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A o.¢|.a TC (silthsvtrI’lV-31 a2 . .c . n ‘ \l . u u. . ¢ . ‘ ‘ c . mm.m".. ‘ . . .l‘ IIIII . , mum J III I ll llllllil .lllll llllllilll H 3 1293 00612 2596 LIBRARY Michigan State University This is to certify that the dissertation entitled Nonlinear Seismic Analysis of Steel Arch Bridges presented by Chung-Ming Lee has been accepted towards fulfillment of the requirements for Ph. D. degree in Civil Mneering 19/17/wa Major professor Date May 1,1999 MSU is an Affirmative Anion/Equal Opportunity Institution 0- 12771 PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE mar—’1 ’1 MSU Is An Afllnnelive Action/Ema! Opportunity |nstitution __‘ “4L..- ‘- ;;"1"'9 NONLINEAR SEISMIC ANALISIS OF STEEL ARCH BRIDGES BY Chung-Hing Lee A DISSERIAIION Sub-itted to Michigan State University in partial fullfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil and Environmental Engineering 1990 211637 (005 ABSTRACT NONLINEAR SEISMIC ANALYSIS OF STEEL ARCH BRIDGES BY Chung-Hing Lee This study presents a method for the nonlinear seismic analysis of steel arch bridges. The effects of either geometric or material nonlinearity have been taken into account. The effects of such nonlinearities enter in the analysis through the computation of the "resistance" of the arch ribs. The elasto-plastic resistance of a curved beam element has been derived using the plastic potential theory as applied to stress resultants. For geometric nonlinearity, a twelve degrees of freedom incremental stiffness matrix was also derived. A computer program was prepared for the implementation of the time-history analysis. Three bridge models: Medium Span Bridge, Short Span Bridge and Long Span Bridge, based on three prototype bridges, Cold Springs Canyon Bridge (700 ft), South Street Bridge (193 ft) and New River Gorge Bridge (1700 ft), respectively, were used to obtain the numerical results. Three-dimensional models were employed to consider the nonlinear inelastic effects. Results for nonlinear elastic solution were based on two-dimensional models. The ground motion used was the artificially generated motion CIT-A2 with different amplification factors applied to induce nonlinear effects. From time histories, it is seen that even for the nonlinear cases the responses were generally dominated by the "fundamental modes" (either in-plane or out-of-plane) except at those points where the fundamental modal response is small. The history curves of nonlinear elastic responses exhibited different periods of vibration. In general, the dominant period increased by 5% to 10% from that of the linear solution. For the maximum force responses involving material nonlinearity, plasticity limited the magnitude of the internal force response to that as defined by the yield function. It follows that if plastic deformations are allowed, the design forces may be reduced from that which would be required if the design is to be done on a linearly elastic basis. Time histories of work and energy distribution indicated that the dissipated damping energy (for a 1.5% damping ratio) is over 70% of the work done for all models. When the damping ratio was varied from 0.25% to 5%, the percentage of damping energy to work done changed from 50% to 96%. These observations emphasize the importance of damping in the response. The writer wishes to express his gratitude to his academic advisor, Dr. Robert K. Wen, Professor of Civil and Environmental Engineering, Michigan State University, for his supervision and encouragement and also to members of the writer's guidance committee, Dr. George E. Mase, Dr. David H. Yen and Dr. Parviz Soroushian for their help. Appreciation is extended to the Division of Engineering Research and the Department of Civil and Environmental Engineering at Michigan State University as well as the National Science Foundation for their generous support. The writer also owes his appreciation to his parents, Pi-Yao Lee and Chang S. J. Lee, for their financial support. Special thanks is also due his wife, Huang Shin-Hwei Lee, for her patience and spiritual support. iv TABLE OF CONTENTS Page LIST OF TABLES .................................................... vii LIST OF FIGURES ................................................... ‘viii CHAPTER I: INTRODUCTION ........................................ l 1.1 GENERAL ................................................. 1 1.2 OBJECT AND SCOPE ........................................ 2 1.3 PREVIOUS STUDIES ........................................ 6 1.4 NOTATION ................................................ 10 CHAPTER II ANALYSIS AND METHOD OF SOLUTION ..................... 16 2.1 GENERAL ................................................. 16 2.2 EQUATIONS OF MOTION ..................................... 17 2.3 METHOD OF SOLUTION ...................................... 18 2.4 USE OF CONSTRAINTS AMONG NODAL DEGREES OF FREEDOM ....... 22 2.5 RESISTANCE OF STRUCTURAL ELEMENTS ....................... 24 2.5.1 General .......................................... 24 2.5.2 Linear Resistance of Curved Beam Elements ........ 25 2.5.3 Nonlinear Elastic Resistance of Curved Beam Elements .......................... 26 2.5.4 Elasto-Plastic Resistance of Curved Beam Elements .......................... 27 2.5.4.1 General ................................. 27 2.5.4.2 Plastic Potential Function .............. 28 2.5.4.3 Plastic Deformation of Member Ends ...... 29 2.5.4.4 Tangent Stiffness Matrix of Member ...... 31 2.5.4.5 Elastic Return .......................... 31 2.5.4.6 Incremental Resistance and Deformations of Elasto-Plastic Element ............... 32 2.6 PLASTIC WORK DENSITIES AND DUCTILITY FACTORS ............ 36 2.6.1 Plastic Werk Densities ........................... 36 2.6.2 Ductility Factors ................................ 38 2.7 WORK AND ENERGY DISTRIBUTION CHECK ...................... 39 CHAPTER III: APPLICATIONS AND NUMERICAL RESULTS .................. 50 3.1 GENERAL ................................................. 50 3.2 COMPUTER PROGRAM ........................................ 50 3.3 MODELS OF BRIDGE AND GROUND MOTION USED ................. 51 3.4 MATERIAL NONLINEAR PROBLEMS ............................. 53 MW um WW 3 3 .9. .4. E:::§§- GUI 7 Typical Displacement Time Histories ............... 3.4.2.1 Medium Span Bridge (MSB) .................. .2 Short Span Bridge (SSB) ................... 3 Long Span Bridge (LSB) .................... Displacements ............................. Force Time Histories ...................... Medium Span Bridge (MSB) .................. Short Span Bridge (SSB) ................... Long Span Bridge (LSB) .................... Forces and Force Reduction Factors ........ Typical Work and Energy Distribution Time Histories .................................... Variation of Energy Distribution with Different Damping Ratio ...................... Inelastic Responses Versus Linear Response Factor ............................ "E3 #b-fifl 52 fl WNHH 3.5 GEOMETRIC NONLINEAR PROBLEMS ............................ 3.5.1 General ........................................... 3.5.2 Typical Displacement Time Histories ............... WU Uh) 3. CHAPTER IV: 4. 4. 5. MM UIUI l-‘u GUI p. O D H 7 3.5.2.1 Medium Span Bridge (MSB) .................. .2 Short Span Bridge (SSB) ................... .3 Long Span Bridge (LSB) .................... Displacements ............................. Force Time Histories ...................... Medium Span Bridge (MSB) .................. Short Span Bridge (SSB) ................... Long Span Bridge (LSB) .................... Forces .................................... Work and Energy Distribution Time Histories .................................... Instability Effects ............................... woo UIUI gNN so” bbcfifl UMP." Ewan» EMU!“ SUMMARY AND CONCLUSION ............................. 1 SUMMARY ................................................. 2 CONCLUSION .............................................. LIST OF REFERENCES ................................................ APPENDIX: PROPERTIES OF BRIDGE MODELS ............................ vi Page 53 53 54 55 56 56 58 58 59 59 61 62 64 64 66 66 66 66 67 68 68 69 69 70 70 71 Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table 3-1. 3-2. 3-3. 3-4. 3-5. 3-6. 3-7. 3-8. 3-9. 3-10. 3-11. 3-12. 3-13. 3-14. 3-15. LIST OF TABLES Natural Periods of Vibration' ......................... Elastic and Elastic and Elastic and Elastic and Elastic and Elastic and Inelastic Maximum Inelastic Maximum Inelastic Maximum Inelastic Maximum Inelastic Maximum Inelastic Maximum Maximum Responses for Various of Damping Ratio ..................................... Linear and Nonlinear Elastic Maximum Displacements of MSB ......................... Linear and Nonlinear Elastic Maximum Displacements of 888 ......................... Linear and Nonlinear Elastic Maximum Displacements of LSB ......................... Linear and Nonlinear Elastic Maximum End Forces of MSB Linear and Nonlinear Elastic Maximum End Forces of $88 Linear and Nonlinear Elastic Maximum End Forces of LSB Maximum Displacements for Different Values of Initial Load vii Displacements Displacements Displacements End Forces of End Forces of End Forces of Values of MSB ... of SSB ... of LSB ... 73 74 75 76 77 77 78 79 80 80 81 82 82 82 83 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 2-1. 2-2. 2-3. 2-4. 2-5. 2-6. 2-7. 2-8. 2-9. 2-10. 2-11. 3-1. 3-2. 3-3. 3-4. 3-5. 3-6. 3-6. 3-7. LIST OF FIGURES Arch Bridge Considered .............................. Nodal Coordinates for Deck and Arch Ribs Nodes ...... Constraint for Infinitely Large Axial Stiffnesses Treatment of "Rigid Column" in Mixed Nodal Coordinates system ............................ Typical Curved Beam Element ........................ Curved Beam Element End Displacements ............... Spherical Yield Surface ............................. Member End Forces Corresponding to Displacement Changes ............................. Preliminary Definitions and Basic Operations ........ Definition of Cross-Section ......................... Typical Member End Force Path For An Elasto-Plastic Element ....................... Medium Span Bridge Model (A ”True" Three-Dimensional Model) ......... I ......... Short Span Bridge Model (A ”True" Three-Dimensional Model) .................. Long Span Bridge Model (A "One-Plane" Model) ........ Medium Span Bridge Medal (Two-Dimensional Model) In-Plane Mode Shapes for MSB ........................ Out-of-Plane Mode Shapes for MSB and 888 ............ (Continued) Out-of-Plane Mode Shapes for MSB and $88 ..................................... In-Plane Mode Shapes for $88 ........................ viii 42 42 43 43 44 45 45 46 47 48 49 84 85 86 87 88 89 90 91 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 3-8. 3-9. 3-10. 3-11. 3-12. 3-13. 3-14. 3-15. 3-16. 3-17. 3-18. 3-19. 3-20. 3-21. 3-22. 3-23. 3-24. 3-25. 3-26. 3-27. In-Plane Mode Shapes for LSB ........................ 92 Out-of—Plane Mode Shapes for LSB .................... 93 In-Plane Mede Shapes for Two-Dimensional MSB Model ....................... 94 CIT-A2 Ground Mbtion ................................ 95 Time Intervals with Inelastic Response (MSB) ........ 96 Time Intervals with Inelastic Response (SSB) ........ 97 Time Intervals with Inelastic Response (LSB) ........ 98 Node 5 Horizontal (X) Displacement Time History for MSB ................................ 99 Node l7 Horizontal (X) Displacement Time History for MSB ................................ 100 Node 31 Horizontal (X) Displacement Time History for MSB ................................ 101 Node 5 Vertical (Y) Displacement Time History for MSB ................................ 102 Node 17 Vertical (Y) Displacement Time History for MSB ................................ 103 Node 31 Vertical (Y) Displacement Time History for MSB ................................ 104 Node 5 Lateral (2) Displacement Time History for MSB ................................ 105 Node l7 Lateral (2) Displacement Time History for MSB ................................ 106 Node 31 Lateral (2) Displacement Time History for MSB ................................ 107 Node ll Horizontal (X) Displacement Time History for SSB ................................ 108 Node 13 Horizontal (X) Displacement Time History for SSB ................................ 109 Node 23 Horizontal (X) Displacement Time History for SSB ................................ 110 Node 11 Vertical (Y) Displacement Time History for SSB ................................ 111 ix Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 3-28. 3-29. 3-30. 3-31. 3-32. 3-33. 3-34. 3-35. 3-36. 3-37. 3-38. 3-39. 3-40. 3-41. 3-42. 3-43. 3-44. 3-45. Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time 13 Vertical (Y) Displacement History for SSB ................................ 23 Vertical (Y) Displacement History for SSB ................................ 11 Lateral (2) Displacement History for SSB ................................ l3 Lateral (2) Displacement History for SSB ................................ 23 Lateral (2) Displacement History for SSB ................................ 12 Horizontal (K) Displacement History for LSB ................................ 18 Horizontal (X) Displacement History for LSB ................................ 24 Horizontal (X) Displacement History for LSB ................................ 12 Vertical (Y) Displacement History for LSB ................................ 18 Vertical History for (Y) Displacement LSB OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 24 Vertical History for (Y) Displacement LSB ................................ 12 Lateral (2) Displacement History for LSB ................................ 18 Lateral (2) Displacement History for LSB ................................ 24 Lateral (2) Displacement History for LSB ................................ Member 1 End J (Node 5) Axial Force Time History for MSB ................................ Member 16 End I (Node 31) Axial Force Time History for MSB ................................ Member 1 End J (Node 5) In-Plane Bending Time History for MSB ................................ Member 16 End I (Node 31) In-Plane Bending Time History for MSB ................................ 3 112 113 114 115 116 117 118 119 120 123 124 125 126 127 128 129 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 3-46. 3-47. 3-48. 3-49. 3-50. 3—51. 3-52. 3-53. 3-54. 3-55. 3-56. 3-57. 3-58. 3-59. 3-60. 3-61. 3-62. 3-63. Member 1 End J (Node 5) Out-of-Plane Bending Time History for MSB ................................ Member 16 End I (Node 31) Out-of-Plane Bending Time History for MSB ................................ Member 9 End J (Node 11) Axial Force Time History for SSB ................................ Member 13 End I (Node 23) Axial Force Time History for SSB ................................ Member 9 End J (Node ll) In-Plane Bending Time History for SSB ................................ Member 13 End I (Node 23) In-Plane Bending Time History for SSB ................................ Member 9 End J (Node ll) Out-of-Plane Bending Time History for SSB ................................ Member 13 End I (Node 23) Out-of-Plane Bending Time History for SSB ................................ Member 5 End I (Node 12) Axial Force Time History for LSB ................................ Member 10 End J (Node 24) Axial Force Time History for LSB ................................ Member 5 End I (Node 12) In-Plane Bendingv Time History for LSB ................................ Member 10 End J (Node 24) In-Plane Bending Time History for LSB ................................ Member 5 End I (Node 12) Out-of-Plane Bending Time History for LSB ................................ Member 10 End J (Node 24) Out-of-Plane Bending Time History for LSB ................................ Member 4 End I (Node 10) Out-of-Plane Bending Time History for LSB ................................ Member 5 End J (Node 14) Out-of-Plane Bending Time History for LSB ................................ Work and Energy Distribution Time History for MSB (Linear Elastic) ............................ Work and Energy Distribution Time History for MSB (Inelastic) 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 3-64. 3-65. 3-66. 3-67. 3-68. 3-69. 3-70. 3-71. 3-72. 3-73. 3-74. 3-75. 3-76. 3-77. 3-78. Work and Energy Distribution Time History for SSB (Linear Elastic) ............................ Work and Energy Distribution Time History for SSB (Inelastic) ................................. Work and Energy Distribution Time History for LSB (Linear Elastic) ............................ Work and Energy Distribution Time History for LSB (Inelastic) ................................. Work and Energy Distribution Time History for MSB (2-D Model) with 0.0% Damping Ratio (Inelastic) ......................................... Werk and Energy Distribution Time History for MSB (2-D Model) with 0.25% Damping Ratio (Inelastic) ......................................... Work and Energy Distribution Time History for MSB (2-D Model) with 0.5% Damping Ratio (Inelastic) ......................................... Work and Energy Distribution Time History for MSB (2-D Model) with 1.0% Damping Ratio (Inelastic) ......................................... Work and Energy Distribution Time History for MSB (2-D Model) with 2.0% Damping Ratio (Inelastic) ......................................... Work and Energy Distribution Time History for MSB (2-D Model) with 3.0% Damping Ratio (Inelastic) ......................................... Work and Energy Distribution Time History for MSB (2-D Model) with 5.0% Damping Ratio (Inelastic) ......................................... Inelastic Responses Versus Linear Response Factor for MSB ...................... 159 Inelastic Responses Versus Linear Response Factor for SSB ...................... 160 Inelastic Responses Versus Linear Response Factor for LSB ...................... 161 Node 3 Horizontal (K) Displacement Time History for MSB ................................ 162 xii Figure 3-64. Work and Energy Distribution Time History for SSB (Linear Elastic) ............................ 148 Figure 3-65. Work and Energy Distribution Time History for SSB (Inelastic) ................................. 149 Figure 3-66. Work and Energy Distribution Time History for LSB (Linear Elastic) ............................ 150 Figure 3-67. Work and Energy Distribution Time History for LSB (Inelastic) ................................. 151 Figure 3-68. Work and Energy Distribution Time History for MSB (2-D Model) with 0.0% Damping Ratio (Inelastic) ......................................... 152 Figure 3-69. Work and Energy Distribution Time History for MSB (2-D Model) with 0.25% Damping Ratio (Inelastic) ......................................... 153 Work and Energy Distribution Time History for MSB (2-D Model) with 0.5% Damping Ratio (Inelastic) ......................................... 154 and Energy Distribution Time History : (2-D Model) with 1.0% Damping Ratio -rgy Distribution Time History Model) with 2.0% Damping Ratio ribution Time History th 3.0% Damping Ratio ................... .............. Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 3-79. 3-80. 3-81. 3-82. 3-83. 3-86. 3-87. 3-88. 3-89. 3-90. 3-91. 3-92. 3-93. 3-94. 3-95. 3-96. Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Node Time Member 1 End Time Member 4 End Time Member 1 End Time Member 4 End Time Member 2 End Time 5 Horizontal (X) Displacement History for MSB ................................ 3 Vertical (Y) Displacement History for MSB ................................ 5 Vertical (Y) Displacement History for MSB ................................ 