constant coefficient. To estimate the two parameters, i.e., the system's natural frequency, co, and the system damping ratio, we need to obtain the complex frequency response of the structure from the input base acceleration and the output top relative displacement; i.e., we need X(j(o) I Afjoa) versus co. The computer program MATLAB was used for this purpose. The results of this analysis for the shake table test are summarized in Table 3.2. Table 3.2 Measured Damping. Test 10% 40% 60% 80% 120% 150% Damping, E, 8.7 10.2 5.9 6.8 12.2 13.1 3.2. J Filter Analysis Filters are mostly used to enhance signals by removing unwanted components. The most common filter of this type is the frequency selective filter, which allows only certain desired frequency bands to pass through. The MATLAB computer program was used for this purpose. Low-pass filtering was accomplished with 4-pole two-pass (backward and forward) Butterworth filters set at a cut-off frequency of 10 Hz, which was chosen to be above the first natural frequency of the system. The sampling rate was 200 samples per second per channel. Chapter 3 32 To show how filtering helped in noise reduction from the signals, the power spectrum of the bent accelerometer and one of the strain gauges (T5) for two extreme cases (5% and 150%) are shown in Figs. 3.9 to 3.11. From these figures it can be realized that noise is more of a factor in low-amplitude excitations. Fig. 3.9 shows some clear peaks at power line frequencies with a magnitude of about 100 times as big as the original signals. Although in the 150% run (Fig. 3.10) this peak exists, the amplitude is about 1000 times as low as the original signals. The filtering process was done for all data; the attenuation of noise amplitude is an indicator of good filtration. Fig. 3.9 5 % Run - Power Spectrum of T5 Sensor, Strain at the top of the Beam. Chapter 3 33 Frequency (Hz) Fig. 3.11 150% Run - Power Spectrum of Bent Acceleration. Chapter 3 34 3.2.4 Time Histories Typical time histories of the main response parameters of the 0.27 scale model to the earthquake excitation are discussed in this section. Fig. 3.12 presents the absolute acceleration time histories at die shake table level and the cap beam level for the 10%, 60%, and 150% tests. In general, the recorded accelerations on the shake table are characterized by two strong phases at approximately seven second intervals. At the beginning of the testing, die natural frequency of the specimen was close to the peak of the acceleration response spectrum of the scaled Joshua earthquake. Therefore the absolute acceleration at the cap beam was magnified significantly to almost three times as high as that of the shake table. As previously discussed, after each test the specimen degraded and die frequency of the bent decreased. Henceforth, the peak response spectrum of the shake table and die bent separated from each other, and die amplification of die acceleration was reduced gradually in successive tests. In the final 150% test, die bent frequency decreased to 55% compared to that obtained from the 10% test, and there was almost no magnification of acceleration was observed. The response during the shake table test can also be evaluated by studying plots of the lateral displacement relative to die table. Fig. 3.13 shows that die specimen generally behaved as a single-degree-of-freedom (SDOF) system. The displacements of the cap beam and the top of the mass block relative to the shake table for three different runs (10%, 60%, and 150%) are also shown in Fig. 3.13. Obviously, in each test, both points moved at the same frequency and the specimen response was characterized by a single-degree-of-freedom system. The gradual yielding and reduction of the stiffness of the specimen in successive runs is also reflected in Fig. 3.13. A comparison between die 10% and the 150% tests shows that the frequency of the time history response is reduced dramatically. Chapter 3 35 The base shear time histories for the same three runs are presented in Fig. 3.14. The base shear of the structure was evaluated by multiplying the acceleration at center of gravity by the total mass of tire specimen. The maximum base shear in the 60% test was increased almost 64% compared to the one in the 10% test. By tire final 150% test, the stiffness of the specimen had degraded significantly, therefore Hie base shear did not increase. A general trend of yielding in successive tests can also be observed in base shear time histories. The maximum values of shake table acceleration, bent acceleration, bent relative displacement, and base shear are summarized in Table 3.3. Table 3.3 Maximum Values of Acceleration, Displacement, and Base Shear. Test Table Absolute Bent Absolute Bent Relative Base Shear Acceleration (g) Acceleration (g) Displacement (mm) (kN) + - + - + - + 10% 0.20 -0.19 0.56 -0.55 4.3 -4.1 59.2 -57.1 40% 0.33 -0.31 0.76 -0.83 8.4 -10.7 77.1 -88.7 60% 0.47 -0.41 0.80 -0.87 9.7 -11.8 79.8 -93.5 80% 0.59 -0.53 0.85 -0.96 11.2 -16.0 81.7 -102.8 120% 0.97 -0.82 0.88 -1.19 17.3 -25.4 90.4 -111.1 150% 0.98 -1.13 0.80 -1.01 35.3 -23.8 76.1 -99.1 Chapter 3 36 Fig. 3.12 Absolute Acceleration Time Histories at Shake Table & Bent. Chapter 3 37 (a) 1 0 % Test E E. c CD E CD O JO 8-b 10 -10 4 6 sec Sol id Line: Beam Dashed Line: M a s s Block 10 E E. c ai E CD o a CL co E E^ c E CD O a b 20 r -20 40 -40 (b) 6 0 % Test sec (c) 150% Test • y 10 10 sec Fig. 3.13 Relative Displacement Time Histories at the Top of Beam & Mass Block. Chapter 3 (a) 10% Test 38 Fig. 3.14 Base Shear Time Histories. Chapter 3 39 3.3 Material Characteristics An interesting study may be carried out by examing the hysteresis characteristics of the specimen, the strain gauge recorded data, and the distribution of forces during subsequent tests. 3.3./ Hystersis curves To get an overall view of the behavior of the specimen during die tests, the lateral force-defonnation relationship was plotted. To obtain the total base shear, the acceleration at center of gravity was multiplied by the mass of the specimen. Since die accelerometers were installed at die top of the steel plates and at die cap beam level (ALU and ALB), a linear interpolation between these two data sets was performed in order to obtain die acceleration at the center of gravity, which was 411 mm above the cap beam. Figs. 3.15a to 3.15f show the total base shear force vs. the relative displacement at the cap beam level for different, levels of excitation. To evaluate the nature of these hystersis loops, the response during successive short duration intervals was studied. The results are shown in Figs. 3.16 tiirough 3.21. As well, die whole set of hystersis loops from all the tests are plotted in Fig. 3.22. From this figure, the yielding and dissipation of energy due to inelastic behavior and also the gradual degradation in subsequent: tests can be observed. Fig. 3.15a shows almost linear behavior at die 10% test. At the 40% test the force-defonnation relationship shows a mild hystertic component due, probably, to the opening and closing of the concrete cracks. This behavior is more pronounced in the 60% test. The reduction of die fundamental frequency compared to die 10% test is 14% and 22%, respectively. This indicates the specimen suffered degradation in its lateral stiffness. The hysteretic energy dissipation of the specimen increased in the 80% test, compared to die previous tests. Pinching behavior was observed more clearly in this test. The fundamental frequency was Chapter 3 40 reduced by 9% from the previous test which shows stiffness degradation was not so dramatic compared to previous tests. As long as the yielding was not significant, the force deformation relationships kept the same pattern, although there was more stiffness deterioration in successive tests. The complexity of the response of reinforced concrete structures under severe seismic excitation for the 120% and 150% tests is evident in Figs. 3.15e and 3.15f. The hysteretic loops indicate more energy absorption and a significant pinching effect due to shear deformation and bar slip. The interaction of various effects such as the yielding of reinforcement bars, severe concrete cracking, and steel-concrete bond deterioration is more pronounced in these two tests. The reduction of the fundamental frequency for the 120% test compared to the 10% test was 35%, while for the 150% test, it increased to 55%. It is evident that the specimen suffered a noticeable degradation in its stiffness. For the 120% test, larger displacements were observed and the specimen experienced its maximum force capacity. Beyond this point, the specimen lost its strength: a very large displacement occurred, but did not develop a large force. This stage was considered a failure situation. Chapter 3 4 1 (a) Test 10% (b) Test 40% -60 -40 -20 0 20 40 60 Relative Displacement at Cap Beam Level (mm) 150 100 50 0 -50 -100 -150 -60 -40 -20 0 20 40 60 Relative Displacement at Cap Beam Level (mm) (c) Test 60% -60 -40 -20 0 20 40 60 Relative Displacement at Cap Beam Level (mm) (d) test 80% 150 100 50 o -50 -100 -150 -60 -40 -20 0 20 40 60 Relative Displacement at Cap Beam Level (mm) Fig. 3.15 Hysteresis Curves -10%, 40%, 60%, 80%, 120%, and 150% Tests. Chapter 3 42 (e) Time: 20 - 25 sec. 60 ^ 30 -30 -60 -5 -2.5 0 2.5 5 Relative Displacement at Cap Beam Level (mm) (f) Time: 25 - 30 sec. 60 5" 30 -30 -60 -5 -2.5 0 2.5 5 Relative Displacement at Cap beam Level (mm) Fig. 3.16 Hysteresis Curves During 5 Sec. Intervals -10% Test. Chapter 3 43 (c) Time: 10-15 sec. -15 -10 -5 0 5 10 15 Relative Displacement at Cap Beam Level (mm) 100 ^ 50 ra m -50 -100 (d) Time: 15 -20 sec. -15 -10 -5 0 5 10 15 Relative Displacement at Cap Beam Level (mm) (e) Time: 20 - 25 sec. (f) Time: 25 - 30 sec. -15 -10 -5 0 5 10 15 Relative Displacement at Cap Beam Level (mm) cu in 100 ^ 50 -50 -100 -15 -10 -5 0 5 10 15 Relative Displacement at Cap beam Level (mm) Fig. 3.17 Hysteresis Curves During 5 Sec. Intervals - 40% Test. Chapter 3 44 (e) Time: 20 - 25 sec. (f) Time: 25 - 30 sec. 100 -50 -100 -15 -10 -5 0 5 10 15 Relative Displacement at Cap Beam Level (mm) CO <1> 100 ^ 50 -50 -100 -15 -10 -5 0 5 10 15 Relative Displacement at Cap beam Level (mm) Fig. 3.18 Hysteresis Curves During 5 Sec. Intervals - 60% Test. Chapter 3 45 (e) Time: 20 - 25 sec. (f) Time: 25 - 30 sec. -20 -15 -10 -5 0 5 10 15 20 Relative Displacement at Cap Beam Level (mm) 100 50 a m -50 -100 -20 -15 -10 -5 0 5 10 15 20 Relative Displacement at Cap beam Level (mm) Fig. 3.19 Hysteresis Curves During 5 Sec. Intervals - 80% Test. Chapter 3 46 (a) Time: 0.0 - 5 sec. 120 80 40 -40 -80 -120 7* -25 -20 -15 -10 -5 0 5 10 15 20 25 Relative Displacement at Cap Beam Level (mm) co 120 80 i 40 -40 -80 -120 (b) Time: 5 - 1 0 sec. -25 -20 -15 -10 -5 0 5 10 15 20 25 Relative Displacement at Cap Beam Level (mm) 120 -120 (c) Time: 10-15 sec. -25 -20 -15 -10 -5 0 5 10 15 20 25 Relative Displacement at Cap Beam Level (mm) co CO 120 80 H 40 -40 -80 -120 (d) Time: 15 -20 sec. -25 -20 -15 -10 -5 0 5 10 15 20 25 Relative Displacement at Cap Beam Level (mm) (e) Time: 20 - 25 sec. 120 80 i 40 -40 -80 -120 -25 -20 -15 -10 -5 0 5 10 15 20 25 Relative Displacement at Cap Beam Level (mm) (f) Time: 25 - 30 sec. 120 80 i 40 -40 -80 -120 -25 -20 -15 -10 -5 0 5 10 15 20 25 Relative Displacement at Cap beam Level (mm) Fig. 3.20 Hysteresis Curves During 5 Sec. Intervals -120% Test. Chapter 3 47 (e) Time: 20 - 25 sec. 120 80 a. 40 -40 -80 -120 -40 -30 -20 -10 0 10 20 30 40 Relative Displacement at Cap Beam Level (mm) (f) Time: 25 - 30 sec. m ra m 120 80 40 0 -40 -80 -120 -40 -30 -20 -10 0 10 20 30 40 Relative Displacement at Cap beam Level (mm) Fig. 3.21 Hysteresis Curves During 5 Sec. Intervals -150% Test. Chapter 3 4 8 120 80 40 cu to CO m -40 -80 -120 Relative Displacement at Cap Beam Level (mm) Fig. 3.22 Comparison of Hysteresis Curves for Critical Duration. Chapter 3 4 9 Table 3.3 (refer to Section 3.2.4) presents the maximum values of the base shear at each test and the corresponding relative displacements. The specimen showed a more significant stiffness degradation and pinching effect toward die end of successive tests, indicating the increasing importance of shear deformation, bond deterioration, and cracking. In general the specimen showed brittle behavior due to the shear failure at the cap beam. The shear cracks developed in die expected regions, i.e., near both ends of the cap beam, outside of the haunch regions. In those locations, the stirrups were inadequate and the shear demand was quite high. 3.3.2 Strain Gauge Evaluation The specimen was instrumented with 24 strain gauges. The voltage reading from each strain gauge record was calibrated and converted into strain. As mentioned earlier, since the records contained high frequency noise, tfiese were filtered using the Butterworth filters with cut-off frequency of 10 Hz. The exact strain gauge locations and all the strain gauge time history records can be found in Davey (1996). Only strains in the longitudinal direction were used for tiiis analysis. Fig. 3.23 shows die position of the strain gauges. Most of the strain gauges, in general, functioned properly during the tests. Table 3.4 indicates that as the testing progressed, tiiere were more local yieldings and residual stresses in die strain gauge readings. Whereas no yielding was observed in the 10% test, tiiere was some obvious yielding at. some points in die cap beam and at top of the east: column in the 40% test. During the initial tuning up of the shake table, a jolt occured due to a sudden movement of the shake table actuators. This caused shear cracks near the two ends of die cap beam. The amount of the yield and Chapter 3 50 residual strain were recorded in the cracked regions. Locations B1, B2, B6, and T6 in Fig. 2.4 recorded the maximum yielding. The strain gauges showed almost consistent readings throughout the tests except for the final test, in which the strain gauges located near the regions of the two big cracks saturated and those located in the middle or the east of the cap beam recorded no yielding. The east column showed yielding on both faces after the 40% test. The west column started yielding on the outer face after the 60% test. To evaluate the data from the strain gauges, it was decided to obtain the curvature distribution over the cap beam and columns. Since the earthquake load was the source of excitation, plotting of curvatures at all time intervals was not manageable. Therefore curvatures were found at selected times, i.e., at 6 sec. and 10 sees before and between the two major peaks of the excitation and at the time when the maximum acceleration occurred at the center of gravity. The curvature was found by reading the strains across each section dividing them by the distance between top and bottom reinforcements. As long as the behavior of the specimen is elastic, this curvature distribution can represent the moment distribution along the cap beam and columns. Figs. 3.24 to 3.27 present the curvature distribution along the cap beam. From the graphs it can be clearly seen that at the locations Tl -Bl the cracks caused the curvatures to have sharp peaks. As mentioned earlier, these strain gauges were installed at the places where the large shear cracks occured. The curvature magnitudes calculated from the strain gauge readings during the 120% and 150% runs were not reasonable; consequently, the data from these two tests was not used in the analysis. Fig. 3.28 shows the column curvature obtained from the strain gauge records. Since there is only one set of strain gauges in each column, the curvatures are approximate. The opposite direction of curvature show that when one of them is in tension the other one is in compression. In the last two runs, the column records showed inconsistent results, which could have been caused by malfunctioning strain gauges. Chapter 3 51 T l T2 T3 T4 T8 T7 T6 T5 • • • • 5T~ 3uTT 78( .5 485 i 59( 675 365 1 k 5 900 k Fig. 3.23 Strain Gauge Locations. Table 3.4 Progressing of Local Yielding. 10% 40% 60% 80% 120% 150% T l Bl yield * yield * yield * yield * * saturate T2 B2 yield yield yield yield yield yield yield saturate yield T3 B3 T4 B4 T5 B5 saturate T6 B6 yield yield yield yield* yield yield* yield* saturate * T7 B7 T8 B8 yield* yield* yield*1 saturate C2 CI yield* yield* yield yield* yield* yield* C4 C3 yield* yield yield* yield yield* yield* yield* yield yield* yield *: residual strain Chapter 3 52 E £ c cs 3 3 o at 12.955 sees at 10 sees at 6 sees at 7.505 sees at 6 Seconds at 7.505 Seconds 0.010 r 0.005 I 0.000 o -0.005 -0.010 -1000 -500 0 500 1000 Distance from the Cap Beam Centerline 3 o 0.010 0.005 a 0.000 -0.005 -0.010 -1000 -500 0 500 1000 Distance from the Cap Beam Centerline 3 o 0.010 0.005 -0.005 -0.010 at 10 Seconds -1000 -500 0 500 1000 Distance from the Cap Beam Centerline 3 o 0.010 0.005 0.000 \--0.005 at 12.955 Seconds -0.010 -1000 -500 0 500 1000 Distance from the Cap Beam Centerline Fig. 3.24 Experimental Curvatures (milli strain/mm) -10% Test. Chapter 3 53 E E c «> i_ % CD 3 13 3 at 7.875 sees at 10 sees at 6 sees at 7.99 sees at 6.0 Seconds at 7.875 Seconds 3 o 0 .010 0 .005 - 0 . 0 0 5 - 0 . 0 1 0 - 1 0 0 0 - 5 0 0 0 5 0 0 1000 Distance from the Cap Beam Centerline 0 .010 r 0 .005 § 0 .000 3 o - 0 . 0 0 5 - 0 . 0 1 0 - 1 0 0 0 - 5 0 0 0 5 0 0 1 0 0 0 Distance from the Cap Beam Centerline 3 o 0.010 0 .005 S 0 .000 - 0 . 0 0 5 -0 .010 at 7.99 Seconds - 1 0 0 0 - 500 0 5 0 0 1000 Distance from the Cap Beam Centerline 0 .010 0 .005 S 0 .000 3 o - 0 . 0 0 5 -0 .010 at 10.0 Seconds - 1 0 0 0 - 500 0 5 0 0 1 0 0 0 Distance from the Cap Beam Centerline Fig. 3.25 Experimental Curvatures (mili strain/mm) - 40% Test. Chapter 3 54 at 13.32 sees E E <5 13 <3 0.030 |— 0.020 at 10 sees at 6 sees at 13.445 sees at 6.0 Seconds at 1 0.0 Seconds 3 I 3 0.030 0.020 0.010 0.000 h« -0.010 -0.020 -0.030 -1000 -500 0 500 1000 Distance from the Cap Beam Centerline 0.030 0.020 h 0.010 0.000 -0.010 --0.020 --0.030 -1000 -500 0 500 1000 Distance from the Cap Beam Centerline at 13.32 Seconds 0.030 0.020 £ 0.010 <5 0.000 3 o -0.010 -0.020 -0.030 -1000 -500 0 500 1000 Distance from the Cap Beam Centerline at 13.445 Seconds 0.030 0.020 £ 0.010 3 co 0.000 -0.010 -0.020 -0.030 -1000 -500 0 500 1000 Distance from the Cap Beam Centerline Fig. 3.26 Experimental Curvatures ( milli strain/mm) - 60% Test. Chapter 3 55 at 11.86 sees E E c <5 km 3 3 o at 6.0 Seconds at 7.73 Seconds 3 o 0.050 0.025 | 0.000 -0.025 -0.050 -1000 -500 0 500 1000 Distance from the Cap Beam Centerline 3 o 0.050 0.025 0.000 -0.025 -0.050 -1000 -500 0 500 1000 Distance from the Cap Beam Centerline 3 3 o 0.050 r 0.025 0.000 -0.025 at 10.0 Seconds -0.050 -1000 -500 0 500 1000 Distance from the Cap Beam Centerline 3 15 2 3 o 0.050 r 0.025 h 0.000 -0.025 at 11.86 Seconds -0.050 -1000 -500 0 500 1000 Distance from the Cap Beam Centerline Fig. 3.27 Experimental Curvatures (miili strain/mm) - 80% Test. Chapter 3 56 Fig. 3.28 Experimental curvature of the column. 3.3.3 Evaluation of Forces To obtain the distribution of forces in the specimen two approaches were tried. In the first approach the acceleration at the center of gravity was found. Two accelerometers were installed almost at the same location, one at the top of the steel plates and the other one at the top of the bent beam. To obtain the acceleration at the center of gravity, a linear interpolation was done. By multiplying this acceleration by the mass of the specimen the inertia force was calculated. This force was applied at the center of gravity as an equivalent static load. Consequently, the distribution of forces in the beam and columns was found. Since finding the inertial forces for the whole duration of excitation was not manageable, in each test the maximum inertial forces (equivalent static load) in both the positive and negative directions were Chapter 3 57 considered. The results are shown in Fig. 3.29. The analysis was performed by using the computer program CMAP(Ha,1993). W/2 W W W W/2 10301 0.06W • 1037 1037 M M M 2W 1030 0.06W i < 2W -a 616'--... 866.5 2531 866.5 J ^ ^ ) 0.5P t 1510.5 0.5P 4 i 0.84P T 0.84P Shear (kN) 1.30 Moment (kN-m) 10.8 8.9 2.0 8.3 33.3 11.1 1.30 10.8  . .8.9 12.0 0.84P 0.59P 0.84P 0.5P 0.5P 0.56P 0.76P ^ \ 0.76P Fig. 3.29 Elastic Analysis Approach. Chapter 3 58 To be able to compare these values to tire ones obtained from analytical approaches, these maximum forces were computed in elements of the analytical model (refer to Fig. 4.10). Tables 3.5 to 3.16 have listed the maximum shear forces and bending moments in the elements of the specimen. It should be noted that the approach explained above considers only the elastic behavior of the structure, and, therefore, is approximate. Since the specimen's behavior was mostly nonlinear, this approach can only give a rough idea about the distribution of bending moments, shear forces and axial loads under gravity and lateral load. In the second approach, the strain gauge records were used to obtain the strain and stress distributions at strain gauge locations. By having the strains at tire top and bottom of a section, and by finding the stress-strain relationships, the stress distribution was obtained, and the forces and moment at those sections computed. The computer program RESPONSE (Felber,1990) was used for this purpose. For the 10% test, the moments computed at the cap beam from the two different approaches are compared in Table 3.17. Moments were found at the strain gauge locations when tire maximum acceleration at the center of gravity occured. By studying Table 3.17, it can be seen that the moment values did not match at all. Where tire equivalent static load provided more consistent values, the strain gauges produced numbers seemingly out of the blue. It was therefore concluded that the recorded data from strain gauges was not reliable. Saturation, debonding from reinforcement, the sensitivity of the results to the strain gauge locations, and gauge installation were the major factors that contributed to such unreliable results. Chapter 3 59 Table 3.5 Maximum Shear From the Elastic Approach -10% Run. Member (1) Shear (kN) Gravity Load (2) Shear (kN) P = 59.2 (3) (2) + (3) (kN) (4) Shear (kN) P = -57.1 (5) (2) + (5) (kN) (6) Max. Value (kN) (7) 1 -11.1 0.0 -11.1 0.0 -11.1 11.1 2 33.3 -49.7 -16.4 48.0 81.3 81.3 3 11.1 -34.9 -23.8 33.7 44.8 44.8 4 11.1 -34.9 -23.8 33.7 44.8 44.8 5 -11.1 -34.9 -46.0 33.7 22.6 46.0 6 -11.1 -34.9 -46.0 33.7 22.6 46.0 7 -33.3 -49.7 -83.0 48.0 -14.7 83.0 8 11.1 0.0 11.1 0.0 11.1 11.1 9 -1.3 29.6 28.3 -28.6 -29.9 29.9 10 1.3 29.6 30.9 -28.6 -27.3 30.9 Table 3.6 Maximum Moment From the Elastic Approach -10% Run. Member Moment Moment (2) + (3) Moment (2) + (5) Max. Value (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) Gravity Load P = 59.2 P = -57.1 + -(1) (2) (3) (4) (5) (6) (7) 1- Left 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Right -8.9 0.0 -8.9 0.0 -8.9 - -8.9 2- Left 10.8 -45.0 -34.2 30.0 43.4 54.2 54.2 -34.2 Right -3.2 +33.2 -32.0 -35.2 30.0 -35.2 3- Left 3.2 -33.2 -30.0 32.0 35.2 35.2 -30.0 Right 2.5 +16.6 19.1 -16.0 -13.5 19.1 -13.5 4- Left -2.5 -16.6 -19.1 16.0 13.5 13.5 -19.1 Right 8.3 0.0 8.3 0.0 8.3 8.3 -5- Left -8.3 0.0 -8.3 0.0 -8.3 - -8.3 Right 2.5 -16.6 -14.1 16.0 18.5 18.5 -14.1 6- Left -2.5 16.6 -14.1 -16.0 -18.5 - -18.5 Right -3.2 -33.2 -36.4 32.0 28.8 28.8 -36.4 7- Left 3.2 33.2 36.4 -32.0 -28.8 36.4 -28.8 Right -10.8 -45.0 -55.8 43.4 32.6 32.6 -55.8 8- Left 8.9 0.0 8.9 0.0 8.9 8.9 -Right 0.0 0.0 0.0 0.0 0.0 0.0 -9- Left 0.0 0.0 0.0 0.0 0.0 0.0 -Right -2.0 45.0 43.0 -43.4 -45.4 43.0 -45.4 10- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right 2.0 45.0 47.0 -43.4 -41.4 47.0 -41.4 Chapter 3 60 Table 3.7 Maximum Shear From the Elastic Approach - 40% Run. Member (1) Shear (kN) Gravity Load (2) Shear (kN) P = 77.1 (3) (2) + (3) (kN) (4) Shear (kN) P = -88.7 (5) (2) + (5) (kN) (6) Max. Value (kN) (7) 1 -11.1 0.0 -11.1 0.0 -11.1 11.1 2 33.3 -64.8 -31.5 74.5 107.8 107.8 3 11.1 -45.5 -34.4 52.3 63.4 63.4 4 11.1 -45.5 -34.4 52.3 63.4 63.4 5 -11.1 -45.5 -56.6 52.3 41.2 56.6 6 -11.1 -45.5 -56.6 52.3 41.2 56.6 7 -33.3 -64.8 -98.1 74.5 41.2 98.1 8 11.1 0.0 11.1 0.0 11.1 11.1 9 -1.3 38.6 37.3 -44.4 -45.7 45.7 10 1.3 38.6 39.9 -44.4 -43.1 43.1 Table 3.8 Maximum Moment From the Elastic Approach - 40% Run. Member Moment Moment (2) + (3) Moment (2) + (5) Max. Value (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) Gravity Load P = 77.1 P = -88.7 + -(1) (2) (3) (4) (5) (6) (7) 1- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right -8.9 0.0 -8.9 0.0 -8.9 - -8.9 2- Left 10.8 -58.6 -47.8 67.4 78.2 78.2 -47.8 Right -3.2 . 