9 Horizontal (X) Displacement History for SSB ................................ 21 Horizontal (K) Displacement History for SSB ................................ 9 Vertical (Y) Displacement History for SSB ................................ 21 Vertical (Y) Displacement History for SSB ................................ 12 Horizontal (K) Displacement History for LSB ................................ 18 Horizontal (K) Displacement‘ History for LSB ................................ 24 Horizontal (X) Displacement History for LSB ................................ 12 Vertical (Y) Displacement History for LSB ................................ 18 Vertical (Y) History for LSB Displacement 00000000000000000000000000000000 24 Vertical (Y) History for LSB ................................ J (Node History for MSB I (Node History for MSB J (Node History for MSB I (Node History for MSB J (Node History for SSB OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 163 164 165 166 167 168 169 170 171 172 173 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 3-97. 3-98. 3-99. 3—100. 3-101. 3-102. 3-103. 3-104. 3-105. 3-106. 3-107. 3-108. 3-109. 3-110. Member 6 End I (Node 21) Axial Force Time History for SSB ................................ Member 2 End J (Node 9) In-Plane Bending Time History for SSB ................................ Member 6 End I (Node 21) In-Plane bending Time History for SSB ................................ Member 5 End I (Node 12) Axial Force Time History for LSB ............................... Member 10 End J (Node 24) Axial Force Time History for LSB ............................... Member 5 End I (Node 12) In-Plane Bending Time History for LSB ............................... Member 10 End J (Node 24) In-Plane Bending Time History for LSB ............................... Work and Energy Distribution Time for MSB (Linear Elastic) ........................... Werk and Energy Distribution Time for MSB (Nonlinear Inelastic) ...................... Work and Energy Distribution Time for SSB (Linear Elastic) ........................... Work and Energy Distribution Time for 888 (Nonlinear Inelastic) ...................... Work and Energy Distribution Time for LSB (Linear Elastic) ........................... Work and Energy Distribution Time for LSB (Nonlinear Inelastic) OOOOOOOOOOOOOOOOOOOOOO ”Instability" Effects for LSB ...................... xiv 181 182 183 184 185 186 187 194 CHAPTER I INTRODUCTION 1-1 GENERAL Bridges form vital links in land transportation systems. The recent Lama Prieta Earthquake caused the collapse of more than a mile of elevated highway (essentially bridge-like structures) on I-880 and serious damage to the Bay Bridge in San Francisco. It again bespeaks the importance of seismic effects on the safety of bridges. Actually, significant damage to bridges had occurred in the 1971 San Fernando . earthquake and highlighted the need for reassessment of existing seismic design practice for bridges. A series of studies had since been conducted on the safety of highway bridges under earthquake loads. Those studies have been reported for long multiple span highway bridges ([16], [35], [36], and [37]), suspension bridges ([1], [2], [3], [4], [5], and [17]), and steel deck-type arch highway bridges ([10], [11], [12], and [13]). The study reported here represents an effort to develop a method of analysis for the nonlinear behavior of arch bridges subjected seismic loading. The two nonlinearities of structural behavior are ”material nonlinearity” that originates from the plasticity of the material, and "geometric nonlinearity" (nonlinear elastic effects) which represents the effects of the distortion of the structure on its response. 1-2 W The major objective of the study is to present a method of analysis for the seismic response of arch bridges that takes into account the effects of geometric nonlinearity or material nonlinearity. Included in this objective is the development of a computer program that carries out the necessary computations for the analysis. The second objective is to use the computer program to obtain certain numerical results based on a reasonable modelling of several real steel deck-type arch bridges subjected to earthquake motions. Included in this objective is the interpretation of the numerical results with a view to greater understanding of the seismic behavior of such bridges. It is hoped that the analysis and the computer program developed may serve as tools for further research in this area, and the numerical results presented may point to directions leading to the development of improved design procedure for such structures. In Chapter II the structural system is introduced and the method of analysis is presented. For the analysis, the mass matrix is formulated by the lumped mass approach and the damping matrix is of the Rayleigh type. The structure system model features "mixed nodal coordinate systems”: cartesian coordinates for deck nodes and curvilinear coordinates for arch rib nodes. In consequence, no coordinate transformations are needed for the beam elements as their element coordinates are the same as the system nodal coordinates. Another means employed to make the computational procedure more effective is the use of constraints that correspond to the assumption that the axial stiffnesses of certain members are infinitely large, thus reducing the number of degrees of freedom. 3 The effects of either geometric or material nonlinearity are included in the analysis. The computational procedures of the nonlinear I'resistance" for both nonlinearities are derived. For material nonlinearity, a method of analysis is developed for the elasto-plastic resistance of a curved beam element based on the “plastic hinge“ concept. The yield function is based on three stress resultants: P (axial force), M.y (in-plane bending moment) and Mx (out-of-plane bending moment). For geometric nonlinearity, a twelve degrees of freedom incremental stiffness matrix is derived. For the solution of the equations of motion, the method is one of a step-by-step numerical integration in the time domain coupled with a Newton-Raphson scheme implying an outer loop of iteration for the equilibrium of the system. For each elasto-plastic element, an inner loop of iteration is needed because of the material nonlinearity. Measures of the inelastic response such as "curvature ductility factors" and ”plastic work densities" are defined. As a check on the validity of the procedure, a work and energy balance for the system is considered. A computer program is prepared to implement the three-dimensional nonlinear seismic analysis described above. In Chapter III three real steel deck-type arch bridges: the Cold Springs Canyon Bridge (CSCB) in California, the South Street Bridge (SSB) in Connecticut, and the New River Gorge Bridge (NRGB) in West Virginia are chosen to be the prototypes for the medium span bridge (MSB), the short span bridge (SSB), and the long span bridge (LSB) used for numerical studies. The bridges are modelled by using curyed beam elements for the arch ribs, straight beam.and truss elements for the deck system, columns, and bracing systems. It should be noted that while 4 CSCB and SSB each has two ribs with a solid box cross-section, NRGB has a single box rib with each side a truss. “True" three-dimensional models for MSB and SSB and "one-plane" three-dimensional model [12] for LSB are used to consider material nonlinearity. For studies on geometric nonlinearity, only motions in the vertical plane of the rib(s) are considered and simplified two- dimensional models used. The ground motion used is the artificially generated motion CIT-A2 [18] with amplification factors applied to induce nonlinear effects. Nonlinear behavior due to material inelasticity is presented first, followed by behavior due to geometric nonlinearity. For comparision purposes, numerical results of the linear response are also obtained. The time history curves are presented for displacements (horizontal-X, vertical-Y, and lateral-Z), and for internal forces (axial force P, in—plane bending moment My, and out-of- plane bending moment Mx) at selected points. From time histories, it can be seen that even for the nonlinear cases the responses were generally dominated, as expected, by the “fundamental modes” (either in-plane or out-of-plane) except at those points where the fundamental modal response is small. However, the higher modes, up to the fourth mode, can be significant. For example, the response of the vertical displacement time history at the crown point for LSB was primarily in the second and fourth in-plane mode. This fourth in-plane mode is overall the ninth mode (i.e., counting in-plane and out-of-plane modes together) for the LSB model. For the displacement time history curves, the wave patterns for the linear elastic and nonlinear inelastic cases are quite similar. But 5 the wave patterns for geometric nonlinear case differ more (about 9% for‘ the fundamental mode and lower for higher modes) because of the changes in the natural periods due to stiffness differences. The maximum displacement responses, obtained by considering either material or geometric nonlinearity, are not much different from those of the linearly elastic analysis. For all models, the maximum displacement was about 1% of the arch height in the horizontal direction, 2.5% in the vertical direction. In the lateral direction, the maximum displacement was about 8% of the arch height for MSB and 3% for SSB and LSB. Although the wave patterns of the internal force (stress resultants) time histories are quite similar between the elastic and inelastic cases, the magnitudes of the maximum internal forces differed considerably. If inelastic behavior is allowed in the structural system, because of the definition of yield function, the internal forces (P, M y and Mx) are bounded by the fully plastic values (PO, and Mx0)' The “yo maximum values of such forces were about 20% less than those obtained from a linearly elastic analysis. This ”force reduction“ was realized at a price of plastic deformations in the structure. This led to the consideration that, if one is willing to accept such plastic deformations, the structure could be idesigned" with a ”reduced load." For the geometric nonlinear cases, the wave patterns of internal force time histories, as in the case of displacement time histories, differed from the linear elastic ones. However, there is no appreciable difference in the maximum values of the internal forces obtained from the linear elastic and nonlinear elastic solutions. During an earthquake energy is fed through the base of the structure. It is instructive to know how the energy is distributed in 6 the structural system during the earthquake loading. The work and energy balance equation of the system is written by setting the work done by the support reactions and by gravity loads equal to the sum of the recoverable strain energy, irrecoverable (plastic) strain energy, kinetic energy, and the energy dissipated by damping. From the work and energy distribution time histories, the dissipated damping energy (for a 1.5% damping ratio) was over 70% of the work done for all cases. When the damping ratios were varied from 0.25% to 5%, the percentages of damping energy to work done changed from 50% to 96%. In the inelastic analyses, about 15% of the work done was dissipated by irrecoverable strain energy. 1-3 W A Response of arch ribs (with no deck system) to earthquake shaking was reported by Thakkar and Arya [33]. The study was limited to linearly elastic behavior of a single rib subjected to in-plane motion only. A study of the in-plane strength of deck arch bridges under longitudinal ground motion was reported by Kuranishi and Nakajima [19]. Dusseau and Wen ([10] and [11]) have reported the elastic seismic responses of two existing arch bridges: the Cold Springs Canyon Bridge (CSCB) in California and the South Street Bridge (SSB) in Connecticut. Free vibration characteristics were studied. Seismic responses in all three dimensions were estimated from a "normalized rock spectra” [16] using an input ground acceleration level of 0.50 g and 0.09 g, respectively for the California bridge and Connecticut bridge. Dusseau and Wen ([12], [13]) have also studied the effects of unequal motion at the supports of three deck arch bridges: CSCB, SSB, and New 7 River Gorge Bridge (NRGB) in West Virginia. In using time history analysis with step-by-step numerical integration, a ”one-plane model" was introduced to decrease the number of degrees of freedom. In this model a transverse cross-section of the arch rib(s) is modelled as that of a single beam element. Thus for a two rib bridge, in lateral response, each rib would act as a flange of a beam. A key factor in the nonlinear dynamic analysis of a bridge is the structural stiffness. Studies on the material nonlinear static behavior of a beam-column member by various investigators have been discussed in a treatise by Chen and Atsuta [6]. In particular, the yield surface for a cross-section and the behavior of a segment have been described in some detail. For elasto-plastic statical problems the ..concept of plastic potential theory using stress resultants as generalized stresses has been adopted by Hodge [15], Morris and Fenves [21], and others. A method for the small displacements analysis of three- dimensional inelastic frames subjected to static loads has been described by Morris and Fenves [22]. The elements are assumed to be elastic-plastic and to yield at generalized plastic hinges, the behavior of which is governed by four dimensional curved yield surfaces. To insure that the point representing the end forces on any element does not travel outside the yield surface, when such a point reaches the surface it is drawn back a small distance and constrained to move tangent to the surface. The dynamic response of three dimensional frames with elasto- plastic elements has been studied by Nigam [23]. The elements are assumed to yield at generalized plastic hinges governed by two 8 dimensional circular yield surfaces. For each load increment the increments in member and forces at each plastic hinge are constrained to move in the tangent plane to yield surface, and a new tangent stiffness is formulated for each load step. Nigam's formulation of the problem was not in matrix algebra, and consequently the equations involved are somewhat complex. A concise and general matrix form was presented by Porter and Powell [26]. A more general form of the stiffness of a beam element was derived by Wen and Farhoomand [38]. They carried out dynamic analysis of three dimensional frames in which the yielded regions were assumed to have finite length. The elasto-plastic elements are assumed to be governed by a four dimensional parabolic or elliptic plastic potential function. An iterative procedure was described that would keep the force vector on the curved yield surface during yielding. To keep the force point on the yield surface during yielding, several approaches including iterative procedures [38] and one-step approximate force corrections [29] have been proposed. Chang and Kitipitayangkul [7] adopted a different approach from the plastic potential theory to handle the elasto-inelastic analysis of building frames. The force-deformation relationship for each pair of generalized stress and strain was taken to be of the Ramberg-Osgood hysteretic type. However, the yield values of the various generalized stresses were governed by appropriate interaction equations such as given by Tebedge and Chen [31]. The effect of torsional moment was also included in the interactive behavior on the basis of Von Miss' yield criterion as was used in Ref. [21]. Powell and his co-workers [28] had employed two and three parallel elements to represent a single element 9 and thus strain hardening effects may be considered. A number of investigators (See, for example, in [30], [19], and [42]) had formulated and analyzed the nonlinear problem at the stress, rather than stress-resultant level. Such approach, although more refined than the stress-resultant formulation, generally requires considerably more computational resource for a given physical problem. For the consideration of geometric nonlinearities, the tangent stiffness matrix may be formed by adding the initial and geometric stiffness matrices. Many studies have been reported on the subject (e.g., [8], [24], [27], [32], and [40]). Among them Oran [24] has presented the nonlinear elastic tangent stiffness matrix of a straight beam element which is exact within the framework of the elastic beam- column theory. The expressions involve the axial force as a parameter that requires iterations for its determination. A nonlinear elastic tangent stiffness matrix based on a finite element approach [40] has been shown to be quite accurate for problems that do not involve very large displacements (for example, of the order of the dimension of the structure itself). It would cover the great majority of civil engineering structures. A method of analysis for investigating the stability of complex structures has been described by Toridis and Khozeimeh [34]. The general approach is based on the finite element method and incremental numerical solution techniques. This incremental loading approach has been used with no equilibrium check. In the incremental solution process, the stiffness properties of the structure are continuously updated in order to properly account for large changes in the geometry of the structure (i.e., to take into account the effect of the geometry of the deformed 10 structure on the instantaneous stiffness matrix). Both material and geometric nonlinearities were considered for the frame structures by Porter and Powell [26]. The geometric nonlinearity was considered in two-dimensional problems and only static loads were applied. The elasto-plastic stiffness and geometric stiffness of straight member have been derived. A general computational procedure has been described for the collapse load analysis of statically loaded plane frames and the analysis of dynamically loaded inelastic frames. For arch buckling analysis, the linear and incremental stiffneSs matrices of a curved beam element deformable in three dimensional space have been described by Wen and Lange [39]. In developing a nonlinear curved beam element of general shape, the geometry of the curved axis of the element is represented by a fourth-order polynomials in terms of the inclination angle with the tangent at a member end, and the displacement functions are approximated by cubic polynomials in the same variable. The linear stiffness matrix [k] and the first and second order incremental stiffness matrices, [n1] and [n2] were derived by differentiating the strain energy. 1.4 NQIAIIQN The notations shown below has been used in this report: A - cross-sectional area; B - width of the cross-section; b2,b3,b4 - curved beam element geometry coefficients; [C] - viscous damping matrix; c - subscript denoting "constrained"; D - depth of the cross-section; 11 E - Young's modulus of elasticity; ED - energy dissipated by damping; EK - kinetic energy; ESE - recoverable strain energy; ESP - irrecoverable (plastic) strain energy; e - superscript denoting ”elastic part"; e - subscript denoting “elastic part"; Aes - incremental strain energy; Fe - maximum force by elastic analysis; F1e - maximum force by inelastic analysis; Fy - yield stress; g - gravitational acceleration; h - time interval; Ix - moment of inertia about the x-axis; Iy - moment of inertia about the y-axis; [J] - Jacobian matrix; [kt] - elasto-plastic tangent stiffness matrix; [k]; [k0] - linear elastic stiffness matrix; [ke] - elastic stiffness matrix; [M] - lumped mass matrix of the entire structure; Mi - bending moment about x-axis; My - bending moment about y-axis; "x0 - fully plastic bending moment about x-axis; M&O - fully plastic bending moment about y-axis; [m] - lumped mass matrix of an element; [n1] - first order incremental stiffness matrix; [n2] - second order incremental stiffness matrix; (P) (P) {Q} {dql 12 axial force; fully plastic axial force; superscript denoting “plastic part"; subscript denoting ”plastic part"; external load vector; external static load (dead load) vector; member end force vector; incremental displacement vector; radius of curvature; X component of reaction at support 1; Y component of reaction at support 1; 2 component of reaction at support i; radii of curvature at ends of an element; resistance vector; section modulus about the x-x axis; section modulus about the y-y axis; longitudinal axis of curved beam element; superscript denoting matrix transposition; time; thickness of flange; thickness of web; beginning and end time of the time interval h; ground displacement component in x direction for node R; ground displacement component in y direction for node R; ground displacement component in z direction for node R; subscript denoting "unconstrained"; ground motion component; 13 displacements along x, y, z axes, respectively; incremental support or ground displacement vector; vertical displacement at node D and E; hinge volume for Mx; hinge volume for My; hinge volume for P; relative displacement vector; incremental relative displacement vector; incremental relative velocity vector; incremental relative acceleration vector; work done by gravity loads; work done by support reactions; total displacement vector for unconstrained degrees of freedom; plastic work done by M.x in time interval h; plastic work done by My in time interval h; plastic work done by P in time interval h; dimensionless plastic work quantity; strain energy per unit volume of material at yield; incremental displacement vector; incremental velocity vector; incremental acceleration vector; cartesian coordinate system; ground displacement components in cartesian coordinates; curvilinear coordinate system; incremental horizontal ground displacement at support 1; incremental vertical ground displacement at support i; 14 incremental vertical displacement of node point j; incremental lateral ground displacement at support 1; parameters used for definition of displacement functions; constants for Rayleigh type damping matrix; transformation matrix; tolerance for yield function; yield strain; tolerance vector of force or moment; tolerance vector of displacement or rotation; scalar tolerance of work; angle that the tangent at the node makes with the global X- axis; rotations about x, y, z axes, respectively; plastic rotation about x-axis; plastic rotation about y-axis; plastic axial displacement; flow constant, a positive scalar; transformation matrix; curvature ductility factors for out-of-plane bending; curvature ductility factors for in-plane bending; ductility factors for axial strain; yield function; linear response factor; angle of the tangent measured with respect to the tangent at a reference and (Fig. 2-5); incremental operator; prefix denoting "gradient"; l l [J [ 1 (') (") 15 column vector; row vector; rectagular matrix; d( )/dt; d2( )/dt2; CHAPTER II ANALYSIS AND METHOD OF SOLUTION 2-1 emu For purposes of analysis, the bridge structure is modelled by finite elements: truss, straight beam, and curved beam elements. A three-dimensional version of the model is shown in Figure 2-1. Generally, the bridge consists of two arch ribs (modelled by curved beam elements) and a deck (by straight beam elements) and columns (by truss elements) between the ribs and deck. The cross-beams between the ribs and the deck girders are modelled by straight beam elements. The other bracings are modelled by truss elements. For the analysis, mass is lumped at the nodes. The system equation consists of the equations of motion for the unconstrained degrees of freedom of the nodes. These degrees of freedom are described in cartesian coordinates for the nodes on the deck, and curvilinear coordinates for nodes on the ribs. Thus there are two kinds of nodal coordinates: one cartesian, the other curvilinear (See Figure 2-2). The analysis considers response due to either geometric or material nonlinearity that are associated with the curved beam elements. The straight beam and truss elements are presumed to be linearly elastic. The method of analysis is one of a step-by-step numerical integration in the time domain. Within a time increment, the method of solution is essentially one of the Newton-Raphson type which calls for 16 17 an "outer loop" of iteration for the dynamic equilibrium of the nodal degrees of freedom. In the case of elasto-plastic behavior an "inner loop" of iteration is necessary to satisfy the constitutive equation of the element involved. In this Chapter, in addition to the method of analysis and solution, certain measures of the inelastic responses, i.e., "plastic work densities” and ”ductility factors”, are presented. The computation of work and energy balance is also described. 2.2 W Let {wu} denote the unconstrained degrees of freedom (generally those of the interior nodes of the structure) and {we} the constrained degrees of freedom (generally those of the external nodes or supports that are constrained to be equal to prescribed ground displacements). As noted previously, for a node on the deck, the equation of motion is written in the usual cartesian coordinates, and for a node on the ribs, it is written in curvilinear coordinates; See Figure 2-2. The assembled set of equations of motion or dynamic equilibrium may be written as follows: [Mllfllfiil [Cllclh'vl {R} {P} [Mcu] [Moe] {we} [Ccu] [Ccc] {we} {Rc} {Po} in which the [M]'s are mass submatrices, the [C]'s are damping submatrices, {Ru} and {Re} are resistances resulting from deformations of structural elements, {Pu} and {Po} are external loads. The subscripts "u” and ”c" denote ”unconstrained" and "constrained"; the superscripts of ”dots" denote time derivatives. 18 For lumped mass inertia, the cross submatrices [Muc] and [Men] are null. It is assumed also that damping is due entirely to the velocities associated with the unconstrained degrees of freedom, i.e., [Cue] - [Ccu] - [0]. Writing [M] for [Maul’ [C] for [Cuu]’ {R} for {Ru}, {P} for (Pu), and (w) for [wfi}, the equations of motion for the unconstrained degrees of freedom are [ulna + [61167) + {R} - {Pl - {0) am In an earthquake response problem, the real dynamic load {P} is in general null. The driving mechanism is contained in the resistance term [R] which depends on the and displacements of the structural members. For members that are connected to the ground, the displacements at those ends are constrained to be equal to that of the ground motion. As those member forces change, they would disturb the equilibrium of the structure. The detailed procedure of analysis is given in the next section. 2.3 W The equations of motion for the unconstrained degrees of freedom may be represented as: tin" - [mm + [01m + {m - {P} - {0) (2-3) For a numerical analysis of the problem, consider the solution involved in a typical time interval h from time t to time t - t + h. 0 l 0 The equations of motion will be integrated numerically using a stable scheme, e.g., the Newmark fl - 1/4 method. ml - mo + (h/2)({ii}o + ml) ””1 - mo + (rt/mono + m1) (2-4) in which the subscripts “0” and "1” denote the beginning and end of the 19 time interval. Substituting Eqs. 2-4 into Eq. 2-3 for t - t one 1’ obtains when,» - [Mlmm - h MONA/1:2) - mo] + [C][2{Aw}/h - {wlol + (Rio + {AR} - {P10 - {AP} - {0} (2-5) in which the symbol A is used as a modifier signifying the change of the modified quantity from t to t thus, {Aw} - {W}1 ' {w}01 o 1‘ {AR} - (R11 - (Rio. and {AP} - {P11 - (Plo- In order to compute {Aw}, the Newton-Raphson method of iteration is employed. For the (k+l)th iteration, the procedure calls for a solution of 8({Aw})k+1 - {Aw)k+1 - {Aw}k from the following linear equation .111" 5 11* 26 l (l l .{Awl)k] ({Awl)k+1 - -l l k ( ' ) in which [J({H}*,{Aw})k] is the Jacobian matrix of {H}* with respect to {Aw}. i.e., its i-j entry - 8H*1/6ij with H*i denoting the ith element of {H)* and Aw} the jth element of {Aw}, evaluated at {Aw} - {Aw}k, and —[H}*k is evaluated from Eq. 2-5 with {Aw} - {Aw}k. It follows from Eq. 2-5 that IJ in which [k*] is the usual beam stiffness in degrees of freedom {d*}. Of course, [k] is then assembled into the system stiffness matrix in the usual fashion. The constraints also affect the mass matrix. Denoting by {d*} and [m*] the “old" velocity vector and "old” mass matrix, and {d} and [m] the "new" velocity vector and “new" mass matrix, let the old and new velocities be related by the transformation {d*} - {A}{d}. By equating the kinetic energy expressed in the old system to that expressed in the 24 T a new system, one has [m] - {A} [m ]{A}. Consider the effect of the lumped mass at D, mD. In this case, .* . x . . . T T {d } - VD’ [m ] - ED. {d} - L “R "R J and {A} - L -cos€R sinoR ], one obtains 2 cos 0R ~sin0RcosOR [m] - HID 2 (2-16) -sin0RcosoR sin 0R This submatrix is assembled into the system mass matrix (to rows and columns uR and wk). (Hence the system mass matrix is not diagonal.) For the other constraints involving the same coordinate systems, the masses of the slave degrees of freedom are added to those of the master degrees of freedom. 2.5 W 2.5.1 cm In the preceding, a method of analysis has been develOped for the linear and nonlinear response of a structure in three dimensional space. In the analysis, the structural resistance vector {R} plays a key role as it is through this vector that the nonlinear behavior is accounted for in this study. However, the behavior of straight beam and truss elements is assumed to be linearly elastic during the entire loading period. The resistance is {R} - [R] {q} . (2-17) in which {q} is the element displacement vector and [k] is the usual linear elastic stiffness and need not be presented here. In the following, resistance of the curved beam elements, making up of the major components of the bridge, is discussed first for linear behavior 25 and then for nonlinear elastic and elasto-plastic behavior. 2.5.2 W The curved beam finite element model used is illustrated in Figure 2-5, in which A and B denote the two and nodes, x, y, and 2 represent the curvilinear coordinates. The stiffness of this element has been presented in Ref. [39], in which the curved axis of the element was represented by a fourth order polynomials in d, the angle of the tangent measured with respect to the tangent at end A. The coefficients of the polynomial are determined by end slopes and curvatures as well as the coordinates of end B. Thus at a common node of two elements, continuities of slopes and curvatures can be maintained. Four independent displacement functions are considered: u (radial), v (transverse), w (tangential) translations, and 02 (twist about tangential axis), each represented by a third order polynomials in ¢- Each of the four displacement functions involve four coefficients. The sixteen coefficients were determined by eight degrees of freedom at each end. For example, at end A, the three translations, uA, VA’ “A and three rotations oxA’ oyA, 02A plus (ddz/ds)A and (dw/ds)A. The last two degrees of freedom are "nonessential". The twelve essential degrees of freedom are illustrated in Figure 2-6. In this work, the nonessential degrees of freedom are condensed out before assembling into the system stiffness matrix with the usual six degrees of freedom per node in three dimensional space behavior. The linear elastic resistance is of course simply the linear elastic stiffness (k0) multiplied by the end displacement vector. 26 2.5.3 W The resistance of a nonlinear elastic element may be written as (Ref. [20]) + -— ) {q} (2-18) in which [n1] and [n2] are the first and second order incremental stiffness matrices. The matrices [k0], [n1], and [n2] are obtained as the second derivatives of the quadratic, cubic, and quartic parts, respectively, of the strain energy expression. The incremental or nonlinear stiffness matrices [n1] and [n2] based on the interpolation functions and sixteen degrees of freedom mentioned previously are also given in Ref. [39]. In this study, the second order incremental stiffness matrix [n2] is not used because numerical experience showed that it would make the element too stiff with unacceptable errors. In order to fit into the twelve degrees of freedom per node scheme of analysis, the 16 by 16 [n1] matrix need be condensed to 12 by 12. Since it is linear in the displacement variables, the condensed matrix is valid only for the displacement state at which the condensation is executed. For a time history analysis, this means that a condensation is needed for every time increment. In order to save computing time, this condensation is avoided by using a twelve degrees of freedom incremental stiffness matrix. It is based on third order polynomials for the radial and transverse displacements and first order polynomials for the tangential displacement and twist about the tangential axis. They are: 2 3 u - a1 + “2‘ + a3¢ + “4‘ 27 2 3 v - as + “6‘ + 07¢ + asp w - a9 + “10‘ (2-19) 0 + a z ' “11 12‘ As mentioned previously, the nonlinear elastic stiffness matrix [n1] was derived as the second derivatives of the cubic part of the strain energy expression (now corresponding to the preceding interpolation functions). Numerical experience (Ref. [39]) also indicated that in general more accurate results would be obtained if the terms containing rotations were dropped. The rotation terms include rotation about y-axis - (du/ds) + (w/R) and rotation about x-axis - (dv/ds). This modified form of [n1] was used to obtain the numerical results presented in Section 3.5. 2.5.4 Walkman 2.5.4.1 angxgl The elasto-plastic resistance is calculated using the plastic potential theory as applied to stress resultants (Ref. [15]). Material yielding is assumed to take place only at either or both ends of the member. This is warranted if there is no load between the ends, and the offset is small in comparison with the length of the chord. The part of the element between the two ends would remain linearly elastic. Although the plastic potential function is defined individually for each end, the tangential stiffness is derived for the entire member. The incremental resistance and the elastic and plastic parts of the deformation are obtained by an iteration process. The details of the above are given in the following sections. 28 2.5.4.2 z1astis_£2tential_£unctien It is assumed that the material is "associative". Thus the plastic potential function and the "yield function" are the same. The yield function defines the combination of the force components necessary to initiate yielding at a cross-section. A shape factor of 1.0 is assumed for the end sections, i.e., the cross section is assumed to make an abrupt transition from a completely elastic state to a plastic state in which unrestricted plastic flow can occur. Thus plastic yielding is confined to an individual cross section with zero "hinge length." The yield function is assumed to remain the same as yielding progresses, i.e., there is no strain hardening. For the numerical results presented herein, the yield function O for a cross-section is written as: P 2 M 2 Mx2 o-(—)+(—1)+(——)-1-o (2-20) Po Myo MxO in which P is the axial force, My the bending moment in the plane of the rib, M.x the out-of-plane bending moment or moment about the radial axis, and Po, MyO’ M.x0 are the fully plastic force components corresponding to P, My, Mx, respectively. For this study, they were computed as: Po - Fy A u-rs yOyyy “x0 - F& Sxx (2-21) in which F is the yield stress, A, S , S are the cross-sectional Y yy xx area, section moduli about the y-y and x-x axis, respectively. This yield surface, though quite idealized, provides a convenient 29 approximation for the complex phenomenon of inelastic behavior in the three dimensional space. Figure 2-7 shows the spherical yield surface. Thus, in accordance with the plastic potential theory, if the ”force point" (P/PO, "y/Myo' Mx/Mxo) is inside the surface, the section is regarded as linearly elastic. If it is on the yield surface, it is plastic. It cannot go outside the yield surface. As mentioned previously, the force-deformation properties at the end sections are not entirely independent because the elastic parts of the responses are governed by the member elastic stiffness. The member elasto-plastic properties are described in the following. 2.5-4-3 We The displacement increment vector at a member end can be expressed as the sum of an elastic and a plastic part: (den - {dqe} + {qu1 (Hz) in which {dqe} is the vector of the elastic part of the member and displacement increments and {dqp} is the vector of the plastic part of the increments. The member end force increments are related to the elastic member end displacement increments as follows: {do} - [kelldqel <2-23) in which {dQ} is the vector of member and force increments and [ke] is the elastic stiffness matrix. Following Drucker's normality criterion (Ref. [9]), at a point on the yield surface the incremental plastic deformation vector has the direction of the outwardly directed normal. 30 {dqp} - A{V¢} <2-24> in which A is a positive scalar, called the flow constant, which defines the magnitude of the plastic deformation at the point (plastic hinge); and {Vi} (gradient of function O) is the outwardly direction normal vector at the point on the yield surface. As the force vector can not extend beyond the yield surface, any force increment vector {dQ} corresponding to a plastic deformation at the cross section must move on the surface (or on the tangent plane). This requirement is expressed by the normality condition. {vstTtdoi - 0 (2-25) in which the superscrpt T denots "transpose". Substituting Eq. 2-23 into Eq. 2-25, one obtains tvcileeltdqei - o <2-26) Then, from Eqs. 2-22, and 2-24, one has tvciTtkelttdqi-tdqpi> - o <2-27> {V0}T[k61(ldql-A{V¢l) - o (2-28> Solve for A: {veiTtkeltdqi A - T (2-29) {vs} [ke]{V§l substituting Eq. 2-29 into Eq. 2-24, one obtains the plastic part of the incremental displacement as: {V¢}{V¢}T[ke] {dqp} - {dq} <2-30> {VfilrlkellVél 31 2.5.4.4 Ianssnt_Stiffness_uatrix_cflflemher The following sequence of equations can be obtained from Eqs. 2-23, 2-22, and 2-24 {d0} - [kelldqel (2-31) - [ke](tdq}-tdqp}) (2-32) - [ke]({dq}-A{V¢}) (2-33) Then, from Eq. 2-30 {V§l{V¢}T[ke] {do} - [ke]({dql- {dq}) (2-34) {V¢}T[k8]{V¢} {V¢}{V¢}T[ke] {dQl - [ke]([I]- T )ldq} <2-3s> (vs) [ke]{V¢} Therefore, the elasto-plastic tangent stiffness matrix is {V¢l{V¢}T[ke] [kt] - [ke]([I]- > <2-36) {vetleeltvei 2 5.4.5 Elastis_Retnrn As a structure is being deformed, a change in load distribution may cause one or more plastic hinges to unload and become elastic again. This phenomenon is referred to as an "elastic return.“ It occurs whenever there is a reversal in the direction of the incremental displacement at a plastic hinge. It follows from Eq. 2-24 that the elastic return at any plastic hinge is signalled by a negative flow constant A at the hinge. 