43.2 40.0 -49.7 -52.9 40.0 -52.9 3- Left 3.2 -43.2 -40.0 49.7 52.9 52.9 -40.0 Right 2.5 21.6 24.1 -24.8 -22.3 24.1 -22.3 4- Left -2.5 -21.6 -24.1 24.8 22.3 22.3 -24.1 Right 8.3 0.0 8.3 0.0 8.3 8.3 -5- Left -8.3 0.0 -8.3 0.0 -8.3 - -8.3 Right 2.5 -21.6 -19.1 24.8 27.3 27.3 -19.1 6- Left -2.5 21.6 19.1 -24.8 -27.3 19.1 -27.3 Right -3.2 -43.2 -46.4 49.7 46.5 46.5 -46.4 7- Left 3.2 43.2 46.4 -49.7 -46.5 46.4 -46.5 Right -10.8 -58.6 -69.4 67.4 56.6 56.6 -69.4 8- Left 8.9 0.0 8.9 0.0 8.9 8.9 -Right 0.0 0.0 0.0 0.0 0.0 0.0 -9- Left 0.0 0.0 0.0 0.0 0.0 0.0 -Right -2.0 58.6 56.6 -67.4 -69.4 56.6 -69.4 10- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right 2.0 58.6 60.6 -67.4 -65.4 60.6 -65.4 Chapter 3 61 Table 3.9 Maximum Shear From the Elastic Approach - 60% Run. Member (1) Shear (kN) Gravity Load (2) Shear (kN) P =79.8 (3) (2) + (3) (kN) (4) Shear (kN) P = -93.5 (5) (2) +(5) (kN) (6) Max. Value (kN) (7) 1 -11.1 0.0 -11.1 0.0 -11.1 11.1 2 33.3 -67.0 -33.7 78.5 111.8 111.8 3 11.1 -47.1 -36.0 55.2 66.3 66.3 4 11.1 -47.1 -36.0 55.2 66.3 66.3 5 -11.1 -47.1 -58.2 55.2 44.1 58.2 6 -11.1 -47.1 -58.2 55.2 44.1 58.2 7 -33.3 -67.0 -100.3 48.5 45.2 100.3 8 11.1 0.0 11.1 0.0 11.1 11.1 9 -1.3 39.9 38.6 -46.8 -48.1 48.1 10 1.3 39.9 41.2 -46.8 -45.5 45.5 Table 3.10 Maximum Moment From the Elastic Approach - 60% Run. Member Moment Moment (2) + (3) Moment (2) + (5) Max. Value (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) Gravity Load P = 79.8 P = -93.5 + (1) (2) (3) (4) (5) (6) (7) 1- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right -8.9 0.0 -8.9 0.0 -8.9 -8.9 2- Left 10.8 -60.6 -49.8 71.1 81.9 81.9 -49.8 Right -3.2 44.7 41.5 -52.4 -55.6 41.5 -55.6 3- Left 3.2 -44.7 -41.5 52.4 55.6 55.6 -41.5 Right 2.5 22.3 24.8 -26.2 -23.7 24.8 -23.7 4- Left -2.5 -22.3 -24.8 26.2 23.7 23.7 -24.8 Right 8.3 0.0 8.3 0.0 8.3 8.3 5- Left -8.3 0.0 -8.3 0.0 -8.3 -8.3 Right 2.5 -22.3 -19.8 26.2 28.7 28.7 -19.8 6- Left -2.5 22.3 19.8 -26.2 -28.7 19.8 -28.7 Right -3.2 -44.7 -47.9 52.4 49.2 49.2 -47.9 7- Left 3.2 44.7 47.9 -52.4 -49.2 47.9 -49.2 Right -10.8 -60.6 -71.4 71.1 60.3 60.3 -71.4 8- Left 8.9 0.0 8.9 0.0 8.9 8.9 -Right 0.0 0.0 0.0 0.0 0.0 0.0 9- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right -2.0 60.6 58.6 -71.1 -73.1 58.6 -73.1 10- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right 2.0 60.6 62.6 -71.1 -69.1 62.6 -71.1 Chapter 3 Table 3.11 Maximum Shear From the Elastic Approach - 80% Run. Member (1) Shear (kN) Gravity Load (2) Shear (kN) P =81.7 (3) (2) + (3) (kN) (4) Shear (kN) P = -102.8 (5) (2) + (5) (kN) (6) Max. Value (kN) (7) 1 -11.1 0.0 -11.1 0.0 -11.1 11.1 2 33.3 -68.6 -35.3 86.4 119.7 119.7 3 11.1 -48.2 -37.1 60.7 71.8 71.8 4 11.1 -48.2 -37.1 60.7 71.8 71.8 5 -11.1 -48.2 -59.3 60.7 49.6 59.3 6 -11.1 -48.2 -59.3 60.7 49.6 59.3 7 -33.3 -68.6 -101.9 86.4 53.1 101.9 8 11.1 0.0 11.1 0.0 11.1 11.1 9 -1.3 40.9 39.3 -51.4 52.7 52.7 10 1.3 40.9 42.2 -51.4 -50.1 50.1 Table 3.12 Maximum Moment From the Elastic Approach - 80% Run. Member Moment Moment (2) + (3) Moment (2) + (5) Max. Value (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) Gravity Load P =81.7 P = -102.8 + (1) (2) (3) (4) (5) (6) (7) 1- Left 0.0 0.0 0.0 0.0 0.0 0.0 -Right -8.9 0.0 -8.9 0.0 -8.9 -8.9 2- Left 10.8 -62.1 -51.3 78.1 88.9 88.9 -51.3 Right -3.2 45.8 42.6 -57.6 -60.8 42.6 -60.8 3- Left 3.2 -45.8 -42.6 57.6 60.8 60.8 -42.6 Right 2.5 22.9 25.4 -28.8 -26.3 25.4 -26.3 4- Left -2.5 -22.9 -25.4 28.8 26.3 26.3 -25.4 Right 8.3 0.0 8.3 0.0 8.3 8.3 -5- Left -8.3 0.0 -8.3 0.0 -8.3 -8.3 Right 2.5 -22.9 -20.4 28.8 31.3 31.3 -20.4 6- Left -2.5 22.9 20.4 -28.8 -31.3 20.4 -31.3 Right -3.2 -45.8 -49.0 57.6 54.4 54.4 .49.0 7- Left 3.2 45.8 49.0 -57.6 -54.4 49.0 -54.4 Right -10.8 -62.1 -72.9 78.1 67.3 67.3 -72.9 8- Left 8.9 0.0 8.9 0.0 8.9 8.9 -Right 0.0 0.0 0.0 0.0 0.0 0.0 9- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right -2.0 62.1 60.1 -78.1 -80.1 60.1 -78.1 10- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right 2.0 62.1 64.1 -78.1 -76.1 64.1 -76.1 Chapter 3 63 Table 3.13 Maximum Shear From the Elastic Approach -120% Run. Member (1) Shear (kN) Gravity Load (2) Shear (kN) P = 90.4 (3) (2) + (3) (kN) (4) Shear (kN) P = -111.1 (5) (2) + (5) (kN) (6) Max. Value (kN) (7) 1 -11.1 0.0 -11.1 0.0 -11.1 11.1 2 33.3 -75.9 -42.6 93.3 126.6 126.6 3 11.1 -53.3 -42.2 65.5 76.6 76.6 4 11.1 -53.3 -42.2 65.5 76.6 76.6 5 -11.1 -53.3 -64.4 65.5 54.4 64.4 6 -11.1 -53.3 -64.4 65.5 54.4 64.4 7 -33.3 -75.9 -109.2 93.3 60.0 109.2 8 11.1 0.0 11.1 0.0 11.1 11.1 9 -1.3 45.2 43.9 -55.6 -56.9 56.9 10 1.3 45.2 46.5 -55.6 -54.3 54.3 Table 3.14 Maximum Moment From the Elastic Approach -120% Run. Member Moment Moment (2) + (3) Moment (2) + (5) Max. Value (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) Gravity Load P =90.4 P =-111.1 + -(1) (2) (3) (4) (5) (6) (7) 1- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right -8.9 0.0 -8.9 0.0 -8.9 - -8.9 2- Left 10.8 -68.7 -57.9 84.4 95.2 95.2 -57.9 Right -3.2 50.6 47.4 -62.2 -65.4 47.4 -65.4 3- Left 3.2 -50.6 -47.4 62.2 65.4 65.4 -47.4 Right 2.5 25.3 27.8 -31.1 -28.6 27.8 -28.6 4- Left -2.5 -25.3 -27.8 31.1 28.6 28.6 -27.8 Right 8.3 0.0 8.3 0.0 8.3 8.3 -5- Left -8.3 0.0 -8.3 0.0 -8.3 - -8.3 Right 2.5 -25.3 -22.8 31.1 33.6 33.6 -22.8 6- Left -2.5 25.3 22.8 -31.1 -33.6 22.8 -33.6 Right -3.2 -50.6 -53.8 62.2 59.0 59.0 -53.8 7- Left 3.2 50.6 53.8 -62.2 -59.0 53.8 -59.0 Right -10.8 -68.7 -79.5 84.4 73.6 73.6 -79.5 8- Left 8.9 0.0 8.9 0.0 8.9 8.9 -Right 0.0 0.0 0.0 0.0 0.0 0.0 9- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right -2.0 68.7 66.7 -84.4 -86.4 66.7 -86.4 10- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right 2.0 68.7 70.7 -84.4 -82.4 70.7 -82.4 Chapter 3 64 Table 3.15 Maximum Shear From the Elastic Approach -150% Run. Member (1) Shear (kN) Gravity Load (2) Shear (kN) P = 76.1 (3) (2) + (3) (kN) (4) Shear (kN) P = -99.1 (5) (2) + (5) (kN) (6) Max. Value (kN) (7) 1 -11.1 0.0 -11.1 0.0 0.0 11.1 2 33.3 -63.9 -30.6 83.2 116.5 116.5 3 11.1 -44.9 -33.8 58.5 69.6 69.6 4 11.1 -44.9 -33.8 58.5 69.6 69.6 5 -11.1 -44.9 -56.0 58.5 47.4 56.0 6 -11.1 -44.9 -56.0 58.5 47.4 56.0 7 -33.3 -63.9 -97.2 83.2 49.9 97.2 8 11.1 0.0 11.1 0.0 11.1 11.1 9 -1.3 38.1 36.8 -49.6 -50.9 50.9 10 1.3 38.1 39.4 -49.6 -48.3 48.3 Table 3.16 Maximum Moment From the Elastic Approach -150% Run. Member Moment Moment (2) + (3) Moment (2) + (5) Max. Value (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) (kN-m) (1) Gravity Load (2) P =76.1 (3) (4) P = -99.1 (5) (6) (7) 1- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right -8.9 0.0 -8.9 0.0 -8.9 - -8.9 2- Left 10.8 -57.8 -47.0 75.3 86.1 86.1 -47.0 Right -3.2 42.6 39.4 -55.5 -58.7 39.4 -58.7 3- Left 3.2 -42.6 -39.4 55.5 58.7 58.7 -39.4 Right 2.5 21.3 23.8 -27.7 -25.2 23.8 -25.2 4- Left -2.5 -21.3 -23.8 27.7 25.2 25.2 -23.8 Right 8.3 0.0 8.3 0.0 8.3 8.3 -5- Left -8.3 0.0 -8.3 0.0 -8.3 - -8.3 Right 2.5 -21.3 -18.8 27.7 30.2 30.2 -18.8 6- Left -2.5 21.3 18.8 -27.7 -30.2 18.8 -30.2 Right -3.2 -42.6 -45.8 55.5 52.3 52.3 -45.8 7- Left 3.2 42.6 45.8 -55.5 -52.3 45.8 -52.3 Right -10.8 -57.8 -68.6 75.3 64.5 64.5 -68.6 8- Left 8.9 0.0 8.9 0.0 8.9 8.9 -Right 0.0 0.0 0.0 0.0 0.0 0.0 9- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right -2.0 57.8 55.8 -75.3 -77.3 55.8 -77.3 10- Left 0.0 0.0 0.0 0.0 0.0 0.0 Right 2.0 57.8 59.8 -75.3 -73.3 59.8 -73.3 Chapter 3 65 Table 3.17 Comparison of Moments. Strain Gauge Locations (mm) Moment from Equivalent static load, + (kN-m) Moment from Strain Gauges Record, + (kN-m) Moment from Equivalent static load, -(kN-m) Moment from Strain Gauges Record, -(kN-m) 365.5 27.3 7.5 -35.5 -33.6 590.5 23.3 No positive value -24.4 -13.9 780.5 19.4 0.8 -14.4 -4.8 965.5 15.9 1.7 -5.1 -8.7 C H A P T E R 4 C O M P U T E R PROGRAMS AND PROPOSED A N A L Y T I C A L M O D E L 4.1 Overview In this chapter the features of two computer programs, CANNY-E and RUAUMOKO, are described. The advantages and limitations of each are discussed and the element library, solution techniques, analysis options, hysteresis modeling, and damage indices are reviewed. The general aspects of the proposed analytical model are also discussed. 4.2 The CANNY-E Computer Program CANNY-E (or simply CANNY) is a three-dimensional non-linear structural dynamics analysis computer program. This program was originally developed by Kang-Ning Li during his doctoral studies at the University of Tokyo, Japan, in 1987. The earliest version of die program was written in FORTRAN. CANNY-E is the latest version of this program which is written in C. This "macro finite element" program is capable of doing nonlinear analysis based on material nonlinearity. The structure is idealized as a set of rigid nodes connected by elastic elements. In the program, line elements are primarily used to represent the structural members. Membrane elements and solid elements such as shell elements are not included in the program. One of the advantages of CANNY-E is its ability to model structures of irregular shape and with 66 Chapter 4 67 complicated geometrical configurations. Therefore, although C A N N Y - E was developed to analyze building structures, it can be used to model such structures as bridges as well. The C A N N Y program recognizes two main categories: building and non-building structures. For the purposes of C A N N Y - E , the main difference between the two is the way the coordinate system is introduced into the program. For building structures, the structural geometry is defined by story levels and the location of column lines. In nonbuilding structures, the structural geometry is defined by node positions. At the time this study was conducted, the non-building category had not been completed, therefore, modeling had to be done based on a building-type structure. 4.2.7 The Element Library A single-component model known as the Giberson model (Giberson, 1967) is the main resource of the C A N N Y program. Each structural member is idealized as an elastic line element with inelastic (non-linear) springs attached at both ends. Inelasticity along the member is assumed to be lumped at the springs. The characteristics of flexture springs are detennined by assuming deformed shapes as members. Springs are used to represent inelastic flexure, shear and axial load. The main advantage of this one-component model is its simplicity and the possibility of using it to define any kind of hysteresis law for non-linear springs. The C A N N Y element library consists of live nonlinear element types: beam element, column element, shear panel, link (or truss) element, and cable (or constraint) spring which gives restraint to any displacement component at a specified node (Li, 1995). In this particular study, only beam, column, and link elements have been required. These three elements are explained in the following sections. Chapter 4 68 The Beam Element - The inelastic behavior of a beam member closely approximates the concept of the Giberson's one-component model. Limits are specified for unaxial bending with optional shear and axial deformations. A nonlinear bending spring is placed at botii ends of the beam element to express die flexural properties of the beam. The springs may be assigned different positive and negative crack and yield moments. It is also possible to define different positive and negative post cracking and yielding stiffnesses for each spring. The shear and the axial defonnations of the beam are approximated by independent shear and axial springs placed at its mid span. Shear springs can be elastic or inelastic. The program deals with shear in a manner similar to the way it deals with flexture. The concrete's shear strength, Vc, and the shear capacity of the section, Vu> are proposed to the shear backbone curve as die cracking and yielding capacity of the section. There are various empirical expressions used to estimate the shear strength of the beam section. In this diesis, tiiree different methods have been used to obtain the best estimate of the shear capacity of a section. The model does not include die interaction between the bending, shear and axial defonnations. Each spring is assigned a unique set of values that defines its hysteretic behavior. Fig 4.1 presents die beam modeling. A a M„ spring Fig. 4.1 Beam Modeling. Chapter 4 69 The program claims that it has the ability to predict shear failure as well as flexural failure. Since the shear strength of a beam is modeled by a shear spring, the program has a potential of predicting the shear failure, but as it does not yet include a hysteresis loop, it cannot model the shear failure properly. Predicting shear failure, however, is not easy: while the shear strength can be detennined by means of various empirical expressions, e.g., the Canadian Code (CSA, 1994), shear distortions in the post-cracking and post-yielding regions are difficult to establish. The author of CANNY has indicated that he plans to introduce some hysteresis loops which can properly model the shear behavior in a future version of the program. In the latest version, the force-deformation relationship for die beams is established according to the tliree possible moment distributions on the beams: symmetrical, anti-symmetrical, and triangular, as shown below. CANNY-E includes a section analysis program called "MC", which may be used to calculate the moment-curvature relationsliip of reinforced concrete members under a given constant axial load. This program is based on stress-strain relationships and two auxiliary assumptions: 1) a plane section remains plane, and 2) integrating over the section determines the complete moment-curvature relationship. Positive and negative cracking and yielding moments at each beam-end constitute the output. The program also calculates the post-cracking and yielding stiffness factors based on the linear distributed curvature assumption. The moment-rotation relationship for the bending spring has a bi-linear or a tri-Chapter 4 70 linear curve, as shown in Fig. 4.2. The corresponding cracking and yielding rotations, 9C and 9y, are calculated assuming a curvature distribution along the beam axial line, which is shown in Fig. 4.3. A'yK0 Fig. 4.2 Moment-Rotation Skeleton Curve for Beam Elements. (a) Anti-symmetrical and triangular moment (b) Symmetrical moment Fig. 4.3 Curvature Distribution Assumption. The Column Element - In CANNY-E's manual, a column is defined as an element which is subjected to unaxial or biaxial bending and shear, and an axial load as well (Li, 1995). There are three types of column models in CANNY-E: 1. One-component model for unaxial bending column element, Chapter 4 71 2. Bi-axial bending model, 3. Multi-spring model. From tire above three models, only the multi-spring model can take into account the interaction between the axial load and the bending moment. The shear defonnation of the column is represented with a shear spring. Link Element - A link element can be subjected only to tension or compression; thus, it represents either a truss member or a spring member. 4.2.2 Solution Technique Structural mass is assumed to be concentrated at each node or at the center of gravity of each rigid floor level. The mass matrix, [M], is diagonal, while the viscous damping matrix, [C], is proportional to the mass matrix and the initial or tangent stiffness matrix, [K]. This viscous damping, also known as Rayleigh damping (Clough and Penzien, 1993), is defined as: [C] = a [M] + (3 [K] where a and P are the two Rayleigh damping factors. According to the version of CANNY-E used in this study, viscous damping is the same for all the elements of the structure; a recent version of the program, however, can handle different dampings for different elements. Chapter 4 72 To obtain the dynamic response, at each time step the incremental stiffness matrix is formed and the equation of motion is solved. The Newmark (3-mediod or the Wilson 0-method is used as an integration scheme. In these methods, it is assumed that acceleration during the small time step remains constant. The Equations of Motion - Applying Newton's second law of motion to the structural elements, we. obtain (Li, 1995): [M] {AX} + [C] {AX} + [K] {AX} = {AFe} - {Fu} + {AFt} where [M]: diagonal mass matrix, [C]: damping matrix, { AX }: displacement increment of the structure relative to the base, { AX }: increment of die velocity vector, { AX ): increment of die acceleration vector, [K]: stiffness matrix, {AFe}: increment of earthquake inertia force vector, {Fu }: unbalanced force vector at the beginning of the present time interval, {AF,}: increment of time-varying element load vector, if applicable. Within each integration time step, as long as die elements remain elastic, die incremental displacement, velocity, and acceleration are accurate; therefore, at die end of each time step the equilibrium condition is satisfied. When elements start yielding during an integration time step, the calculated displacement is not quite accurate; thus at the end of the time step, die stiffness value is smaller titan at beginning, and Chapter 4 ; 73 therefore the displacement is underestimated. As a result, at the end of the time step the incremental velocity and acceleration are also incorrect. Consequently, the equilibrium is not satisfied, and the response may diverge from the actual value. The program, however, applies an approximate method to correct this equilibrium error: the displacement increment is assumed to be correct and the unbalanced force is added to the load vector in next time step. Due to this limitation, a very small time step should be assigned, as the smaller the time step, the smaller the unbalance a force, and therefore the more accurate the calculated response. Iteration methods can further reduce numerical error significantly, and the author of the program is trying to add this feature in future versions of the program. 4.2 J Analysis options Analysis options provided by CANNY-E include static analysis, static push-over analysis under a monotonic load, quasi-static analysis under a cyclic lateral load, seismic analysis, and design analysis, hi addition, the ground motion may be applied in three X, Y, and Z directions. Static vertical loads or any "initial load" can be included in all the analysis options. The response under the initial load is calculated prior to the step-by-step analysis. The initial load can be a distributed load or a concentrated load on a beam element. The fixed-end moment and shear caused by distributed gravity load on a beam element are computed by the program. "Initial force" data is another option to introduce the initial bending, shear, and axial load to the members. In such cases, the starting point on the hysteresis curve is not zero any more. 4.2.4 Hysteresis Modeling CANNY has a number of hysteresis models. To represent important characteristics of concrete, such as degradation in stiffness, strength deterioration, pinching, or softening, the CANNY's sophisticated Chapter 4 74 hysteresis model was selected for this study. The backbone curve in this model is a tri-lmear curve which consists of three distinct segments. The first segment represents elastic stiffness up to cracking; the second segment represents cracked stiffness up to yielding; the third segment represents post-yielding stiffness. Behavior under dynamic loading is controlled by seven dimensionless parameters: 8, 9, Xe, Xu, X3, e, and Xs. These parameters are explained in Section 4.4.5. 4.2.5 Damage Index The program is able to calculate local and global damage indices, as defined by Park and Ang (1985). For each nonlinear spring, die damage index is obtained from (Li, 1995): 8 E where the first right-hand term represents the ductility ratio, while the second term is related to hysteretic energy absorbed during loading. In the above fonnula: Di: the spring damage index, 8,„, by: the maximum and yielding deformation, Fy: yielding strength, (3: a control parameter representing the level of strength degradation. It can take on any value between 0.0 and 0.5. A large fl means poor-quality concrete, whereas a small (3 means the damage depends more on ductility. The default value is 0.1, u.: a control parameter representing the ductility capacity of the member. The default Chapter 4 value is 4.0 for beams and 2.0 for columns, the spring-dissipated hysteresis energy. 75 Since CANNY is a multi-story frame-oriented program, it calculates the story damage index, Ds. It is based on the weighted average of member damage indices ! £ . £ > . D = 1 ' The global damage index, D, is calculated based on the weighted average of all member indices. In most cases, the locations which undergo higher damages will also absorb larger amounts of energy and, therefore, are given a liigher weighting in the above formula. However, this is not always the case; there are situations in which such an index cannot show the overall damage to the structure. 4.3 The R U A U M O K O Computer Program RUAUMOKO, named after the Maori God of Volcanoes and Earthquakes, is a nonlinear computer program written in FORTRAN and developed by Athol J. Carr (1996). This program is designed to perform nonlinear analysis for two-dimensional framed structures subjected to a dynamic excitation. It is able to deal with both material and geometrical nonlinearities. Chapter 4 76 4.3. / The Element Library RUAUMOKO's element library classifies structural members into seven categories: frame, spring, structural-wall, dashpot, tendon, contact, and quadrilateral. Since in this particular study only tire frame element has been used, only this element is explained in the following section. The interested reader may refer to the RUAUMOKO's manual (Carr, 1996) for descriptions regarding the rest of the members. Frame Element - Beam and beam-column elements are two categories of frame members. While tire interaction of moment-axial loads can not be considered in beam members, it is possible to do so with beam-column members. Unlike CANNY-E, RUAUMOKO is not able to predict shear failure. Frame members can be modeled in four ways, each of which has its own advantages and drawbacks: 1. Giberson one-component model - This model assumes an elastic member along the length of a member with lumped plasticity at one or both ends (CANNY also uses this model); 2. Two-component model - Two members in parallel, one represents elastic behavior and the other represents plastic behavior of the member (DRAIN-2DX also uses this model); 3. Distributed plasticity along the member hinge - The yielding zone has a flexibility which varies parabolically from the inner elastic zone to a maximum flexibility at the member ends (RUAUMOKO manual); 4. Four-hinge beam member - This model allows for two plastic hinges within the span of tire member in addition to the two hinges at its ends (RUAUMOKO manual). Since this thesis is primarily a comparative study, every effort has been made to keep the methods for modeling and analysis as similar as possible in both programs. As a consequence, since CANNY-E only Chapter 4 11 deals with the Giberson one-component model, it was decided to use die one-component model in RUAUMOKO as well. 4.3.2 Solution Technique Depending on the user's selection, a lumped, diagonal, or consistent mass matrix would be formed. Several options are available to form die damping matrix, among them, die Rayleigh damping matrix, [C], a combination of mass and stiffness proportional damping, is the most popular. Like CANNY-E, at each time step RUAUMOKO forms the incremental tangential stiffness matrix. The equation of dynamic equilibrium is then integrated and solved for a small time interval using either die "Newmark constant average acceleration" method (i.e., (3 = 0.25) or die "explicit central difference" metiiod. If die stiffness of the structure changes during die time-step, the displacements result in internal member forces which do not correspond to the applied loads. Any residual out-of-balance force is thus picked up at the next time-step. The program is able to perfonn iteration, so out-of-balance forces may be met by iterating within the integration time step for further changes in both displacement and member forces, until equilibrium is achieved. Therefore, RUAUMOKO is not as sensitive to the choice of time step as CANNY is. 4.3.3 Analysis Options Static analysis, modal analysis, and dynamic analysis are main analysis options for an earthquake excitation or dynamic force excitation. Push-over analysis is available by tricking the dynamic force Chapter 4 78 excitation analysis option, for instance, using a slow-ramp loading function. An earthquake force may be applied in X, Y, or both directions. Unlike CANNY-E, in cases of distributed static load on beams or on columns, the effect of loading like fixed-end forces should be given as part of tire section data. The program creates one output file which, based on what is required by the user, may contain modal analysis results such as period and mode shapes, or static or dynamic analysis results like node displacements and member forces at any selected time-step interval. At the end of the analysis, the envelope of maximum responses, member ductility, and, if required, damage indices are printed. The program is able to compute and plot the response time histories and the hysteresis loops. The program can also compute and plot the response spectra for a given earthquake excitation. 4.3.4 Hysteresis Modeling RUAUMOKO has a wide range of hysteresis loops, from the simple elasto-plastic model to more complex ones, which represent the stiffness degradation. Strength degradation for most of hysteresis loops can be provided through an input file. 4.3.5 Damage Index The program utilizes seven formulae to calculate the damage index of an element, including the simplest method of calculating the damage index, based on defonnation, and more complicated methods such as the Park and Ang damage indices (Park and Ang, 1985). This program is not able to calculate a global damage index, however. Other limitations of calculating the damage index using RUAUMOKO are related to the Chapter 4 79 type of hysteresis loop selected. For some cases, damage indices cannot be calculated for certain types of hysteresis loops. According to the manual, another limitation is the uncertainty in damage indices resulting when strength degradation in hysteresis behavior is considered. 4.4 General Aspects of the computer model for both C A N N Y - E and R U A U M O K O The goal of modeling is to simulate and understand the physical behavior of a structure in an analytical way. In the case of inelastic reinforced concrete structures, this is not an easy task. There are many questions which must be answered in this regard: What is the correct initial stiffness of a member? What cross sectional dimensions and reinforcing areas are to be used in calculating section properties? Where and how much of the rigid zone should be considered? How should the change of plasticity in an element be considered? What type of hysteresis loop is suitable for the model? These and many other questions must be answered in order to create a realistic model. As mentioned earlier, a good model should simulate the response of the real structure to tire applied load as precisely as possible. This requires that the mass, stiffness, damping, and other properties of the structure be reflected in the model. To simulate all the material and dynamic characteristics of the specimen used in this thesis, a computer model was suggested. The layout and overall dimensions of the analytical model are shown in Fig. 4.4. The specimen layout is shown in Fig. 2.1 To represent the numerous cut-off of the longitudinal reinforcement, four different sections were studied for the cap beam. Section details are shown in Fig. 4.5. The computer model has nine nodes in the cap beam corresponding to the beam reinforcement and the loading points. Table 4.1 presents the position and coordinates of the nodes. It has two nodes for tire column at the column base pin connection and at the beam-column joints. The additional stiffness due to the haunches at the beam-column joints have been ignored. The gravity load from the superstructure is applied at five nodes on the cap beam. The mass block on the top of the beam is connected to the beam through two points. Such an ideal brace system is used to model die overturning effect of the lateral inertial load, which acts on die mass block center. The braces were assumed to be elastic witii relatively large stiffness as compared to the bridge bent, so that the mass block could be considered as part of the bridge bent but not as an additional degree of freedom to die system. W/8 Attached weight W = 88.95 kN (20 kips) Beam self-weight: 0.29x0.411x4.35x24.5=12.7 kN Column self-weight: 0.327x0.327x1.162x24.5=3.04 kN Fig. 4.4 The Computer Model (Dimensions are in mm). Chapter 4 81 25fZ 25 411 poooooq 3 O O O O O b o. • Stirrups @65 wire 8 / 2 wire 8 16-#3 290 H25 Section(j) 25LZ 25 411 p o o oc p o o o o o 14 Stirrups @55 wire 8 2 wire 8 13-#3 290 U 25 Section(2) 411 TT h o o d Stirrups @245 wire 10 / 2 wire 8 9-#3 290 y25 411 IT h n o n o 3 Stirrups @245 wire 10 / 2 wire 8 / 11-#3 290 H25 Sectior(^ 3) 327 Sectior(3) > 3 0 0 C 0 3 0 U c O o P. Stirrups @55, wire 14 8 wire 1 V 8-#3 I 25 327 y 2 5 Column Section(5) Fig. 4.5 Details of Member Sections (dimensions are in mm). Chapter 4 Table 4.1 Computer Model Node Coordinates. 82 Node Number X-Coord. Y-Coord. Node Number X-Coord. Y-Coord. (mm) (mm) (mm) (mm) 1 -801.5 1510.5 7 2302.5 1510.5 2 0.0 1510.5 8 2531.0 1510.5 3 228.5 1510.5 9 3332.5 1510.5 4 747.0 1510.5 10 1265.5 2126.5 5 1265.5 1510.5 11 0.0 0.0 6 1784.0 1510.5 12 2531.0 0.0 4.4.7 Material Properties The steel material properties used in the analysis are listed in Table 4.2. Table 4.2 Material Properties of Steel Bars (Unit: MPa). Bar Type Gy eh Gmax £max #3 461 0.005 680 0.04 0.04 wire 1 568 . . . — 0.02 wire 4 595 — — 0.02 wire 8 580 — . . . 0.02 wire 10 256 0.016 350 0.07 0.07 wire 14 220 — — — 0.02 Young modulus: E s = 2.06E+5 MPa Material Properties of Concrete: Compression strength: F'c = 40.5 MPa, at the strain of 0.003 Tension: 3.8 MPa (Ft = 0.61^), (Ec = y c L 5 0 .043^) yc = 2400 kg/m3, E c = 3.03xl04 MPa, G = 3/7 E =1.3xl04 MPa Stress-strain relationships assumed for section analysis are shown in Fig. 4.6. Chapter 4 83 (2) Steel Fig. 4.6 Material Stress-Strain Curves. 4.4.2 Beam Modeling in CANNY-E The beam of the bridge bent was divided into eight beam elements, as shown in Fig 4.7. In the input tile of the program, beam members 1 to 9 were conveniently named XI-X2, X 2 - X 3 , X 8 - X 9 . XI X2 X3 X4 X5 X6 X7 X8 X9 I I i i > i i i i Fig. 4.7 Node Position (units are in mm). Chapter 4 84 Shear deformation was considered, but the axial deformation of the beam was neglected. In other words, the nodes in each level had the same lateral displacement. Flexural and shear properties of beams are discussed below. Flexural Properties - Using the sectional analysis program of CANNY, the moment-curvature of each section was calculated based on the material properties, the original dimensions, and the reinforcement arrangement. The cracking moment, M c , the yielding moments, M y , and the slopes of post-cracking and post-yielding in both positive and negative directions were determined by running the sectional analysis program. Tire moment-rotation relationship was assumed to have a tri-linear backbone curve (Fig. 4.8). The corresponding cracking and yielding rotations, 6C and 8y, were calculated based on the assumption of the curvature distribution along the beam axial line; i.e., e c = MC/K0 where for anti-symmetrical moment: K0 = 6EI/L, ey = (J>yL/6 for triangular moment: = 3EI/L, e , = <|>yL/3 for symmetrical moment: = 2EI/L, e , The stiffness reduction factors Ac and Ay were then evaluated from My ~ M, My - Mc A e = e y - e c = Eiiy -MC M^My Mu~ M y Ay-ey{iL-i)-Ei$y(\L-i) Chapter 4 85 M A y K 0 M'y e A ' y K 0 Fig. 4.8 Moment-Rotation Skeleton Curve for Beam Element. In order to calculate the post-yielding stiffness reduction factor, a ductility factor p, = 4 was assumed. The sectional analysis was perfomed based on a zero axial load and no interaction between bending and shear. Although it would be possible to trick the program to modify some flexural properties to consider shear-moment interaction, as was demonstrated using a 0.45 scale model and die IDARC computer program (Williams, 1994). This method was not followd in this thesis for two reasons: 1. This method is very approximate; instead, the elastic analysis results were used for that modification. In the 0.27 scale test, there was a sudden jolt at the very beginning of the tests, which had a discernible effect near the beam-column joints just outside the haunches, where shear failure was expected. Using forces from elastic analysis would not be correct, especially with the knowledge that after the big cracks, the distribution of forces would be different from the elastic case. 2. CANNY is able to predict shear failure. Before the actual test, the CANNY program was used to do some predictions. According to die results, failure was of a flexural type, yet experimentally titis was not the case. Reducing the moment due to shear did not produce a better set. of analytical results. Chapter 4 86 Based on the tri-linear primary curve, the hysteresis model #15 from the CANNY's library was selected for analytical studies, which could appropriately model stiffness degradation, strength deterioration, and softening and pinching behaviors. It should be mentioned that the initial stiffness was modified to adjust for the frequency of the computer model in different tests, and to account for initial cracks due to shrinkage and handling, as well as to account for some previous damage due to moderate shaking and the sudden jolt. Shear Properties - Again, a trilinear backbone curve for shear force-shear deformation was applied in the program. V c and V u were introduced as crack and yield points. The hysteresis model #15 was used to model hysteresis behavior; although this model was not quite able to show the shear behavior in loading and unloading, it was the best alternative at the time of this study. As previously mentioned, the program is able to predict shear failure as flexural failure but it does not. have a realistic hysteresis model to do this job. The author of the CANNY program is going to introduce a suitable hysteresis loop to the program which will allow it to predict shear failure. The shear strengths of the beam sections were evaluated by using three methods: the Canadian Code simpified method (CSA, 1994), the ACI/ASCE loint Committee 426 (1978), and the Japanese Code. The Canadian Code simplified method(SI Units) gives: v„=ve + v, s Chapter 4 87 Vc=0.2-b-d-jFj if A v > 0 . 0 6 V ^ - y Jy c 1000 + ^ v c V / , where A v : the area of the shear reinforcement, fy- the yield strength of tlie shear reinforcement, d: the effective depth, s: the stirrup spacing, F'c: the concrete cylinder strength in N/mm2, h: the section width. The ACI/ASCE Committee (SI Units) gives: v„=ve + v. Vc = (0.067 + 10p)-b-d-jFj d s where p is the tension longitudinal steel ratio. The Japanese Code gives: — + 0.12 8 V-d Chapter 4 88 where p, longitudinal bar (in tension) ratio (%), Fc concrete compression strength (kg/cm2), d section effective depth (d = h - shell concrete thickness in tension side), b section width, M/V shear span pw hoop bar ratio, V-d hoop bar yielding strength (kg/cm2), average axial stress over the section (kg/cm2). For beams: a„ = 0. The first term in the parentheses was considered as the contribution of concrete to calculate the cracking shear strength, Vc. The last method was used by Li (1996). The results from these three approaches are listed in Table 4.3. Table 4.3 Shear Strength of Beam Sections (kN). Beam Section Japanese Code Canadian Coc e ACI/ASCE Vc Vu Vs Vc Vu Vc Vu 1 83.8 147.2 88.0 142.5 230.5 56.X 144J& 2 66.(1 150.8 104.0 142.5 246.5 56.8 160.8 3 66.0 88.5 7.5 133.6 141.1 56.X $43 4 66.0 88.5 7.5 133.6 141.1 56.H 643 Note that the ACI/ASCE Code produced rather conservative shear strengths, except in the case of Section 2, where the Japanese Code gave a slightly lower shear strength. In this analysis, the results given by the ACI/ASCE formula were used for the shear strength of the beams. Chapter 4 89 4.4.5 Column Model ing in CANNY-E The two columns of the bridge bent were also modeled using one-component models lumped plastically. This was done by placing a nonlinear bending spring at its top end and a pin-connection at its base end, as shown in Fig. 4.9. Fig. 4.9 Column Modeling. Flexural Properties - Column flexural behavior was evaluated by the same method as that used for beams. The interaction between the bending moment and axial load of the column was approximated by adjusting the column cracking and yielding strengths (both flexural and shear strengths) according to the tension or compression axial load. Note that the column is subjected to a varying axial load caused by rite overturning moment of the lateral inertial load. In the one-component model, the variation in column axial load could not be included. To adjust for this deficiency, a push-over analysis was conducted to obtain the maximum compression and tension forces needed to fonn a "mechanism" in columns. Later, at each sample time these forces are validated with test results. To compute these forces from test results, the equivalent static load was applied at die center of gravity, and the structure was analyzed for gravity loading, lateral loading, and Chapter 4 90 a combination of the two. Once these results were obtained, the maximum forces were calculated (see Chapter 3). The final results from this approach were in agreement with push-over analysis results. Fig. 4.10 shows the effect of axial load variation on the flexural capacity of columns. From die figure it can be concluded that the higher the compressive force, the greater the cracking and yielding moment capacity of a section. Reducing tension forces had also the same effect. According to the test results, flexural yielding did not happen at die columns, and since there was good agreement between the push-over analysis and die test results, it was decided to use the push-over analysis results in the section analysis program to account for the moment-axial interaction. It is worth noting that in die section analysis program, columns can be treated as multi-spring models that can include the moment-axial interaction. Since multi-spring models have some limitations in other aspects of the program (e.g., die global damage index can not be calculated if a multi-spring model is introduced), it was decided to carry out the analysis using a one-component model. Although the method perfonned to consider moment-axial interaction is not strictiy correct, since the height of the specimen is only about two meters and also the applied axial load is much less tiien die axial capacity of columns , it behaves more like beam than beam-column. Then the variation of axial load was approximated by the two extreme cases Shear Properties - The shear strength of the column was evaluated by the Japanese code in order to allow for axial load effects on the shear strength. The results are shown in Table 4.4. Chapter 4 91 Fig. 4.10 Effect of Variation of Axial Load on Flexural Capacity of Columns. Table 4.4 Column Flexural and Shear Strength (kN-m). Section 5 Mc My Ac Ay vc Vu Under tension 10.15 54.00 0.1246 0.0092 62.7 88.6 Under compression 22.19 65.97 0.1378 0.0125 62.7 97.2 Table 4.5 Column Axial Load (kN). By gravity load Tension Compression 51.75 -2.0 106.0 Chapter 4 92 4.4.4 Link (Truss) Element Modeling in CANNY-E This element was used to simulate the inertial force of the whole specimen concentrated at its center of gravity. It was introduced as an almost rigid element which could bear only compression or tension forces. 4.4.5 Hysteresis Modeling in CANNY-E CANNY-E includes a hysteresis model (model #15) that represents the stiffness degradation, strength deterioration, and softening and pinching behavior of the structural member by a series of control parameters, i.e., 8, 6, Xe, K„ ta, £, and Xs, which are explained as follows: Stiffness Degradation - The unloading following a new peak displacement (mostly outside the unloading zone) was directed to a target point as shown in Fig. 4.11(a). The instantaneous stiffness for unloading QF + F QF' + F' branches was given by Ku = ' — at the positive side, and Ku - . — at the negative ®Fy / K0 + "w vFy J K0 + drn side, in which 9 > 1.0. There was no stiffness degradation when 9 = °°, which was represented in tire program by 9 = 0. Fm, dm, F'm, and d'm were the unloading starting points. The unloading stiffnesses, Ku and K'u, were kept constant until a new peak displacement was reached. The unloading from an outside loop came to an end at an inclined axis UU', as shown in Fig. 4.11 (a), followed by reloading or slip branches. The gradient of the UU' axis was equal to SK0., where the parameter 8 may take a small value between 0.0 to 0.05. Chapter 4 93 Strength Deterioration - The model represents strength decay by directing the reloading towards a reduced strength level F m a x at the same displacement corresponding to die previous peak strength F m a x , as shown in Fig. 4.11(b). This method is similar to that used by the program ID ARC, altiiough in ID ARC the strength deterioration is expressed as ^maxO' + 1 ) = F m a x ( 0 dE The second item in parentheses is an energy-related factor not applicable to an asymmetric section with unequal yielding strength and displacement at positive and negative sides. The third item in the equation above is a ductility-related factor which indicates excessive strength loss at large displacements. Therefore, the CANNY program uses the following equation to evaluate F m a x : p — p M i l ax M i l ax F d + F' d' 1 y^max y max i - -U J > 0 IF max — max J where the second item in parentheses represents the energy related strength deterioration, and the third item represents the ductility-related strength decay. Eh is the hysteresis energy, and Fy and dmax are the yielding strength and the maximum displacement at the positive side, while F'y and d'max represent the yielding strength and the maximum displacement at the negative side, u is the ductility factor, u > 1 (u - dm&xjdy at the positive side, and u = d'm&xjd'y at the negative side). Xe and Xu are control parameters that may take a value from 0 to 1. For example, if Xe = 0 and Xu = 0.5, tiiere would be 67% strength remaining at u = 3. Chapter 4 94 Softening - Here softening means reduction in die post-yielding stiffness. The softening effect was represented by a third parameter, X3, that lowers the post-yielding envelope. Thus, we have Kpy Kpy f ! Y\ 1 - - , Kpy>0MK py where K is the post-yielding stiffness of die primary envelope curve, and Kpy is the new post-yielding stiffness. The post-yielding branch remains constant until unloading occurs. Similar to Xu, X3 can take a value from 0 to 1. Pinching Behavior - Rules 12 and 13, as shown in Fig. 4.11(c), were to simulate the pinching behavior caused by the opening and closing of cracks. Target point. (Fe, de) is to control the slip branches, as shown in the figure. Therefore, de = e • du Both parameters Xs and e range from 0 to 1.^ = 0 means no pinching; also, there is no pinching when the following condition is true: (Fe - Fs)l(de - ds) > ( F m a x - Fs)/(dma% - ds) Apparendy, Xs = 1 will not pinch at all. The slip branches took place only after yielding in die corresponding side. In the analysis by the CANNY program, the parameters used were as follows: 5 = 0, 9 = 2, Xe = 0.2, Xu = 0.6, X3 = 0, z = 0.5, Xs = 0.2 Chapter 4 95 Fig. 4.11 CANNY Hysteresis Model #15. C H A P T E R 5 PUSH-OVER ANALYSIS 5.1 Introduction In Oiis chapter, die behavior of the specimen under a monotonically increasing lateral load is studied using a push-over analysis. Push-over or step-by-step analysis is a static analysis technique which can provide a good picture of the global behavior of a structure, including general information regarding its strength and ductility, especially if the structure behaves in a similar fashion to a single-degree-of-freedom system. In die analysis, the response is followed from die first yielding to the collapse mechanism. One advantage of using push-over analysis is that it is inexpensive and takes much less time than time-step analysis. It is also easier than the time-step analysis. However, it should be noted that whde the results of a push-over analysis can be used as background information for a dynamic test, it fails to account for the dynamic characteristics of a structure and the effects of inertial and damping forces. Both computer programs, i.e, CANNY-E (Li, 1995) and RUAUMOKO (Carr, 1996), were used to undertake a push-over analysis of the 0.27 scale model. Fig. 5.1 illustrates the general outline of the model for this analysis (node coordinates are shown in Table 4.1). In both programs, die structural stiffness matrix is modified after the formation of each hinge, and a new load increment is then applied. The effective moments of inertia for the beams and die columns were assumed to be 50% of the gross moments of inertia. The elastic properties and the yield surfaces of the beams used in the analyses are shown in Table 5.1. Column data and other required infonnation can be found in the input files which are given in Appendix C. The analysis results and the comparison studies are described in the sections that follow. 96 Chapter 5 97 A . Fig. 5.1 Model Outline. Table 5.1 Beams properties : E=27400 MPa, 1=0.0008389 m4. Member My (kN-m) End 1 My (kN-m) End 2 Shear(kN) Vu XI - X 2 38.3 -143.9 26.3 -126.9 144.8 X 2 - X 3 26.3 -126.9 26.3 -126.9 150.8 X3 - X 4 26.3 -126.9 50.2 -59. 