32 2.5.4.6 WWW 2f_Elea£2;£lsstie_Elemen£ As mentioned earlier, a key step in the solution procedure is the calculation, for each member, of the new member end force vector (Q2) and possible incremental plastic deformation vector for a given current force vector {Q1} and incremental end displacement vector {dq}. In the Section 2.5.4.4, the tangential stiffness has been derived. The incremental member forces based on the tangent stiffness would necessarily violate the yield condition (going out of the yield surface). The technique of keeping the force point on the yield surface consists of essentially obtaining, by iteration, a "local secant stiffness". Corresponding to a given {dq}, none, one or both ends may yield or return to elastic behavior. The details of the procedures to treat these complex behavior are given below (Ref. [41]). Procedure described in this section combines two features. Firstly, following an iterative process, the new member force vector would be made to stay on the yield surface as required by the theory of plasticity. Secondly, the incremental displacement can be large enough so that the element may undergo the process of changing from a state of total elasticity to having one yielding end and then on to having both ends yielding. This feature allows the solution procedure to use constant load or time increment. It is assumed that during a load or time increment the change of the displacement is linear and one may write {dq} - {dqee} + {dqep} + {dqpp} (2-37) in which {dqee} is the part of {dq} with both ends elastic, and {dqep} and {dqpp} are those with one and two yielding ends, respectively. The 33 corresponding changes in the member and forces are illustrated in Figure 2-8, in which the notation should be self-explanatory. For example, Q1A is the current force point and Q2A,ee is the force point at A at the end of {dqee}. The displacements would take place in the order as given on the right side of Eq. 2-37. However, any one or two of the three incremental terms may be null. Consider the yield surface of a generic yielding end, say C, as shown in Figure 2-9(a). Let Qlc denote the initial force point, (Q2C*) - (Q10) + [kec]{dq}, in which [keC] is a 6 by 12 stiffness matrix (partitioned from [ke], the element elastic stiffness matrix). The intersection of the vector Q1C-QZC* and the yield surface, point is a ”general penetration point," given by **) - (Q10) + 18[kec]{dq} in which 18 is solved from the yield function §((Q1C) + 18[kec]{dq}) - 0. Illustrated in Figure 2-9(a) is also a “radial penetration point.” It is the intersection of the vector O-Q2c* and the yield surface, point QZC,r**’ given by {Q2C,r**} - 1rthc*}, in which 7r is solved from C(1IIQ2C*}) - 0. When yielding takes place at only one end, the initial force point Q1c is on the yield surface. Corresponding to an incremental displacement {Aq}, the force is (QZC*} - {Qlc} + [kc]{Aq} in which the stiffness [kc] depends on the element elastic stiffness and the gradient of Q at Q10, (See Eq. 2-36 and Refs. [22], [25] and [38]). Because of the convexity of the yield surface, ch* is necessarily not inside the surface. To keep Q2 on the surface the following iteration procedure is used (See Figure 2-9(b)). Let Q2Cj denote the value of Q2C* for the jth iteration. For the first iteration, {chl) - (Q1C} + [kc1]{Aq} in which [kcl] is based on 34 the gradient at QlC’ i.e., Vé - VQ({Q1C}). For j > 1, compute J- J J -— {Q2C } {Q10 + [kC ]{Aq} in which [kC ] is based on V¢ VQJ given as follows: V735 - 0.5(vo-1'1 + 6634) if ouQZC-j’ln > e (2-38a) or 66-1 - 0.50%»1 + $334) if ouQZCJ'ln < -5 (2-38b) 1 - V§({Qlc)) and VQJ.1 is the gradient at the In the preceding, 33 radial penetration point for QZCJ-1’ and e is a tolerance. The iteration ends when louozcjm s c. When there are two yielding ends, the procedure is the same except that the gradients at both ends should be simultaneously treated as described in the preceding for the formulation of the elasto-plastic stiffness, and the convergence criterion, of course, applies to both end force vectors. The computation procedures of the incremental resistance and deformations of an elasto-plastic element are: 1. Using the elastic stiffness [ke], compute a tentative end force increment {dQ*) - [{dQA} {dQB}JT - [ke]{dq}, and a tentative new force point {Q2*] - {Q1} + {dQ*). (The superscript ”*" denotes "tentative”.) 2. Compute the values of the yield functions for both ends corresponding to {Q1} and (Q2*}: “Al - §A((Q1A}); ’31 - 95mm»; 52 - ¢A({Q2A*}); s32 - ebuozfn. 3. Locate the general penetration points and compute the quantities (See Figure 2-9(a)): L.A - ¢A2 - CA1; CA - - CA1; 5A - L - aA; and similarly for LB’ A and BB. GB, 10. 35 If 5A 5 0 and 5B s 0, the element response is entirely elastic. Then, {dqee} - {dq}; {dqep} - {dqpp} - null vector, {Q2} - {Q2*). The resistance computation ends. Otherwise, compute 11 - minimum (aA/LA, aB/LB)' Assuming 11 - A/LA (otherwise, switch the subscript from A to B), signifying that end A (or B) yields first. Compute: {dqee} - Vlldq}; {Q2} - {Q1} + [kelidqee} and {den - <1 - 11mm - {dqepn + {dqpp}. Compute the gradient of yield surface at end A, * V§A - VQA ({QZA,ee}) and form the corresponding elasto- plastic tangent stiffness matrix [k*] - [[kA*] [kB*]JT. c t (Q *) - [{Q *} {Q *}JT ompu e 2,ep 2A,ep 2B,ep * _. * - ‘Q2,ee} + [k ]{dq). If ‘X’BUQZB’ep }) > 6, go to step 9. Otherwise, end B is still elastic, and if * |°A(‘Q2A,ep })| s c, then {dqpp} is null. Set * {dqep) - (l - 11)(dq} and {Q2} - {Q2,ep ). The plastic part of {dqep} is {dq )jT with } - [{dq } {dq GPA»? ePBsP epA,p} - AAYCA in which AA is the ep,p {dq p) - null, and {dq epB, flow constant computed as usual (Eq. 2-29). The resistance computation ends. * __ If |§A({Q2A,ep })| > e, compute VQA as per Eqs. 2-38, update [k*] accordingly, and return to step 7. * * * * Compute 12 - aB /LB based on {Q28 ee) and (Q2B ep } (corresponding to Q1C and QZC* in Figure 2-9(a)). Compute ** k* * d ** 1 h h {Qz’ep ) - {Q2,ee}‘+ [ 112 { CU- {Q23,ep } S t us on t e yield surface. If |¢ ** < * ** A({Q2A,ep )>l _ e. set 12 - 12 . {Q2,ep} - {Qz’ep 1, 36 {dqep} - 12(1 - 11)ldq}. The plastic part {dqep,p} is computed as usual. The part of {dq} remaining to be ”accounted for" is [dqpp} - {dq} - {dqee} - {dqep}. Go to ** * .— step 11. If |°A({Q2A,ep })| > 5, update [k ] using VQA ** computed as per Eqs. 2-38 (with Q2A ep taking the place of * O Q2C ). Co to step 7. ll. Compute {Q2 *1 - {Q2 ep} + [k*]{dqpp} in which [k*] is oPP based on the gradients at Q2 ep' Iterate as described previously to have Q2 pp on the yield surface for both ends. * 12. The final stress point is {Q2} - {Q2.ep} + [k ]{dqpp}. The incremental lastic deformations in d , i.e., d , p { q'pp} { qpp.p) B B computed as usual based on the lastest gradients. The total consisting of {dqppA p} - AAVEA and [dqppB p} - A 66 are incremental plastic deformation is ({dqep p) + {dqpp p}). The resistance computation ends. 2.6 W The calculation of the plastic part of the incremental element displacements has been presented above. For purposes of interpreting the inelastic response characteristics of the structure, it is appropriate to further define the quantities “plastic work densities" and ”ductility factors”. They may be considered as approximate measures of “damage" to the structural elements. 2.6.1 Was It is assumed that all plastic work is due to the inelastic straining of the ”longitudinal" fibers of the beam elements. Thus the 37 work would be done by the bending moments: Mx’ fly, and the axial force F. It is further assumed that the beam cross-section has the shape of a rectangular box with two axes of symmetry (See Figure 2-10.) Thus for each member end, the plastic work done by each of the stress resultants in the time interval h may be expressed respectively as: wxp - E Mxiwxpi Z M w - 80 YD 1 y1 ypi wfip - E Pisapi (2-39) in which soxp, soyp, and SAP denote the plastic part of the incremental rotations about x-x and y-y axis and the incremental axial displacement. The subscript "i” denotes the ith time interval. Although in the mathematical analysis, the plastic length is assumed to have zero length, for purposes of engineering interpretation of the results, it is assumed that the plastic deformations are distributed for a finite length. Corresponding to the rotation in a principal plane that length is taken to be the depth of the cross- section in that plane. Consider the case ”x # 0 (signifying rotation in the principal plane y-z), My - P - 0. Assuming that there is no transition to fully plastic moment, i.e., all fibers across the web thickness tw (see Figure 2-10) yield simultaneously, a "plastic hinge volume” for M.x may be defined as th - 2thB, in which D and B are respectively the depth and.width of the section. Similarly, a plastic hinge volume for My is defined as V - 2thD, in which t by f thickness of the flange. For P, a plastic hinge volume, VhA is taken to is the 38 be th + vhy' (If D/B - tf/tw, this volume corresponds to that of the cross-section area times a length equal to (D + B)/2.) The strain energy per unit volume of material at yield is wo - (l/2)F&¢y. The dimensionless measures of the various plastic work quantities may thus be represented as: wxp - wxp/(thwo) w - w V w yp yp/( by 0) fihp - wAp/(VhAVO) (2-40) 2 6.2 Dustili:x_£assers As alternative or additional measures of inelastic response, member and ductilities may be defined as follows. Corresponding to out- of-plane bending, the "curvature ductility factor" is "inelastic curvature" 0 /B - 1 + - 1 + “P (2-41) elastic limit curvature MxO/(Elx) Il¢x in which exp is the plastic rotation, B is the plastic hinge length for out-of-plane bending, MxO’ Ix are the elastic limit moment and moment of inertia about the x-axis, respectively, and E is the Young's modulus. Similarly, for in-plane bending, the "curvature ductility factor" is 9 /D _ 1 + .yp “yo/(Ely) 2-42 u¢y ( ) in which the meaning of the symbols are entirely analogous to those in the preceding expression. The "ductility factor" for axial strain, pA, is defined as the ratio of ”plastic axial strain" to the elastic limit 39 strain. The former is taken as the plastic axial displacement, AP, divided by its ”hinge length." Taking the latter as (D + B)/2, one obtains 2A u - P (2-43) (B + D):y 2.7 HQBK_AHD_EHEBQX.DI§IBIEQIIQE_§BE§K Considering work and energy balance, the following equation should hold for the system analyzed at all times: wk + WC — ESE + ESP + BK + ED (2-44) in which WR and WC denote work done by support reactions and by gravity loads, respectively, E is recoverable strain energy, E is SE SP irrecoverable (plastic) strain energy, ER is kinetic energy and ED is energy dissipated by damping. An examination of the variation of these quantities is instructive. Moreover, it can also serve as a check on the validity of the analysis and numerical computation. In the following for each time interval h, each work or energy item, with the exception of the kinetic energy, is computed as the product of force (taken as the average of those at the beginning and at the end of the time interval) and the incremental displacement. The total amount of a given kind of work or energy at a given time is of course obtained by summing the incremental quantities over all time intervals preceding it. The incremental work done by the support reactions is NS AWR -1§1( Isuzmi + pYiAyi + RZiAZi ) (2-45) 40 in which th, RYi and R21 are the X, Y, 2 components of the reaction and Axi, AYi, and A21 are the corresponding ground displacements at support 1, and N8 is the number of supports. The reactions at a support can be computed from summing the end forces of the structural members incident to that support. The incremental work done by gravity loads is NN - ~46 AVG ng NJ 3 AYJ (2 ) in which H3 is the lumped mass at a non-support node point j, g is the gravitational acceleration, A33 is the incremental vertical displacement of node point j, and NN is the number of free nodes. For a curved or straight beam element the displacement change {dq} over h may be, in the general case, decomposed into: {dq} - {dqee} + {dqep} + {dqpp} (2-47) in which, as defined in Section 2.5.4.6, {dqee} denotes that part of {dq} for both ends being elastic, {dqep} for one end elastic and the other end elasto-plastic, and {dqpp} for both ends plastic. The general case of force displacement change for a generic end, say end C, is illustrated in Figure 2-11 in which {Q1} denotes the force vector at the beginning of h, {Q2,ee}’ {Q2,ep} and {Q2,pp} denote the force vectors at the end of {dqee}, {dqep} and {dqpp}, respectively. The incremental strain energy Ae for the single member consists of an elastic part Ae 8 Se and a plastic part Aesp; i.e., Aes - Aese + Aesp (2-48) with T T AeSe - 0.5[<{Q1}+IQ2 eel) ‘dqee’ + ({Qz’ee)+{Q2’ep}) tdqep e} 41 + ({Q2 ep}+{Q2 pp1)T{dqpp,e}1 (2-49) T AeSp - 0.5[((Q2’ee}+{Q2’ep}) {dqep p} T + + d 2-50 ({Qz’ep} {02 pp!) K qpp p11 < > in which the third subscript "e" or “p” attached to {dq} denotes the "elastic part“ or “plastic part", i.e., {dqep} - {dqep e} + {dqep p} (2-51) d - d + 2-52 { qpp) { qpp.e} ‘dqpp.p} ( ) The incremental recoverable strain energy of the system is AESB - (AeSe)CB + (AeSe)SB + (AeSe)TR (2-53) in which the subscripts CB denote summing over all curved beam elements, SB over all straight beam elements, and (AeSe)TR - Z 0.5(Q1 + oz>dq <2-sa> denotes the incremental elastic strain energy due to all truss elements, with Q1 and Q2 representing the axial forces at the beginning and end of the time interval h, dq is the axial deformation changes taken place in the interval. The incremental irrecoverable strain energy is AESP - (Aesp)CB (2-55) The incremental kinetic energy over the time interval h is AER - 0.5<{é1T[m1{é1)t1 - o.s<{&1Ttm1{v})to <2-56> The incremental energy dissipated by damping is AED - 0.5[([C]{w))t1 + <[C]!é:>tolT{Aw) <2-s7> 42 338—,— ‘ ‘1 ‘ ( ’4 :2r M -m 4"» .32.? ’ 1 _ 7.- uni=!7'"" .«n I I ' I g 33 ’ :l ‘2 ’ ---SL’ : I ’6 I '3 -\ I "\\\l\i :: 2 ‘ "‘~:— 36 5: l- \ I in | 1 a e 31.5' - 700- Q Figure 2-1. Arch Bridge Considered Y a deck node X ( :‘1 fi_{ 2 Z / /\e rib node / 1/ t Figure 2-2. Nodal Coordinates for Deck and Arch Ribs Nodes 43 a transverse panel deck floor beam ’ Figure 2-3. Constraint for Infinitely Large Axial Stiffnesses Figure 2-6. Treatment o£_'Bigid Column” in Mixed Nodal Coordinates System 44 Figure 2-5. Typical Curved Beam lie-ant 45 9 ’3/ 0x3 3 / >5 “3 Figure 2-6. Curved Beam Element Ind Displacements yield surface L h x r: 5 1‘mo P80 ,5 _ 0.0 yield surface Figure 2-7. Spherical Yield Surface 46 eons-cu useaeoeuceun ou amassed-ounce eeouom ecu weasel .a-« eusuum an A A one A .aec x< egos A..euc mesa accede.» use Ace one message» ago sec ago nauseous oz and eo.n«o e\\\\x use so on mac a use an de.-a ao.n~o use use.» an.n~c < ‘60 an ensues. uses» 47 %"/jx V/axpc \ c \ \ \ ”general P.P." \ Q \ \ x Q \ ** "radial r.r.~ 1c Q20.s ch ;** Q . yield surface ’e-oo (a) Penetration point (P.P.) yield surface 'e-om (b) Iteration for d0 - O (staying on yield surface) Figure 2-9. Preliminary Definitions and Basic Operations 48 Figure 2-10. Definition of Cross-Section 49 yield surface 0 - 0.0 Figure 2-11. Typical Bomber End Force Path For An Elasto-Plastic Element CHAPTER III APPLICATIONS AND NUMERICAL RESULTS 3-1 am This chapter presents the numerical results based on the analyses described in the preceding chapter. The results were obtained by use of a computer program that incorporated the method of analysis presented. They cover three model bridges: one each of short span, medium span, and long span, and one ground motion. Nonlinear behavior due to material inelasticity is presented first, followed by behavior due to geometric nonlinearity (nonlinearly elastic behavior). In each case, the time histories of displacements, internal forces (stress resultants) and energy distribution are discussed. 3-2 W The computer program was built from one based on linear analysis using straight beam elements (Ref. [12]). It consists of approximately 6000 lines of FORTRAN statements. The major additions are the nonlinear curved beam elements (material and geometric). The bulk of the numerical results was obtained by use of the Supercomputer CYBERZOS at Purdue University with the linear equations subroutine written for vector processing. After the National Sciense Foundation ceased to support the Purdue Supercomputer operation, the 50 Sl computation was done on a VAX/VMS-ll/750 computer at College of Engineering, Michigan State University. For a problem solved by both computers for comparision, the latter was 56 times slower than the supercomputer. However, the numerical results were within 2% of each other. 3.3 W Three existing steel deck-type arch bridges: the Cold Springs Canyon Bridge (CSCB) in California, the South Street Bridge (SSB) in Connecticut, and the New River Gorge Bridge (NRGB) in West Virginia with arch spans of 700, 193, and 1700 feet, respectively, were chosen to be the prototypes for the medium span bridge (MSB), the short span bridge (SSB), and long span bridge (LSB). A complete real arch bridge system generally contains a large number of degrees of freedom. The expense for analysis of such a system can be kept within reasonable bounds in the case of a linear analysis. Because of the high cost of nonlinear analysis, it seemed necessary to use models that have fewer degrees of freedom than those used for linear analysis. For example, for the CSCB, the number of panels was reduced from eleven to eight. Nonlinear curved beam elements are used for the arch ribs. Straight beam and truss elements are used for the deck, columns, and cross bracing members. They are presumed to be entirely elastic. ”True” three-dimensional models with the two ribs modelled as distinct curved beam elements for MSB (Figure 3-1) and SSB (Figure 3-2), and “one-plane" three-dimensional model (Ref. [12]) with the two ribs (plus bracing) modelled as a single curved element for LSB (Figure 3-3) 52 were used to consider the inelastic effects (material nonlinearity). Yet a third type of model, i.e., a two-dimensional plane model (Figure 3-4) was used to study the nonlinear elastic effects. (Results for the three- dimensional models were obtained on the supercomputer and the latter two-dimensional models on the VAX.) A complete description of the properties and boundary conditions for these bridge models is given in APPENDIX. There are two differences between the true three-dimensional model and the one-plane three-dimensional model. Firstly, the axial force in a rib of a true three-dimensional model would be approximated by the axial force and the bending stress effects in a one—plane three- dimensional model. Secondly, the bending response of the individual ribs in a true three-dimensional model cannot be produced in a one-plane three-dimensional model. The natural periods of linear undamped vibration of.the first four modes for out-of-plane and in—plane motions are listed in Table 3-1 and the mode shapes are shown in Figures 3-5, 3-6, 3-7, 3-8, 3-9 and 3-10. The fundamental out-of-plane natural periods are 3.032, 1.180, and 4.716 seconds for MSB, SSB, and LSB, respectively. The fundamental in-plane natural periods are 2.247, 1.065, 3.565, and 2.514 seconds for MSB, SSB, LSB, and the two-dimensional MSB model, respectively. For this study, the basic ground motion used is an artificially generated motion, CIT-A2 (Figure 3-11) (Ref. [18]), which is intended to represent the ground motion near the epi—center of a magnitude 8 shock. It has a maximum acceleration of 0.39 g, a duration of approximately 120 seconds. In all cases, the first 45 seconds which covers the most intense part of the ground motion were used. The ground motion in all 53 three directions in space were used with the amplitude of the vertical motion being set equal to 75% of the two horizontal ones. The same ground motion was applied to all supports of the structure with no phase difference. 