64.3 X 4 - X 5 50.2 -59.10 104.0 -26.20 64.3 X 5 - X 6 104.0 -26.20 50.2 -59.10 64.3 X 6 - X 7 50.2 -59.10 26.3 -126.9 64.3 X 7 - X 8 26.3 -126.9 26.3 -126.9 150.8 X8 - X 9 26.3 -126.9 38.3 -143.9 144.8 5.2 C A N N Y - E Using CANNY-E, the static-load-to-collapse analysis was conducted for two situations: 1) assuming a tri-linear backbone curve which could represent both cracking and yielding of the model; 2) using a bi-linear model to create an identical situation for both CANNY-E and RUAUMOKO, thus establishing a direct comparison. In both approaches, the hinges were formed in the same sequence. Chapter 5 98 The load was applied from left to right, and the haunch section was ignored. The first flexural hinge was indicated at the beam section X2, i.e., the face section of the beam-column joint, where the beam was subjected to a positive bending moment that caused tension at the bottom and compression at the top. The second hinge was at the beam section X3 of tire bridge bent. After the push-over analysis, it was observed that the beam was poor in positive flexural strength due to insufficient reinforcing bars at the bottom side; as a result, no shear failure occured because of tire low flexural capacity. Since the beam X2-X3 had both hinges at its two ends, the increasing load simply caused a higher moment and a shear force at the column on the compression side (the right column). The monotonic load was increased gradually until the third flexural liinge appeared at the right column's top end. A fourth hinge was also fonned at tire top end of the left column, while the shear in the beams X5-X6, X6-X7, and X7-X8 exceeded the shear crack strength. The columns, however, were quite safe in shear, as the maximum shear demand remained around 44.0 kN in the columns for both analyses, which was well below the concrete shear capacity for the columns, Vc„ which was about 62.7 kN. 5.2.7 The Tri-Linear Model In Fig. 5.2, the lateral load versus the displacement at the cap beam level is plotted using a tri-linear model. Tire response was nearly linear for a short time before flexural cracks occurred either in the beams or at the top of the columns. After the initial cracks occured, the response was characterized by a gradual loss in lateral stiffness due to cracking and yielding effects. Analysis results showed that at about 46.9 kN and 52.1 kN, the first and tire second hinges formed, respectively. Since these hinges occured close to each other at the left end of the beam, the change in the global stiffness of the model was not drastic. However, Chapter 5 99 as soon as the hinges were fonried in the columns the model became much softer and the stiffness of the model decreased dramatically. Fig. 5.3 shows the changes in the specimen's natural period due to flexural and shear cracks, and plastic hinges. The sharp changes in the natural period were associated with the fonnation of plastic hinges at the top of the columns, which occured at about 80.6 kN. 5.2.2 The Bi-linear model Since no cracking strength was introduced into the input file, the response was nearly linear until the first yielding occured. The sequence of hinges was exactly the same as in the tri-linear model, except that X2 and X3 yielded simultaneously at about 52.1 kN. Fig. 5.4 shows a change of slope at this stage. When the third hinge was fonned at top of the right column at 58.3 kN, the model became much softer, and finally the mechanism was fonned at 79.8 kN. The sequence of hinges formed according to the two approaches is listed in Tables 5.2 and 5.3. It is interesting to note that different modeling of the backbone curve produces different results. While the elastic properties, the size of tire applied load, and the way the load was applied were exactly the same in both approaches, and yielding occured at almost the same load, the yielding displacements were quite different for the two models. At the mechanism, the model with a tri-linear load-deformation behavior experienced a displacement more than 1.5 times that of the model with a bi-linear backbone curve. The explanation is that in the bi-linear model the stiffness is assumed to remain unchanged before yielding, whereas in the tri-linear model there is a gradual loss in stiffness almost from the beginning of the analysis. Chapter 5 100 Fig. 5.2 CANNY-E: Static Push-Over Analysis Using a Trilinear Backbone Curve. 1.1 1.0 0.9 0.8 0.7 0.1 1 — I - r : : r ; : : ] ~ \ 0.0 I ! 1 1 1 : 1 1 ! 1 1 1 0 5 10 15 20 25 30 35 40 45 50 55 60 Lateral Displacement at beam Level (mm) Fig. 5.3 Period of the Specimen at Different Stages of Loading. Chapter 5 101 Table 5.2 CANNY-E: Hinge Sequences Using a Tri-linear Backbone Curve. Step of Analysis Location of Hinges Lateral Load (kN) Displacement at the Beam Level (mm) 9 X2 46.9 11.1 10 X3 52.1 13.3 18 Right column- top 60.4 17.4 149 Left column- top 80.6 45.9 Table 5.3 CANNY-E: Hinge sequences Using a Bi-linear Backbone Curve. Step of Analysis Location of Hinges Lateral Load (kN) Displacement at the Beam Level (mm) 10 X2 52.1 3.4 10 X3 52.1 3.4 12 Right column- top 58.3 4.3 128 Left column- top 79.8 29.0 5.3 RUAUMOKO The computer program RUAUMOKO carried out a push-over analysis by assuming a bi-linear load-deformation behavior. The response was determined by imposing a gradual increasing lateral load from Chapter 5 102 zero up to the collapse load. Fig. 5.5 shows the relationship between the lateral load and die displacement at the beam level. The sequence of hinge formation was the same as the one in the analysis performed by CANNY. Yielding commenced at X2 at about 49.9 kN. The second hinge was formed at X3 at 63.6 kN and was closely followed by another yielding at die top of the right column under a 65.4 kN lateral load. The last hinge was formed at the top of the left column at 80.8 kN. Shear values were examined to determine die type of failure, however die results showed the shear capacity calculated from different codes were larger than the shear demand. Just as in the CANNY's results, shear demand of the beams X5-X6, X6-X7, and X7-X8 exceeded the shear crack strength in die RUAUMOKO' analysis, which would be expected under diis level of deformation. The RUAUMOKO program asks for the length of the plastic hinge at the end of each member. By trying different plastic hinge lengths for die beams and columns, it was found that die post-yielding behavior is quite sensitive to plastic hinge length. Load-deflection results for three different plastic hinge lengths are plotted in Fig. 5.6, where it can be seen that increasing the plastic hinge lengtii caused: 1) the structure to became softer, 2) die slope of load deflection to reduce, and 3) die model to experience more displacement. According to Paulay and Priestiey (1992), half of the deptii can be used as a good estimate for plastic hinge length. Consequently, as the depth of the beam was 0.411 m, a plastic hinge lengtii of 0.2 m was assumed for the final run. Table 5.4 RUAUMOKO: Hinge Sequences Using a Bi-linear Backbone Curve. Step of Analysis Location of Hinges Lateral Load ' (kN) Displacement at die Beam Level (mm) 117 X2 . 49.9 3.3 149 X3 63.3 5.0 155 Right column- top 65.4 6.0 187 Left column- top 80.8 22.1 Chapter 5 103 Chapter 5 104 5.4 Comparison Study Fig. 5.5 shows that there is a close agreement between the results obtained from each program. The sequences of hinge formation were the same according to both analyses (as shown in Fig. 5.7), and the results from both programs were identical prior to tire formation of the first liinge. After the first yielding, there was minor difference in the results from the two computer programs. In the analysis perfomed using RUAUMOKO, the second hinge was fonned at a slighdy higher force than the one predicted by CANNY, but the collapse load remained similar, i.e., 79.8 kN from CANNY as compared with 80.8 kN according to RUAUMOKO; the displacement at collapse mechanism was 29.0 mm and 22.1 mm from CANNY and RUAUMOKO, respectively. After tire mechanism occured, computations in both programs continued up to the maximum load of 86.0 kN, which was specified by the input files. As nonlinearity started, it was observed that the model in RUAUMOKO remained stiffer than the model in CANNY, an effect explained by the fact that the two nonlinear computer programs are not exactly the same in the way they solve nonlinear problems. In RUAUMOKO, it is possible to define only one value for strain hardening for an element, while in CANNY-E, not only is it possible to define different strain hardening values at each end of a member, but it is possible to assign different values in the positive and negative directions of the force-displacement relationship. As a result, the average values for strain hardening stiffness used in CANNY-E were also applied for each member in the analysis performed by RUAUMOKO. Another source of discrepancy is that RUAUMOKO considers the effect of axial-moment interaction yield surface for columns while CANNY-E assumes separate springs for different forces in a uniaxial model. Chapter 5 105 XI X2 X3 X4 X5 X6 X7 X8 X9 0 Hinges A Fig. 5.7 Damage Distribution and Mechanism. C H A P T E R 6 C O R R E L A T I O N OF A N A L Y T I C A L AND E X P E R I M E N T A L RESULTS OF 0.27 S C A L E M O D E L 6.1 INTRODUCTION This chapter describes the analytical studies of the 0.27 scale model of the Oak Street Bridge subjected to a series of simulated earthquakes on a shake table. It also investigates the degree of correlation between experimental results and analytical results obtained using two computer programs, CANNY-E and RUAUMOKO. The purpose of the study is to determine the degree to which the computer programs are successful in simulating the experimentally obtained response of the 0.27 scaled specimen. This chapter reviews the excitation records and damping values used in the computer analysis, and compares the natural frequencies of the analytical models from both computer programs with those obtained during the tests. It includes consideration of the elastic and inelastic behavior of the analytical models and compares the time histories obtained from tests with those obtained from analytical approaches. 6.2 Earthquake Record The scaled California 1992 Landers earthquake record from Joshua Tree station in the E-W direction was used as input to the shake table for all test runs. The measured horizontal acceleration time history was used for the analysis. The total duration of the excitation in each test was about 37.5 seconds. The 106 Chapter 6 107 analytical studies and the subsequent comparison with test results were carried out only during the strong motion which lasted for about 10 seconds. The sampling frequency was 200 Hz, meaning that during shake table testing, data was recorded every 0.005 seconds. 6.3 Fundamental Frequencies The fundamental frequency of the specimen at each stage of loading was derived from two types of excitation: shake table test and hammer impact. The undamped fundamental frequency, co, of a single degree of freedom system is given by: co= y/k /m where K is the model stiffness and m is the mass of the system. Since mass is a well-defined parameter, tuning the frequency of the analytical model was possible by modifying the model stiffness, K. This stiffness may be controlled by establishing boundary conditions for the members, the moment of inertia of each section, and the rigid zone length at beam-column connections. To account for the effect of initial cracks due to handling, shrinkage, and the damage sustained by the specimen during consecutive excitations, an effective moment of inertia was used in the analytical study at each stage of excitation. As mentioned in Chapter 3, a sudden jolt occured right after the beginning of the first test, and the specimen was significantly cracked; the analytical model was therefore assumed to be cracked at the beginning of the analysis as well, and this was reflected in the choice of the effective moment of inertia. A trial-and-error procedure to select the effective moment of inertia was used to ensure that the analytical model would have Chapter 6 108 Table 6.1 Fundamental periods Obtained from Test, CANNY, and RUAUMOKO. Run Shake Hammer C A N N Y R U A U M O K O Effective I % Table Test (sec) (sec) for beam & Test (sec) (sec) columns 10% 0.178 0.150 0.168 0.166 50% 40% 0.207 0.174 0.187 0.185 40% 60% 0.229 0.181 0.191 0.190 38% 80% 0.249 0.190 0.204 0.203 33% 120% 0.274 0.224 0.233 0.232 25% 150% 0.392 0.280 0.299 0.298 15% a frequency value close to the one from test. The length of the rigid zone was assumed to be zero in order to achieve closer agreement with die frequency from the analytical approach. With these modifications, the fundamental frequency of the analytical model was obtained using CANNY-E and RUAUMOKO. The experimental and analytical fundamental periods, along with the effective moment of inertia used in die analysis, are summarized in Table 6.1. In the analysis, the gross areas of beams and columns were used. From the results in Table 6.1, it can be seen that the maximum discrepancy between the experimental and analytical fundamental periods was around 5%, except for die 150% run, for which die difference was about 9%. 6.4 Damping The procedure to obtain damping from experimental data is explained in Chapter 3. In both CANNY-E and RUAUMOKO programs, viscous damping can be included as Rayleigh damping coefficients, a, and (3. These parameters were calculated by specifying the fraction of critical damping (Clough and Penzien, Chapter 6 109 1993). It should be noted that while viscous damping is a function of velocity, the dissipated strain energy or hysteretic damping is a function of force-displacement loops. hi each stage of excitation the analysis was started by using the damping measured during the corresponding shake table test. In almost all cases, the analytical response was quite far from the test results. This discrepancy was mainly due to the way damping was assessed in the experiment. For the computation of damping values from recorded data (previously explained in Chapter 3), it was assumed that viscous damping is the source of the damping mechanism. This assumption is not quite correct, because the jolt experienced at the beginning of the test produced some unwanted cracks in the specimen; as a result, the damping was a combination of viscous and hysteretic damping. Due to the difficulty in distinguishing between the viscous and hysteretic portions of damping, a type of "damping juggling" was attempted to obtain a better correlation between the experimental and theoretical responses. The hysteretic damping pronounces itself through the hysteretic loops. Table 6.2 summarizes the damping values obtained from tests and the ones which were used in the analysis. Table 6.2 Damping Ratios. Run Test - Combination of CANNY RUAUMOKO Viscous and Hysteretic Viscous Viscous Damping Damping Damping % % % 10% 8.7 2 2 40% 10.2 5 5 60% 5.9 3 3 80% 6.8 3 3 120% 12.2 3 3 150% 13.1 3 3 Chapter 6 110 6.5 Analytical Response and Comparison Study Structural response is usually separated into linear or non-linear response. Wlule simulating linear behavior is relatively simple, non-linear behavior is more complicated and tiius more difficult to simulate. The same analytical models using different computer programs, under the same excitation, provide the same elastic responses; so, the first step was to obtain similar elastic responses from CANNY and RUAUMOKO. This ensured that the analytical models were the same in both programs, and that the discrepancy in nonlinear responses from the two programs was caused due to some other factors. This study also helped in recognizing the degree of error resulting from a linear analysis. Next, inelastic analyses were conducted using CANNY-E and RUAOMOKO, and the results were compared with shake table test results. Such a comparison, however, is not easy, as the seismic response of reinforced concrete is complicated, and different programs have different hysteresis loop libraries which are not equivalent. The modeling of elements may also be quite different from one program to the next. Tiiere are also different methods for integration and time step analyses. Some programs are able to do iteration while some require to take very small time steps for integration. In this study, modeling and analysis methods were kept as close as possible in order to accurately compare the two computer programs. Both linear and non-linear analyses were performed using CANNY-E and RUAUMOKO. While linear analysis was done for 10% and 40% runs, nonlinear analysis was performed for six levels of excitation: 10%, 40%, 60%, 80%, 120%, and 150%. Note that a 10% ran means the Joshua Tree earthquake was scaled to a peak ground acceleration equal to 10% of 0.7g. hi each stage, the analytical base shear was found by adding the computed shear force at the end of each column. Relative displacement and absolute Chapter 6 111 acceleration time histories were also provided in output files. A sample output file obtained from each program is given in Appendix B. To compare the analytical base shear force with the experimental base shear, the latter was computed by multiplying the measured absolute acceleration at the center of gravity by the mass of the specimen. Here, it was assumed that the measured base acceleration was the same as the shake table acceleration (i.e., rigid connection) and the viscous damping force was negligible. The reason was that the base shear obtained from the computer programs was observed to be very close to the analytical inertia force (see, e.g., Fig. 6.1), so that a fair comparison between the two shear forces could be made. 6.5.7 Linear A nalysis In both programs all the elements were assumed to behave linearly in this phase of analysis, and shear deformation was ignored. We also assumed that Ieff= 0.5 lgross, damping ratio = 5%, and used only 10% of the measured shake table acceleration record as the ground acceleration. The resulting analytical response time histories, compared in Fig. 6.2, are identical, which confirms the similarity of the analytical models used in the two computer programs. Figs. 6.3 and 6.4 compare the computed response time histories with the one obtained experimentally. For the low-amplitude excitation, the peak responses were in general agreement, but the displacements and forces, after 6 seconds, were underestimated in the analytical approaches. The experimental force-displacement relationship at this stage, as given in Chapter 3, showed some hysteresis behavior, indicating that a non-linear analysis would provide a better simulation. Chapter 6 112 The same linear analysis process was repeated for the 40% run, in which Ieff = 0.4 Igross and damping ratio = 5%. Figs. 6.5 and 6.6 compare the time history responses from the analytical and experimental approaches. Here the calculated displacements are in agreement, but the calculated forces are much higher than what was measured during the test. The peak values of forces are +131.0 kN and -115.0 kN in the Chapter 6 113 analytical approaches and +77.1 kN and -88.7 kN in the test. The 40% test suggests that the specimen behaved inelastically and that an elastic computer model is not able to truly reflect the specimen's behavior. Tables 6.3 and 6.4 compare die peak values for die base shear and relative displacement at die cap beam level from each elastic analysis to the experimental values. Table 6.3 Comparison of Base Shear Maximum Values. Run Test CANNY RUAUMOKO (kN) Elastic Analysis (kN) Elastic Analysis (kN) 10% 59.2 65.4 65.5 -57.1 -54.1 -54.1 40% 77.1 131.3 131.6 -88.4 -115.2 -115.6 Table 6.4 Comparison of Displacement Maximum Values. Run Test CANNY RUAUMOKO (mm) Elastic Analysis Elastic Analysis (mm) (mm) 10% 4.3 4.2 4.2 -4.1 -3.5 -3.5 40% 8.4 10.5 10.5 -9.1 -9.2 -9.2 Chapter 6 114 Ruaumoko Fig. 6.2 Elastic Analysis - Results from CANNY and RUAUMOKO using Shake Table Recorded Acceleration from 10% Test. Relative Displacement Time Histories at the Cap Beam Level and Base Shear Time Histories. Chapter 6 115 Test E E, c CD E CD "0. w b 5.0 0.0 -5 .0 Max. value = 4.3 mm. Min. value = -4.1 mm. Canny E E •£ CD E CD o 8-b 5.0 r 0.0 -5 .0 Max value = 4.2 mm. Min value = -3.5 mm. E E, c CD E CD S. b 5.0 0 .0 -5 .0 Ruaumoko Max. value = 4.2mm. Min. value = -3.5 mm. 10 sec Fig. 6.3 Elastic Analysis - Relative Displacement Time Histories at the Cap Beam Level: 10% Test, CANNY-E, and RUAUMOKO. Chapter 6 116 Test 100 .0 50 .0 0.0 -50 .0 h -100 .0 Max. value = 59.2 kN. Min. value = -57.1 kN. sec Canny 100 .0 5 0 . 0 0.0 - 50 .0 - 1 0 0 . 0 Max value = 65.4 kN. Min value = -54.1 kN. sec Ruaumoko 100.0 50 .0 0.0 - 50 .0 -100 .0 Max. value = 65.5 kN. Min. value = -54.1 kN. sec Fig. 6.4 Elastic Analysis - Base Shear Time Histories: 10% Test, CANNY-E, and RUAUMOKO. Chapter 6 117 Test 4 0 % E E c CD E cu o JO % b 10 r Max. value = 8.4 mm. Min. value = -9.1 mm. Canny E E, c CD E CD O JO a en b Max. value = 10.5 mm Min. value = -9.2 mm Ruaumoko E E, c: CD E CD O JO & b Max. Value = 10.5 mm Min. value = -9.2 mm Fig. 6.5 Elastic Analysis - Relative Displacement Time Histories at the Cap Beam Level: 40% Test, CANNY-E, and RUAUMOKO. Chapter 6 118 Test 40% 120 r -120 Max. value = 77.2 kN. Min. value = -88.4 kN. Canny 120 -60 -120 Max. Value = 131.3 kN Min. Value = -115.2kN 120 -120 Ruaumoko Max. value =131.6 kN. Min. value = -115.6 kN. sec Fig. 6.6 Elastic Analysis - Base Shear Time Histories: 40% Test, CANNY-E, and RUAUMOKO. Chapter 6 119 6.5.2 Non-linear Analysis Modeling of sectional moment-curvature relationships is explained in Chapter 4. In the present section the tuning of the proposed models in both CANNY-E and RUAUMOKO is discussed. The correlation between responses from the two analytical approaches to each other and to the experimental results is examined next. The comparisons are carried out for the following response histories: • Relative displacement time histories at the cap beam level, • Absolute acceleration time histories at the cap beam level, • Base shear time histories, • Hysteresis loop - base shear versus relative displacement at the cap beam level. The comparison studies are done for six levels of excitation. Each is discussed in more detail in the following sections. 