3-4 flAIEBIAL_EQHLIEEAR_EEQBLEES 3.4.1 general For the material nonlinear studies, the three-dimensional bridge models were used with ground accelerations applied in all three directions. In order to induce inelastic response, an amplification factor (AF) was applied to the basic ground motion as described in Section 3.3. The typical amplification factors used were 2.0, 1.2, and 2.0 for MSB, SSB, and LSB, respectively. (Using AF - 1.2 for the SSB model induced as much inelastic response as that for the MSB model with AF- 2.0.) ' Under the specified ground motion, Figures 3-12, 3-13, and 3-14 indicated for MSB, SSB, and LSB, respectively, the time intervals during which there was inelastic action (i.e., some elements yielding). One may observe that the first yielding occurred at 12.22 second for MSB, 7.54 second for SSB and 12.45 second for LSB. For the SSB even the AF used was 1.2 only, there were many members yielded within the duration of 45 seconds. 3-4-2 W In this section displacement time histories are presented for three nodes in each bridge model. For each node, three displacement components are plotted. They are X (horizontal), Y (vertical), and Z 54 (lateral; out-of—plane) displacement in the cartesian coordinates system. 3.4.2.1 fledium Span Bridge (MSB) The three nodes selected for this bridge are: node 17, 31 and 5 (See Figure 3-1). Node 17 is at the crown of the front arch rib, node 5 is at the 1/8 span of the front arch rib, and node 31 is at the 7/8 span of the rear arch rib. The results indicate that at both node 5 and 31 there were yielding of the arch ribs. The horizontal (X) displacement time histories for the three nodes are shown in Figures 3-15, 3-16, and 3-17. They indicate that the predominant period is approximately 2.2 seconds. The first in-plane undamped natural period had been calculated to be 2.247 seconds (See Table 3-1). Thus the response was then primarily in the first mode. There were also some contributions, though small, from the third mode (undamped natural period equal to 0.685 seconds). It may be noted that the second mode is symmetric (Figure 3-5) which would not be excited by the horizontal ground motion. Experience indicates that the effects of horizontal ground motion dominate over those of vertical ground motion. Because of antisymmetry in mode shapes, the horizontal displacement time histories of node 5 and 31 are in phase and almost equal. The wave patterns for the elastic and inelastic responses are quite similar. The vertical (Y) displacement time histories are shown in Figures 3-18, 3-19, and 3-20. The predominant period is again approximately 2.2 seconds for time histories of nodes 5 and 31. As above, it indicates that the response was mostly in the first in-plane mode. Similarly, there were appreciable third mode contributions. These 55 vertical displacement time histories of node 5 and 31 are out-of-phase because of antisymmetry in mode shapes. The dominant period for node 17 is approximately 3.0 seconds which corresponds to the first out-of-plane mode. One can observe from Figure 3-5 that the vertical displacement is zero at the crown node in the first in-plane mode. Thus the displacements are apparently excited by the out-of-plane ground motion (See the first out-of-plane mode shapes in Figure 3-6). The wave patterns for the elastic and inelastic responses are similar too. The lateral (Z) displacement time histories are shown in Figures 3-21, 3-22, and 3-23. The predominant period is approximately 3.0 seconds. It indicates that the response was primarily in the first out- of-plane mode. (Table 3-1 shown the first out-of-plane natural period to be 3.032 seconds.) One can also observe that the fourth mode with period equal to 0.942 seconds participated significantly in the overall response of node 5 (Figure 3-21) and node 31 (Figure 3-23). The lateral displacement time histories of node 5 and 31 are in phase and almost equal. The wave patterns for the elastic and inelastic responses are quite similar too. 3.4.2.2 WE). For this bridge the nodes chosen for presenting displacement time histories are the two symmetric points: node 11 and 23. They are at the 2/7 and the 5/7 span of the rear rib (Figure 3-2), respectively. A third point chosen is node 13 at the 3/7 span of the front rib. The displacement time histories in the X, Y, and 2 direction for these nodes are shown in Figures 3-24 through 3-32. In general, they are similar in character to those presented above for the MSB. Hence, no 56 further discussion will be presented for them. 3.4.2.3 W The displacement time histories for this bridge are shown in Figures 3-33 through 3-41. They refer to node 18 at the crown, node 12 at the 4/14 span of the arch rib, and node 24 at the 10/14 span of the arch rib (See Figure 3-3). Node 12 and 24 are symmetric points and there was yielding near them. These time history results are also similar in character to those presented for MSB, except for the following point. For the vertical displacement time history of the crown point (node 18, See Figure 3-37), the response was primarily in the second in- plane mode (1.5 seconds) and fourth in-plane mode (1.1 seconds) as expected. Because the crown node has zero vertical displacement in the first and third in-plane modes (Figure 3-8). It differs from the response of MSB because, as mentioned previously, LSB is a one-plane model. The vertical displacement of the rib in that model corresponds to that of the center line of the bridge. That displacement is essentially unaffected by lateral ground motion. To deduce the actually vertical displacements along the edges of a cross-section of the bridge represented by a one-plane model, one needs to consider the torsional response. 3.4.3 mm For the MSB model, the magnitudes of the maximum displacements at the above selected nodes (Section 3.4.2.1) and certain additional nodes are listed in Table 3-2. The odd numbers correspond to nodes on the ribs and the even numbers on the deck (Figure 3-1). The maximum 57 horizontal displacement occurring at the node next to a support amounts to 1.1% of the arch height (121.25 feet) for the linear elastic case and 1.13% for the inelastic case. The horizontal displacements in the deck are smaller. There are no significant differences between the vertical displacements in the ribs and deck. The maximum vertical displacement occurred at the mid-span and amounted to 2.02% of the arch height for elastic case and 2.12% for inelastic case. The maximum lateral rib displacement also occurred at the mid-span and is considerably larger, 8.13% of the arch height for elastic case and 7.6% for inelastic case. The deck nodes all had even larger lateral displacement. The maximum responses of the rear and front rib were approximately equal. The maximum displacements for the SSB model are presented in Table 3-3. As before, the odd numbers are on the ribs and the even number are on the deck (Figure 3-2). There are no significant differences between the rib and deck for horizontal and vertical displacements. The maximum horizontal displacement was 1.58% of the arch height (28.398 feet) for the elastic case and 1.43% for the inelastic case. The corresponding maximum vertical displacements were 3.17% and 3.12%. The corresponding maximum lateral displacements were 3.48% and 3.29%. For this bridge, the inelastic displacement responses were smaller than the elastic responses. But the differences are relatively small. The maximum displacements for the LSB model are listed in Table 3-4. The even numbers are on the rib and the odd numbers are on the deck (Figure 3-3). The maximum horizontal displacement was 0.98% of the arch height (370 feet) for the elastic case and 0.95% for the inelastic 58 case. The corresponding maximum vertical displacements were 1.65% and 1.61%. The corresponding maximum lateral displacements were 2.73% and 2.54%. 3-4-4 We: In this section the force time histories are presented for two rib points in each bridge. At each point, three components are plotted: Pz (axial force), My (referred to as ”in-plane bending" because the bending takes place in the plane of the rib), and fix (referred to as 'out-of-plane bending” because the bending causes deformation out of the plane of the rib). 3.4.4.1 W The two points selected are: and J of member 1 at node 5 (the 1/8 span of the front arch rib) and end I of member 16 at node 31 (the 7/8 span of the rear arch rib) (See Figure 3-1). They are symmetric points and yielding had been indicated in the inelastic analysis. The P2 time histories are shown in Figures 3-42 and 3-43. The predominant period is approximately 3.0 seconds which indicates that the response was mostly in the first out-of-plane mode. In this mode, the two ribs essentially act as the two opposite flanges of a beam. One can also observe that the fourth in-plane mode (with a period equal to approximately 0.5 seconds.) participated significantly in the overall response. In this mode, the two ribs would respond identically. Because of the sign conventions used in the analysis (See Figure 2-6), the two time histories are seen to be in phase for the first out-of-plane mode and out-of-phase for the fourth in-plane mode. As it was the case for 59 displacement time histories, the wave patterns for the P2 forces for the elastic and inelastic responses are quite similar. The in-plane rib bending moment time histories are shown in Figures 3-44 and 3-45. The predominant period is approximately 3.0 seconds which indicates that the response was again primarily in the first out-of-plane mode. (Note that in-plane bending of ribs may exist in an out-of-plane normal mode. See Figure 3-6) Similarly, there was substantial fourth in-plane mode contribution. The two time histories are in phase for the first out-of-plane mode and out-of-phase for the fourth in-plane mode. The out-of-plane bending time histories are shown in Figures 3-46 and 3-47. The predominant period was measured as 3.0 seconds which indicates that the response was primarily in the first out-of-plane mode. The two time histories are out-of-phase. 3.4.4.2 WE). The two points selected for this structure is and J of member 9 at node 11 (the 2/7 span of the rear arch rib) and end I of member 13 at node 23 (the 5/7 span of the rear arch rib) (See Figure 3-2). The results are shown in Figures 3-48 through 3-53. In general, they are similar in character to those presented above for the MSB. Hence, no further discussion will be presented for them. 3.4.4.3 W The points selected are end I of member 5 at node 12 (the 4/14 span of the arch rib) and end J of member 10 at node 24 (the 10/14 span of the arch rib) (See Figure 3-3). They are symmetric points and 60 yielding did occur at both points. The axial force time histories are shown in Figures 3-54 and 3-55. The predominant period is approximately 1.1 seconds which indicates that the response was mostly in the fourth in-plane mode. It may be noted from Figure 3-8 that the fourth mode is the lowest mode in which the axial force dominates (over bending). The wave patterns for the elastic and the inelastic responses are quite similar. The in-plane bending time histories are shown in Figures 3-56 and 3-57. The predominant period is approximately 3.5 seconds which indicates that the response was primarily in the first in-plane mode. There was also small contributions from higher modes. The two time histories are in phase for the first in-plane mode and out-of-phase for the higher modes. The out-of-plane bending time histories are shown in Figures 3-58 and 3-59. The predominant period is approximately 0.5 seconds which indicates that the response was primarily in the higher mode. The two time histories are out-of-phase. The question arose why there was little first out-of-plane mode response. The time histories of out-of-plane bending at the two nodes (node 10 and 14) adjacent to node 12 were obtained and are shown in Figures 3-60 and 3-61. It can be seen that the responses at both nodes are mostly in the first out-of-plane mode. Thus it may be concluded that the selected points (node 12 and 24) are very close to points of inflection (zero moment) for out-of-plane bending in the fundamental out-of-plane mode. 61 3.4.5 W The maximum values of the internal forces: Pz’ My and M.x for the MSB model are listed in columns 3 and 4 of Table 3-5. It is seen that they differ considerably for the elastic and inelastic cases (although the time history wave patterns are quite similar for the two cases). It may be noted that the maximum values in the inelastic case'are limited by the fully plastic values (See Eq. 2-20 and Table 3-5). No such limits exist in the elastic case. For an elastic design, the design is generally carried out based on the maximum forces as indicated by the elastic analysis. For an inelastic design, if the designer would accept the "damage" due to plastic deformations, the design could be carried out based on the maximum force as indicated by the inelastic analysis. Assuming that the material needed is roughly proportional to the design force, it follows that savings in material are possible if the "inelastic force" is less than the “elastic force". Let the term ”force reduction factor" or simply "reduction factor," be defined as: Fe - Fie reduction factor - (3-1) F e in which, Fe - maximum force by elastic analysis and F1e - maximum force by inelastic analysis. The factor may be regarded as a measure of possible material saving using inelastic design. Its values are listed in Table 3-5 for the data presented above. It is seen that they are of appreciable magnitude. The largest reduction factor for the SSB (listed in Table 3-6) is 0.223 for in plane bending of member 9 end J. It is less than the 62 maximum values of 0.333 of the preceding MSB case. But the latter is based on a ground acceleration amplification factor AF - 2.0, while for the SSB model, AF - 1.2. Thus, the reduction factor per unit of AF is approximately the same for the two cases. The largest reduction factor for the LSB (See Table 3-7) is 0.137 for in-plane bending of member 10 end J. It is smaller than the maximum reduction factors for either the MSB or SSB. It may also be noted that the values of the dimensionless response (displacements) for LSB model are also smaller than the other two bridges. 3 4.6 Ixnissl_E2rk_and_EnsIsx_Distributign_Iims_flist2ries The work and energy balance equation of the system was presented in Section 2.7 as: "g + “c ' ESE + ESP I Ex + En (2’44) in which, as noted in Chapter II, WR and WG denote work done by support reactions and by gravity loads, respectively, ESE is recoverable strain energy, ESP is irrecoverable (plastic) strain energy, BK is kinetic energy and ED is energy dissipated by damping. During an earthquake energy is fed through the base of the structure. It is instructive to know how the energy is distributed among the terms in the work and energy balance equation. There is no irrecoverable (plastic) strain energy for a linear elastic case, therefore, for that case the term ESP should be deleted from the equation. The equation representing work and energy distribution may be rewritten from Eq. 2-44 as follows: ESE ESP Ex ED + -———— + + W W W W 1.0 - (3-2) 63 in which W - WR + WG. The equation becomes 1.0-SE+KE+DE+PW (3-3) where SE - ESE/W, XE - EK/W, DE - ED/W and PW - ESP/W are presented in the figures of the work and energy distribution time histories. A spline technique was used in the graphs to connect the discrete points. The work and energy distribution time histories of the MSB for the elastic and inelastic cases are shown in Figures 3-62 and 3-63. The elastic case has a greater percentage of dissipated damping energy compared to the inelastic case. The dissipated damping energy continued to increase to 87% of the work done for the elastic case and 74% for the inelastic case. For both cases the kinetic and recoverable strain energy remain relatively small throughout the whole 45 seconds and the elastic case has a greater percentage. For the inelastic case, 18% of the work done is shown to be dissipated by irrecoverable strain energy after about 20 seconds. From Figure 3-63, a decrease in PW can be noted. It is because of the fact that the ratio ESP/W is plotted, i.e., a decrease signifies that the denominator, the work done, has increased faster than the numerator. The work and energy distribution time histories of the SSB for the elastic and inelastic cases are shown in Figures 3-64 and 3-65. The dissipated damping energy continued to increase at an almost constant rate to 88% of the work done for the elastic case and 74% for the inelastic case. The percentages of the kinetic and recoverable strain energy were smaller than the corresponding percentages of MSB. According to Figure 3-65, 17% of the work done is shown to be dissipated by irrecoverable strain energy after about 15 seconds. The work and energy distribution time histories of the LSB for 64 the elastic and inelastic cases are shown in Figures 3-66 and 3-67. The dissipated damping energy continued to increase at an almost constant rate to 85% of the work done for the elastic case and 75% for the inelastic case. The percentages of the kinetic and recoverable strain energy were larger than the corresponding percentages of MSB with the same amplification factor. For the inelastic case, 12% of the work done is shown to be dissipated by irrecoverable strain energy after about 23 seconds. 3.4.7 . : 9. - 1:71 . : -_ ., 1 . - .,7 'an- ,. :: . The results of the work and energy distribution time histories, obtained by using a two-dimensional model of MSB (Figure 3-4) with different damping ratios, are shown in Figures 3-68 through 3-74. The . responses are listed in Table 3-8. It is seen that the inelastic responses and reduction factor decreased when the value of damping ratio was increased. The amplification factor used is 2.0 for all cases. There is no inelastic effect when the damping ratio is equal to 5%. In that case, most (96%) of the work done was dissipated by damping. When the damping ratio is equal to 0.25%, 50% of the work done was dissipated by damping and 30% of the work done was dissipated by irrecoverable strain energy. The ratios of irrecoverable strain energy, kinetic energy and recoverable strain energy to the work done increased when the damping ratio decreased. 3.4.8 For a given bridge model and ground motion, one may carry out a linear elastic analysis. The quantity 3;, for the case of "spherical 65 plastic potential function," is defined as: - P 2 "I 2 Mx 2 me - ( ———- ) + ( ) + ( ————‘ ) (3'4) Po Myo MxO (The yielding condition is defined by setting 8; - 1 - 0 (Eq. 2-20).) It is a function of time and position in the structure. Its value can be greater than 1. Let ca be the maximum value of 5; for all points and times considered in the linear analysis. It is called the linear response factor. For the same bridge model and ground motion, an inelastic analysis may be carried out. The maximum inelastic response in terms total plastic work (Eq. 2-39), plastic work density (Eqs. 2-40) and ductility factor (Eqs. 2-41 through 2-43) may be computed. These inelastic response quantities may be used to represent measures of "damages” done to the structure due to inelasticity. They are plotted for the most severely strained member in the bridge MSB, in Figure 3-75 as a function of oe. It is seen that the inelastic response quantities generally increase with the linear response factor. This is expected. Following the concept of force reduction factor as discussed in Section 3.4.5, one might use such a graph as follows. If one would accept a certain level of "damage”, for example, a curvature ductility factor about y-axis equal to 3.5, the corresponding curve in Figure 3-75 would indicate a linear response factor equal to 1.88. This factor may be used as a "load reduction factor“ (to be applied to the "seismic load" based on linear behavior) or an allowable stress increase factor. Similar curves are presented for the other two bridge models in Figures 3-76 and 3-77. 66 3.5 W 3.5.1 General In the preceding the nonlinear effects due to material inelasticity were considered. In this section numerical results on nonlinear elastic effects (“geometric nonlinearity”) are presented. For simplicity, only in-plane behavior is studied. The same bridge models and earthquake loading considered previously are used with the exception that the MSB is simplified to a 4-panel model as depicted in Figure 3-4. The main feature of nonlinear elastic effects is that the axial compressive force in the arch ribs would lower the stiffness of the structure (analogous to the case of a beam-column). That would change the response characteristic and could even lead to instability. 3.5.2 W 3.5.2.1 W For this bridge, typical displacement time histories are presented for node 3 (at the 1/4 point of the rib) and node 5 (at the crown). The horizontal displacement time histories are shown in Figures 3.78 and 3-79 for both the linear and nonlinear analyses. The predominant period for the linear elastic case is approximately 2.5 seconds which corresponds to the first mode (Table 3-1). For the nonlinear elastic case, the measured period is approximately 2.8 seconds which represents approximately a 10% increase from the linear case. This, of course, is due to the effects of the decreased stiffness of the ribs resulting from dead load compression. The vertical displacement time histories are shown in Figures 67 3-80 and 3-81. For the quarter point (node 3), the predominant period is that of the first mode (2.5 seconds) for the linear case. The crown point (node 5) has zero vertical displacement in the first mode (See Figure 3-10), which explains the fact that the response shows mainly a combination of second and third modes (1.15 and 0.50 seconds, respectively). For the nonlinear response, one can again notice an increase in the values of the dominant period. For the quarter point, it is approximately 10%, as for the preceding case of horizontal displacement. For the crown point, the change is from 1.15 seconds to 1.2 seconds representing an increase of about 5% in the second mode. 35-2-2 WEE). This bridge model has no node at the crown. The displacements of the two nodes symmetric with respect to it are considered. They are node 9 (at the 2/7 span) and node 21 (at the 5/7 span) (See Figure 3-2). The horizontal displacement time histories are shown in Figures 3-82 and 3-83. For the linear elastic case, the predominant period is approximately 1.1 seconds which corresponds to the first mode. The period for the nonlinear elastic case is approximately 1.2 seconds which represents approximately a 9% increase. The responses of the two nodes are in phase. The vertical displacement time histories are shown in Figures 3-84 and 3-85. The features of dominant periods of these responses are similar to those discussed above. The responses of the two nodes are out -of-phase. 68 3.5-2.3 mm For this bridge model, the displacements are considered at node 18 (the crown), and two nodes symmetric to it: node 12 (at the 4/14 span) and node 24 (at the 10/14 span) (See Figure 3-3). The horizontal displacement time histories are shown in Figures 3-86, 3-87, and 3-88. Again, the response was primarily in the first mode (with a period of about 3.5 seconds). There was approximately a 8% increase of the dominant period for the nonlinear elastic case (3.8 seconds) from the linear elastic case. The vertical displacement time histories are shown in Figures 3-89, 3-90 and 3-91. The responses of the two symmetric nodes, nodes 12 and 24, were mostly in the first mode and there was little contribution from higher modes. The response of the crown node, node 18, was primarily in the second mode (1.5 seconds) and fourth mode (1.1 seconds). The change in the dominant period is approximately a 7% increase for these nodal displacements. 3.5-3 mm The maximum displacements are listed in Tables 3-9, 3-10 and 3-11 for MSB, SSB and LSB, respectively. The odd numbers represent nodes on the rib and the even numbers on the deck for MSB and SSB (Figures 3-4 and 3-2). For LSB, the even numbers are on the rib and the odd numbers are on the deck (Figure 3-3). In Table 3-9, It can be seen that the maximum horizontal displacements for MSB are 0.77% of the arch height (121.25 feet) for the linear elastic case and 0.94% for the nonlinear elastic case. The maximum vertical displacements are 1.69% of the arch height for the 69 linear elastic case and 1.79% for the nonlinear elastic case. As listed in Table 3-10, the maximum horizontal displacements for SSB are 1.56% of the arch height (28.398 feet) for the linear elastic case and 1.19% for the nonlinear elastic case. The corresponding maximum vertical displacements are 3.22% and 2.59%. As listed in Table 3-11, the maximum horizontal displacements for LSB are 0.98% of the arch height (370 feet) for the linear elastic case and 0.92% for the nonlinear elastic case. The corresponding maximum vertical displacements are 1.65% and 1.6%. 3-5-4 MW Two points in each bridge model are selected for presentation of force time histories. At each point, two components are plotted: Pz (axial force) and My (in-plane bending). 3.5.4.1 W The two points selected are end J of member 1 at node 3 (the 1/4 span of the arch rib) and end I of member 4 at node 7 (the 3/4 span of the arch rib) (See Figure 3-4). They are symmetric with respect to the crown. The axial force time histories are shown in Figures 3-92 and 3-93. The predominant period is approximately 0.5 seconds which corresponds to the third mode. As was noted previously in Section 3.4.4.3, this dominance appears to be due to the fact that this third mode is the lowest mode in which axial force dominates (over bending) (See Figure 3-10). The wave patterns for the linear elastic and nonlinear elastic responses are quite similar. 70 The in-plane bending time histories are shown in Figures 3-94 and 3-95. In this case, it appears that the responses were mainly in the first and third modes. (These points, at which the responses are being considered, appear to be close to points of counter-flexure for the second mode.) 3.5-4.2 WEB). The two points selected are and J of member 2 at node 9 (the 2/7 span of the arch rib) and end I of member 6 at node 21 (the 5/7 span of the arch rib) (Figure 3-2). The results on this bridge are shown in Figures 3-96 through 3-99. They are generally similar in character to those presented in the preceding. However, the in-plane bending response was primarily in its first mode. Unlike the preceding case of MSB, there was little contribution from the third mode. It should be noted that the points, at which the responses are being considered, do not quite correspond for the two bridge models. 3.5.4.3 W The two points selected are and I of member 5 at node 12 (the 4/14 span of the arch rib) and end J of member 10 at node 24 (the 10/14 span of the arch rib) (See Figure 3-3). They are the same points from the same ("one-plane") model in Section 3.4.4.3 for material nonlinearity. The results on this bridge are shown in Figures 3-100 through 3-103. It can be seen that the results are very close to those presented in Section 3.4.4.3. While in that section ground motions in three 71 directions were applied, here only in-plane motions are applied. These results thus suggest that the in-plane and out-of-plane responses were largely uncoupled for this ("one-plane") bridge model. 3.5.5 W The maximum internal forces are listed in Tables 3-12, 3-13 and 3-14 for MSB, SSB, and LSB, respectively. There is no appreciable difference between linear elastic and nonlinear elastic solution. 3.5-6 WWW As was done in Section 3.4.6 for the case of material nonlinearity, the work and energy distribution time histories of the linear elastic and nonlinear elastic case of MSB, SSB, and LSB were calculated and are shown in Figures 3-104 through 3-109. It may be seen that the differences between the linear and nonlinear elastic cases are very small for all three bridge models. The values of the ratios plotted for the kinetic and strain energy of SSB are larger than those of MSB and less than those of LSB. However, for both cases the kinetic and recoverable strain energy remain relatively small throughout the whole 45 seconds. This points to the importance of the role of damping. 357 111W The results in Sec. 3.5.3 indicated that a consideration of the nonlinear elastic behavior increased the maximum displacement by 10% to 20% for MSB. But for the other two bridge models, such effects of the nonlinear behavior were much smaller (even negative). To consider 72 possible instability effects, the static buckling loads were computed as linear eigenvalue problems. They were found to be 4.415 D.L., 7.185 D.L. and 8.557 D.L. (D.L. stands for ”dead load" for the respective models) for MSB, SSB and LSB, respectively. The seismic responses were computed by increasing the initial load (initially applied static load) to be a factor times D.L.. No instability effects were observed until the initial load was closed to the static buckling load. This is illustrated in Table 3-15, in which are listed the maximum horizontal displacements at the crown for MSB and LSB, and at node 7 for SSB. It is seen that the maximum displacements remain quite moderate until the dead load placed almost reached the magnitude of the static buckling load. The responses for MSB and SSB became “infinite" at 99% of the buckling load. However, for LSB (See also Figure 3-110), even after the initial load exceeded the static buckling load the responses remained "finite” (but at larger rates of increase). The reason may lie in the difference in the boundary conditions between LSB and the other two bridge models (See Figures 3-2, 3-3 and 3-4.) Table 3-1. Natural Periods of Vibration 73 Out-of-Plane (second) Mode No. MSB SSB LSB 1 3.032 1.180 4.716 2 1.204 1.127 3.677 3 1.174 0.379 2.612 4 0.942 0.370 1.424 In-Plane (second) Mode No. MSB SSB LSB 1 2.247 1.065 3.565 2 1.276 0.484 1.534 3 0.685 0.304 1.221 4 0.469 0.242 1.094 Two-Dimensional model of MSB (4 panels) (second) Mode No. 1 2 3 4 MSB 2.514 1.151 0.503 0.173 74 Table 3-2. Elastic and Inelastic Maximum Displacements of MSB Nodal Displacements(ft) No. Horizontal Vertical Lateral No. 1.082 1.637 0.856 5 (1.034) (1.782) ( 0.834) No. 0.383 1.625 3.747 6 (0.359) (1.795) (3.528) No. 1.237 1.867 0.856 7 (1.370) (2.073) (0.834) No. 0.401 1.855 3.747 8 (0.376) (2.049) (3.528) No. 0.455 2.437 7.772 17 (0.541) (2.267) (7.239) No. 0.177 2.449 9.858 18 (0.169) (2.267) (9.215) No. 0.456 2.389 7.772 19 (0.534) (2.558) (7.239) No. 0.178 2.389 9.858 20 (0.173) (2.571) (9.215) No. 1.334 2.073 0.867 29 (1.225) (1.976) (0.732) No. 0.588 2.098 4.765 30 (0.552) (2.001) (4.401) No. 1.124 1.843 0.867 31 (1.225) (1.855) (0.732) No. 0.567 1.879 4.765 32 (0.531) (1.843) (4.401) ( ..... ) : Inelastic Displacement 75 Table 3-3. Elastic and Inelastic Maximum Displacements of SSB Nodal Displacements(ft) No. Horizontal Vertical Lateral No. 0.426 0.898 0.520' 9 (0.378) (0.835) (0.503) No. 0.388 0.900 0.987 10 (0.355) (0.837) (0.922) No. 0.449 0.884 0.520 11 (0.400) (0.884) (0.503) No. 0.386 0.886 0.987 12 (0.360) (0.886) (0.922) No. 0.369 0.686 0.851 13 (0.334) (0.771) (0.804) No. 0.383 0.689 0.950 14 (0.350) (0.771) (0.893) No. 0.371 0.726 0.851 15 (0.339) (0.830) (0.804) No. 0.383 0.729 0.950 16 (0.357) (0.832) (0.893) No. 0.447 0.830 0.520 21 (0.400) (0.853) (0.487) No. 0.386 0.835 0.987 22 (0.353) (0.856) (0.933) No. 0.421 0.835 0.520 23 (0.407) (0.766) (0.487) No. 0.388 0.837 0.987 24 (0.360) (0.771) (0.933) ( ..... ) : Inelastic Displacement 76 Table 3-4. Elastic and Inelastic Maximum Displacements of LSB Nodal Displacements(ft) No. Horizontal Vertical Lateral No. 3.619 6.068 5.365 12 (3.156) (5.439) (5.180) No. 2.723 6.105 10.101 13 (2.560) (5.476) (9.250) No. 2.868 2.320 8.658 18 (2.538) (2.213) (8.362) No. 2.150 2.331 7.548 19 (2.146) (2.224) (7.141) No. 3.530 5.883 5.365 24 (3.497) (5.920) (5.254) No. 2.609 5.920 10.101 25 (2.560) (5.957) (9.398) ( ..... ) : Inelastic Displacement 77 Table 3-5. Elastic and Inelastic Maximun End Forces of MSB Member Linear Nonlinear Reduction Fully Plastic End Elastic Inelastic Factor Force P2 8907 8635 0.031 10064.34 1-J My 40547 29584 0.270 30034.75 Mx 4369 4989 ' -0.142 12169.45 P2 9088 7890 0.132 10064.34 16-I My 43851 29344 0.331 30034.75 Mx 4381 4077 0.069 12169.45 Pz : kips ; M& : ft-kips ; Mg : ft-kips Table 3-6. Elastic and Inelastic Maximun End Forces of SSB Member Linear Nonlinear Reduction Fully Plastic End Elastic Inelastic Factor Force P2 2669 2455 0.080 4554.000 9-J M.y 7235 5623 0.223 5696.625 MX 185 193 -0.043 3112.313 P2 2910 2637 0.094 4554.000 13-1 M.y 6779 5577 0.177 5696.625 Mx 185 247 -0.335 3112.313 -Pz : kips ; M? : ft-kips ; M; : ft-kips Table 3-7. Elastic and Inelastic Maximun End Forces of LSB Member Linear Nonlinear Reduction Fully Plastic End Elastic Inelastic Factor Force Pz 68350 67605 0.011 82849 5-1 M.y 1667632 1554821 0.068 1634933 Mx 322119 322119 0.000 2982580 Pz 66610 66610 0.000 82849 10-J My 1765728 1523757 0.137 1634933 M.x 322119 283643 0.119 2982580 P: : hips ; M& : ft-kips ; M¥ : ft-kips 79 Table 3-8. Maximum Responses for Various Values of Damping Ratio Damping Node 5 Displacement(ft) Ductility Plastic Work Total ratio Factor Density by Plastic (%) Horizontal Vertical M.y M.y Work 0.25 0.558 0.825 17.560 527.33 11094 0.5 0.571 0.808 16.049 330.56 6772 1.0 0.761 0.764 14.591 140.72 2599 2.0 0.618 0.732 5.909 56.92 884 3.0 0.557 0.755 2.083 20.31 272 5.0 0.468 0.743 ----------------- Damping Member 1 End J Ratio Axial Force (Pz) In-Plane Bending (My) (%) Inelastic Elastic Reduction Inelastic Elastic Reduction 0.25 9197 15420 0.404 48087 112430 0.572 0.5 8922 11841 0.246 48136 98802 0.513 1.0 7972 9982 0.201 48330 79334 0.391 2.0 6870 7724 0.110 44923 61325 0.263 3.0 6182 6499 0.049 45313 52565 0.138 5.0 5259 5259 0.000 43123 43123 0.000 Pz : kips ; M 7 : ft-kips 80 Table 3-9. Linear and Nonlinear Elastic Maximum Displacements of MSB Nodal Linear Elastic (ft) Nonlinear Elastic (ft) No. Horizontal Vertical Horizontal Vertical 3 0.918 1.831 1.135 2.037 4 0.409 1.879 0.409 2.086 5 0.650 0.955 0.833 1.135 6 0.663 0.968 0.663 1.146 7 0.935 2.001 1.062 2.134 8 0.832 2.049 0.832 2.170 Table 3—10. Linear and Nonlinear Elastic Maximum Displacements of SSB Nodal Linear Elastic (ft) Nonlinear Elastic (ft) No. Horizontal Vertical Horizontal Vertical 5 0.443 0.912 0.329 0.684 6 0.389 0.914 0.312 0.687 7 0.378 0.525 0.278 0.491 8 0.386 0.528 0.307 0.491 11 0.437 0.849 0.338 0.733 12 0.389 0.852 0.312 0.736 81 Table 3-11. Linear and Nonlinear Elastic Maximum Displacements of LSB Nodal Linear Elastic (ft) Nonlinear Elastic (ft) No. Horizontal Vertical Horizontal Vertical 12 3.619 6.068 3.414 5.920 13 2.723 6.105 2.542 5.920 18 2.868 2.320 2.642 2.301 19 2.150 2.331 1.865 2.309 24 3.530 5.883 3.230 5.587 25 2.609 5.920 2.290 5.624 Table 3-12. Linear and Nonlinear Elastic Maximum End Forces of MSB Linear Elastic Nonlinear Elastic Member Axial Force In-Plane Axial Force In-Plane End Pz Bending M Pz Bending M kips ft-kips y kips ft-kips y 1-J 8633 67653 8564 71546 4-1 8784 68626 8756 69600 Table 3-13. Linear and Nonlinear Elastic Maximum End Forces of SSB Linear Elastic Nonlinear Elastic Member Axial Force In-Plane Axial Force In-Plane End Pz Bending M Pz Bending M kips ft-kips y kips ft-kips y 2-J 2268 7064 2218 6152 6-I 2026 6551 2077 6665 Table 3-14. Linear and Nonlinear Elastic Maximum End Forces of LSB Linear Elastic Nonlinear Elastic Member Axial Force In-Plane Axial Force In-Plane End Pz Bending M Pz Bending M kips ft-kips y kips ft-kips y 5-1 68350 1667632 68599 1623488 10-J 66610 1765728 69593 1554821 83 Table 3-15. Maximum Displacements for Different Values of Initial Load MSB SSB (Buckling Load - 4.415 D.L.) (Buckling Load - 7.185 D.L.) Initial Node 5 Initial Node 7 Load Horizontal Displ. Load Horizontal Displ. 4.300 D.L. 0.3492 ft 7.000 D.L. 0.5992 ft 4.350 D.L. 0.4450 ft 7.100 D.L. 0.4089 ft 4.360 D.L. 0.4729 ft 7.150 D.L. 0.4742 ft 4.365 D.L. 0.4874 ft 7.155 D.L. 0.5311 ft 4.370 D.L. 2.0E+10 ft 7.160 D.L. 6.5031 ft LSB (Buckling Load - 8.557 D.L.) Initial Node 18 Initial Node 18 Load Horizontal Displ. Load Horizontal Displ. 8.45 D.L. 1.2136 ft 8.56 D.L. 2.7454 ft 8.46 D.L. 1.2506 ft 8.57 D.L. 3.7740 ft 8.47 D.L. 1.2987 ft 8.58 D.L. 5.2540 ft 8.48 D.L. 1.3579 ft 8.59 D.L. 7.2890 ft 8.49 D.L. 1.4356 ft 8.60 D.L. 9.5090 ft 8.50 D.L. 1.5281 ft 8.61 D.L. 13.801 ft 8.51 D.L. 1.6428 ft 8.62 D.L. 18.944 ft 8.52 D.L. 1.7760 ft 8.63 D.L. 25.567 ft 8.53 D.L. 1.9499 ft 8.64 D.L. 33.078 ft 8.54 D.L. 2.1645 ft 8.65 D.L. 43.290 ft 8.55 D.L. 2.4235 ft 8.66 D.L. 54.760 ft ll 84 Auovox mmaodmfloflun-oounb tombs: 4v devoi Anmlv autumn comm law‘s! 17.." 85m: .8” n .93 w a \. 85 Aeneas Heeoaneoaan-eouae gonna. 4V doves sunny census cede eaten .~-n ensues .QN 1— .nGfl I .GN W n a 91’] aOLS L. F‘ “Hove: .eeeaa-oeo. do some: Amuse oueatn team used ‘3 b O F. \n m .n.n .teenn .eona n .ne.a~a w ea an: 87 sauces announce-ea-otav seeds engage teem Issues .eqn gunman aS'EEI .eee u .nea w e Front and Rear Deck Front and Rear Rib Mode 1 (2*) Front and Rear Deck Front and Rear Rib and. 3 (3*) Front and Rear Deck ——»x Front and Rear Rib and. 2 (3*) “7!-xv 1“? Front and Rear Deck -—~x Front and Rear Rib and. 4 (12*) (* denotes overall mode number.) Figure 3-5. In-Plane Mode Shapes for MSB 89 x Y Y ‘ ll H A“ J-\ 21! =:~x :g_x Front and Rear Deck Front Rib Front Deck 1 ‘ Y Y II A /”‘\A ‘\ ’1' ‘4 ‘7 Front and Rear Rib Rear Rib Rear Deck and. 1 (1*) ._ x Y Y * 1| fl /‘-\\ 4“ /\ ‘\_../’ ‘~-f \ 7 Front and Rear Deck Front Rib Front Deck x ‘Y Y " A n s/«\ 1"\ /A\ \\_../r \J' \J 21! ==:x -—~x Front and Rear Rib Rear Rib Rear Deck * and. 2 (e*-uss; 3 -833) (* denotes overall mode number.) Figure 3-6. Out-of-Plane Mode Shapes for MSB and SSB [ 2 Front and Rear Deck ——»1 21! Front and Rear Rib 211 Front and Rear Deck ‘r’x 21! Front and Rear Rib 90 X Front Rib X Rear Rib Mode 3 (5*) --X Front Rib Y A :1 Rear Rib and. 4 (6*) 1 LV \__/’ ::;X Front Deck 4/‘.:; ‘\-'/r : x Rear Deck 1-\\~/1‘-\ v—ex Front Deck z”‘\ T x Rear Deck (* denotes overall mode number.) Figure 3-6. (Continued) Out-of-Plane Mode Shapes for M38 and SSB 91 Front and Rear Deck I: x Front and Rear Rib Mode 1 (2*) l/‘\ 4"\ \~./r \~/' Front and Rear Deck Y :— x Front and Rear Rib Mode 3 (7*) Front and Rear Deck 2: x Front and Rear Rib * Mode 2 (4 ) J”w—~\ Front and Rear Deck Y t-—-x Front and Rear Rib Mode 4 (9*) (* denotes overall mode number.) Figure 3-7. In-Plane Mode Shapes for SSB 92 Front and Rear Deck Front and Rear Deck Y A Front and Rear Rib Front and Rear Rib * Mode l (2 ) Mode 2 (5*) //""\\ ’,__ _‘ \\v/"— 1 \ Front and Rear Deck Front and Rear Deck Y Front and Rear Rib Front and Rear Rib Mode 3 (3*) 3°40 4 (9*) (* denotes overall mode number.) Figure 3-8. In-Plane Mode Shapes for LSB r \ ’ Front and Rear Deck 1“~ ’,7 Front and Rear Rib Mode 1 (1*) N “—r-“* W... Front and Rear Deck ’M‘ \‘ Front and Rear Rib and. 3 (4*) Front and Rear Deck ”-\ Front and Rear Rib Mode 2 (3*) ——' Front and Rear Deck Front and Rear Rib and. 4 (5*) (* denotes overall mode number.) Figure 3-9. Out-of-Plane Mode Shapes for LSB 94 Front and Rear Deck Front and Rear Rib Front and Rear Deck Front and Rear Rib Mode 3 Figure 3-10. Front and Rear Deck Y H : 1 Front and Rear Rib Mode 2 1"- Lax <‘I T\.