6.5.2.1 Run 10%: CANNY - A tri-linear backbone curve modelled the concrete cracking and steel yielding characteristics of a concrete section. CANNY-E has 7 tri-linear hysteresis models, of which 4 were more appropriate to model concrete behavior. Hysteresis models number 4, 6, 7, and 15 were tried. As mentioned earlier, for beams and columns we set Ieff= 0.5 Igross and damping ratio = 5%. The first 10 seconds of die shake table recorded acceleration time history for the 10% run was used as input signals. Since the program was not capable of iteration, to prevent accumulation error, the analysis was carried out using a very small integration time step equal to 0.0004 sec, with results given every 12 time steps. Chapter 6 120 Hysteresis model #4 provided very poor results. Hysteresis models #6 and #7 provided almost identical results, and though they provided time histories which were comparable to those obtained during the tests, the peaks were far from the experimental values. The next attempt used the hysteresis model #15, which is designed to represent a stiffness degratlation, strength deterioration, and pinching behavior suitable for modeling concrete behavior. The displacement and base shear time histories followed the same pattern as that provided by experimental data, while peak displacements were +3.6 mm and -5.7 mm, compared to the values of +4.3 mm and -4.1 mm obtained through the experiment, amounting to a +16% and -39% discrepancies in peak values. In Fig. 6.7, during the last three seconds of the analysis, a persistent shift in the displacement is observed, which at this stage of excitation was not expected. The main problem with this particular analysis was the values of the base shear: the peak base shears from experiments were +59.2 kN and -57.1 kN, while from analysis they were +27.3 kN, and -33.7 kN, i.e., a 53% error! The experimental base shear for the whole duration of excitation was larger than the analytical response. Since the hysteresis model #15 provided the same pattern as that derived through experiment, it was decided to use the model and to try to modify it. During the next run, columns were assumed to behave elastically, while the beam elements were assumed to behave inelastically following the hysteresis model #15. This was a valid assumption since for the 10% test, the columns did not experience any damage (Davey, 1996). As a result, a considerable improvement in the analytical response was observed and Hie base shear improved dramatically: its peak values were almost doubled (the new base shear peak values were +46.0 kN and -47.2 kN), reducing the discrepancy to 22%, compared to the previous 53% error. Besides, shift in the displacement time history vanished and a better correlation to the test results was obtained: the displacement peak values became +3.8 mm, and -4.0 mm. Fig. 6.7 compares the response of Chapter 6 121 the analytical model for the case when columns behaved linearly witii die case when tiiey behaved nonlinearly. Since die base shear peak values were still less dian those measured during the test, it was decided to reduce the damping ratio in order to further reduce the discrepancy and achieve a better correlation. In the next run, a 2% damping ratio was assumed for the computer model. As discussed in Section 6.4, the measured damping was a combination of viscous and hysteretic dampings, therefore there was a possibility that the assumption of a 5% viscous damping was not quite correct. Running the CANNY-E program with a new 2% viscous damping ratio, we obtained an improved reponse for the model: the maximum discrepancy was reduced to 10% and 7% for the force and displacement, respectively. Since tiiis analysis could provide a reasonable time-history pattern and a good correlation with the experimental peak values, this run was assumed as the final run. Table 6.5 summarizes the results from this analysis. Figs. 6.8 to 6.10 present the time histories obtained from experiments and analyses (results from RUAUMOKO are discussed later in diis section). Table 6.5 10% Test - CANNY Results from Different Runs. Assumptions at Different Runs Relative Displacement at the Cap Beam Level (mm) Base Shear (kN) Beam Inelastic, Column Inelastic Damping = 5% 3.6 -5.7 27.3 -33.7 Beam Inelastic, Column Elastic Damping = 5% 3.8 -4.0 46.0 -47.2 Beam Inelastic, Column Elastic Damping = 2% 4.6 -4.4 55.0 -51.4 Chapter 6 122 6.0 r E E CD E CD O re o. CO ro CD .c CD CD CO ro m sec Fig. 6.7 10% Run - Analytical Response Time Histories: CANNY-E: Inelastic Beam, Inelastic Columns, Damping Ratio = 5%. Inelastic Beam, Elastic Columns, Damping Ratio = 5%. Chapter 6 123 Test 5.0 E E. c a E o CO b 0.0 -5 .0 4 6 sec Max. value = 4.3 mm. Min. value = -4.1 mm. 10 5.0 r E E. c CD E CD o _CB Cl w b 0.0 -5 .0 Canny (Damping 2%) Max value = 4.6 mm. Min value = -4.4 mm. 5.0 E E c CD E CD O CO T L cn 0.0 -5 .0 Ruaumoko (Damping 2%) Max. value = 5.3mm. Min. value = -4.7 mm. Fig. 6.8 Relative Displacement Time Histories at the Cap Beam Level: 10% Test, CANNY-E, and RUAUMOKO. Chapter 6 124 Test z 100 .0 50 .0 h 0.0 -50 .0 - 100 .0 Max. value = 59.2 kN. Min. value = -57.1 kN. Canny (Damping 2%) 100 .0 50 .0 0.0 - 50 .0 - 100 .0 4 6 sec Max value = 55.0 kN. Min value = -51.4 kN. 10 Ruaumoko (Damping 2%) 100 .0 50 .0 0.0 -50 .0 -100 .0 Max. value = 6 6 . 9 kN. Min. value = - 6 0 . 0 kN. 4 6 sec 10 Fig. 6.9 Base Shear Time Histories: 10% Test, CANNY-E, and RUAUMOKO. Chapter 6 125 Test 10 .0 c o (0 \— 0) d) o o < 0.0 -10 .0 Max. value = 0.56g Min. value = -0.55g c o co Q) O O < 1.0 0.0 -1 .0 Canny (Damping 2%) 4 6 sec Max value = 0.50g Min value = -0.53g 10 Ruaumoko (Damping 2%) c o 05 0) d) o o < sec Max. value = 0.61 g Min. value = -0.69g Fig. 6.10 Absolute Acceleration Time histories at the Cap Beam Level: 10% Test, CANNY-E, and RUAUMOKO. Chapter 6 126 Base shear versus beam relative displacement hysteresis loops are plotted in Fig. 6.11, in which a general agreement between the measured and analytical approach is obvious. Visually, the stiffness degradation is similar; if the hysteresis loops are approximated with lines, the slopes of the line will be very close. Maximum and minimum values are also in good agreement, though there is some pinching behavior in the experimental response that was not predicted by CANNY. Since at this stage the pinching is very small, improvements in the hysteresis loop will be discussed in relation to later runs. In the next stage of analysis, the shear deformation effects on the model response was considered by neglecting the shear deformation in the analysis. No significant difference in the results was observed, and the computed peak values for displacement and force were +4.4 mm and -4.4 mm, and +54.1 kN and -51.6 kN, respectively. These values were close to those calculated by assuming a shear deformation - in fact, the displacement and base shear time histories for each assumption were almost identical. Table 6.6 compares the peak values when shear deformation was assumed and when it was ignored in the analysis. To check the effect of integration time step in the model response, different integration time steps of 0.001 sec. and 0.0004 sec. were tried in the analysis. However, no significant change was observed in the results, indicating that local yielding did not occur at this stage. Table 6.6 10% Test - CANNY-E Results from Different Runs. Assumptions at Different Runs Relative Displacement at the Cap Beam Level (mm) Base Shear (kN) Assuming the Shear Deformation 4.6 -4.4 55.0 -51.4 Ignoring the Shear Deformation 4.4 -4.4 54.1 -51.6 Chapter 6 127 RUAUMOKO - Similar to the CANNY's runs, we assumed Ieg = 0.5 Igross, and viscous damping = 2%. The first 10 seconds of the shake table recorded acceleration time history for the 10% run was used as base excitation. RUAUMOKO has several tri-linear hysteresis models, most of which have the limitation that the slope of cracking to yielding and post-yielding (for negative and positive moment-rotation backbone curves) must be the same, which was not the case in this study. From the RUAUMOKO's library, the hysteresis loop #3 was tried with different values for r (r is the Ramberg-Osgood factor that controls abruptness of loss of stiffness), as well as hysteresis loops #4, #8, #11, and #23. Hysteresis loop #3 provided a very large base shear, while hysteresis loops #8, #11, and #23 did not produce satisfactory results. Hysteresis loop #4, also known as "modified Takeda hysteresis model", has a bi-linear backbone curve and provided the best result. Unlike CANNY-E, RUAUMOKO allows only the introduction of one number as the bi-linear (or post-yielding) factor for an element. It was therefore decided to apply the average value of positive and negative post-yielding slopes at botii ends of an element, as used in the analysis by CANNY. Also, as in CANNY, the tangent stiffness Rayleigh damping was used in the analysis. To keep die analytical models in both programs as close as possible, the columns were assumed elastic in behavior. The plastic hinge length was set to 0.2 m for each beam element, except elements #2 and #7, which were assumed to be 0.05 m long. The hysteresis loop #4 selected for the analysis has two parameters, a and p\ which control the stiffness during unloading and reloading periods. To obtain a better correlation between the experimental and analytical results, sensitivity analysis was carried out using different values for a and p\ It was found that a = (3 = 0.1 provided the best correlation. While strength degradation based on ductility or number of Chapter 6 128 cycles could have been introduced in tire data file, no strength reduction was assumed, since nonlinearity effects were not strong at this stage of excitation. The response time histories and the hysteresis loop are presented in Figs. 6.8 to 6.10. Since a translational slaving was imposed on the nodes of the beam, they all possessed the same horizontal displacement, and a response time history for any selected node provided the same result. A similar assumption was made when using CANNY, and, as can be seen, the results are in agreement with those obtained by CANNY and those measured during the test. The CPU time for running the RUAUMOKO program was about 21 minutes when an integration time step of 0.001 sec. was selected, which was about 10.5 times as much as the CPU time required by CANNY-E. At this stage, the analysis was not sensitive to integration time step, nor a smaller time step did cause a significant change in the model response. Results obtained from analysis and experiment are summarized in Table 6.7. Table 6.7 Maximum Values from 10% Test, CANNY-E, and RUAUMOKO. Max. Relative Max. Base Shear Bent Max. Absolute Displacement (mm) (kN) Acceleration (g) Test 4.3 -4.1 59.2 -57.1 0.56 -0.55 Canny 4.6 -4.4 55.0 -51.4 0.50 -0.53 Ruaumoko 5.3 -4.7 66.9 -60.0 0.61 -0.69 Test 6.0 1 0 0 . 0 6 .0 - 1 0 0 . 0 Canny (Damping 2%) 1 0 0 . 0 r kN 6 .0 mm 6 .0 - 1 0 0 . 0 Ruaumoko (Damping 2%) 100.0 ^ kN mm 6.0 - 1 0 0 . 0 Fig. 6.11 Base Shear vs. Relative Displacement at the Cap Beam Level: 10% Test, CANNY-E, and RUAUMOKO. Chapter 6 130 6.5.2.2 Run 40% CANNY - From previous analysis results, it was recognized that the hysteresis loop #15 was the most appropriate hysteresis model for beams and columns. Forty percent of the moment of inertia was used as the effective moment of inertia for all the beams and columns, and the damping ratio was assumed to be 5%. The first 10 seconds of the shake table recorded acceleration time history for the 40% run was used as input signals. As in the 10% run, the response time histories obtained from analysis closely matched the experimental results, and although the computed peak displacements were not far from the experimental values, the base shear peak values were significantly smaller than test data. While experimental response showed the peak values of +8.4 mm and -9.1 mm for displacement and +77.1 kN and -88.7 kN for base shear force, analytical response gave the peak values of +6.4 mm and -7.3 mm for displacement and +33.4 kN and -33.1 kN for base shear force. In the next step, it was assumed that the columns remained elastic during the excitation period. As a result, the analytical response was considerably improved and the time histories, especially for the first 7 seconds, followed the same pattern as the recorded data. The peak displacements changed to +7.8 mm and -9.1 mm, while the peak base shear increased to +70.6 kN and -75.0 kN (see Table 6.8). In other words, the displacements reached 93% and 100% of the maximum and minimum of the measured responses, respectively, while the analytical base shear reached 92% and 85% of the maximum and minimum of the experimental values, respectively, which all indicate close agreements. The analytical and experimental time histories are plotted in Figs. 6.12 to 6.14. It should be noted that the observations during the 40% test showed some cracks at columns, therefore tire assumption that columns remained elastic is not quite correct although it did provide responses in good agreement with those measured during testing. Chapter 6 131 Table 6.8 40% Test - CANNY-E Results from Different Runs. Assumptions at Different Runs Relative Displacement at the Cap Beam Level (mm) Base Shear (kN) Beam Inelastic, Column Inelastic Damping = 5 % 6.4 -7.3 33.4 -33.1 Beam Inelastic, Column Elastic Damping = 5 % 7.8 -9.1 70.6 -75.0 The plots of base shear versus relative displacement are presented in Fig. 6.15, where a general agreement is observed and the stiffness degradation curves look similar dispite some discrepancies. Since the specimen and the analytical model have underwent an earthquake excitation, it is difficult to compare their strength deterioration characteristics although they visually appear to be similar. In Fig. 6.15, some pinching behavior is observed in the test, though this cannot be directly related to the analytical response. In CANNY, tiiere are two parameters which control the pinching behavior: Xs and e. In a sensitivity analysis, different values for Xs and e were tried but did not yield different results. The two parameters, Xs and £, can reflect the pinching behavior caused by the opening and closing of cracks. There might be different sources for pinching behavior, e.g., pinching due to bar slip or shear (Saatcioglu, 1991). It is therefore difficult to express why CANNY-E appeared to be insensitive to Xs and e. It is possible that these parameters are only dummy parameters in the program, or that CANNY-E may not be able to consider other sources of pinching. For the integration time steps of 0.0004 sec. and 0.001 sec, the results were almost identical. Also, it was found that shear deformation had no significant effect on the model response, as the peak displacements did not change by ignoring the shear deformation. The peak base shears were +70.1 kN and -75.4 kN, which were almost the same as the peak values obtained by considering the shear deformation (see Table 6.9). Chapter 6 132 Finally, comparing the analytical and the experimental results, we found some slight discrepancies in both force and displacement time histories. Table 6.9 40% Test - CANNY Results from Different Runs. Assumptions at Different Runs Relative Displacement at the Cap Beam Level (mm) Base Shear (kN) Including Shear Deformation 7.8 -9.1 70.6 -75.0 Ignoring the Shear Deformation 7.8 -9.1 70.1 -75.4 RUAUMOKO - The analysis was carried out by using the hysteresis loop #4 for beam elements and the simple bi-linear hysteresis loop #2 for column elements. As with CANNY, we set Ieff - 0.4 Igross and damping ratio = 5%. The first 10 seconds of the shake table recorded acceleration time history for die 40% run was used as the base excitation for the analytical model. As in the 10% run, the plastic hinge length was assumed to be 0.2 m for all beam elements except elements #2 and #7, for which a 0.05 m length was assumed. The response time histories were computed and then compared to the experimental results. Except for shifting that existed in the calculated displacement, the analytical and test results were in agreement. After a number of loops, a jump was observed that had no correlation with the experimental response. In the next run, columns were assumed to behave elastically. In this case, although the agreement improved, the jump in the hysteresis loop existed persistently. To reduce the error, different values for a (the unloading stiffness parameter) and p (the reloading stiffness parameter) were tried. Improved results were obtained by changing the values of a and P from a = 0.1 and p = 0.1 to a = 0.0 and P = 0.6. Chapter 6 133 Test 40% E E, c CD E CD O JS 8-b 10 - 1 0 4 6 s e c M a x . v a l u e = 8 . 4 m m . M i n . v a l u e = - 9 . 1 m m . 10 E E. c CD E CD O CO TL <H b 10 - 1 0 Canny (Damping 5%) 4 6 s e c M a x . v a l u e = 7 . 8 m m M i n . v a l u e = - 9 . 1 m m 10 Ruaumoko (Damping 5%) M a x . V a l u e = 7 . 2 m m 0 2 4 6 8 10 s e c Fig. 6.12 Relative Displacement Time Histories at the Cap Beam Level: 40% Test, CANNY-E, and RUAUMOKO. Chapter 6 134 Test (40%) 1 2 0 - 1 2 0 Max. value = 77.2 kN. Min. value = -88.4 kN. 1 2 0 6 0 - 6 0 - 1 2 0 Canny (Damping 5%) Max. Value = 70.6 kN Min. Value = -75.0kN 4 6 sec 1 0 120 60 - 6 0 - 1 2 0 Ruaumoko (Damping 5%) Max. value =73.2 kN. Min. value = -78.5 kN. 4 6 sec 1 0 Fig. 6.13 Base Shear Time Histories: 40% Test, CANNY-E, and RUAUMOKO. Chapter 6 1 3 5 Test (40%) 1.2 r c g +-» 05 t_ 0) CD o o < Max. value = 0.76g Min. value = -0.83g Canny (Damping 5%) c o 2 CD O O < 1.2 0.6 h 0.0 -0.6 -1.2 Max. value = 0.73g Min. value = -0.69g 10 sec Ruaumoko (Damping 5%) g 2 CD O O < 1.2 0.6 0.0 -0.6 -1.2 sec Max. Value = 0.70g Min. value =-0.75g 10 Fig. 6.14 Absolute Acceleration Time Histories at the Cap Beam Level: 40% Test, CANNY-E, and RUAUMOKO. Chapter 6 136 Test (40%) 120 -i -120 J Canny (Damping 5%) 120 - i -120 J Ruaumoko (Damping 5%) 120 - i -120 J Fig. 6.15 Base Shear vs. Relative Displacement at the Cap Beam Level: 40% Test, CANNY-E, and RUAUMOKO. Chapter 6 137 This way, by reducing a and increasing p, the strength of the model structure increased, and, as a result, the jumps in the hysteresis loop and displacement time history vanished. The maximum values for the relative displacement at the cap beam level, the base shear, and the bent absolute acceleration are listed in Table 6.10. Table 6.10 Maximum Values from 40% Test, CANNY-E, and RUAUMOKO. Max. Relative Displacement (mm) Max. Base Shear (kN) Bent Max. Absolute Acceleration (g) Test 8.4 -9.1 77.2 -88.4 0.76 -0.83 CANNY-E 7.8 -9.1 70.6 -75.0 0.73 -0.69 RUAUMOKO 7.2 -9.1 73.2 -78.5 0.70 -0.75 6.5.2.3 Run 60% CANNY - The analysis was carried out by assuming that leff= 0.38 Isross, A = Agross for beam and column elements, and viscous damping = 5%. The first 10 seconds of the shake table recorded acceleration time history for the 60% run was used as input signals, and the hysteresis loop #15 for the beam elements and elastic columns was selected, as previous experience showed that this hysteresis model provided die best correlation. As a result, the analytical and experimental responses were found to be in agreement: the peak values were +8.4 mm and -10.6 mm for the displacement, and +73.3 kN and -76.7 kN for the base shear. To obtain a better correlation, die viscous damping was then reduced from 5% to 3%. From Figs. 6.16 to 6.18 it can be seen that the analytical response followed the same pattern as did die test results. The peak values were +8.9 mm and -11.6 mm for the displacement, and +74.0 kN and -77.7 kN for the base shear. In other words, the peak values for the displacement and base shear from analysis achieved at least 92% and 83% of their experimental counterparts, respectively. Table 6.11 summarizes the results. Chapter 6 138 Table 6.11 60% Run - CANNY Results from Different Runs compared to the Experiment. Relative Displacement at the Cap Beam Level (mm) Base Shear (kN) CANNY, Damping = 5% 8.4 -10.6 73.3 -76.7 CANNY, Damping = 3% 8.9 -11.6 74.0 -77.7 Test 9.7 -11.8 79.8 -93.5 The hysteresis loops plotted in Fig. 6.19 show that the discrepancy is increased compared to the 10% and 40% runs. Fig. 6.20 shows the hysteresis loops from CANNY and test results for smaller time intervals, where it can be seen that around the first peak of the excitation the analytical response revealed a flexural yielding (see Fig. 6.17(b)), whereas the experimental results showed evidence of pinching, probably due to shear effects. At the second peak of excitation, however, the specimen also showed some flexural yielding (see Fig. 6.20(d)). Stiffness degradation was more obvious from the CANNY results than than from the experimental results. Any speculation about strength deterioration is difficult unless the loops are carefully traced one after anotiier. In general, however, the analytical approach showed more flexural behavior and yielding than the physical model did. One reason is that shear phenomenon is too complicated to be accurately modelled, and although the CANNY program has the potential of predicting shear failure, this function has yet to be completed and the parameters of the hysteresis loop used in shear modeling need to be further adjusted. Next, the program was run using two different integration time steps, i.e., 0.0004166 sec. and 0.001 sec. It was found that although the peak values were identical for both integration time steps, the bigger time step caused a shift, in the displacement time history after the second excitation peak occured. This was probably as a result of accumulation error due to yielding in the elements of the analytical model. Chapter 6 139 Finally, shear deformation effects were examined. By ignoring shear defomiation we found that peak displacements increased by a small amount (+9.1 mm and -12.5 mm compared to +8.9 mm and -11.6 nun) and peak forces decreased marginally (+70 kN and -77.1 kN compared to +74 kN and -77.7 kN). RUAUMOKO - As in CANNY, we set Ieff= 0.38 Igross, A = Agross, and viscous damping = 5%. The first 10 seconds of the shake table recorded acceleration time history for the 60% run was used as the base excitation. Similar to previous runs, the plastic hinge length was assumed to be 0.2 m for all beam elements except elements #2 and #7, for which a length of 0.05 m was assumed. The hysteresis loops #2, #3, and #8 were tried first, but none of them could provide reasonable results. As in previous runs, we carried out the analysis using the hysteresis loop #4 for beams, with a = 0.1 and (3 = 0.6, and assumed that columns remained elastic during the analysis. The analysis resulted in a shift in the displacement time history, and although a new set of hysteresis parameter values, i.e., a = 0.0 and (3 = 0.6, were tried, the shift persistently existed. Even changing the magnitude of the reloading factor, NF, from 1 to 10, or reducing the integration time step to 0.0004166 sec. (to decrease the accumulation error) did not improve the model response. As the program is able to do iteration, this was tried too, yet again the results did not improve. In the next runs, the variation of response with the viscous damping was studied, and it was found that, the results were quite sensitive to the damping value. Using an 8% damping ratio caused a big jump and a shift in the displacement time history. Attempting to achieve a better result, we then lowered the damping ratio value down to 3%, which improved the results dramatically. Figs. 6.16 to 6.18 show strong correlations between the results obtained by RUAUMOKO with those given by CANNY and the 60% test. The peak values for displacement time history are +9.2 mm and -12.0 mm compared to +9.7 mm and -11.8 mm Chapter 6 140 derived from experimental results, indicating that the ratio between analytical and experimental results is around 95%. The peak values of base shear from analytical response are +83.4 kN and -92.4 kN, compared to +79.8 kN and -93.5 kN derived from experimental results: the program overestimated die positive peak value by 5% and underestimated the negative peak response by only 1%, which suggests an excellent correlation. Again, like CANNY, RUAUMOKO's analytical hysteresis loop produced more flexural yielding, probably due to existing problems in shear modeling. The stiffness degradation, however, is similar to the experimental response. Table 6.12 compares die maximum values from CANNY-E, RUAUMOKO, and test results. Table 6.12 Maximum Values from 60% Test, CANNY-E, and RUAUMOKO. Max. Relative Max. Base Shear Max. Absolute Acceleration Displacement (mm) (kN) (g) Test 9.7 -11.8 79.8 -93.5 0.80 -0.87 CANNY-E 8.9 -11.6 74.0 -77.7 0.75 -0.72 RUAUMOKO 9.2 -12.0 83.4 -92.4 0.92 -0.80 Chapter 6 141 E E, c CD E CO o w Q_ w b 12 -6 h - 12 Test (60%) Max. value = 9.7 mm. Min. value = -11.8 mm. 4 6 sec 10 Canny E E. c CD E CD O a CO 12 -12 Max. value Min. value = 8.9 mm -11.6 mm 4 6 sec 10 Ruaumoko E E, c CD E CD O JS a cn b Max. Value = 9.2 mm Min. value = -12.0 mm Fig. 6.16 Relative Displacement Time Histories at the Cap Beam Level: 60% Test, CANNY-E, and RUAUMOKO. Chapter 6 142 Test (60%) 150 100 5 0 0 - 50 - 1 0 0 - 1 5 0 4 6 sec Max. value =79.8 kN. Min. value = -93.5 kN. 10 Z 1 5 0 1 0 0 5 0 0 - 50 - 1 0 0 - 1 5 0 Canny 4 6 sec Max. Value = 74.0 kN Min. Value = -77.7 kN 10 Ruaumoko Max. value =83.4 kN Min. value =-92.4 kN Fig. 6.17 Base Shear Time Histories: 60% Test, CANNY-E, and RUAUMOKO. Chapter 6 143 Test (60%) c g to _C0 CD O C q CO o 3 1-5 r 1.0 0.5 0.0 -0 .5 -1 .0 -1 .5 1.5 1.0 0.5 0.0 -0 .5 -1 .0 -1 .5 sec Canny Max. value = 0 . 8 0 g Min. value = - 0 . 8 7 g 10 Max. value = 0 . 7 5 g Min. value = - 0 . 7 2 g 10 Ruaumoko c o to J) CD O o < Max. Value = 0 . 9 2 g Min. value = - 0 . 