—v’ Front and Rear Deck Y [I *::-x Front and Rear Rib Mode 4 In-Plane Mode Shapes for Two-Dimensional MSB 95 cease: eeaoue «<-aHe .Ha-n enemas A.oouv ones gl‘ 0:. 60-. a. 00. e. I.“ a. a! a. GM. 6.. a. NI< ux<301h¢ \ > > . a)?! \/ 7 K/ 11.. >3 > > .h> \/ (k < < r\ << G E3 <<< < < £< . 3.5.0. 274‘ E- on: on. an. 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I. a. ...nvlo . . null.-- .oe-a magnum n— o— IrrL+r+*|DLLL‘|m+rh|r 00. n 00‘-0 I ' ‘l .00. it"- O ,' '.l,o! ‘4' it . ..LO.L' tL'tkL IIL llr"L¢. ‘1-’-L'[l‘}.rl[ tr. r..L|.rll[‘ LL ’ L! b b b p b p '1 ..IL I III ¢ '01.: - I'!.c’t' l.‘ - J ' .Prlblarllr .aFl:L|I.$II-F.llpll?.lhlls tutti 'u'CrL o oal‘rllbl o. o.|. .. .5 1:91.03. gut-Ir Ibl-rtl 3 (IdIa-33) vi fiva—v—‘rv 131 mm: Ham muouaaz «ads «cannon o:.~m-uo-u=o “an oeozv.~ on» ad nanny: .ho-n ouauum ouuuuuocn any A.uomv «can ILLIL I IL I I + I’LL VI II. .III IIIIIOI no. a. a ---. ‘.i :11:|:..|us|l:.l-|lu..a| ,Illlr I. ...nlb IPI [III-Ia o. Inllbllbrlnbllbvltpn LII? ...? ..IIIYIP... [III .IFIIPILIIPIIPIpIIFII-l I; 39:: ...-.5.— A3 ' ‘1 b ' r, b. b. b b b - b x, b b. PL! ’ ’L Tb LI. P ’ . .L ’ ’ ’ b ' i x, ’ rb l’lL ‘ lb ’ I..I II IIIIIIoIInI o - I III. IIJ IOIIIIII‘.OI - I (II. .nI II I. can c.n—HHIIIL I :-.I...|r..rIII.rI-DIIOI vr.|r 01.51... .b.IrlrI[.I ... LI . ..I .r. Ir. . .I.I..III . . . I. In... -PIn. IrllIDAILI-Ir.l[u1r|[ - In .aI .n'I. IIIII I I4' fivvi (3511-33) 132 mum you huouuus «ads couch Huux< add ovozv n can a Hosea: .oe-n gunman ouunuaocn any .oomv oaas coonl oocul VVIII clhllb (Ill?) 7, I5! occ— uauunuu unusua any (8d?!) ooonl cow N.A.LVL Til oooul coo—I 0.... III. .II. 1’ Oqlo‘ . o..‘I|II.|& 1. Illf‘bl.’ 1.. T. Lt - ‘u d 133 GM“ HON hhcunH—n GEAR. HOMO“ HIH: ANN OQOZV H uu»..~o=H any can «a wanna: .oe-n ounuam A.oouv onus n. mw mm an on on n. a. m = ? b b L LL h b b h I n n » > p -L IL I P L b h F - . t b L L L P b b > h b L SOFI -- u : In « ‘ . -... iltl- I -1 c I -....lln. it'lllLrO _ - I ,_ _ _ g . . coo. t.an: - s A i ‘.; - - .:-;1.ooo« umm «27.04 T n p b I I I D I D D b n p I h D n b I DL F I k) b u I I D I D \b I I coon aqua-an nag-.3 n3 ,I’II‘IIII‘IIIII. LII ‘VL.-I I.DIIII[III[fi °°°PI . ‘ OOON ‘ OIL! I»vb l.- - ~‘|-.\ - ail o‘VI..vol|o ,‘vrtvl- ‘LI °°°n 134 mam uom muouauz oaua mauecon o:.am-¢~ add oeozv a new a unsung .on-n «gang» cauauaocu Aav “.33 3:. a. o— 069.1 I .. ..‘1:II.9I.I ... . x .11.--.nll-.- 0111-!!! gt! a oaov can «in Iuv‘ .l .. It It? :0. ago 0‘“ “n 33.3 .3054 A3 4 u” LLLLLLLLL 88! d I ( II .I ll....l!l...lc--ll!-xl|.,- . ‘ . . , gnu .... .I..III ’ LT oou N.H 0 IO?! OIOIII’IIOIIIOI. .0 1|.. III!:TIICIIOIEOII§I!.I¢TIOI.D am you .333: as; 9.2.5.. 232-5 8“ 8on H can 3 woe-o: Ana 9:8: cunning; Aav Tommy 3:. coon... coon... o conn 5 3 1. each LR \l 1. Qua-nan ...-2:.— AS 41. of 82... .m ( canal 09a N.HIVI— ‘0 .IAIIO E-IIA h. Li‘a. 5|..O¢.b. II tlzix‘l'IlffillhluYIPIDIIPIItL’ui D OS” 136 nmm you myoyayz oEHa mayooon oooym-yo-yoo Ayy ooozv a can a yoaaox .un-n oyouyy oyyyoyooy any .uowv T cowl § x H Ugandan unocua any can u.a.14 8 N I (sdIa-=;) can 137 now you yyoyouz oaus mouooon ooouu-uo-yoo Any ooozv u can my yon-ox .an-n oyouuu ouyyoyooy “no A . 2.3 on: av at on on ma on n— 3 rr’ ’ ’IPLleLL I ’L Ilbr b I ' Ill. I’ILIPLlin I rL -Irb ILL’I blrm ’L P .oon N.~ I n ‘ .0 o I. .o I. >>>>.f <54 I'll- YIII Ito : . a . I. V . ‘ o. I u o _I. 0‘ u. l .T . . I.I|.t-. nlo4 033.:— unocua A3 'rLIILp IIILL ILLLLILLI} IL’ I I I I IL, I I’lrIIy’fibl’ PI I P+b ilbllfllollbL Ir-o-l.oi.o- .bnif- .o .I i? .ih, -o‘IrIIIPIIPIblbpl.[.L :IPUOrIPIIDIIbLIPQIDIIPIIP'PILIIPibiuPIIOIII oil: (IdIa-as) on. ~.d 2 VII-II oOIt. 1.....- . 1|. . II,.I.OIILOII'Io O ..' ... 3.. ’II.. .Vt...... O. ...- 0 L. OI. . I. . IIIIO.I‘I.I.I OILII'.-oCIr O.If I.II.II¢‘.I'IIO.I'L i T. .- -_ ...i 138 may you myoyoua oaua ooyoy yoyy< “Nu ooozv u can n yooaoa .on-n ouunouosu any .oomv uou H.H iIT gnu-dam unocua Anv ouawum 3 d ('d11) 139 no; you myoyoua onus ooyoy youy< no" oooav n eon oy yoonoz Ouagadocu Aav A.oomv club ooa a.u:&l_ ouuunun undead Adv .nn-n oyouuu J (id?!) 140 one you yyoyoux onus mouooon osoyu-ou Auu ooozv u oou n yon-o: .on-n oyouuy ouyooyoou “Av A.oomv onus can n.n.1fll1!+ uuuuudn Hanged any -+> r+L \P‘Lllrbrl’ ‘ ’bx’" IPF‘L'L k’ ’ ’ L L ILL r‘ ’L I by ’ bL ’LLL f‘Tv'v‘v—f‘f'vvfjrv'vj - I (sdIa-as) 141 mag uom macaw“: cage unaccou o:.~m-:~ Ac“ ouozv a can ed gonzo: .sn-n ouamuu uduaofiouu Aav I’ll. I P I L r r r I +0'P‘.P P I I I I I I I I I I .I I l.|-. l A.uomv onus $-,+ 8--,-%rL,.fi--rLfi---afi>-br@-g.-£-- -m--r-a : TIL can n n fip88: ul. #4 > > > I jug! # I z ulcu~z1uny .unu . =. f889- m Pb»DrlblbbP-bhhbnbbb>bhhDPPLp-DL+bP}DL[P+PDLIPDg" 03am:— u-oga A3 .aaopnfu .aOOQUI o 99999: .L (tdwa-=;) 142 and How huoumum uEwB unavcun u:n~m-mo-u:o Aug ovozv H vcfl n Mensa: uuuaudosu any A.uomv oaua .onun ousmuh 3.--.3. .-fl---nfi-¥Lr%Lf..fi.?¥.£--- 8....» .--.33! »-—. . —r_ —L Bus—l «2..-77 .7... ....7K.._..»77 .;._7_.7. _- ___ .... r .. .o ... ... . .... .... . I 4 . — q——J +30— .-....+ . .--...--......._..n.....w..T.. ..t. oauuudm anon“; Aug LrbL-».- . ..-LlplflblllrlrlrltL.. .. . ..-»r»& 888i .Illltlzais: 2- . r :zr.9xan¢. ll 5 7 7 p- ’- > < < ('7— g < < C) (sdIa-n;) 143 am; now macaw“: oaas ucducon u:.~m-uo-u=o Ac“ ovozv a can ca yuan.» .on-n ouauam ouuuuuocn any .uomv cage ooa n.o.+._|1 ouuauau unocua any (can-3;) I’VII’PIr E 144 am; you muounaz gags «cannon o:a~m-uo-uao Aod «oozv H can e non-o: .oo-n ouauum 03:: H.223 A.uomv onus 9- - --.».-- - knit? {$3qu - --...- - --m.-- (M... ...8§a- j . I: - .xaaao.n \\s/7 1);); 7\<17 . . r .w . - . ...... .. ;....... .o . . a- .l:.. .zxaao. . [It'r P I’ I I I ’ W I II iill i a a 5.... IIIIlI’lPPIrPIrIbIIbrIIIIIPIIrI’PIIIL'r’II-lfiltf (IdIa-:;) 145 am; you muouaaz «ads ucfiucou o:¢~m-mo-u=o neg ovozv H van m uoaao: .do-n unauum Ufl fluid” HIOp—ufl A.uomv onus 0* 0' on an 8 ON 0. O— m ... I I I b I I I I LP I I I I b I I I I I II b I P P I LP I I I I D I I I I I I I I g'l‘ ‘ ‘ j if». ... :ss--- , - ..ooooon (tdxx-a;) . l'tittltl‘rrollfl Irblloll o .lflIIIbUIII! out. .lrlflrlbllb‘art. IL.I?¢¢O§IIIIIOIIYI.5|IIIIIIIOIII u. g. 146 nouumnau unusuav am: you known“: oaua noduaauuuuaa huuocn an. duo: .33 3:. .No-n ouauum IIIMHIIIIRIIIIIBI IIII‘IIrIiprILIIII p»».»»7777»>29> 7 . r an LIIIPIIII’PI’PI’IPI'I’IIDIIII» NI 7 . um .I‘O‘I Sm+mo+mm+uxi «L mo+um+mx msm 5+9. 0...... 7' ‘lll‘lll.a"llil'l mwv_ mm mm r6; .vi v7 axon / £33011: 147 33230:: on: now 5.39:: clan. saunas—«.33: hang—u and Juan .34" shaman 11‘114111d“3‘1344 11‘14‘ A. oouv 0.59 g up 0.. Or.- III IrlrbI IIIPIlI I I .. ma 11‘““1“1““““‘1‘I““‘1““ E+mo+mm+mx «an mo+mm+ux mm mm+mx o o I D I I I I .- I I’ r} II I D I I 3.9 r... fi axon / axon: 148 33:3 H2255 can you 533: 25a. 60:33:2— huuocu can Juan .3...” 9.5»: A . oouv clan. 9 ‘3--L-mm--L-.wnp-rn.«--- 2 +-n..i---.w. 75;; gfifip #7 7T? 3. . ”a . 2E+ma+mm+mx «in . mo+mm+mx mum . mw+mx $10 149. a» x . axon / anus 149 Auxumuaocuv nmm uom mucuaxa gags coau:Auuu¢«a huuocm on. «no: .no-n gunman A.oomv oada 9 1‘3r- - -o.n- - - -..er [-8 cu .... 2 M . _ ---.---- .od _. . . :— . NW 100 an . a ................................................ 111:1::::1::::::::::: o . sm+mo+mm+mx «In uo+um+mx m1m mm+mx 91¢ 9. 2T... . axon / axon: 150 332:. 22:5 23 you #33: 3:. 533223 325 ....- auon .ooé 35»: ’Il I’lmll. I I I r m. FII‘IPIIPI; .ILII. lloiiolllbnavaiPuiolLI! I b I I I A. 2.3 on: a an D II I I b I ”M I III IIII’IIIIIDDIIIII 3 L, I I I I ‘ III .3m+mo+mm+mx «in mo+um+mx 0&3 mm+ux 0:0 MXI L I I Lula-..Ibuloq' lIllI-uILIvIIIIIOIIIL axon / axon: 1.51 .411. v v Auxuuadocnv am; you uuou.x= oaua coxu:a«uu..a muuosu an. sacs .he-n enough .. --- A603 3:. -mn----fi----m«----om- p.-L--P.--.- .L 5— am an an ..... - . . 141.334.3133.:<«11313ioo11341313o4d11 _ gm+mo+mm+mx «In ma+mm+wx aim. um+mx 96 UV. *3% I I blIlIlllo...’ ' IIIIIIIIIII hr III?! II. P I mun/nuns 152 Auuuuaxocnv ouuuu acumen: ao.c sud: Axouoz a-~V an: no“ muouauz cage coxuaaduuaxn manage on. sue: .oo-n gunman A.uomv ends a 2 a ...... m ....n m. .... m- .. IIIIIIIIIPIIIII IIII III IIII IlIII IIII III fig 1 . mu I I I I D I I. I 3a+uo+mm+mx «In . mo+mm+mx arm . mm+mx olo I -. mx I . axon / axon: 153 . “aqua-doauv eggs: mafia-.n an«.c and: Agave: a.«. am: you muouaqa onus coduaaduuuxn uuuosu on. anon .oo-n .uauxu A.oomv quH fi»--ue-r-rmrI»-Q-L--fl-LL-fl--- é 3 v x ”a . ”G t f In Agii..§i§‘§‘§ 7 3m+mo+mm+mx arid . mo+mm+mx arm. mm+mx Olé . mx xix .66 . .06 i m. . m . / A n . a. .9.— . r 154 33232.: ouuam moi—son on... 5.... Ayoooz o-~v no: you yyoyyym onus oouyonuyyouo yuyoou on. xyoa .oy-n oyauuu A.ooav onus .. 8. a. 2 Nil III 3m 1d§§91C£¢§§§.§1:Q1= v . 3m+ma+mm+mx old mo+mm+mx mlm mm+ux ole III I I n I IIIPI '0' I PII PIbrI I II FII . I. oI {I I O . .66 . . .8 . m . a . / a . m» .9.— . . 155 ,TQIDII... 9.3!... n r . nuuuuogocnv o.».¢ mafia-an .o.d nag: “deco: n-~v am: you nuouaum onus coausa«uuadn uuuosu ea} «you .fih-a gunman b ’ P ’ b A.oouv oaas 3m .— a: . .. II " LL! ? [Irl’yblb ’l‘bl’. no gifldas:gfldfidd‘ififl‘ddgii:9di‘ififiifizifli ‘ Ba+mo+mm+mx «10 mo+mm+mx aim mm+wx 09¢ Ii--lllll. at- 'o.‘-- D D D P PII...’ PL- DIL P r. F D h axon / £31011; 156 t¥r j Angus-doc: 399m 9395: ac.“ Aunt 33.0: 9-3 an: new buoyed: ulna coda—53:3 huuocu can snot. .2...“ as»: 9. an . _ . a {1.144 _TI.IMK@,11.,IMHMr‘.flIl11 A.oouv onus D D b r P P - by .. he II ‘ [P Pl?’ ‘ h hDIL 5 ma 3m+mo+wm+wx mo+mm+ux wm+w¥ mx IIEI - r ’ by ’ x- . b’ ’ bL LIL b L’ I: gzoa / Liana 157 a l“ a. v .y ‘ . . a .. l . 33:13.: 0.33— 95955 3.." :35 33.0: 93 am: you .333: alum. 60393.33: hung—n v:- xuou .2...” 9.5»: ......rvfri P p ..ud17fi. 4.3.114 ,. A. 003 clan. a a . p 3 ma 1 {.1 «Hau‘a¢:i§¢d= 3m+mo+mm+mx file uo+mm+ux mlm mm+mx olo ,li UX : } b ’L b D L...’ ’ D l. r ’ L I r L. b ’ ’ k. F’ ’ D ’ axon / £33011: 158 Auuuaudocnv oduum magnaoa ac.“ can: Agave: a-~v am: you huouaun ones coau:Aguu.un annucu ea. anon ..s-n gunman A.oomv cage 8 . . H .. on up up - a - - -L 1 . 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E . nu --v'fvvvvv --. - V N v fv fv r7 § Vi v'v 8 - Kw :uounsaa scans Kq £311:an mom ogcold vv‘ .oan vvf‘vth' r pus 9 Jaqwow (00m x sdm-u) mom ansold tom 161 and you Hannah aacoanom unocau uaouob nouuoauou ouuundonn .sh-n ouauum Aoov nououm oncomnom unocud ;flo \ , l" PUB 01 460mm 8.. 0910—5 111qu 101005 £11.1an ammuno fi‘ . ..I" 0'- 0.11 I -... . .0! lo.-.‘ ‘0‘--. ---. . --- {03 0500.0 .33. «Mm... \ 33:00 {03 03020 E36362 mim K 323030 9.30230 E3E_x02 020 a; M momma ms M 018000 mm 91mm r 003 01 Amman (00001 x Bdm-u) you 00801.: mm 162 am: you huouuun club acolooudmuua any dauGOuuuoz n 0602 .an-n ouauuh anon-nu unocudcoz any 738 .5... I. "l’l‘!0'|-i| > > 3>> ‘ A rtllr'bllrtlr-tll[lrurrbilplf b b rL unannuu unocad adv <3N <<< oo- c.u > 003 n.~ I.[.1I[Ilrlarlr‘[LI-Lll1rr.oLn'r.[l[lLII‘lL’[ . b > < :> a one: .o».« .uauam ouunann unocqucoz any A.oomv onus l‘ -"ll| - 'I' I'.!I'I ouuuuuu unocqa any 00“ “.0 00¢ n.NI It'lblL'b bl[ul|ln.llI II III I I I I I I I I II I I I I I QQ0I .0ATu :30 «0A7. p0dYu —0A. «0A. aqfijou qazv / :ulInaItdstq 165 am: now mucus: 05:. acoaou-Hn—uua A: «donut—2r n one: uuuunnn unocuucoz Aav A.oomv onus .ao-n ouauwu muting....m..-£°.r..Lt£_.§.? 5%? .. ‘_ — 4 23.0 1..-! III]! I- 11:. l -41- TI 000.0 IIII-IIIIIII’IIbDI.upIIIIIDIIIp-IIII-IIII-II-I °P°o° ouuunum uuoflun any . ....,.....-......--..-...-........,,...LL.03.0! . . . _ . . 1‘ fl - l Ix.‘ I I ‘l. . _ 'llv‘llvDSOl 11 q~ T ... 086 ii 23.1-9: : _ I . 1T i a: 08.0 :qSI-n uazv / :uausavIdSIa 166 mam you mucuauz oaua acoaoocaauun any Huucouuuoz a 0602 .Nn-n Guyana aquundu unocuncoz Aav A.oouv onus i. " . 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The effects of such nonlinearities enter in the analysis through the computation of the ”resistance" of the arch ribs. For the inelastic effects. a method of analysis based on the "plastic hinge” concept has been developed for the elasto-plastic resistance of a curved beam element in the three-dimensional space. The yield function is based on the three stress resultants (axial force P, in-plane bending moment fly. and out-of-plane bending moment Hx). For the analysis of nonlinear elastic effects, a twelve degrees of freedom incremental stiffness matrix [n1] was also derived. Other features of the analysis included the use of ”mixed nodal coordinate systems”: cartesian coordinates for the nodes on the deck, and curvilinear coordinates for the nodes on the arch ribs, and the use of various constraints such as those that would result in infinitely large axial stiffness for the cross-beams and/or columns. All these features were motivated by a desire to make the model more computationally effective by reducing the number of degrees of freedom. For seismic analysis, the load input is a uniform ground acceleration in 195 196 all three directions in space. The system equation consists of the equations of motion for the unconstrained degrees of freedom of the nodes. The method of analysis is one of a step-by-step numerical integration in the time domain. Within a time increment, the solution is essentially of the Newton-Raphson type implying an outer loop of iteration for the equilibrium of the system equation. For each elasto- plastic element (curved beam element), an inner loop of iteration is needed because of the material nonlinearity. The validity of the analysis was corroborated by checking the balance of the various energies in the system and the work done on the system in each time increment. For interpretation of the results on inelastic behavior, the quantities "ductility factors” and "plastic work densities" are defined. A computer program has been developed for the implementation of the three-dimensional nonlinear seismic analyses described above. Three bridge models, MSB, SSB and LSB, based on three prototype bridges: CSCB (700 ft), SSB (193 ft) and NRGB (1700 ft), respectively, were used to obtain the numerical results. Three-dimensional models were employed to consider the nonlinear inelastic effects. Results for nonlinear elastic solution were based on two-dimensional models. The ground motion used were the artificially generated motion CIT-A2 with different amplification factors applied to the amplitude of the ground acceleration. For the maximum displacement responses involving material nonlinearity, there is no appreciable difference between the linear and nonlinear analysis. But plasticity limited the magnitude of the internal 197 force response to that as defined by the yield function. It follows that if plastic deformations are allowed, the design forces may be reduced from that which would be required if the design is to be done on a linearly elastic basis. For example, this reduction factor, as discussed in Section 3.4.8 and illustrated in Figures 3-75 through 3-77 and applied to the linear response factor, would be 1.0/1.88 if the "damage" represented by a curvature ductility factor of 3.5 was accepted. Both the maximum displacements and internal forces of nonlinear elastic responses did not vary a great deal from the linear solution. The history curves, however, exhibited different periods of vibration. In general, the dominant period increased by 5% to 10%. According to work and energy distribution time histories, the dissipated damping energy (for a 1.5% damping ratio) is over 70% of the work done for three bridge models. From Section 3.4.7, the results show that almost 50% of the work done was dissipated by damping if the damping ratio is equal to 0.25%. There was no nonlinear inelastic effects if the damping ratio is equal to (or, presumably, exceeds) 5%. These observations emphasize the importance of damping in the response. 4.2 W This study as summarized in the previous section has developed a method of analysis, prepared a computer program and obtained a significant amount of numerical results that provide much understanding of the behavior of deck-type arch bridges subjected to earthquake ground motions. Because of the capabilities of supercomputer, the arch bridge structures could be modelled in sufficient detail and subjected to 198 strong earthquake ground motions of a realistic duration. The nonlinear analyses had yielded responses that are more realistic than those obtainable using linear models of earlier studies, particularly when applied to models of the true three-dimensional kind. The method of analysis described herein can provide a good basis for the development of design procedures. For consideration of design, the following improvements seem.worthwhile for future study. The results on nonlinear behavior obtained in this report are either for geometric nonlinearity (nonlinear elastic effects) only, or for material nonlinearity (nonlinear inelastic effects) only. Obviously, it is more realistic to consider both nonlinearities simultaneously. For applications to the design of steel structures, the "octahedronal yield surface” (which is linear or consists of planar surfaces) seems more appropriate than the spherical yield surface. The latter may over-estimate the strength of the element, and hence could err on the unconservative side. The numerical results of this study have concentrated on steel deck-type arch bridges. They are only one kind of arch bridges. The other two types of steel arch bridges are tied through and tied half- through steel arch bridges. The responses of these types of arch bridges could be quite different from the responses of the deck-type. The analysis and the computer program developed in this study can, of course, be applied to these types of arch bridges. The artificially generated ground motion CIT-A2 was used in this study. For purposes of design studies, additional ground motions should be used. A variety of other ground acceleration histories such as the type B, C and D artificially generated accelerograms (Ref. [18]) and the 199 records of actual earthquake ground motion such as the 1941 El Centro and 1971 San Fernando earthquakes may be used. Another limitation of the study presented here is the assumption that the motions of all bridge supports are the same. The validity of this assumption obviously decreases with an increase in the span length of the bridge. Thus, effects of non-uniform motion of the supports would be a significant topic for future study. In Ref. [12], these effects were presented in a linearly elastic setting. Future work may consider non-uniform motion of the supports in a nonlinear-response framework. LIST OF REFERENCES 10. 11. LIST OF REFERENCES . Abdel-Shaffer, A. M., ”Dynamic Analysis of Suspension Bridge Structures," California Institute of Technology, Report No. EERL 76- Ol, Earthquake Engineering Research Laboratory, Pasadena, California, May 1976. . Abdel-Shaffer, A. M., "Free Torsional Vibrations of Suspension Bridges," Journal of the Structural Division, ASCE, April, 1979, pp. 767-788. . Abdel-Ghaffar, A. M., and Rood, J. D., "Simplified Earthquake Analysis of Suspension Bridge Towers," Journal of Engineering Mechanics Division, ASCE, April, 1982, pp. 291-308. . Abdel-Ghaffar, A. M., and Rubin, L. I., "Suspension Bridge Response to Multiple-Support Excitations," Journal of Engineering Mechanics Division, ASCE, April, 1982, pp. 419-435. . Abdel-Shaffer, A. M., and Rubin, L. 1., "Lateral Earthquake Response of Suspension Bridges,” Preprint 82-051, American Society of Civil Engineers Convention, April, 1982, Las Vegas, Nevada. . Chen, W. F., and Atsuta, T., "Theory of Beam-Columns, Volume II: Space Behavior and Design," McGraw-Hill Book Company, 1977. . Cheng, F. Y., and Ritipitayangkul, P., "Investigation of the Effect -of 3-D Parametric Earthquake Motions on Stability of Elastic and Inelastic Building Systems," Report to the National Sci. Foun., Dept. of Civil Eng., Univ. of Missouri-Rolla, Missouri, August, 1979. . Conner, J. J., Logcher, R. D., and Chan, 8., "Nonlinear Analysis of Elastic Framed Structures," Journal of the Structural Division, ASCE, Vol. 94, No. 8T6, June, 1968, pp. 1525-1547. . Drucker, D. C., ”A More Fundamental Approach to Plastic Stress Strain Relations," Proceedings, lst U. 8. National Congress on Applied Mechanics, Chicago, 1951, pp. 487-491. Dusseau, R. A., "Seismic Analysis of Two Steel Deck Arch Bridges," M.S. Thesis, Department of Civil Engineering, Michigan State University, 1981. Dusseau, R. A., and Wen, R. R., "Seismic Responses of Two Deck Arch Bridges," Pre-print 82-010, American Society of Civil Engineers 200 12. 