8 0 g Fig. 6.18 Absolute Acceleration Time Histories at the Cap Beam Level: 60% Test, CANNY-E, and RUAUMOKO. Chapter 6 144 Test (60%) 1 0 0 -i - 1 0 0 J Fig. 6.19 Base Shear vs. Relative Displacement at the Cap Beam Level: 60% Test, CANNY-E, and RUAUMOKO. Chapter 6 145 Fig. 6.20 Hysteresis Loops at Different Time Intervals: 60% Test and CANNY-E. CANNY-E Test 6.5.2.4 Run 80% CANNY-E - Again, the hysteresis loop #15 was used for beam elements, whde columns were assumed to remain elastic during the analysis. We also set Ieff= 0.33 Igross, A = Asross, and viscous damping = 5%. The integration time step was 0.0004166 sec. The first 10 seconds of the shake table recorded acceleration time history for the 80% run was used in the analysis as base excitation. Chapter 6 146 The analytical results showed that although the computed displacement and base shear time histories had the same shape as tire ones provided by the experiment, the base shear peak values were well underestimated. The computed peak values for displacement were +10.3 mm and -15.9 mm, compared to the experimental values of +11.2 mm and -16.0 mm, and the computed base shear peak values were +71.9 kN and -75.3 kN, compared to the test values of +81.7 kN and -102.8 kN. To improve the base shear results, again, the viscous damping ratio was reduced from 5% to 3%; the results did not improve much, however. Believing that reducing the strengtir deterioration could increase the base shear, we changed the hysteresis loop parameters, Xe and X^, which controlled the strength deterioration (see Chapter 4), from Xe = 1.0 and K = 1.0 to Xe = 0.0 and X» = 0.5, respectively. The program was run with these new values, and the peak base shear increased considerably. A run using X.e = 0.0 and Xu = 0.2 produced the best results: the computed peak values were +11.0 mm and -16.8 mm for displacement, and +73.9 kN and -83.9 kN for base shear, resulting in a maximum discrepancy of 5% for displacement and 18% for base shear. The final results are listed in Table 6.13, and the time histories for this final analysis are presented in Figs. 6.21 to 6.23. Table 6.13 80% Run - CANNY-E Results from Different Runs. Assumptions at Different Runs Relative Displacement at the Cap Beam Level (mm) Base Shear (kN) xe = l .o, K = l .o 10.3 -15.9 71.9 -75.3 Xe = 0.0, Xu = 0.2 11.0 -16.8 73.9 -83.9 Analytical base shear is plotted versus relative displacement in Fig. 6.24, which indicates a general agreement, except for the pinching behavior that is not seen in the analytical response. Also, The specimen reveals more shear behavior than flexural behavior but the analytical model indicates the opposite. As Chapter 6 147 discussed in Section 6.5.2.3, uncertainties in non-linear modeling and difficulties in shear modeling may be responsible for this discrepancy. Fig. 6.25 presents the results of a parametric study, which recognizes the fact diat the change of hysteresis parameters may help to improve the model response, but only marginally. We also found that ignoring shear deformation in the analysis had the same effect. It is worth noting that a small integration time step equal to 0.0004166 sec. was used in the analysis. RUAUMOKO - The analysis was started by assuming that Ieff= 0.33 Igross, A = Agross, viscous damping = 3%, and integration time step = 0.001 sec. The computed peak values from analysis were reasonable: + 11.7 mm and -15.8 mm for displacement, and +85.6 kN and -99.3 kN for base shear. There was a shift in the displacement time history, and the hysteresis loop was not in good agreement with the experimental response. In next stage of analysis, a very small integration time step equal to 0.0005 sec. was used. The peak values were unchanged but the unwanted shift in displacement was reduced, indicating diat a smaller time step for integration was necessary at this stage, because of yielding, in order to reduce accumulation error. In die final run, sensitivity of response to the unloading stiffness factor, oc, was studied. Raising the magnitude of a from 0.0 up to 3.0 did not increase pinching in the hysteresis loop significantly, although it did reduce the unwanted shift in the displacement time history. In this run, with a = 3.0, which was taken as the final run, the computed peak values were +11.9 mm and -17.3 mm for displacement, and +84.0 kN and -98.9 kN for base shear, respectively As a result, the maximum discrepancy was 8% for displacement and 4% for base shear. Table 6.14 summarizes the results, while measured and analytical responses are compared in Figs. 6.21 to 6.24. Chapter 6 148 Table 6.14 80% Run - RUAUMOKO Results from Different Runs. Assumptions at Different Runs Relative Displacement at the Cap Beam Level (mm) Base Shear (kN) a = 0.0 11.7 -15.8 85.6 -99.3 a = 3.0 11.9 -17.3 84.0 -98.9 a: unloading stiffness factor As with CANNY, the hysteresis loop did not predict the shear behavior of the specimen (see Fig. 6.24). Shear modeling and other uncertainties are responsible for this failing, as will be discussed later in this chapter. Visually, the stiffness degradation was similar to the one obtained from the test results. Table 6.15 compares the maximum values from the analytical and experimental approachs. Table 6.15 Maximum Values from 80% Test, CANNY-E, and RUAUMOKO. Max. Relative Max. Base Shear Max. Absolute Acceleration Displacement (mm) (kN) (g) Test 11.2 -16.0 81.7 -102.8 0.85 -0.96 CANNY-E 11.0 -16.8 73.9 -83.9 0.83 -0.72 RUAUMKO 11.9 -17.3 84.0 -98.9 1.0 -0.80 Chapter 6 149 Test 80% E E , c CD E CD o re Q. w b Max. value = 11.2 mm. Min. value = -16.0 mm. Canny E E , c CD E CD o re o_ co b Max. value = 11.0mm Min. value = -16.8 mm Ruaumoko CD E CD O re Q_ b Max. Value = 11.9 mm Min. value = -17.3 mm Fig. 6.21 Relative Displacement Time Histories at the Cap Beam Level: 80% Test, CANNY-E, and RUAUMOKO. Chapter 6 150 Test 80% z XL 120.0 60.0 0.0 -60.0 -120.0 Max. value =81.7 kN. Min. value = -102.8 kN. sec Canny 120.0 60.0 0.0 -60.0 -120.0 Max. Value = 73.9kN Min. Value = -83.9 kN 10 sec Ruaumoko 120.0 60.0 -60.0 -120.0 Max. value =84.0 kN. Min. value = -98.9 kN. Fig. 6.22 Base Shear Time Histories, 80% Test, CANNY-E, and RUAUMOKO. Chapter 6 151 Test 80% c g ro i_ a> CD o o < Max. value = 0.85g Min. value = -0.96g sec Canny c g i _CD CD O O < Max. value = 0.83g Min. value = -0.72g sec Ruaumoko c g i _CD CD O O < 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 Max. Value =1.0g Min. value =-0.8g 10 sec Fig. 6.23 Absolute Acceleration Time Histories at the Cap Beam Level: 80% Test, CANNY-E, and RUAUMOKO. Chapter 6 152 Test 80% 1 0 0 -I - 1 2 0 J Ruaumoko 1 2 0 - i - 1 2 0 J Fig. 6.24 Base Shear vs. Relative Displacement at the Cap Beam Level: 80% Test, CANNY-E, and RUAUMOKO. Chapter 6 153 120 n -120 J Fig. 6.25 Parametric Study of CANNY-E - 80% Run. X.e = 1 A.u - 1 A,e = 0 X,u = 0.2 6.5.2.5 Run 120% CANNY - As before, the hysteresis loop #15 was used for beam elements while columns were assumed to remain elastic throughout the analysis. In the analysis, we set Ieff = 0.25 lgross, A = Agross, damping ratio = 3%, and the integration time step = 0.0004166 sec. The first 10 seconds of the shake table recorded acceleration time history for the 120% run was used as base excitation. The output results from CANNY for the first run were not satisfactory. As mentioned earlier, only 25% of the gross moment of inertia was used in the analysis, and it is assumed that this might have caused the analytical model to soften excessively, so that when the elements started yielding, they collapsed and produced unreasonable numbers. It was therefore decided to do a parametric study to see if by changing the hysteresis parameters Xe and Xu, which control strength deterioration, the results could be improved. By Chapter 6 154 reducing their magnitudes from Xe = Xu = 1.0 to Xe = Xa = 0.2, satisfactory results were obtained: the computed peak values were +20.9 mm and -26.1 mm for displacement, and +77.4 kN and -95.9 kN for base shear force. This indicated that the analytical and experimental peak displacements and base shears were at most 21% and 14% apart, respectively. Measured and analytical time histories, compared in Figs. 6.26 to 6.28, show the time histories followed the same pattern in each method. In an attempt to get a better correlation between experimental and analytical results, the values of seven parameters of the hysteresis loop #15 were modified and die subsequent responses were examined. At this stage of excitation, the overall response was quite sensitive to some parameters, especially 6, which controlled the stiffness degradation. With 9 = 0, i.e., no stiffness degradation, unacceptable results were produced. Setting 9 = 5 provided peak values of +23.9 mm and -27.0 mm for displacement and +83.2 kN arid -96.4 kN for base shear, suggesting a better correlation for forces but not for displacements. The parameter sensitivity study results for 9 are summarized in Table 6.16. While there were many combinations of hysteretic parameters which could be used to provide better correlations between experimental and analytical responses, it was felt that there was little purpose in juggling too many parameters for the current study. The hysteresis loops plotted in Fig. 6.29 reveal the fact that wlule the experimental loop showed a combination of shear and flexural characteristics, die analytical loop only reflected a flexural behavior. This was due to a weakness in the shear modeling which existed in CANNY-E. Ignoring shear springs was also considered, which provided +23.8 mm and -31.2 mm for displacement peak values and +84.2 kN and -106.9 kN for base shear peak values. After 6 seconds, a major shift in Chapter 6 155 displacement was observed, which was not comparable to the test results. Obviously, including shear deformation provided better results. Table 6.16 120% Run - CANNY-E Results from Different Runs. Assumptions at Different Runs Relative Displacement at the Cap Beam Level (mm) Base Shear (kN) 0 = 2.0 11.7 -15.8 85.6 -99.3 0 = 5.0 11.9 -17.3 84.0 -98.9 0: stiffness degradation parameter. RUAUMOKO - As in CANNY, we set Ieff= 0.25 Igross, A = Agross, and damping ratio = 3%. The first 10 seconds of the shake table recorded acceleration time history for the 120% run was used as input signals. From the experience gained in the previous runs, we decided to use the hysteresis loop #4 (from RUAUMOKO's manual) to model the beam elements, while the columns were assumed to remain elastic during the excitation period. The program was run with the hysteresis parameters a = 0.3 and (3 = 0.6. The computed peak values for displacement were +28.5 mm and -26.2 mm (compared to +17.3 mm and -25.4 mm from the test results), while the computed peak values for the base shear were +88.2 kN and -92.6 kN (compared to +90.4 kN and -111.1 kN). Apparently, the displacement correlation between analysis and experiment was poor, especially in the positive direction. As a result, it was decided to try other possibilities in an attempt to achieve a better analytical response. The hysteresis loop #4, provides two options for unloading, referred to as "kkk" in the program's manual. The value of the "kkk" parameter was upgraded from kkk = 1 to kkk = 2, which resulted in a better correlation. Chapter 6 156 The new computed peak values were +19.4 mm and -26.8 mm for displacement, and +90.2 kN and -101.7 kN for base shear, amounting to a correlation of at least 89% and 91% for peak displacements and base shears, respectively (see Table 6.17). Figs. 6.26 to 6.28 show that the analytical time histories have a similar shape to the ones obtained experimentally. Table 6.17 120% Test - RUAUMOKO Results from Different Runs. Assumptions at Different Runs Relative Displacement at the Cap Beam Level (mm) Base Shear (kN) kkk = 1, a= 0.3, p= 0.6 28.5 -26.2 88.2 -92.6 kkk = 2, oc= 0.3, p= 0.6 19.4 -26.8 90.2 -101.7 kkk = unloading pattern, a = unloading stiffness, p = reloading stiffness. Comparing hysteresis curves from Fig. 6.29 indicated that the analytical results did only reflect the flexural characteristics of the experimental hysteresis loop, while the latter showed a combination of shear and flexural yielding. Table 6.18 has listed the maximum values obtained from the analytical and experimental approaches. Table 6.18 Maximum Values from 120% Test, CANNY-E, and RUAUMOKO. Max. Relative Max. Base Shear Max. Absolute Acceleration Displacement (mm) (kN) (g) Test 17.3 -25.4 90.4 -111.1 0.88 -1.19 Canny 20.8 -26.1 77.4 -95.9 0.95 -0.76 Ruaumoko 19.4 -26.8 90.2 -101.7 1.04 0.87 Chapter 6 157 Test 120% E E . c CD E CD o Q. y> Q Max. value Min. value 17.3 mm. -25.4 mm. sec Canny E E . c CD E CD O w Q_ CO b Max. value = 20.8 mm Min. value = -26.1 mm sec Ruaumoko E E , c CD E CD O « Q. b sec Max. Value =19.4mm Min. value =-26.8 mm Fig. 6.26 Relative Displacement Time Histories at the Cap Beam Level: 120% Test, CANNY-E, and RUAUMOKO. Chapter 6 158 Test 120% Max. value =90.4 kN. Min. value = -111.1 kN. sec Canny Max. Value = 77.4 kN Min. Value = -95.9 kN sec Ruaumoko 150 100 50 0 -50 - 100 - 1 5 0 Max. value =90.2 kN. Min. value = -101.7 kN. 4 6 sec 10 Fig. 6.27 Base Shear Time Histories, 120% Test, CANNY-E, and RUAUMOKO. Chapter 6 159 Test 120% c g jtD CD O 3 1.5 1.0 0.5 0.0 -0 .5 -1 .0 -1 .5 4 6 sec Max. value = 0.88g Min. value = -1.19g 10 Canny c g o 1.5 1.0 0.5 0.0 -0 .5 -1 .0 -1 .5 4 6 sec Max. value = 0.95g Min. value = -0.76g 10 Ruaumoko c o a> 0) o o < Max. Value =1.04g Min. value =-0.87g sec Fig. 6.28 Absolute Acceleration Time Histories at the Cap Beam Level: 120% Test, CANNY-E, and RUAUMOKO. Chapter 6 160 Test (120%) 1 2 0 -i - 1 2 0 J Fig. 6.29 Base Shear vs. Relative Displacement at the Cap Beam Level: 120% Test, CANNY-E, and RUAUMOKO. Chapter 6 161 6.5.2.6 Run 150% CANNY - As in previous runs, the hysteresis loop #15 was used for beam elements, with Ieff= 0.15 Isross, A = Agross, damping ratio = 3%, and integration time step = 0.0004166 sec. The first 10 seconds of the shake table recorded acceleration time history for the 150% run was used as base excitation in the analysis. Fig. 6.30 shows the displacement time histories, where it can be seen that the analytical displacement followed the measured displacement for about 7 seconds, after which an obvious shift in the analytical response occured. The computed peak values for displacement were found to be +34.9 mm and -37.5 mm, compared to +35.3 mm and -23.8 mm from measured response. Although the discrepancy in the positive direction was quite low (about 1%), in the negative direction, the correlation was poor, with a discrepancy of about 58%. Analytical response at this level of excitation was anticipated to be poor, however. It was also found that while the specimen suffered from both shear and flexural failures, the analytical model only experienced a flexural failure. On the other hand, the base shear and acceleration time histories, which are plotted in Figs. 6.31 and 6.32, showed that the analytical base shear closely follows the experimental base shear. The computed peak values for base shear were +89.5 kN and -93.9 kN, compared to +76.1 kN and -99.1 kN, and the correlation was much better than the displacement time histories: the maximum discrepancy in peak values was only 18%. Fig. 6.33 shows the hysteresis loops, in which the correlation, except for the flexural behavior, is generally poor. Chapter 6 162 RUAUMOKO - The analysis was carried out using the hysteresis loop #4 for beam elements. As in CANNY, leff= 0.15 Igross, A = Agross, damping ratio = 3%, and the integration time step was set to 0.001 sec. The first 10 seconds of the shake table recorded acceleration time history for the 150% run was used as base excitation in the analysis, and the hysteresis parameters for the hysteresis loop #4 were set to a = 0.3 and (3 = 0.6. The analytical and experimental responses are plotted in Figs. 6.30 to 6.33. Surprisingly, die displacement time history appeared to have the same shape as the experimental one for the first 7 seconds. The computed peak values for displacement were +30.6 mm and -36.6 mm, compared to +35.3 mm and -23.8 mm from test results, indicating a poor correlation, especially in die negative direction (die discrepancy was about 54%). Nevertheless, the analytical peak values from the two computer programs were in close agreement. As in analytical displacement results, die analytical base shear time histories closely followed die experimental values for the first 7 seconds of die excitation period. The computed peak values for base shear were +90.2 kN and -101.0 kN, compared to +76.1 kN and -99.1 kN from test results. The maximum discrepancy here was about 19%. The analytical and experimental hysteresis loops are compared in Fig. 6.33, which shows a good correlation between the two analytical loops, but a poor correlation between the analytical and die experimental loops. Table 6.19 Maximum Values from 150% Test, CANNY-E, and RUAUMOKO. Max. Relative Max. Base Max. Absolute Acceleration Displacement (mm) Shear (kN) (e) Test 35.3 -23.8 76.1 -99.1 0.80 -1.01 CANNY 34.9 -37.5 89.5 -93.9 0.95 -0.89 RUAUMOKO 30.6 -36.6 90.2 -101.0 1.02 -0.87 Chapter 6 163 Test (150%) Max. value = 35.3 mm. - 30 y 0 2 4 6 8 10 sec Canny Max. value = 34.9 mm Min. value = -37.5 mm 2 4 6 8 10 sec Ruaumoko Max. Value =30.6mm Min. value =-36.6 mm ro 0 2 4 6 8 10 sec Fig. 6.30 Relative Displacement Time Histories at the Cap Beam Level: 150% Test, CANNY-E, and RUAUMOKO. Chapter 6 164 Test (150%) 150 100 5 0 0 - 50 - 1 0 0 -150 Max. value =76.1 kN. Min. value = -99.1 kN. 10 sec Canny Max. Value = 89.5 kN Min. Value = -93.9 kN Ruaumoko 1 5 0 1 0 0 5 0 0 -50 - 100 -150 Max. value =90.2 kN. Min. value = -101.0 kN. 10 sec Fig. 6.31 Base Shear Time Histories, 150% Test, CANNY-E, and RUAUMOKO. Chapter 6 165 Test (150%) c o ro i _CD CD O O < Max. value = 0.8g Min. value = -1.01 g sec Canny c o ro aj CD o o < Max. value = 0.95g Min. value = -0.89g Ruaumoko c o ro a> CD O O < Max. Value =1.02g Min. value =-0.87g Fig. 6.32 Absolute Acceleration Time Histories at the Cap Beam Level: 150% Test, CANNY-E, and RUAUMOKO. Chapter 6 166 Test (150%) 120 -i -120 J Ruaumoko 120 - i -120 J Fig. 6.33 Base Shear vs. Relative Displacement at the Cap Beam Level: 150% Test, CANNY-E, and RUAUMOKO. C H A P T E R 7 CONCLUSIONS The purpose of this thesis was to assess two nonlinear structural analysis programs, CANNEY-E and RUAUMOKO, by performing a correlation study between an analytical model and shake table test results. The outcome of this assessment gives us a better insight into the nonlinear behavior of reinforced concrete structures under severe shaking conditions, as well as the ability of these computer programs to accurately predict such nonlinear behavior. A detailed study and interpretation of the recorded data was performed from a shake table test of a 0.27 scale model of the Oak Street Bridge bent at six different levels of excitation. The dynamic and material characteristics of the specimen through different stages of excitation were studied. Subsequently, an analytical model for the bent was proposed that consisted of 8 beam elements and 2 column elements. The linear and nonlinear responses of the analytical model subjected to monotonically increasing lateral load, as well as the shake table earthquake excitations, were investigated using the CANNEY-E and RUAUMOKO computer programs. Finally, the ability of the two computer programs to predict reliably the structural behavior of the reinforced concrete bridge bent was assessed by comparing analytical with experimental results. The following conclusions have be drawn from this process: 167 Chapter 7 168 7.1 Important Features of the Computer Programs: • It was fount! that CANNY-E is generally a more powerful nonlinear computer program than RUAUMOKO. For instance, in CANNEY-E we were allowed to define different springs for flexure, shear, and axial loads. This feature enabled us to further define any type of hysteretic law for different non-linear springs. RUAUMOKO (and DRAIN-2DX), in contrast, limited us to the use of only one type of hysteresis loop for a given element. CANNEY-E is thus the only program we tested that has tire potential to predict the shear failure value for a structural member. This feature remains only a potential, however. The specific version of CANNY-E we used did not have a hysteresis loop that adequately reflected tire shear distortions in the post-cracking and post-yielding phases, and therefore could not model the shear failure. • Although both the CANNEY-E and RUAUMOKO programs can model any type of regularly or irregularly shaped structure, CANNEY-E has the advantage diat it automatically computes tire fixed-end moment and the shear load applied by the distributed gravity load on a beam element regardless of the overall shape of the structure. • One main problem with CANNY-E-E is the accumulation error due to the way it corrects the equilibrium condition at the end of each integration time step. Specifically, whenever an element yields, the program corrects the equilibrium at the end of the integration time step by applying an impulsive load to the load factor for the subsequent time step. It was shown that the best way to avoid this type of error was to choose a very small time step, effectively minimizing the equilibrium corrections. • During each test run, it was found that RUAUMOKO is a well organized program and more user-friendly than CANNEY-E. It has more powerful graphics capabilities, and is more versatile in choosing the type of element models, damping, mass, etc. Since RUAUMOKO is able to do iterations, Chapter 7 169 the solution obtained was not very sensitive to the chosen integration time step. In terms of computation time, however, RUAUMOKO is almost 10 times slower than CANNY-E for the same model. • Neither CANNY-E nor RUAUMOKO was able to consider moment-shear interaction or shear failure demonstrated by the specimen during the tests. 7.2 Validity of the Analytical Results: • In spite of limitations such as the need to adjust the model's damping ratio and hysteresis parameters, both CANNY-E and RUAUMOKO were able to provide a good representation of the dynamic response of the specimen in terms of its fundamental frequency, relative displacement time histories at the cap beam level, base shear time histories, absolute acceleration time histories, and the overall hysteresis responses. It was found that in general predictive accuracy of the analytical models reduced as the excitation intensity and the system nonlinearities increased. The main sources of error in die nonlinear analyses were due to the estimation of damping and stiffness of the structure, the modeling of the elements, the global modeling, and the numerical methods applied to calculate the response. • The hysteresis loops from CANNY-E and RUAUMOKO provided almost the same hysteresis patterns, and were in good agreement with the test results in the sense that they reflected observed flexural behavior. However, the pinching behavior observed in experiment could not be reflected in either analytical approach. • A linear elastic analysis was conducted for 10% and 40% runs to ensure the similarity of the modeling in both programs and also to recognize how far the elastic analysis was from the real behavior, hi the elastic analysis, both programs provided exactly the same results, confirming the similarity of modeling Chapter 7 170 in both programs. In the 10% run, the base shear peak values from elastic analysis were overestimated by a maximum of 10%, while they were overestimated by more than 70% in the 40% run. This confirmed that for low amplitude excitation, elastic and inelastic analysis results are quite close, but that elastic analysis is increasingly unreliable at higher-amplitude excitations. • For the 40% run, the maximum displacements from the elastic analysis, unlike the shear forces, were close to the test results. This confirmed the "Equal Displacement" rule which is applied in seismic design force by the NBCC (National Building Code of Canada, 1995). • Although for high amplitude excitations the actual specimen became inelastic, the assumption that the columns remained elastic for the model used in the CANNY-E program yielded good analytical results that matched with the test results. 7.3 Comments on Analysis: • The push-over analysis results from both computer programs were in very good agreement with each other in terms of the sequences of hinges and tire collapse load. The collapse load obtained from the two computer programs differed by only 1%, while the displacement at collapse mechanism differed by 24%. The results from CANNY-E show more displacement, which was due to the different element models and strain-hardening values used in CANNY-E and RUAUMOKO. While in CANNY-E, each end of an element could be assigned different propoerties, in RUAUMOKO, only one type of property was allowed to be defined for each individual element. • The sensitivity analysis for the hysteresis parameter showed that the results from RUAUMOKO are in general quite sensitive to the stiffness of unloading and reloading. Chapter 7 171 • During the analyses, it was found that using the experimental damping ratio did not yield acceptable analytical results in both the CANNY-E and RUAUMOKO programs. This was due to the way damping was assessed from the experiment, in which it was assumed that viscous damping was the only damping source in the system. In reality, however, structural damping is due to both viscous and hysteretic effects, which cannot easily be separated from each other. • While there are many uncertainties in a nonlinear analysis, we showed that if the dynamic and material properties of a specimen are available, it is possible to obtain a reasonably good correlation. Without prior knowledge of the experimental response, however, it would not be possible for us to reliably predict the nonlinear dynamic response of a specific structure. 7.4 Recommendations for Future Work We suggest that the future studies seeking to continue the present work consider the following: • Performing seismic analysis of the 0.45 scale model and comparing the analytical response to the test results from the slow cyclic test. This is to check the consistency of the analysis using the computer programs, and also to see if the results obtained from the 0.27 scale model can be extended to another scaled model. • Using CANNY-E and RUAUMOKO to investigate the seismic response of rite bridge bent prototype. • Studying the damage indices from the two computer programs CANNY-E and RUAUMOKO, and comparing these to the damage observed during the test. Chapter 7 112 • Carrying out more systematic analyses to investigate the effects of different element models, interaction of shear-moment, and the parametric study of hysteresis modeling. BIBLIOGRAPHY ACI/ASCE Committee 426, 1978 "Suggested Revisions to Shear Provisions for Building Codes," American Concrete Institute, Farmington Hill, MI, pp. 88. Aktan, A. E., Bertero, V. V., Chowdhury, A. A., and Nagashima T., 1983 "Experimental and Analytical Predictions of the Mechanical Characteristics of a 1/5-Scale Model of a 7-Story RC Frame-Wall Building Structure," Report no. 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G., 1998 "Shake Table Test and Performance Evaluation of a Post-Seismic Repaired and Retrofitted Concrete Bridge Bent with Fiber Glass Wrapping," Proceedings of 6th East Asian-Pacific Conference on Structural Engineering and Construction, National Taiwan University, Taipei, Taiwan. Clough, R. W., Benuska, K. L., and Wilson, E. L., 1965 "Inelastic Earthquake Response of Tall Buildings," In Proceedings of the 3rd World Conference on Earthquake Engineering, National Committee on Earthquake Engineering, New Zealand, vol. 11. 173 Clough, R. W. and Penzien, J., 1993. Dynamics of Structures, Second Edition, McGraw-Hill, New York, NY. Crippen International Ltd., 1993. Two-Column Bent Test Program - Comparative Review of Piers of Five Major Bridges, Report no. Q 117/7.2.0, Vancouver, BC. Davey, E., 1996. Shake table Testing of an Oak Street Bridge Bent Model, M.A.Sc. Thesis, Department of Civil Engineering, University of British Columbia, Vancouver, BC. Dolan, J. D., 1989. 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Gibson, M. F., 1967. The Response of Nonlinear Multi-Story Structures Subjected to Earthquake Excitation. Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena, CA. Grapher, Ver. 1.2, 1993, Golden Software Inc., Golden, CO. Ha, K. H., 1993. CMAP Users Manual. Version 4.7 for IBM-PC and Compatible, Concordia University, Montreal, Quebec. Habibullah, A., 1992. ETABS - Three Dimensional Analysis of Building Systems User Manual. Horyna, T., 1995: Dynamic Analysis of Bridges with Laminated Wood Girders, M.A.Sc. Thesis, Department of Civil Engineering, University of British Columbia, Vancouver, BC. Inadui, J. A., and De La Llera, J. C , 1992 "Dynamic Analysis of Nonlinear Structures Using State-Space Fromulation and Partitioned Integration Schemes," Report no. UCBIEERC-92, Earthquake Engineering Research Center, College of Engineering, University of California, Berkely, CA. Kaanan, A. E., and Powell, G. H., 1973 "DRAIN-2D, A General Purpose Computer Program for Dynamic Analysis of Planar Structures," Report no. UCB/EERC-73/6, Earthquake Engineering Research Center, College of Engineering, University of California, Berkely, CA. 174 Kunnath, S. K., Reinliorn, A. M., and Lobo, R.F., 1992, "IDARC Version 3.0 - A Program for the Inelastic Damage Analysis of RC Structures," Technical Report no. NCEER-92-0022, National Center for Earthquake Engineering Research, State University of New York, Buffalo, NY. Li, K. N., 1995. CANNY-E and User's Manual - Nonlinear Dynamic Structural Analysis Computer Program Package, Canny Consultants PTE Ltd., Singapore. Little, J. N., and Shure, L., 1990. Signal Processing Toolbox for use with MATLAB, The Math Works Inc., Natick, MA. Naeim, F., and Anderson, J. C , 1993. Classification and Evaluation of Earthquake Records for Design, The 1993 NEHRP Professional Fellowship Report, Earthquake Engineering Research Institute, Oakland, CA. Park, Y. J., and Ang, A. H. S., 1985 "Mechanistic Seismic Damage Model for Reinforced Concrete," ASCE Journal of Structural Engineering, vol. 111, no. ST4, pp. 722-739. Pauley, T., and Priestly, M. J. N., 1992. Seismic Design of Reinforced Concrete and Masonry Buildings, A John Wiley Interscience Publication, John Wiley and Sons Inc., New York, NY. Prakash, V., Powell, G. H., and Campbell, S., 1993. DRAIN-2DX - Base Program Description and User Guide, Version 1.10, Earthquake Engineering Research Center, College of Engineering, University of California, Berkely, CA. Saatcioglu, M., 1991 "Modeling Hysteretic Force-Deformation Relationships for Reinforced Concrete Elements," in Earthquake Resistant Concrete Structures Inelastic Response and Design, ACI SP 127-5, American Concrete Institute, Farmington Hills, MI, pp. 153-198. Seethaler, M., 1995. Cyclic Response of Oak Street Bridge Bents, M.A.Sc. Thesis, Department of Civil Engineering, University of British Columbia, Vancouver, BC. Ventura, C , 1992. DFT123 - Discrete Fourier Transform Program User Manual, Department of Civil Engineering, University of British Columbia, Vancouver, BC. Williams, M. S., 1994. Inelastic Damage Analysis of the Oak Street and Queensborough Bridge Bents, Technical Report no. 94-03, Earthquake Engineering Research Facility, Department of Civil Engineering, University of British Columbia, Vancouver, BC. Wilson, E. L., and Habibullah, A., 1992. SAP90 Structural Analysis User Manual. 175 APPENDIX A 0.27 Scale Oak Street Bridge Bent Drawing 176 APPENDIX B Sample Input-Output Files for CANNY-E and R U A U M O K O 1996.25.6, CANNY-E input data - n o n l i n e a r a n a l y s i s a n a l y s i s of the specimen f o r the OAK STREET BRIDGE usi n g 8—beam model // a n a l y s i s assumptions and options t i t l e : Oak S t r e e t Bridge Specimen (0.27 i n sc a l e ) t i t l e : EQ. = 10% fo r c e u n i t = kN length u n i t = m time u n i t = sec model i n X - d i r e c t i o n g r a v i t y a c c e l e r a t i o n i s 9.805 ( d e f a u l t 9.8) output of p e r i o d time h i s t o r y output of o v e r a l l responses at f l o o r l e v e l s output of column response output of beam response // c o n t r o l data f o r dynamic response / ^ i n t e g r a t i o n 12—step i n the time i n t e r v a l of the input a c c e l e r a t i o n data i n t e g r a t i o n time i n t e r v a l 0.0004166 s t a r t time 5 end time 15 check peak displacement from "0.1 di v e r g e n t t o l e r a n c e 0.5 check displacement a t 2 — f l o o r l e v e l b i n a r y output of response r e s u l t s at every 12-step Newmark method, Beta-value 0.25 /*Wilson method, Theta-value 1.4 /* d e t e r m i n a t i o n of the damping c o e f f i c i e n t , am and ak: [C] = am[M]+ak[K] /* u s i n g the c i r c u l a r frequency of i — t h and j — t h mode: wi = 4.373, wj = 6.877 /* assuming the damping f a c t o r a c c o r d i n g . t o the mode equal to 5 percent: h i = 0. /* h i = am/(2wi) + ak*wi/2 /* hj = am/(2wj) + ak*wj/2 /* am=2*wi*wj*(hi*wj - hj*wi)/(wj*wj-wi*wi) /* ak=2*(hj*wj—hi*wi)/(wj*wi—wi*wi) /* f o r t h i s model: T l = 0.163 sec, wl = 38.54, w2 = i n f i n i t e /* assuming am/(2wl) = O . l h i , ak*wl/2 = 0.9hi /* h i = 0.02: am - 0.154, ak = 0.0009 damping c o e f f i c i e n t 0.154 p r o p o r t i o n a l to mass matrix [M] damping c o e f f i c i e n t 0.0009 p r o p o r t i o n a l to instantaneous s t i f f n e s s [K] damping c o e f f i c i e n t 0.0 p r o p o r t i o n a l to i n i t i a l s t i f f n e s s [K0] f a c t o r 9.8 to X-EQ f i l e =c:\canny\eqlO.dat f a c t o r 0.01 to Y-EQ f i l e = f a c t o r 0.01 to Z-EQ f i l e = // ========= node l o c a t i o n s ================== degrees of freedom: X - t r a n s l a t i o n , Z - t r a n s l a t i o n , X-Z r o t a t i o n node a t X5 Y l 3F /* i n e r t i a l o a d p o i n t (mass-block center) node at XI to X9 Y l , 2F node at X2 Y l IF node at X8 Y l IF \ 177 // ====== f l o o r l e v e l data ============= /* s e l f w e i g h t = 12.2+2*3=18.2 kN (beam+columns), Mass = 20 kips (88.95 kN) 3 F ( r i g i d f l o o r , above 2F) Z=2.1265, G(l.2655,0) W=88.95, L f = 1 2 F ( r i g i d f l o o r , above IF) Z=1.5105, G(l.2655,0) W=15.20, L f = 0 1F ( f o o t i n g f l o o r , p i n - r o l l e r support) Z= 0, G(l.2655,0), W=0.0, Lf = 0 // ========= frame l o c a t i o n s =========== frame XI :X0 = -0 . 8015 frame X2 :X0 0 frame X3 :X0 - 0. 2285 frame X4 :X0 = 0 . 7470 frame X5 :X0 = 1. 2655 frame X6 :X0 = 1. 7840 frame X7 :X0 = 2 . 3025 frame X8 :X0 = 2 . 531 frame X9 :X0 = 3. 3325 frame Y l :Y0 = 0 //==== amendment of node degrees of freedom node at X5 Y l 3F e l i m i n a t e a l l r o t a t i o n s node a t XI Y l 2F e l i m i n a t e a l l r o t a t i o n s node at X9 Y l 2F e l i m i n a t e a l l r o t a t i o n s node at X2 Y l IF e l i m i n a t e a l l components node at X8 Y l IF e l i m i n a t e a l l components /* I changed the r i g i d zone a c c o r d i n g to Japanese Code from 0.15 to 0.06m out frame Y l XI to X2 2F RU1 SU81 0 0 /* l e f t c a n t i l e v r beam out frame Y l X2 to X3 2F LU1 RU2 SU82 0 0 /* l e f t short-beam out frame Y l X3 to X4 2F LU2 RU3 SU83 0 0 out frame Y l X4 to X5 2F LU3 RU4 SU84 0 0 /* middle-span beam out frame Y l X5 to X6 2F LU4 RU3 SU84 0 0 out frame Y l X6 to X7 2F LU3 RU2 SU83 0 0 out frame Y l X7 to X8 2F LU2 RU1 SU82 0 0 /* r i g h t s h o r t beam out frame Y l X8 to X9 2F LU1 SU81 0 0 /* r i g h t c a n t i l e v e r // ======== column data ============= out X2 Y l IF to 2F TU5 SU85 AU95 0 0 out X8 Y l IF to 2F TU6 SU86 AU95 0 0 // ======== l i n k element data =============== out Y l X3-X5 2F-3F U99 /* assumed l a r g e s t i f f n e s s d i a g o n a l l i n k out Y l X5-X7 3F-2F U99 // ====== s t i f f n e s s and h y s t e r e s i s parameters /* f o r beams 1 So U l 15 2.74e+7 0.0008389 22.37 10.18 143.89 38.298 0.321 0.0827 0.0130 0.0084 0 2 U2 15 2.74e+7 0.0008389 20.32 8.00 126.86 26.27 0.2630 0.0544 0.0060 0.0094 0 2 U3 15 2.74e+7 0.0008389 13.31 12.08 59.09 50.231 0.1311 0.1108 0.0059 0.0061 0 U4 15 2.74e+7 0.0008389 8.03 18.54 26.185 103.99 0.0551 0.2318 0.0079 0.0083 0 2 U81 15 1.17e+7 0.1192 83.8 83.8 147.2 147.2 0.16 0.16 0.001 0.001 0.001 2 1 1 0 U82 15 1.17e+7 0.1192 66.0 66.0 150.8 150.8 0.16 0.16 0.001 0.001 0.001 2 1 1 0 U83 15 1.17e+7 0.1192 66.3 66.3 74.6 74.6 0.16 0.16 0.001 0.001 0.001 2 0 1 1 0 U84 15 1.17e+7 0.1192 66.0 66.0 88.5 88.5 0.16 0.16 0.001 0.001 0.001 2 0 1 1 0 / * f o r columns U5 2 2.74e+7 0.000476 9.88 22.53 48.75 62.88 0.1509 0.1543 0.0194 0.0191 0.0 1 U6 2 2.74e+7 0.000476 22.53 9.88 62.88 48.75 0.1543 0.1509 0.0191 0.0194 0.0 1 U85 2 1.17e+7 0.1069 62.7 62.7 88.6 97.2 0.16 0.16 0.001 0.001 0.0 1 1 U86 2 1.17e+7 0.1069 62.7 62.7 97.2 88.6 0.16 0.16 0.001 0.001 0.0 1 1 U95 1 2.74e+7 0.1069 / * f o r c o n n e c t i o n between mass -b lock and b r i d g e - b e n t U99 1 2e+8 0.001096 / * two s t e e l bars a x i a l s t i f f n e s s (A=2x548 mm2) //===== i n i t i a l l o a d da ta ============ / * beam s e l f w e i g h t /*beam frame Y l XI to X9 2F ( l o a d 2.86 2.86) / * mass b l o c k weight node XI Y l 2 F , Pz = 11.12 node X9 Y l 2 F , Pz = 11.12 node X3 Y l 2F , Pz = 22.24 node X5 Y l 2F , Pz = 22.24 node X7 Y l 2 F , Pz = 22.24 / * h a l f column s e l f weight /*node X2 Y l 2 F , Pz = 1.50 V*node X8 Y l 2F , Pz = 1.50 181 1996 .25 .6 , CANNY-E i n p u t da ta a n a l y s i s of the specimen f o r the OAK STREET BRIDGE u s i n g 8-beam model / / a n a l y s i s assumptions and o p t i o n s t i t l e : Oak S t r e e t B r i d g e Specimen (0.27 i n s c a l e ) t i t l e : EQ. =40% f o r c e u n i t = kN l e n g t h u n i t = m time u n i t = sec model i n X - d i r e c t i o n g r a v i t y a c c e l e r a t i o n i s 9.805 ( d e f a u l t 9.8) output o f p e r i o d t ime h i s t o r y output of o v e r a l l responses a t f l o o r l e v e l s output of column response output of beam response / / c o n t r o l da ta f o r dynamic response / ^ i n t e g r a t i o n 12-s tep i n the t ime i n t e r v a l o f the i n p u t a c c e l e r a t i o n da ta i n t e g r a t i o n t ime i n t e r v a l 0.0004166 s t a r t t ime 5 end t ime 15 check peak d i s p l a c e m e n t from 0.1 d i v e r g e n t t o l e r a n c e 0.5 check d i s p l a c e m e n t a t 2 - f l o o r l e v e l b i n a r y output of response r e s u l t s a t every 12—step Newmark method, B e t a - v a l u e 0.25 / * W i l s o n method, T h e t a - v a l u e 1.4 / * d e t e r m i n a t i o n o f the damping c o e f f i c i e n t , am and ak: [C] = am[M]+ak[K] / * u s i n g the c i r c u l a r frequency of i - t h and j - t h mode: wi = 4 .373 , wj - 6.877 / * assuming the damping f a c t o r a c c o r d i n g to the mode equa l t o 5 p e r c e n t : h i = 0 / * h i = am/(2wi) + ak*wi /2 / * h j = am/(2wj) + ak*wj/2 / * am=2*wi*wj*(hi*wj - h j * w i ) / ( w j * w j - w i * w i ) / * ak=2*(h j*wj -h i*wi ) / (wj*wi -wi*wi ) / * f o r t h i s model: T l = 0.1890 s e c , wl = 33 .24 , w2 = i n f i n i t e / * assuming am/(2wl) = O . l h i , ak*wl /2 = 0 . 9 h i / * h i = 0.05:^111 = 0.332, ak = 0.00271 damping c o e f f i c i e n t 0.332 p r o p o r t i o n a l to mass m a t r i x [M] damping c o e f f i c i e n t 0.00271 p r o p o r t i o n a l to i n s t a n t a n e o u s s t i f f n e s s [K] damping c o e f f i c i e n t 0.0 p r o p o r t i o n a l to i n i t i a l s t i f f n e s s [K0] f a c t o r 9.8 to X-EQ f i l e = c : \ c a n n y \ e q 4 0 . d a t f a c t o r 0.01 to Y—EQ f i l e = f a c t o r 0.01 to Z-EQ f i l e = / / ========= node l o c a t i o n s ================== degrees of freedom: X - t r a n s l a t i o n , Z - t r a n s l a t i o n , X - Z r o t a t i o n node a t X5 Y l 3F / * i n e r t i a l o a d p o i n t (mass -b lock c e n t e r ) node a t XI to X9 Y l , 2F node a t X2 Y l IF node a t X8 Y l IF • / / ====== f l o o r l e v e l da ta ============= 18X /* selfweight = 12.2+2*3=18.2 kN (beam+columns), Mass = 20 kips (88.95 kN) 3F(ri g i d f l o o r , above 2F) Z=2.1265, G(l.2655,0) W=88.95, 2F(ri g i d f l o o r , above IF) Z=1.5105, G(l.2655,0) W=15.20, lF(footing f l o o r , p i n - r o l l e r support) Z= 0, G(1.2655,0), // ========= frame locations =========== Lf = 1 Lf = 0 W=0.0, Lf = 0 frame frame frame frame frame frame frame frame frame frame XI :X0 X2:X0 X3 :X0 X4 :X0 X5 :X0 X6 :X0 X7 :X0 X8 :X0 X9 :X0 Y l : Y0 -0.8015 0 0.2285 0.7470 1.2655 1.7840 2 .3025 2.531 3 .3325 0 //==== amendment of node degrees of freedom ======= node at X5 Yl 3F eliminate a l l rotations node at XI Y l 2F eliminate a l l rotations node at X9 Yl. 2F eliminate a l l rotations node at X2 Yl IF eliminate a l l components node at X8 Yl IF eliminate a l l components // == beam data out frame Yl XI to X2 2F RU1 SU81 0 0 /* out frame Yl X2 to X3 2F LU2 RU2 SU82 0 0 . /* out frame Yl X3 to X4 2F LU2 RU3 SU83 0 0 out frame Yl X4 to X5 2F LU3 RU4 SU84 0 0 out frame Yl X5 to X6 2F LU4 RU3 SU84 0 0 out frame Yl X6 to X7 2F LU3 RU2 SU83 0 0 out frame Yl X7 to X8 2F LU2 RU2 SU82 0 0 /* out frame Yl X8 to X9 2F LU1 SU81 0 0 /* /* middle-span beam // ======== column data ==================== out X2 Yl IF to 2F TU5 SU85 AU95 . 0 0 out X8 Yl IF to 2F TU6 SU86 AU95 0 0 II out Yl out Yl ====== li n k element data =============== X3-X5 X5-X7 2F-3F U99 3F-2F U99 /* assumed large s t i f f n e s s diagonal l i n k s t i f f n e s s and hysteresis parameters /* for beams Ul 15 2.74e+7 0.00067112 22.37 10.18 143.89 38.298 0.321 0.0827 0.0130 0.0084 0 U2 15 2.74e+7 0.00067112 20.32 8.00 126.86 26.27 0.2630 0.0544 0.0060 0.0094 0 U3 15 2.74e+7 0.00067112 1.3.31 12.08 59.09 50.231 0.1311 0.1108 0.0059 0.0061 0 U4 15 2.74e+7 0.00067112 8.03 18.54 26.185 103.99 0.0551 0.2318 0.0079 0.0083 0 I S3 U81 15 1.17e+7 0.1192 83.8 83.8 147.2 147.2 0.16 0.16 0.001 0.001 0.001 2 1 1 0 U82 15 1.17e+7 0.1192 66.0 66.0 150.8 150.8 0.16 0.16 0.001 0.001 0.001 2 1 1 0 U83 15 1.17e+7 0.1192 66.3 66.3 74.6 74.6 0.16 0.16 0.001 0.001 0.001 2 0 1 1 0 U84 15 1.17e+7 0.1192 66.0 66.0 88.5 88.5 0.16 0.16 0.001 0.001 0.001 2 0 1 1 0 7* f o r columns U5 1 2.74e+7 0.0003811 U6 1 2.74e+7 0.0003811 U85 1 1.17e+7 0.1069 U86 1 1.17e+7 0.1069 U95 1 2.73e+7 0.1069 /* f o r co n n e c t i o n between mass—block and bridge-bent U99 1 2e+8 0.001096 /* two s t e e l bars a x i a l s t i f f n e s s (A=2x548 mm2) //===== i n i t i a l l o a d data ============ /* beam s e l f w e i g h t /*beam frame Y l XI to X9 2F (l o a d 2.86 2.86) /* mass b l o c k weight node XI Y l 2F, Pz = 11.12 node X9 Y l 2F, Pz = 11.12 node X3 Y l 2F, Pz = 22.24 node X5 Y l 2F, Pz = 22.24 node X7 Y l 2F, Pz = 22.24 /* h a l f column s e l f weight /*node X2 Y l 2F, Pz = 1.50 /*node X8 Y l 2F, Pz = 1.50 40% Run - Oak S t r e e t Bridge, 0.27 s c a l e , U n i t s are m, N, sec, damp=5% 2 0 1 0 1 0 0 0 0 12 12 10 1 0 0 9.81 0.332 0.00271 0.001 10.0 1 0 5 30 0 1 1 0.7 0.1 0 0 0 C o n t r o l Parameters Frame and T i m e — h i s t o r y Output and P l o t t i n g Optio I t e r a t i o n C o n t r o l NODES 1 -0.8015 1 .5105 0 0 0 0 0 0 2 0.0 1 .5105 0 0 0 1 0 0 3 0.2285 1 .5105 0 0 0 1 0 0 4 0.7470 1 .5105 0 0 0 1 0 0 5 1.2655 1 .5105 0 0 0 1 0 0 6 1.7840 1 .5105 0 0 0 1 0 0 7 2.3025 1 .5105 0 0 0 1 0 0 8 2.5310 1 .5105 0 0 0 1 0 0 9 3.3325 1 .5105 0 0 0 1 0 0 10 1.2655 2 . 1265 0 0 1 0 0 0 11 0.0 0 .0 1 1 1 0 0 0 12 2.5310 0 .0 1 1 1 0 0 0 ELEMENTS 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 11 2 10 9 12 8 11 10 3 10 12 10 7 10 PROPS FRAME 1 0 0 4 0 0 2.74el0 1.17el0 0.1192 0.0 0.00067112 0.0 0.0 0.0 0. 0. 0. 0.0107 0.0 0.05 10e8. 10e8. 38298 -143890 26270 -126860 0. 0.6 1 1 FRAME 1 0 0 4 0 0 2.74el0 1.17eI0 0.1192. 0.0 0.00067112 0. 0.0 0. 0. 0. 0. 0.0077 0.05 0.05 10e8. 10e8. 26270 -126860 26270 -126860 0. 0.6 1 1 FRAME 1 0 0 4 0 0 2.74el0 1.17el0 0 0. 0.0069 0.2 0.2 10e8. 10e8. 26270 -126860 50231 -59090 0. 0.6 1 1 FRAME 1 0 0 4 0 0 2.74el0 1.17el0 0 0. 0.0070 0.2 0.2 1192 0.0 0.00067112 0. 0. 0. 0. 0 1192 0.0 0.00067112 0. 0. 0. 0. 0. 10e8. 10e8, 0. 0.6 1 1 FRAME 50231 -59090 103990 -26185 ! beam ! Parameters ! E l a s t i c p r o p e r t i e s ! beam ! Parameters ! E l a s t i c P r o p e r t i e s ! beam ! Parameters ! E l a s t i c p r o p e r t i e s ! B i - l i n e a r & Hinge ! beam ! Parameters ! E l a s t i c P r o p e r t i e s ! B i - l i n e a r & Hinge ! beam 1£5 1 0 0 4 0 0 2 .74e l0 1 .17el0 0.1192 0.0 0.00067112 0. 0. 0. 0. 0. 0. 0.0070 0.2 0.2 10e8. 10e8. 103990 -26185 50231 -59090 0. 0.6 1 1 6 FRAME 1 0 0 4 0 0 2 .74e l0 1 .17el0 0.1192 0.0 0.00067112 0. 0. 0. 0. 0. 0. 0.0069 0.2 0.2 10ei3. 10e8. 50231 -59090 26270 -126860 0. 0.6 1 1 7 FRAME 1 0 0 4 0 0 2 .74e l0 1 .17el0 0.1192 0.0 0.00067112 0. 0.0 0. 0. 0 0. 0.0077 0.05 0.05 10e8. 10e8. 26270 -126860 26270 -126860 0. 0.6 1 1 8 FRAME 1 0 0 4 0 0 2 .74e l0 1 .17el0 0.1192 0.0 0.00067112 0.0 0.0 0.0 0. 0. 0.0107 0.05 0.0 10e8. 10e8. 26270 -126860 26270 -126860 0. 0.6 1 1 9 FRAME 2 1 0 0 0 0 2 .74e l0 1 .17el0 0.1069 0.0 0.0003811 0. 0. 0.0 0. 0. 10 FRAME 1 0 0 0 0 0 2 . 0 e l l 7 . 6 e l l 0.001096 0.0 0. 0. Parameters E l a s t i c P r o p e r t i e s B i - l i n e a r & Hinge beam Parameters E l a s t i c p r o p e r t i e s B i - l i n e a r & Hinge beam Parameters E l a s t i c P r o p e r t i e s ! beam ! Parameters ! E l a s t i c p r o p e r t i e s ! column ! E l a s t i c P r o p e r t i e s ! c o n n e c t i o n ! E l a s t i c P r o p e r t i e s WEIGHT 1 2 3 4 5 6 7 8 9 10 11 12 1074 . 2901. 898. 9 1318. 2801. 1318. 898.7 2901. 1074 . 88950 0.0 0 0.0 0 1074.9 2901.7 898.94 1318 .7 2801 1318 898.7 7 2901 9 1074 88950 .0 .0 LOADS 1 0 .0 -11120.0 0.0 3 0 .0 -22240.0 0.0 5 0 .0 -22240.0 0.0 7 0 . 0 -22240.0 0.0 9 0 .0 -11120.0 0.0 EQUAKE 3 1001 0.0 1.0 166 <CANNY> REPORT - 10% Run, Damping 5%, and Columns stay e l a s t i c Sat Sep 27 13:56:19 1997 Oak Street Bridge Specimen (0.27 i n scale) EQ. = 10% 2-dimensional dynamic analysis i n X-direction Unit system:kN,m,sec,rad. 1. (1) a) Bl : B2 : B3: B4 : B5 : B6: B7 : B8: b) CI: C2 : c) LI: L2: (2) a) B l : B2: B3: B4: B5: B6: B7 : B8: b) CI: C2 : d) LI: L2: MAXIMAL and MINIMAL RESPONSE Member Forces Beam (Ml)Ql,Qr(Mr)AxialF X1-X2 Y l 2F (0)11.120~11.120,-11.120~-11.120(8.913~8.913) X2-X3 Yl 2F (51.128~-18.364)-58.794~23.813,58.794~-23.813(40.037~-15.175) X3-X4 Yl 2F (40.096~-14.887)-69.552~2.724,69.552~-2.724(10.135~-21.581) X4-X5 Yl 2F (10.557~-18.401)-37.451~14.295,37.451~-14.295(-6.040~-20.430) X5-X6 Y l 2F (-5.960~-20.331)-12.310~37.292,12.310~-37.292(13.263~-18.799) X6-X7 Yl 2F (11.983~-22.175)-3.560*67.431,3.560~-67.431(42.218~-13.823) X7-X8 Y l 2F (42.192~-14.032)-23.687~63.738,23.687~-63.738(53.500~-16.748) X8-X9 Yl 2F (8.913~8.913)-11.120~-11.120,11.120~11.120(0) Column (Mj)[x:(Mb)Q(Mt)][y:(Mb)Q(Mt)]AxialF X2 Yl 1F-2F [x:(0)22.624~-22.733(34.338~-34.173)]81.492~6.457 X8 Y l 1F-2F [x:(0)26.463~-23.271(35.151~-39.972)]82.503~7.468 Link Element X3-X5 Y l 2F-3F 22.836~-23.447 X5-X7 Yl 3F-2F 23.447~-22.836 Member D u c t i l i t y and F i r s t Yielding Step Beam [YieldStep]leftR(, ShearD)(, AxialD), rightRfYieldStep] 0]0, S-0.0232~-0.0232, 0 . 0222~0.0222[0] 0]0.5084~-0.3117, SO.0986"-0.0400, 0.5149~-0.4068[0] 0]0.5272~-0.3914, SO.7330"-0.0231, 0.0280~-0.2745[0] 0]0.0291~-0.1940, S0.1813~-0.0692, -0.0156~-0.0689[0] 0]-0.0154~-0.0678, SO.0596~-0.1805, 0.0365~-0.2040[0] 0-J0.0283~-0-.2895, SO . 0301"—0-. 6209 , 0 . 4358~-0 . 