13. 14. 15. l6. 17. 18. 19. 20. 21. 22. 23. 24. 201 Convention, Las Vegas, Nevada, April, 1982. Dusseau, R. A., ”Unequal Seismic Support of Steel Deck Arch Bridges," Ph.D. Thesis, Department of Civil and Environmental Engineering, Michigan State University, 1985. Dusseau, R. A., and Wen, R. K., “Seismic Responses of Deck-Type Arch Bridges,” Earthquake Engineering and Structural Dynamics, Vol. 18, 1989, pp. 701-715. Gates, J. M., “Factors Considered in the Development of the California Seismic Design Criteria,” Proceedings of a Workshop on Earthquake Resistance of Highway Bridges, Applied Technological Council, Palo Alto, California, January, 1979. Hodge, H. C., “Plastic Analysis of Structures,“ McGraw-Hill Book Company, 1959. Imbsen, R. A., Nutt, R. V., and Penzien, J., "Seismic Response of Bridges--Case Studies,” UCB/EERC - 78/14, Earthquake Engineering Research Center, University of California, Berkeley, June, 1978. Irvine, H. M., "The Estimation of Earthquake Generated Additional Tension in a Suspension Bridge cable," Earthquake Engineering and Structural Dynamics, 1980, pp. 267-273. Jennings, P. C., Housner, G. W., and Tsai, N. C., ”Simulated Earthquake Motions," Report to the National Science Foundation, California Institute of Technology, Earthquake Engineering Research Laboratory, Pasadena, California, April, 1968. Ruranishi, S., and Nakajima, A., “Strength Characteristics of Steel Arch Bridges Subjected to Longitudinal acceleration," Structural Engineering Earthquake Engineering, JSCE, Vol. 3, 1986, 287s-295s. Mallett, R. M., and Marcal, P. V., "Finite Element Analyses of Nonlinear structures," Journal of the Structural Division, ASCE, Vol. 94, No. 8T9, Sept., 1968, pp. 2081-2105. Morris, G. A., and Fenves, S. J., "Approximate Yield Surface Equations," Journal of the Engineering Mechanics Division, ASCE, Vol. 95, No. EM4, August, 1969, pp. 937-954. Morris, C. A., and Fenves, S. J., "Elastic-Plastic Analysis of Frameworks," Journal of the Structural Division, ASCE, Vol. 96, No. 8T5, May, 1970, pp. 931-946. Nigam, N. C., "Yielding in Framed Structures under Dynamic Loads," Journal of the Engineering Mechanics Division, ASCE, Vol. 96, No. EMS, October, 1970, pp. 687-709. Oran, C., ”Tangent Stiffness in Space Frames," Journal of the Structural Division, ASCE, Vol. 99, No. 8T6, June, 1973, pp. 987- 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 202 1001. Orbison, J., McQuire, W., and Abel, J., ”Yield Surface Applications in Nonlinear Steel Frame Analysis," Computer Methods in Applied Mechanics and engineering, Vol. 33, 1982, pp. 557-573. Porter, F. L., and Powell, G. H., “Static and Dynamic Analysis of Inelastic Frame Structures,” Report No. EERC 71-3, Earthquake Engineering Research Center, University of California, Berkeley, California, 1971. Powell, G. H., ”Theory of Nonlinear Elastic Structures," Journal of the Structural Division, ASCE, Vol. 95, No. ST12, December, 1969, pp. 2687-2701. Riahi, A., Powell, G. H., and Mondkar, D. P., '3d Beam-Column Element (Type 2-Parallel Element Theory) for ANSR-II Program," Report No. UCB/EERC-79/31, Earthquake Engineering Research Center, University of California, Berkeley, California, 1980. Row, D. C., Powell, G. H., and Mondkar, D. P., ”2D Beam-Column Element (Type 5-Para11e1 Element Theory) for the ANSR-II Program," Report No. UCB/EERC-79/30, Earthquake Engineering Research Center, University of California, Berkeley, California, 1979. Sakimoto, T., and Komatsu, 8., "Ultimate Strength of Arches with Bracing Systems," Journal of the Structural Division, ASCE, Vol. 108, No. 8T5, May, 1982. PP. 1064-1076. Tebedge, N., and Chen, W. F., “Design Criteria for H-columns under Bi-axial Loading,“ Journal of the Structural Division, ASCE, Vol. 100, No. 8T3, March, 1974, pp. 579-598. Tezcan, S. S., and Mahapatra, B. C., ”Tangent Stiffness Matrix for Space Frame Members,” Journal of the Structural Division, ASCE, Vol. 95, No. 8T6, June, 1969, pp. 1257-1270. Thakkar, S. M., and Arya, A., "Dynamic Response of Arches Under Seismic Forces," proceedings, Fifth World Conference on Earthquake Engineering, Rome, Italy, 1973, pp. 952-956. Toridis, T. C., and Khozeimeh, K., "Bifurcation, Pre- and Post- Buckling Analysis of Frame Structures," Computers and Structures, Vol. 8, 1978, pp. 667-678. Tseng, W. 8., and Penzien, J., "Analytical Investigations of the Seismic Response of Long Multiple Span Highway Bridges,” Earthquake Engineering Research Center, Report No. EERC 73-12, June, 1973, University of California, Berkeley, California. Tseng, W. S., and Penzien, J., “Seismic Analysis of Long Multi-span Highway Bridges,“ Earthquake Engineering and Structural Dynamics, Vol. 4, 1975, pp. 1-24. 37. 38. 39. 40. 41. 42. 203 Tseng, W. S., and Penzien, J., "Seismic Response of Long Multi-span Highway Bridges,“ Earthquake Engineering and Structural Dynamics, Vol. 4, 1975, pp. 25-48. Wen, R. R., and Farhoomand, F., ”Dynamic Analysis of Inelastic Space Frames,” Journal of the Engineering Mechanics Division, ASCE, Vol. 96, No. EMS, October, 1970, pp. 667-686. Wen, R. M., and Lange, J., “Curved Beam Element for Arch Buckling Analysis," Journal of the Structural Division, ASCE, November, 1981, pp. 2053-2069. Wen, R. M., and Rahimzadeh, J., "Nonlinear Elastic Frame Analysis by Finite Element," Journal of the Structural Division, ASCE, Vol. 109, No. 8T8, August, 1983, pp. 1952-1971. Wen, R. H., Lee, C. M., and Alhamad, A., "Incremental Resistance and Deformation of Elasto-Plastic Beams," Technical Notes, Journal of the Structural Division, ASCE, Vol.115, No. 5, May, 1989, pp. 1267-1271. Yabuki, T., and Vinnakota, 8., "Stability of Steel Arch-Bridges a State-of-the-Art Report,“ S.M. Archives, Vol. 9, 1984, pp. 115-158. APPENDIX PROPERTIES OF BRIDGE MODELS APPENDIX PROPERTIES OF BRIDGE MODELS Aol am This appendix includes the description of the three bridge models in both the three- and two-dimensional cases as depicted in Figures 3-1 through 3-4. In conjunction with these figures, the data presented below completely define the bridge models used. A-2 W 11.2.1 EQDALDAIA In each line, the first value is node number (see Figure 3-1). The next six values are the boundary condition codes. (a "0" (zero) denotes "free" and "1" denotes "restrained".) The order is X translation, Y translation, 2 translation, rotation about X, rotation about Y, and rotation about 2 in standard cartesian coordinates system. The following three values are X, Y, and Z coordinate in feet. 1 1 l l l O 0 0.0 0.0 26.0 2 0 O O l O O 0.0 133.5 26.0 3 1 l 1 1 O 0 0.0 0.0 0.0 4 O 0 O 1 O 0 0.0 133.5 0.0 5 0 O O 0 O 0 87.5 53.047 26.0 6 0 O 0 O 0 O 87.5 133.5 26.0 7 O O O O O O 87.5 53.047 0.0 8 0 0 0 O 0 0 87.5 133.5 0.0 9 0 0 0 0 O 0 175.0 90.937 26.0 10 0 0 O 0 O 0 175.0 133.5 26.0 11 0 0 0 0 0 0 175.0 90.937 0.0 12 O O 0 O 0 0 175.0 133.5 0.0 13 0 0 0 O 0 0 262.5 113.672 26.0 14 0 0 0 0 0 0 262.5 133.5 26.0 15 0 O O 0 O 0 262.5 113.672 0.0 204 205 16 O 0 0 0 0 0 262.5 133.5 0.0 17 O 0 O O 0 0 350.0 121.25 26.0 18 O 0 0 0 0 0 350.0 133.5 26.0 19 0 0 0 0 0 0 350.0 121.25 0.0 20 0 0 O O 0 0 350.0 133.5 0.0 21 0 0 0 0 0 0 437.5 113.672 26.0 22 0 O 0 0 O 0 437.5 133.5 26.0 23 O 0 0 0 0 0 437.5 113.672 0.0 24 O 0 0 O 0 0 437.5 133.5 0.0 25 0 0 0 0 0 0 525.0 90.937 26.0 26 0 0 O 0 O 0 525.0 133.5 26.0 27 0 0 0 O O 0 525.0 90.937 0.0 28 0 0 0 0 0 0 525.0 133.5 0.0 29 0 0 0 0 O 0 612.5 53.047 26.0 30 O 0 0 0 0 0 612.5 133.5 26.0 31 0 0 O O 0 0 612.5 53.047 0.0 32 0 O 0 O 0 0 612.5 133.5 0.0 33 1 1 l 1 O 0 700.0 0.0 26.0 34 0 0 0 l 0 0 700.0 133.5 26.0 35 1 1 1 1 O 0 700.0 0.0 0.0 36 0 0 0 1 0 0 700.0 133.5 0.0 37 1 l 1 l 1 1 0.0 133.5 -20.0 38 1 1 1 1 1 1 700.0 133.5 -20.0 39 1 1 l l l 1 -182.0 133.5 26.0 40 1 1 l 1 1 l ~182.0 133.5 0.0 A-2.2 ms. The four values in each line of this group are element number, global nodal number of end 1, global nodal number of end J, and cross section area (ft2). 1 2 8 3.78 2 4 6 3.78 3 6 12 3.78 4 8 10 3.78 5 10 16 3.78 6 12 14 3.78 7 14 20 3.78 8 16 18 3.78 9 18 24 3.78 10 20 22 3.78 11 22 28 3.78 12 24 26 3.78 13 26 32 3.78 14 28 30 3.78 15 3O 36 3.78 16 32 34 3.78 17 1 7 0.3881 18 3 5 0.3881 19 5 11 0.3881 206 20 7 9 0.3881 21 9 15 0.3881 22 ll 13 0.3881 23 l3 19 0.3881 24 15 17 0.3881 25 17 23 0.3881 26 19 21 0.3881 27 21 27 0.3881 28 23 25 0.3881 29 25 31 0.3881 30 27 29 0.3881 31 29 35 0.3881 32 31 33 0.3881 33 37 4 0.00489 34 38 36 0.00489 35 1 2 1.623 36 5 6 0.3264 37 9 10 0.3264 38 13 14 0.3264 39 17 18 0.3264 40 21 22 0.3264 41 25 26 0.3264 42 29 30 0.3264 43 34 33 1.623 44 3 4 1.623 45 7 8 0.3264 46 ll 12 0.3264 47 15 16 0.3264 48 19 20 0.3264 49 23 24 0.3264 50 27 28 0.3264 51 31 32 0.3264 52 36 35 1.623 53 l4 17 0.0144 54 17 22 0.0144 55 16 19 0.0144 56 19 24 0.0144 57 17 20 0.0144 58 18 19 0.0144 59 39 2 2.44 60 40 4 2.44 A.2.3 W The five values are property set number, axial area (ftz), local x-x moment of inertia (fta), local y-y moment of inertia (fta), and torsion constant (fth). 1 0.78 2.18 3.72 1.36 The values in each line are element number, global nodal number 207 of end 1, global nodal number of end J, and corresponding property set number of the straight beam element. 1 2 6 1 2 6 1o 1 3 1o 14 1 4 14 13 1 5 13 22 1 6 22 26 1 7 26 30 1 s 30 3a 1 9 a 3 1 1o 8 12 1 11 12 16 1 12 16 20 1 13 20 24 1 1a 24 28 1 15 28 32 1 16 32 36 1 A-2-4 W In this group of data, every four lines form a set of one kind of properties for a curved beam element. In each set, the four values in the first line are property set number, axial area (ftz), local x-x moment of inertia (fta), local y-y moment of inertia (fth), and torsion constant (fta). The three values in the second line are axial yield force (kips), local y-y yield moment (kips-ft), and local x-x yield moment (kips-ft), respectively equal to P M and Mx in Eq. 2-20. The 0’ yo 0 three values in the third line are local x-x yield rotation, local y-y yield rotation, and axial yield displacement (ft), respectively equal to exp, oyp and AP in Section 2.6.2. The two value in the last line are volume of the plastic hinge (ft3) corresponding to local x-x bending and that to local y-y bending (ft3), respectively equal to th and Vby in Section 2.6.1. 1 2.1180 3.4830 23.497 24.70 10064.34 30034.75 12169.45 0.0025102 0.0027548 0.0068276 6.750 4.221 208 2 2.5920 3.8020 32.397 25.010 12316.52 39794.21 13770.57 0.0026019 0.0026473 0.0068276 11.250 4.221 3 2.8290 3.9620 36.719 25.170 13442.62 44603.56 14571.13 0.0026422 0.0026179 0.0068276 13.500 4.221 4 2.3550 3.6420 27.990 24.850 11190.43 34937.93 12970.01 0.0025581 0.0026902 0.0068276 9.000 4.221 The values in each of the following lines are element number, global nodal number of end 1, global nodal number of and J, and corresponding property set number of the curved beam element. 1 1 5 1 2 5 9 2 3 9 13 3 4 13 17 a 5 17 21 a 6 21 25 3 7 25 29 2 8 29 33 1 9 3 7 1 10 7 11 2 11 11 15 3 12 15 19 a 13 19 23 a .14 23 27 3 15 27 31 2 16 31 35 1 A.2.5 nag: The four values in each line are global nodal number, mass in X direction, mass in Y direction, and mass in 2 direction (kip- secondz/ft). 2 3.8825 3.8825 3.8825 4 3.8825 3.8825 3.8825 6 7.765 7.765 7.765 8 7.765 7.765 7.765 10 7.765 7.765 7.765 12 7.765 7.765 7.765 14 7.765 7.765 7.765 16 7.765 7.765 7.765 18 7.765 7.765 7.765 20 7.765 7.765 22 7.765 7.765 24 7.765 7.765 26 7.765 7.765 28 7.765 7.765 30 7.765 7.765 32 7.765 7.765 34 3.8825 3.8825 36 3.8825 3.8825 5 5.125 5.125 7 5.125 5.125 9 5.125 5.125 11 5.125 5.125 13 5.125 5.125 15 5.125 5.125 17 5.125 5.125 19 5.125 5.125 21 5.125 5.125 23 5.125 5.125 25 5.125 5.125 27 5.125 5.125 29 5.125 5.125 31 5.125 5.125 A.2.6 W5. 209 .765 .765 .765 .765 .765 .765 .765 3.8825 3.8825 .125 .125 .125 .125 .125 .125 .125 .125 .125 .125 .125 .125 .125 .125 \IVNNNNN U'IU'UUIU'UU‘U'U‘U‘UIU'U'UI The Rayleigh damping constants c and 8 are equal respectively, 0.0483000 and 0.0039000. A-3 W A-3.1 EQDALDAIA 1 1 1 1 1 0 0 0.0 2 0 0 0 1 0 0 0.0 3 l 1 1 1 0 0 0.0 4 0 0 0 1 0 0 0.0 5 0 0 0 0 0 0 24.0 6 0 0 0 0 0 0 24.0 7 0 0 0 0 0 0 24.0 8 0 0 0 0 0 0 24.0 9 0 0 0 0 0 0 53.0 10 0 0 0 0 0 0 53.0 11 0 0 0 0 0 0 53.0 12 0 0 0 0 0 0 53.0 13 0 0 0 0 0 0 82.0 14 0 0 0 0 0 0 82.0 15 0 0 0 0 0 0 82.0 16 0 0 0 0 0 0 82.0 17 0 0 0 0 0 0 111.0 18 0 0 0 0 0 0 111.0 19 0 0 0 0 0 0 111.0 20 0 0 0 0 0 0 111.0 0.0 30.75 0.0 30.75 13.281 30.75 13.281 30.75 23.509 30.75 23.509 30.75 28.398 30.75 28.398 30.75 28.398 30.75 28.398 30.75 22. 22. 22. 22. 22. 22. 22. 22. 22. 22. c>c>c>c>c>t>c>c>t>t>c>c>t>t>c>c>t>t>c>c> t0, A.3.2 21 0 0 0 0 22 0 0 0 0 23 0 0 0 0 24 0 0 0 0 25 0 0 0 0 26 0 0 0 0 27 0 0 0 0 28 0 0 O 0 29 1 1 l 1 30 0 0 0 1 31 1 1 l 1 32 0 0 0 1 33 1 1 1 1 34 1 1 1 1 W 1 1 7 2 3 5 3 5 11 4 7 9 5 9 15 6 11 13 7 13 19 8 15 17 9 17 23 10 19 21 11 21 27 12 23 25 13 25 31 14 27 29 15 2 8 16 4 6 17 6 12 18 8 10 19 10 16 20 12 14 21 14 20 22 16 18 23 18 24 24 20 22 25 22 28 26 24 26 27 26 32 28 28 30 29 13 18 30 14 17 31 15 20 32 16 19 33 1 2 34 5 6 35 9 10 36 13 14 37 17 18 HH000000000000 HH000000000000 .064 .064 .064 .064 .064 .064 .064 .064 .064 .064 .064 .064 .064 .064 .303 .303 .303 .303 .303 .303 .303 .303 .303 .303 .303 .303 .303 .303 .048 .048 .048 .048 0.1662 0.1662 0.1662 0.1662 0.1662 00000000000000000000000000000000 140. 140. 140. 140. 169. 169. 169. 169. 193. 193. 193. 193. 193. c>c>c>c>c>c><>c>c>c><>c>c>c> 210 23.509 30.75 23.509 30.75 13.281 30.75 13.281 30.75 0.0 30.75 0.0 30.75 30.75 30.75 22. 22. 22. 22. 22. 22. -10. -10. 00000000000000 A.3.3 A.3.4 38 21 22 0.1662 39 25 26 0.1662 40 30 29 0.1662 41 3 4 0.1662 42 7 8 0.1662 43 11 12 0.1662 44 15 16 0.1662 45 19 20 0.1662 46 23 24 0.1662 47 27 28 0.1662 48 32 31 0.1662 49 14 15 0.303 50 13 16 0.303 51 17 20 0.303 52 18 19 0.303 53 33 4 0.000288 54 34 32 0.000288 W 1 0.746 0.10 1 2 6 1 2 6 10 1 3 10 14 1 4 14 18 1 5 18 22 1 6 22 26 1 7 26 30 1 8 4 8 1 9 8 12 l 10 12 16 1 ll 16 20 l 12 20 24 1 13 24 28 1 14 28 32 1 W5. 1 0.9583 0.5465 4554.000 5696.625 0.0027275 0.0026023 1.9201390 0.8229166 l 1 5 1 2 5 9 1 3 9 l3 1 4 13 17 1 5 17 21 1 6 21 25 1 7 25 29 1 8 3 7 1 9 7 11 1 10 11 15 1 ll 15 19 1 211 0.24 1.726 3112.313 0.0030108 0.0002 1.168 212 1.91 1.91 4.21 4.21 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.21 4.21 1.91 1.91 1.08 1.08 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.08 1.08 1.91 1.91 4.21 4.21 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.21 4.21 1.91 1.91 1.08 1.08 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.08 1.08 23 27 31 1.91 1.91 4.21 4.21 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.21 4.21 1.91 1.91 1.08 1.08 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.08 1.08 19 23 27 12 13 14 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 5 7 9 11 13 15 17 19 21 23 25 27 A.3.5 A.3.6 0.0028000 0.115000 A.4 A.4.l 0000000000000 000m000000000 00000140606090 a a e a a6 02 08 03 e 0666016261606 1.1.1. .1. .1. .1. .1. 44]... 8414942]... 9 8 a 0 1 2 3 0000033669911 . o . . .44882277 00000 e o e e s e 1 11224455 . 224466008 0011100000000 1011100000000 1011100000000 1011100000000 1011100000000 1011100000000 1.9.1.4.5.5.7.nuo,nv1.o.1. 1.1.1.1. A.4.2 A.4.3 14 0 0 0 0 0 0 15 0 0 0 0 0 0 16 0 0 0 0 0 0 17 0 0 0. 0 0 0 18 0 0 0 0 0 0 19 0 0 0 0 0 0 20 0 0 0 0 0 0 21 0 0 0 0 0 0 22 0 0 0 0 0 0 23 0 0 0 0 0 0 24 0 0 0 0 0 0 25 0 0 0 0 0 0 26 0 0 0 0 0 0 27 0 0 0 0 0 O 28 0 0 0 0 0 0 29 0 0 0 0 0 0 30 0 0 0 0 0 0 31 0 0 0 0 0 0 32 1 1 1 1 1 0 33 0 0 0 0 0 0 34 1 1 1 1 1 1 35 1 1 1 1 1 1 36 1 1 1 1 1 1 W 1 6 7 3.529 2 8 9 3.437 3 10 11 3.395 4 12 13 4.061 5 14 15 4.914 6 16 17 5.922 7 18 19 3.042 8 20 21 5.922 9 22 23 4.914 10 24 25 4.061 11 26 27 3.395 12 28 29 3.437 13 30 31 3.529 14 17 18 1.956 15 18 21 1.956 16 1 2 3.6279 17 33 32 3.6279 18 4 2 0.0028 19 35 33 0.0028 1 1.00 1304.52 2 0.00 0.00 3 3.04 12024.00 1 2 7 1 2 7 9 1 3 9 11 1 607. .14 728. 728. 850. 850. .43 .43 1092. 1092. 1214. 1214. .71 1335. 1457. .14 1578. 1578. 1700. 1700. 1710. 1700. 1700. 607 971 971 1335 1457 213 14 57 57 00 00 86 86 29 29 71 14 57 57 00 00 00 00 00 339.795 416.0 362.449 416.0 370.0 416.0 362.449 416.0 339.795 416.0 302.039 416.0 249.186 416.0 181.226 416.0 98.164 416.0 0.0 416.0 416.0 416.0 0.0 80.6544 0.00 0.00 12.804 347.475 3960.96 -1 . OOOOOOOOOOOOOOCOOOOOOOO OOOOCOOOOOOOOOOOOOOOOOO A.4.4 ‘DQNO‘U‘b 10 12 13 14 15 16 17 1 GNOU©wNH 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27 27 29 29 31 31 33 3 2 33 34 19 18 14.428 103620.0 0.00258 1385.9 13.840 97932.0 0.00254 1328.9 13.080 92643.0 0.00254 1256.0 12.370 87643.0 0.00255 1187.7 11.690 82849.0 0.00255 1122.6 11.040 78198.0 0.00254 1059.8 10.400 73624.0 0.00254 998.5 1 6 6 8 8 10 10 12 12 14 14 16 16 18 18 20 uSNDNDP‘P‘F‘P‘P‘P‘F‘P‘F‘P‘F‘ 24947.0 2678184.0 0.0033 777.3 23921.0 2368498.0 0.0031 741.4 22608.0 2096060.0 0.0030 677.6 21378.0 1853374.0 0.0029 612.7 20206.0 1634933.0 0.0029 544.6 19077.0 1435822.0 0.0029 470.6 17973.0 1251613.0 0.0031 387.9 \lVO‘U‘kWNH 214 9775.0 3730340.0 0.1054 8823.0 3525540.0 0.1031 7606.0 3335160.0 0.1008 6464.0 3155160.0 0.0984 5378.0 2982580.0 0.0961 4330.0 2815140.0 0.0938 3307.0 2650480.0 0.0915 1355. 1277. 1177. 1083. 994. 908. 824. A.4.5 A.4.6 A.5 A.5.1 9 20 22 6 10 22 24 5 11 24 26 4 12 26 28 3 13 28 30 2 14 30 32 1 2 30.9315 30.9315 7 61.863 61.863 9 61.863 61.863 11 61.863 61.863 13 61.863 61.863 15 61.863 61.863 17 61.863 61.863 19 61.863 61.863 21 61.863 61.863 23 61.863 61.863 25 61.863 61.863 27 61.863 61.863 29 61.863 61.863 31 61.863 61.863 33 30.9315 30.9315 6 55.963 55.963 8 55.963 55.963 10 55.963 55.963 12 55.963 55.963 14 55.963 55.963 16 55.963 55.963 18 55.963 55.963 20 55.963 55.963 22 55.963 55.963 24 55.963 55.963 26 55.963 55.963 28 55.963 55.963 30 55.963 55.963 0.0406 0.0086 W EQDALDAIA 1 1 1 1 1 1 0 2 0 0 1 1 1 0 3 0 0 1 1 1 0 4 0 0 1 1 1 0 5 0 0 1 1 1 0 6 0 0 1 1 1 0 7 0 0 1 1 1 0 8 0 0 1 1 1 0 9 1 1 1 1 1 0 10 0 0 1 1 1 0 215 30.9315 61. 61. 61. 61. 61. .863 61 61. 61. 61. 61. 61. 61. 61. 863 863 863 863 863 863 863 863 863 863 863 863 30.9315 55. 55. 55. 55. .963 55 55. 55. 55. 55. 55. 55. 55. 55. 175. 175. 350. 350. 525. 525. 700. 700. c>c>c>c>c>c>c>c>2>23 963 963 963 963 963 963 963 963 963 963 963 963 0.0 133.5 90.937 133.5 121.25 133.5 90.937 133.5 0.0 133.5 0000000000 0000060000 216 11 1 1 1 1 1 1 -182.0 133.5 0.0 A-5-2 W5. 1 1 2 1.623 2 3 4 0.3264 3 5 6 0.3264 4 7 8 0.3264 5 10 9 1.623 6 11 2 2.44 A53 W 1 0.78 2.18 3.72 1.36 1 2 4 1 2 4 6 1 3 6 8 1 4 8 10 1 A.5.4 1 2.6559 3.939 35.990 21.01 13768.186 48670.998 14413.960 0.0024828 0.0024828 0.0065172 11.250 4.221 2 2.9058 4.126 41.580 24.75 15063.667 56230.623 15098.248 0.0024828 0.0024828 0.0065172 11.81 4.430 1 1 3 1 2 3 5 2 3 5 7 2 4 7 9 1 A.5.5 MASS 2 7.765 7.765 7.765 3 10.25 10.25 10.25 4 15.53 15.53 15.53 5 10.25 10.25 10.25 6 15.53 15.53 15.53 7 10.25 10.25 10.25 8 15.53 15.53 15.53 10 7.765 7.765 7.765 A.5.6 0.048300 0.0039000