3346 [ 0 ] 0]0.4428~-0.3457, SO.0397~-0.1069, 0.4344~-0.2559[0] 0]0.0222~0.0222, SO.0232~0.0232, 0[0] [YieldStep]baseR(, ShearD[YieldStep])(, AxialD), topR[YieldStep]-0]0, S0[0], AO, 0[0] _Q]0, S0[0], AO, 0[0] Link-Element [YieldStep]Ductility X3-X5 Y l 2F-3F [0]0 X5-X7 Yl 3F-2F [0]0 X1-X2"Y1 2F X2-X3 Y l 2F X3-X4 Y l 2F X4-X5 Y l 2F X5-X6 Yl 2F X6-X7 Yl 2F X7-X8 Y l 2F X8-X9 Y l 2F Column X2 Yl 1F-2F X8 Yl 1F-2F 2. FLOOR MAXIMUM RESPONSE (1) Maximum Story Shear UnderFloor Xmax Xmin Xymax 3F: 40.32 -39.27 0.00 0.00 2F: 47.27 -46.00 0.00 0.00 (2) Inter-Story Displacement (% /Hi) BetweenFloor Xmax Xmin Xymax 2F-3F: 0.0139 -0.0098 0.0000 0.0000 0.0000 (3) Maximum Displacement, Absolute Acceleration and Velocity at 2F Dx(0.00380402 ~ -0.00396231), Ax(4.3560 ~ -4.5032), Vx(0.1245 Xymin Ymax 0.00 0.00 Xymin Ymax Ymin 0.00 0.00 Ymin 0.0000 Yxmax 0.00 0.00 Yxmax 0.0000 Yxmin 0.00 0.00 Yxmin 0.0000 ' -0.1116) 187 <CANNY> REPORT - 10% Run, Damping 5%, and Columns follow Hysteresis #15. Sat Sep 27 15:01:38 1997 Oak Street Bridge Specimen (0.27 i n scale) EQ. = 10% 2—dimensional dynamic analysis i n X-direction Unit system:kN,m,sec,rad. 1. (1) a) BI: B2: B3 : B4 : B5: B6: B7 : B8: b) CI: C2 : c) LI: L2 : (2) a) BI: B2 : B3: B4 : B5: B6: B7 : B8: b) CI: C2 : d) LI: L2: MAXIMAL and MINIMAL RESPONSE Member Forces Beam (Ml)Ql,Qr(Mr)AxialF X1-X2 Yl 2F X2-X3 Yl 2F X3-X4 Yl 2F X4-X5 Yl 2F X5-X6 Y l 2F X6-X7 Yl 2F X7-X8 Y l 2F X8-X9 Yl 2F Column X2 Yl 1F-2F X8 Yl 1F-2F 0)11.120~11.120,-11.120~-ll.120(8.913"8.913) 34.966"-13.130)-51.967"-1.284,51.967"1.284(23.407"-14.543) 23.424"-14.426)-35.708"1.627,35.708~-l.627(4.909"-13.583) 5.575~-13.226)-29.743~4.394,29.743~-4.394(-7.991~-11.723) -8.047"-11.812)-6.455"27.761,6.455"-27.761(4.299"-13.253) 3.776"-13.950)-1.301~31.090,1.301"-31.090(19.896"-14.427) 19.799~-14.554)2.136"54.979,-2.136"-54.979(31.371~-14.066) 8.913"8.913)-11.120"-11.120,11.120"11.120(0) (Mj)[x:(Mb)Q(Mt)][y:(Mb)Q(Mt)]AxialF x:(0)15.501"-14.337(21.656"-23.414)]71.297"21.076 x:(0)11.768"-19.376(29.268"-17.776)]67.884~17.663 Link Element X3-X5 Y l 2F-3F 16.545~-14.440 X5-X7 Yl 3F-2F 14.440~-16.545 Member D u c t i l i t y and F i r s t Yielding Step Beam [YieldStep]leftR(, ShearD)(, AxialD), rightR[YieldStep] 0]0, S-0.0232~-0.0232, 0.0222"0.0222[0] 0]0.1537~-0.1309, SO.0872"0.0022, 0.2326"-0.3731[0] 0j0.2272~-0.3668, SO.3022~-0.0138, 0.0053~-0.0719[0] 0]0.0056"-0.0629, SO.1439~-0.0213, -0.0206~-0.0303[0] S0.0312~-0.1344, 0.0012~-0.0636[0] 0.0813, S0.0110~-0.2631, 0.217l~-0.3668[0] 0.3736, S-0.0036~-0.0922, 0.1257~-0.1632[0] 0]0.0222~0.0222, SO.0232~0.0232, 0[0] [YieldStep]baseR(, ShearD[YieldStep])(, AxialD) 0]0, S0.0690~-0.0515[0], AO, 0.3287~-0.0995[0] 0]0, S0.0423~-0.0863[0], AO, 0.2331~-0.2326[0] Link-Element [YieldStep]Ductility X3-X5 Yl 2F-3F [0]0 X5-X7 Yl 3F-2F [0]0 X1-X2 Y l 2F X2-X3 Yl 2F X3-X4 Yl 2F X4-X5 Yl 2F X5-X6 Y l 2F X6-X7 Yl 2F X7-X8 Y l 2F X8-X9 Yl 2F Column X2 Yl 1F-2F X8 Y l 1F-2F 0]-0 .0208~-0 .0305, 0]0.0011" 0J0.2143' topR[ YieldStep.] 2. FLOOR MAXIMUM RESPONSE (1) Maximum Story Shear UnderFloor Xmax Xmin Xymax 3F: 24.83 -28.45 0.00 0.00 2F: 27.27 -33.71 0.00 0.00 (2) Inter-Story Displacement (% /Hi) BetweenFloor Xmax Xmin Xymax 2F-3F: 0.0081 -0.0062 0.0000 Xymin Ymax 0.00 0.00 Xymin Ymax 0.0000 0.0000 Ymin 0.00 0.00 Ymin 0.0000 Yxmax 0.00 0.00 Yxmax 0.0000 Yxmin 0.00 0.00 Yxmin 0.0000 (3) Maximum Displacement, Absolute Acceleration and Ve l o c i t y at 2F Dx(0.00357006 -0.00574510), Ax(3.1590 -2.7518), Vx(0.1191 ~ -0.0998) 18g <CANNY> REPORT- 10% Final Run F r i Oct 3 15:28:22 1997 Oak Street Bridge Specimen (0.27 i n scale) EQ. =10% , 2-dimensional dynamic analysis i n X-direction Unit system:kN,m/sec,rad. 0)11.120~11.120,-11.120~-11.120(8.913~8.913) 56.935"-21.883)-67.710"24.536,67.710"-24.536(43.578' 43.683~-17.630)-62.386"10.745,62.386"-10.7 45(14.735' 15.076"-15.076)-39.331"18.669,39.331"-18.669(-3.697' -3.638"-15.926)-16.86 6"41.129,16.866"-41.12 9(16.018' 15.107"-17.479)-10.115"62.405,10.115~-62.405(46.533' 46.374"-17.302)-22.225"68.466,22.225"-68.466(60.881' 8.913~8.913)-11.120~-11.120,11.120~11.120(0) (Mj)[x:(Mb)Q(Mt)][y:(Mb)Q(Mt)]AxialF x:(0)24.254"-28.029(42.337"-36.636)]85.862"0.240 x:(0)30.744"-23.407(35.357"-4 6.439)]88.720"3 . 098 -17.627) -17.079) -15.966) -15.535) -17.204) -20.587) 1. MAXIMAL and MINIMAL RESPONSE (1) Member Forces a) Beam (Ml)Ql,Qr(Mr)AxialF B1:X1-X2 Y l 2F B2:X2-X3 Yl 2F B3:X3-X4 Yl 2F B4:X4-X5 Yl 2F B5:X5-X6 Y l 2F B6:X6-X7 Yl 2F B7:X7-X8 Y l 2F B8:X8-X9 Yl 2F b) Column C1:X2 Yl 1F-2F C2:X8 Y l 1F-2F c) Link Element L1:X3-X5 Y l 2F-3F 25.532~-27.281 L2:X5-X7 Yl 3F-2F 27.281~-25.532 (2) Member D u c t i l i t y and F i r s t Y ielding Step a) Beam [YieldStep]leftR(, ShearD)(, AxialD), rightRfYieldStep] B1:X1-X2 Y l 2F [0]0, S~0.0232~-0.0232, 0 . 0222~0.0222[0] B2:X2-X3 Yl 2F [0]0.4395~-0.4388, SO.1287~-0.0318, 0.3493~-0.5379[0] B3:X3-X4 Y l 2F [0]0.3539~-0.5381, SO.5279~-0.0909, 0.0667~-0.1605[0] B4:X4-X5 Yl 2F [0]0.0739~~0.1098, SO.1903~-0.0904, -0.0095~-0.0412[0] B5:X5-X6 Y l 2F [0]-0.0094~-0.0411, SO.0816"—0.1990, 0.0937~-0.1214[0] B6:X6-X7 Yl 2F [0j0.0745~-0.1706, SO.0856~-0.5281, 0.3077~-0.5153[0] B7:X7-X8 Y l 2F [0]0.3043~-0.5205, SO.024l~-0.1366, 0.4791~~0.4509[0] B8:X8-X9 Yl 2F [0]0.0222~0.0222, SO.0232~0.0232, 0[0] b) Column [YieldStep]baseR(, ShearDfYieldStep])(, AxialD), topR[YieldStep] C1:X2 Yl 1F-2F [0]0, S0[0], AO, 0[0] C2:X8 Y l 1F-2F [0]0, S0[0], AO, 0[0] d) Link-Element [YieldStep]Ductility L1:X3-X5 Yl 2F-3F [0]0 L2:X5-X7 Yl 3F-2F [0]0 2. FLOOR MAXIMUM RESPONSE (1) Maximum Story Shear UnderFloor Xmax Xmin Xymax 3F: 46.91 -43.90 0.00 0.00 2F: 55.00 -51.44 0.00 0.00 (2) Inter-Story Displacement (% /Hi) BetweenFloor Xmax Xmin Xymax 2F-3F: 0.0131 -0.0121 0.0000 0.0000 0.0000 (3) Maximum Displacement, Absolute Acceleration and Velocity at 2F Dx(0.00456399 ~ -0.00439323), Ax(4.8644 ~ -5.2273), Vx(0.1337 Xymin Ymax 0.00 0.00 Xymin Ymax Ymin 0.00 0.00 Ymin 0.0000 Yxmax 0.00 0.00 Yxmax 0.0000 Yxmin 0.00 0.00 Yxmin 0.0000 ' -0.1388) <CANNY> REPORT - 40% F i n a l Run F r i Oct 3 15:45:23 1997 Oak S t r e e t Bridge Specimen (0.27 i n s c a l e ) EQ. = 40% 2-dimensional dynamic a n a l y s i s i n X - d i r e c t i o n U n i t system:kN,m /sec /rad. 1. (1) a) BI: B2 : B3: B4 : B5 : B6: B7 : B8: b) CI: C2 : c) L I : L2 : (2) a) BI: B2 : B3 : B4 : B5: B6: B7 : B8: b) CI: C2 : d) L I : L2 : MAXIMAL and MINIMAL RESPONSE Member Forces Beam (M l ) Q l , Q r ( M r ) A x i a l F X1-X2 Y l 2F X2-X3 Y l 2F X3-X4 Y l 2F X4-X5 Y l 2F X5-X6 Y l 2F X6-X7 Y l 2F X7-X8 Y l 2F X8-X9 Y l 2F Column X2 Y l 1F-2F X8 Y l 1F-2F L i n k Element X3-X5 Y l 2F-3F 37.543~-34.984 X5-X7 Y l 3F-2F 34.984~-37.543 Member D u c t i l i t y and F i r s t Y i e l d i n g Step Beam [ Y i e l d S t e p ] l e f t R ( , ShearD)(, A x i a l D ) , r i g h t R [ Y i e l d S t e p ] 0)11.120"11.120,-11.120"-11.120(8.913"8.913) 80.538~-26.787)-92.162"21.295,92.162"-21.295(59.394"-22.053) 58.936~-22.661)-79.381~11.459,79.381~-11.459(24.304~-21.024) 25.074"-20.021)-49.064"26.377,49.064"-26.377(-0.365~-22.550) -1.149"-22.955)-27.957"48.552,27.957"-4 8.552(23.403"-21.249) 22.372"-21.900)-13.666"81.127,13.666"-81.127(57.942"-24.138) 58.490~-24.101)-23.697"97.269,23.697"-97.269(80.729"-29.537) 8.913~8.913)-11.120"-11.120,11.120"11.120(0) (Mj)[x:(Mb)Q(Mt)][y:(Mb)Q(Mt)]AxialF x:(0)29.790"-44.091(66.599~-44.997)]105.378"-12.270 x: (0)40. 77 0 "'-31.624(47. 769"-61. 583) ] 101. 230 "-16 . 4 18 X1-X2 Y l 2F X2-X3 Y l 2F X3-X4 Y l 2F X4-X5 Y l 2F X5-X6 Y l 2F X6-X7 Y l 2F X7-X8 Y l 2F X8-X9 Y l 2F Column X2 Y l 1F-2F X8 Y l 1F-2F 0]0, S-0.0232~-0.0232, 0.0222"0.0222[0] 7200].0.5157~-1.1600, SO . 3851~-0 . 0185 , 0 . 6382 ~-0 . 7746 [ 0 ] 0]0.6369~-0.8071, S41.4560~-0.0876, 0.2681~-0.2604[0] 0]0.2843~-0.2350, SO.2375~-0.1277, -0.0009' 0]-0.0030"-0.0971, SO.1353~-0.2350 0]0.2274~-0.2826, SO.1156"-56.2282, 0]0.6396~-0.8840, SO.0012~-0.4386, 0.5126~-3.2995[6935] 0]0.0222~0.0222, SO.0232~0.0232, 0[0] [YieldStep]baseR(, S h e a r D f Y i e l d S t e p ] ) ( , A x i a l D ) , topR[YieldStep] 0]0, S0[0], AO, 0[0] -0.0926[0] 0.2491~-0.2661[0] 0.6318~-0.8860[0] 0]0, S0[0], AO, 0[0] Link-Element [ Y i e l d S t e p ] D u c t i l i t y X3-X5 Y l 2F-3F [0]0 X5-X7 Y l 3F-2F [0]0 2. FLOOR MAXIMUM RESPONSE (1) Maximum St o r y Shear UnderFloor Xmax Xmin Xymax 3F: 60.15 -64.56 0.00 0.00 2F: 70.56 -75.72 0.00 0.00 (2) I n t e r — S t o r y Displacement (% /Hi) BetweenFloor Xmax Xmin Xymax 2F-3F: 0.0222 -0.0149 0.0000 0.0000 0.0000 (3) Maximum Displacement, Absolute A c c e l e r a t i o n and V e l o c i t y at 2F Dx(0.00780650 ~ -0.00914272), Ax(7.2243 ~ -6.8181), Vx(0.2691 Xymin Ymax 0.00 0.00 Xymin Ymax Ymin 0.00 0.00 Ymin 0.0000 Yxmax 0.00 0.00 Yxmax 0.0000 Yxmin 0.00 0.00 Yxmin 0.0000 ' -0.2245) <CANNY> REPORT 60% F i n a l Run F r i Oct 3 16:02:53 1997 Oak S t r e e t Bridge Specimen (0.27 i n s c a l e ) EQ. = 60% 2-dimensional dynamic a n a l y s i s i n X - d i r e c t i o n U n i t system:kN,m,sec,rad. 1. MAXIMAL and MINIMAL RESPONSE (1) Member Forces a) Beam (Ml)Q1,Qr(Mr)AxialF B1:X1-X2 Y l 2F (0)11.120~11.120,-11.120~-11.120(8.913~8.913) B2:X2-X3 Y l 2F (85.604~-28.979)-92.133~25.694,92.133~-25.694(65.502~-23.089) B3:X3-X4 Y l 2F (65.454~-23.774)-84.858~15.718,84.858~-15.718(27.028~-18.992) B4:X4-X5 Y l 2F (27.944~-16.899)-51.375~28.779,51.375~-28.779(1.306~-15.985) B5:X5-X6 Y l 2F (0.817~-15.926)-30.309~50.529,30.309~-50.529(26.637~-17.102) B6:X6-X7 Y l 2F (25.656~-19.244)-19.428~80.158,19.428~-80.158(59.775~-26.091) B7:X7-X8 Y l 2F (60.314~-26.172)-24.806~100.474,24.806~-100.474(83.278~-31.924) B8:X8-X9 Y l 2F (8.913~8.913)-11.120~-l1.120,11.120~11.120(0) b) Column (Mj)[x:(Mb)Q(Mt)][y:(Mb)Q(Mt)]AxialF Cl:-X2 Y l 1F-2F [x:(0)30.117~-46.276(69.899~-45.491)]106.965~-15.142 C2:X8 Y l 1F-2F [x:(0)44.018~-31.407(47.440~-66.488)]104.102~-18.005 c) L i n k Element L1:X3-X5 Y l 2F-3F 38.529~-36.748 L2.-X5-X7 Y l 3F-2F 36 . 748~-38 . 529 (2) Member D u c t i l i t y and F i r s t Y i e l d i n g Step a) Beam [ Y i e l d S t e p ] l e f t R ( , ShearD)(, A x i a l D ) , r i g h t R f Y i e l d S t e p ] B1:X1-X2 Y l 2F [0]0, S-0.0232"-0.0232, 0.0222~0.0222[0] B2:X2-X3 Y l 2F [8777]0.5739~-3.5663, SO.3848~-0.0268, 0.6010~-0.8299[0] B3:X3-X4 Y l 2F [0]0.6052~-0.8666, S87.8055~-0.1231, 0.3254~-0.2089[0] B4:X4-X5 Y l 2F [0]0.3446~-0.6836, SO.2486~-0.1393, 0.0039~-0.0413[0] B5:X5-X6 Y l 2F [0]0.0024~-0.0411, SO.1467~-0.2445, 0.317l~-0.1611[0] B6:X6-X7 Y l 2F [0]0.2965~-0.2153, SO.1644~-87.7981, 0.6253~-0.9904[0] B7:X7-X8 Y l 2F [0]0.5836~-0.9948, S-0.0023~-0.4723, 0.5025~-5.7614[8517] B8:X8-X9 Y l 2F [0 ] 0.0222~0.0222, SO.0232~0.0232, 0[0] b) Column [YieldStep]baseR(, S h e a r D f Y i e l d S t e p ] ) ( , A x i a l D ) , t o p R [ Y i e l d S t e p ] C1:X2 Y l 1F-2F [0]0, SO[0], AO, 0[0] C2:X8 Y l 1F-2F [0]0, S0[0], AO, 0[0] d) Link-Element [ Y i e l d S t e p J D u c t i l i t y L1:X3-X5 Y l 2F-3F [0]0 L2:X5-X7 Y l 3F-2F [0]0 2. FLOOR MAXIMUM RESPONSE (1) Maximum Story Shear UnderFloor Xmax Xmin Xymax 3F: 63.19 -66.25 0.00 0.00 2F: 74.13 -77.68 0.00 0.00 (2) I n t e r - S t o r y Displacement (% /Hi) BetweenFloor Xmax Xmin Xymin Ymax 0.00 0.00 Xymin Ymax Ymin 0.00 0.00 Yxmax 0.00 0.00 Xymax Ymin Yxmax 2F-3F: 0.0337 -0.0172 0.0000 0.0000 0.0000 0.0000 0.0000 (3) Maximum Displacement, Absolute A c c e l e r a t i o n and V e l o c i t y at 2F Dx(0.00894175 ~ -0.01155788), Ax(7.3871 ~ -7.0719), Vx(0.3269 Yxmin 0.00 0.00 Yxmin 0.0000 ' -0.2456) 191 <CANNY> REPORT- 80% F i n a l run F r i Oct 17 12:22:50 1997 Oak S t r e e t Bridge Specimen (0.27 i n s c a l e ) EQ. = 80% 2—dimensional dynamic a n a l y s i s i n X - d i r e c t i o n U n i t system:kN,m,sec,rad. 1. MAXIMAL and MINIMAL RESPONSE (1) Member Forces a) Beam ( M l ) Q l , Q r ( M r ) A x i a l F B1:X1-X2 Y l 2F (0)11.120~11.120,-11.120~-11.120(8.913~8.913) B2:X2-X3 Y l 2F (98.806~-30.642)-96.871~23.239,96.871~-23.239(77.345~-25.305) B3:X3-X4 Y l 2F (77.592~-25.188)-87.462~20.617,87.462~-20.617(32.041~-16.758) B4:X4-X5 Y l 2F (35.443~-l6.662)-61.865~27.991,61.865~-27.991(4.578~-18.436) B5:X5-X6 Y l 2F (5.281~-17.432)-27.570~49.264,27.570~-49.264(28.057~-19.402) B6:X6-X7 Y l 2F (2 6.964~-19.689)-13.648~81.765,13.648~-81.765(60.027~-2 3.672) B7:X7-X8 Y l 2F (60.010~-23.793)-38.818~90.085,38.818~-90.085(79.239~-28.713) B8:X8-X9 Y l 2F (8.913~8.913)-l1.120~-ll.120,11.120~11.120(0) b) Column (Mj)[x:(Mb)Q(Mt)][y:(Mb)Q(Mt)]AxialF C1:X2 Y l 1F-2F [x:(0)29.463~-47.268(71.398~-44.504)]104.661~-14.975 C2:X8 Y l 1F-2F [x:(0)44.458~-29.724(44.898~-67.154)]103.935~-15.701 c) L i n k Element L1:X3-X5 Y l 2F-3F 37.024~-36.653 L2:X5-X7 Y l 3F-2F 36.653~-37.024 (2) Member D u c t i l i t y and F i r s t Y i e l d i n g Step a) Beam [ Y i e l d S t e p ] l e f t R ( , ShearD)(, A x i a l D ) , r i g h t R [ Y i e l d S t e p ] B1:X1-X2 Y l 2F [0]0, S-0.0232~-0.0232, 0 . 0222~0.0222[0] B2:X2-X3 Y l 2F [4334]0.3625~-7.6517, SO.4345~0.0010, 0.7239~-0.9484[0] B3:X3-X4 Y l 2F [0]0.7359~-0.9422, S109.8370~-0.1745, 0.4309~-0.5678[0] B4:X4-X5 Y l 2F [0]0.5024~-0.5118, SO.2994~-0.1355, 0.0136~-0.0476[0] B5:X5-X6 Y l 2F [0]0.0156~-0.0450, SO.1334~-0.2384, 0.3470~-0.6759[0] B6:X6-X7 Y l 2F [0]0.3240~-0.2266, SO.1155~-76.0705, 0.6598~-0.8611[0] B7:X7-X8 Y l 2F [0]0.6535~-0.8676, SO.0252~-0.3633, 0.4693~-8.8303[5046] B8:X8-X9 Y l 2F [0]0.0222~0.0222, SO.0232~0.0232, 0[0] b) Column [Y i e l d S t e p ] b a s e R ( , S h e a r D [ Y i e l d S t e p ] ) ( , A x i a l D ) , t o p R [ Y i e l d S t e p ] C1:X2 Y l 1F-2F [0]0, S0[0], AO, 0[0] C2:X8 Y l 1F-2F [0]0, S0[0], AO, 0[0] d) Link-Element [ Y i e l d S t e p ] D u c t i l i t y L1:X3-X5 Y l 2F-3F [0]0 L2:X5-X7 Y l 3F-2F [0]0 2. FLOOR MAXIMUM RESPONSE (1) Maximum St o r y Shear UriderFloor Xmax Xmin Xymax 3F: 63.02 -63.66 0.00 0.00 2F: 73.92 -74.88 0.00 0.00 (2) I n t e r - S t o r y Displacement (% /Hi) BetweenFloor Xmax Xmin Xymax 2F-3F: 0.0524 -0.0210 0.0000 0.0000 0.0000 (3) Maximum Displacement, Absolute A c c e l e r a t i o n and V e l o c i t y at 2F Dx(0.01103544 ~ -0.01719763), Ax(7.2537 ~ -7.0599), Vx(0.3264 Xymin Ymax 0.00 0 .00 Xymin Ymax Ymin 0.00 0.00 Ymin 0.0000 Yxmax 0.00 0.00 Yxmax 0.0000 Yxmin 0.00 0.00 Yxmin 0.0000 ' -0.3076) <CANNY> REPORT- 120% F i n a l Run F r i Oct 3 16:14:16 1997 Oak S t r e e t Bridge Specimen (0.27 i n s c a l e ) EQ. = 80% 2-dimensional dynamic a n a l y s i s i n X - d i r e c t i o n U n i t system:kN,m,sec,rad. 1. (1) a) B l : B2: B3: B4: B5 : B6: B7 : B8: b) CI: C2 : c) L I : L2 : (2) a) B l : B2 : B3: B4: B5: B6: B7 : B8: b) CI: C2 : d) L I : L2 : MAXIMAL and MINIMAL RESPONSE Member Forces Beam (M l ) Q l , Q r ( M r ) A x i a l F X1-X2 Y l 2F X2-X3 Y l 2F X3-X4 Y l 2F X4-X5 Y l 2F X5-X6 Y l 2F X6-X7 Y l 2F X7-X8 Y l 2F X8-X9 Y l 2F Column X2 Y l 1F-2F X8 Y l 1F-2F 0)11.120"11.12 0,-11.120"-11.120 (8 .913"8.913) 98.806"-30.642)-96.871"23.239,96.871"-23.239(77.345"-25.305) 77.592"-25.188)-87.462"20.617,87.462"-20.617(32.041"-16.758) 35.443"-16.662)-61.865"27.991,61.865"-27.991(4.578"-18.436) 5.2 81~-17.432)-27.57 0"49.264,27.570"-49.264(28.057"-19.402) 26.964"-19.689)-13.648"81.765,13.648~-81.765(60.027"-23.672) 60.010"-23.793)-38.818"90.085,38.818"-90.085(79.239"-2 8.713) 8.913~8.913)-11.120"-11.120,11.120"11.120(0) (Mj)[x:(Mb)Q(Mt)][y:(Mb)Q(Mt)]AxialF x:(0)29.463~-47.268(71.398"-44.504)]104.661"-14.975 x:(0)44.458"-29.724(44.898"-67.154)]103.935"-15.701 Lin k Element X3-X5 Y l 2F-3F 37.024~-36.653 X5-X7 Y l 3F-2F 36.653~-37.024 Member D u c t i l i t y and F i r s t Y i e l d i n g Step Beam [ Y i e l d S t e p ] l e f t R ( , ShearD)(, A x i a l D ) , r i g h t R [ Y i e l d S t e p ] X1-X2 Y l 2F [0]0, S-0.0232~-0.0232, 0.0222"0.0222[0] X2-X3 Y l 2F [4334]0.3625~-7.6517, SO.4345~0.0010, 0.7239~-0.9484[0] X3-X4 Y l 2F [0]0.7359~-0.9422, S109.8370~-0.1745, 0.4309~-0.5678[0] X4-X5 Y l 2F [0]0.5024~-0.5118, S0.2994~-0.1355, 0.0136~-0.0476[0] X5-X6 Y l 2F [0]0.0156~-0.0450, SO.1334~-0.2384, 0.3470~-0.6759[0] X6-X7 Y l 2F [0]0.3240~-0.2266, SO.1155"-76.0705, 0.6598~-0.8611[0] X7-X8 Y l 2F [0]0.6535~-0.8676, SO.0252~-0.3633, 0.4693~-8.8303[5046] X8-X9 Y l 2F [0]0.0222~0.0222, SO.0232~0.0232, 0[0] Column [YieldStep]baseR(, S h e a r D f Y i e l d S t e p ] ) ( , A x i a l D ) , topR[YieldStep]. X2 Y l 1F-2F [0]0, S0[0], AO, 0[0] X8 Y l 1F-2F [0]0, S0[0], AO, 0[0] Link-Element [ Y i e l d S t e p J D u c t i l i t y X3-X5 Y l 2F-3F [0]0 X5-X7 Y l 3F-2F [0]0 2. FLOOR MAXIMUM RESPONSE (1) Maximum St o r y Shear UnderFloor Xmax Xmin Xymax 3F: 63.02 -63.66 0.00 0.00 2F: 73.92 -74.88 0.00 0.00 (2) I n t e r - S t o r y Displacement (% /Hi) BetweenFloor Xmax Xmin Xymax 2F-3F: 0.0524 -0.0210 0.0000 0.0000 0.0000 (3) Maximum Displacement, Absolute A c c e l e r a t i o n and V e l o c i t y a t 2F Dx(0.01103544 ~ -0.01719763), Ax(7.2537 ~ -7.0599), Vx(0.3264 Xymin Ymax 0.00 0.00 Xymin Ymax Ymin 1.00 J.00 Ymin 0.0000 Yxmax 0.00 0.00 Yxmax 0.0000 Yxmin 0.00 0.00 Yxmin 0.0000 " -0.3076) 193 <CANNY> REPORT - 150% Run Sat Sep 27 12:21:22 1997 Oak S t r e e t Bridge Specimen (0.27 i n s c a l e ) EQ. = 150% 2-dimensional dynamic a n a l y s i s i n X - d i r e c t i o n U n i t system:kN,m,sec,rad. 1. (1) a) BI: B2: B3: B4 : B5 : B6 : B7 : B8: b) CI: C2 : c) L I : L2 : (2) a) BI: B2 : B3: B4 : B5 : B6: B7 : B8: b) CI: C2: d) L I : L2 : MAXIMAL and MINIMAL RESPONSE Member Forces Beam (M l ) Q l , Q r ( M r ) A x i a l F X1-X2 Y l 2F X2-X3 Y l 2F X3-X4 Y l 2F X4-X5 Y l 2F X5-X6 Y l 2F X6-X7 Y l 2F X7-X8 Y l 2F X8-X9 Y l 2F Column X2 Y l 1F-2F X8 Y l 1F-2F 0)11.120"11.120,-11.120"-11.120(8.913~8.913) 109.762~-36.683)-117.384~39.064,117.384~-39.064(83.463~-27.490) 82.942~-26.974)-80.791~25.367,80.791~-25.367(40.913~-14.316) 41.215"-13.371)-69.505"36.773,69.505"-36.773(9.650"-18.838) 9.255"-19.019)-39.557"57.657,39.557"-57.657(36.533"-18.950) 54.743"-14.790)-32.585"87.139,32.585"-87.139(83.228"-29.197) 73.630"-29.212)-39.235"113.865,39.235"-113.865(99.039"-38.194) 8.913~8.913)-11.120~-11.12 0,11.120"11.120(0) (Mj)[x:(Mb)Q(Mt)][y:(Mb)Q(Mt)]AxialF x:(0)33.759"-60.081(90.753"-50.993)]119.84 9"-27.417 x:(0)55.7 09"-33.776(51.018"-84.149)]116.377"-30.890 L i n k Element X3-X5 Y l 2F-3F 46.258"-44.214 X5-X7 Y l 3F-2F 44.214"-46.258 Member D u c t i l i t y and F i r s t Y i e l d i n g Step Beam [ Y i e l d S t e p ] l e f t R ( , ShearD)(, A x i a l D ) , r i g h t R [ Y i e l d S t e p ] X1-X2 Y l 2F [0]0, S-0.0232~-0.0232, 0.0222~0.0222[0] X2-X3 Y l 2F [5835]0.7272~-6.2474, SO.6496~-0.0122, 0.5337~-l.3774[5912] X3-X4 Y l 2F [5928]0.5319~-1.2179, S53.3870~-0.0670, 0.6175~-0.3130[0] X4-X5 Y l 2F [0]0.6239~-0.3603, SO.4254~-0.1780, 0.1109~-0.0512[0] X5-X6 Y l 2F [0]0.0896~-0.0532, SO.1914~-0.2790, 0.5254~-0.5685[0] X6-X7 Y l 2F [0]0.9253~-0.3370, SO.2757~-341.4685, 0.4101"-1.9056[5428] X7-X8 Y l 2F [5427]0.4292~-1.9102, SO.0374~-0.6127, 0.4986~-6.7053[5391] X8-X9 Y l 2F [0]0.0222~0.0222, SO.0232~0.0232, 0[0] Column [YieldStep]baseR(, S h e a r D [ Y i e l d S t e p ] ) ( , A x i a l D ) , topR[YieldStep] X2 Y l 1F-2F [0]0, S0[0], AO, 0[0] X8 Y l 1F-2F [0]0, S0[0], AO., 0[0] Link-Element [ Y i e l d S t e p ] D u c t i l i t y X3-X5 Y l 2F-3F [0]0 X5-X7 Y l 3F-2F [0]0 2. FLOOR MAXIMUM RESPONSE (1) Maximum St o r y Shear UnderFloor Xmax Xmin Xymax 3F: 76.03 -79.54 0.00 0.00 2F: 89.47 -93.86 0.00 0.00 (2) I n t e r — S t o r y Displacement (% /Hi) BetweenFloor Xmax Xmin Xymax 2F-3F: 0.1596 -0.1828 0.0000 0.0000 0.0000 (3) Maximum Displacement, Absolute A c c e l e r a t i o n and V e l o c i t y at 2F Dx(0.03488984 ~ -0.03754637), Ax(9.2978 " -8.7216), Vx(0.7293 Xymin Ymax 0.00 0.00 Xymin Ymax Ymin 0.00 0.00 Ymin 0.0000 Yxmax 0.00 0.00 Yxmax 0.0000 Yxmin 0.00 0.00 Yxmin 0.0000 ' -0.5756) RUAUMOKO -10% RUN MEMBERS POSITIVE ENVELOPE MEMBER ACTION TIME 1 Moment-1 5 .047E-12 2 .471 1 Moment—2 -8 .913E+03 1 .488 1 Shear-1 -1 .112E+04 1 .493 1 Shear-2 -1 .112E+04 1 .493 2 Moment—1 3 .282E+04 2 .510 2 Moment—2 2 . 523E+04 1 .442 2 Shear-1 7 .677E+04 1 .537 2 Shear-2 7 . 677E+04 1 .537 3 Moment-1 2 .523E+04 1 .442 3 Moment-2 1 .699E+04 1 .443 3 Shear-1 4 .467E+04 2 .423 3 Shear-2 4 .467E+04 2 .423 4 Moment—1 1 •699E+04 1 .443 4 Moment—2 8 .947E+03 1 .445 4 Shear-1 4 .467E+04 2 .423 4 Shear-2 4 .467E+04 2 .423 5 Moment—1 8 .947E+03 1 .445 5 Moment—2 1 .660E+04 1 .369 5 Shear-1 2 .307E+04 2 .413 5 Shear-2 2 .307E+04 2 .413 6 Moment-1 1 .660E+04 1 .369 6 Moment—2 2 .497E+04 1 .531 6 Shear-1 2 .307E+04 2 .413 6 Shear-2 2 .307E+04 2 .413 7 Moment—1 2 .497E+04 1 .531 7 Moment—2 3 •314E+04 7 . 167 7 Shear-1 1 .294E+05 9 .693 7 Shear-2 1 •294E+05 9 .693 8 Moment-1 -8 .913E+03 1 .534 8 Moment—2 8 .169E-12 7 . 121 8 Shear-1 1 . 112E+04 2 .434 8 Shear-2 1 . 112E+04 2 .434 9 Force-Ax 1 .058E+04 2 .510 9 Moment—2 3 .597E+04 2 .510 9 Shear-1 2 .381E+04 2 .510 9 Shear-2 2 .381E+04 2 .510 10 Force-Ax 4 .235E+03 2 .419 10 Moment-2 6 •561E+04 2 .513 10 Shear-1 4 .343E+04 2 .513 10 Shear-2 4 .343E+04 2 .513 11 Force—Ax 3 .695E+04 2 .517 12 Force—Ax 3 •221E+04 2 .425 NEGATIVE ENVELOPE ACTION TIME 7 •980E-12 7 .612 8 .913E+03 2 .488 1 . 112E+04 5 .720 1 .112E+04 5 .720 6 . 168E+04 2 .421 5 . 553E+04 7 . 168 1 .176E+05 9 .600 1 .176E+05 9 .600 4 .932E+04 7 . 168 2 .782E+04 7 . 167 2 .745E+04 2 .505 2 .745E+04 2 .505 2 .782E+04 7 . 167 7 .202E+03 7 .268 2 .745E+04 2 .505 2 .745E+04 2 .505 7 .202E+03 7 .268 3 .004E+04 2 .508 4 .889E+04 2 .515 4 . 889E+04 2 .515 3 .004E+04 2 .508 5 .474E+04 2 .513 4 .889E+04 2 .515 4 .889E+04 2 .515 5 .822E+04 7 .275 6 .909E+04 2 .513 7 •570E+04 1 .448 7 .570E+04 1 .448 8 .913E+03 7 .015 5 .09 3E-12 2 .473 1 .112E+04 .000 1 .112E+04 .000 9 .230E+04 2 .421 5 .813E+04 7 . 168 •3 .848E+04 7 . 168 3 .848E+04 7 . 168 •9 •770E+04 2 .513 3 . 603E+04 1 .547 •2 .385E+04 1 .547 •2 •385E+04 1 .547 -3 .141E+04 2 .411 •3 .468E+04 2 .503 /75 RUAUMOKO - 4 0% Run MEMBERS POSITIVE ENVELOPE MBER ACTION TIME 1 Moment—1 1 .774E-11 7 .731 1 Moment-2 -7 .559E+03 7 .733 1 Shear-1 -9 .431E+03 7 .733 1 Shear-2 -9 .431E+03 7 . 733 2 Moment—1 3 .533E+04 2 .996 2 Moment-2 3 .035E+04 7 .756 2 Shear-1 2 .800E+05 8 .414 2 Shear-2 2 .800E+05 8 .414 3 Moment-1 2 .533E+04 2 .574 3 Moment—2 1 .711E+04 2 .574 3 Shear-1 5 •439E+04 2 .897 3 Shear-2 5 .439E+04 2 . 897 4 Moment-1 . 1 .711E+04 2 .574 4 Moment-2 8 .677E+03 2 .574 4 Shear-1 5 .451E+04 2 .897 4 Shear-2 5 .451E+04 2 . 897 5 Moment—1 8 .677E+03 2 .574 5 Moment-2 1 .514E+04 2 .684 5 Shear-1 3 .254E+04 2 .897 5 Shear-2 3 .254E+04 2 .897 6 Moment-1 1 .514E+04 2 .684 6 Moment—2 2 .445E+04 2 .684 6 Shear-1 3 .263E+04 2 .897 6 Shear-2 3 .263E+04 2 .897 7 Moment-1 2 .796E+04 7 .646 7 Moment-2 3 .819E+04 2 .897 7 Shear-1 9 .153E+04 2 .897 7 Shear-2 9 .153E+04 2 .897 8 Moment—1 -7 .165E+03 7 .609 8 Moment-2 2 .303E-11 2 .863 8 Shear-1 1 .356E+04 2 .858 8 Shear-2 1 .356E+04 2 .858 9 Force-Ax 1 .617E+04 2 .998 9 Moment-2 3 .614E+04 2 .574 9 Shear-1 2 .393E+04 2 .574 9 Shear-2 2 .393E+04 2 .574 10 Force-Ax 2 .322E+04 2 .898 10 Moment-2 7 .658E+04 2 .995 10 Shear-1 5 .070E+04 2 .995 10 Shear-2 5 .070E+04 2 .995 11 Force—Ax 3 .951E+04 3 .002 12 Force—Ax 4 .615E+04 2 .899 NEGATIVE ENVELOPE ACTION TIME 1 .197E-11 2 .881 1 .050E+04 7 .728 1 .310E+04 7 .728 1 .310E+04 7 .728 9 •314E+04 8 .411 6 . 945E+04 2 .898 9 .177E+04 2 .996 9 .177E+04 2 .996 6 .858E+04 2 .898 4 .039E+04 2 .898 2 .954E+04 2 .996 2 .954E+04 2 .996 4 .039E+04 2 .898 1 .213E+04 2 .898 2 .950E+04 2 .996 2 . 950E+04 2 .996 1 .213E+04 2 .898 3 .805E+04 2 .997 5 .159E+04 2 .995 5 .159E+04 2 .995 3 .805E+04 2 .997 6 .475E+04. 2 .996 5 .152E+04 2 .995 5 .152E+04 2 .995 6 .545E+04 2 .997 8 .441E+04 2 .995 1 .743E+05 9 .877 1 .743E+05 9 .877 1 . 08.7E+04 2 .858 1 .879E-11 2 .944 8 .940E+03 7 .609 8 .940E+03 7 .609 1 .049E+05 2 .889 8 .074E+04 2 .897 5 . 345E+04 2 .897 5 .345E+04 2 .897 1 .040E+05 2 .989 3 .801E+04 2 .897 2 .516E+04 2 .897 2 .516E+04 2 .897 3 .873E+04 2 .874 3 .879E+04 2 .983 196 

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