ABSTRACT HERRICK, CHRISTOPHER KELLY. An

ABSTRACT

HERRICK, CHRISTOPHER KELLY. An Analysis of Local Out-of-Plane Buckling of Ductile Reinforced Structural Walls Due to In-Plane Loading. (Under direction of Mervyn J. Kowalsky.)

Reinforced structural walls are often implemented as an effective lateral force resisting system in multi-story buildings, and despite often being referred to as “shear walls,” they are usually designed as cantilevers that deform elastically under wind loads and form a plastic hinge at their base due to seismic loadings. Past research suggests that these plastic tensile demands and subsequent load reversals cause a plastic, localized lateral instability in walls. While lateral stability is addressed by some building codes, and good engineering judgment often prevents overly slender walls, plastic buckling is rarely directly addressed in current design standards. Even where design codes attempt to prevent local lateral instability, the effectiveness of such measures has not been fully examined. In 2010 and 2011, New

Zealand experienced earthquakes that damaged many structural wall buildings, and plastic buckling was observed to occur. This thesis re-examines two existing local buckling models using a range of data sources. A review of prior experimental work assesses the models' accuracy at predicting plastic buckling capacities. A parametric study on a range of walls examines the variables most influential in determining the point at which a structural wall is likely to develop lateral instability and buckle. Additionally, results from three non-linear time history analyses of buildings that experienced the 2010 and 2011 New Zealand earthquakes are presented and compared with damage observed in the field to assess each buckling model's accuracy. The impacts of these findings are discussed and revisions to the existing buckling models are suggested and recommendations are provided for future research on local instability of structural walls.

An Analysis of Local Out-of-Plane Buckling of Ductile Reinforced Structural Walls Due to In-Plane Loading

by Christopher Kelly Herrick

A thesis submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the degree of Master of Science

Civil Engineering

Raleigh, North Carolina

2014

APPROVED BY:

______Mervyn J. Kowalsky Committee Chair

______James M. Nau Rudolf Seracino

DEDICATION

I dedicate this thesis to my better half, Christine Nguyen. Thank you for your boundless assistance, never-ending encouragement and seemingly infinite patience and love that led to this being possible.

I also dedicate this thesis to my parents, Kerry and Joseph Herrick, whom spent many days, much of their sanity and all of their patience raising me and making me the person I am today.

BIOGRAPHY

Christopher Kelly Herrick was born and raised in Matthews, North Carolina. He graduated as Valedictorian, receiving his Bachelors of Science in Civil Engineering from

North Carolina State University in 2011. He continued his education at North Carolina State

University to earn his Masters of Science in Civil Engineering in 2013, with an emphasis on structural design.

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ACKNOWLEDGMENTS

I would like to thank my graduate advisor Dr. Mervyn Kowalsky for his time, effort and patience throughout my time as a graduate student. I am also grateful to Dr. James Nau, whose endless words of advice and support have guided my educational career to date.

Towards Dr. Rudolf Seracino I also extend a special thank you, for being member on my committee and providing me his expertise on reinforced concrete.

I would like to thank Dr. Sri Sritharan for providing many of the pictures within this thesis. I would also like to thank Dr. Murthy Guddati for providing his insight on analytical methods for modeling buckling.

I must also extend my gratitude to my friends at the Constructed Facilities

Laboratory: Yuhao Feng, Steven Fulmer, Chad Goodnight, Nicole King, Easa Khan and many others, all of whom who not only put up with my extended, over-complicated structural engineering discussions, thought experiments and “what-if” questions, but also who shared their time, knowledge, and experience with me despite this.

I would also like to thank my friend, Jason Hite, who provided help with

Mathematica and Matlab programming issues.

TABLE OF CONTENTS

LIST OF TABLES ...... viii LIST OF FIGURES ...... ix CHAPTER 1. INTRODUCTION ...... 1 1.1 Background ...... 1 1.2 Motivation and Objective ...... 4 1.3 Research Goals and Scope ...... 5 CHAPTER 2. BACKGROUND ...... 7 2.1 General Information ...... 7 2.1.1 Wall Background ...... 7 2.2 Literature Review...... 9 [17] 2.2.1 Design of Coupled Wall-Frame Structures for Seismic Actions, Goodsir 9 [46] 2.2.2 Stability of Ductile Structural Walls, Paulay and Priestley ...... 16 2.2.3 Lateral Stability of Reinforced Concrete Columns under Axial Reversed [7] Cyclic Tension and Compression, Chai and Elayer ...... 21 [8] 2.2.4 Minimum thickness of ductile RC structural walls, Chai and Kunnath ... 23 2.3 Design Code Considerations ...... 24 [51] 2.3.1 Uniform Building Code 1996 Vol. 2 ...... 25 [1] 2.3.2 American Concrete Institute Building Code Requirements (ACI 318-08) 26 2.3.3 Chile Structural Code, Normal Chilena Oficial (NCh 430-2008 and 433- [32][33] 1996) 26 [36][46] 2.3.4 New Zealand Concrete Structures Standard (NZS 3101:1995,2006) 27 CHAPTER 3. EARTHQUAKES OF INTEREST ...... 30 3.1 Introduction ...... 30 3.2 Darfield Earthquake (September 4th, 2010) ...... 31 3.3 Darfield Aftershocks ...... 33 3.4 Christchurch Earthquake ...... 35 3.5 Christchurch Aftershocks...... 39 CHAPTER 4. RESEARCH METHODS ...... 40 4.1 General Discussion ...... 40 4.2 Cumbia ...... 40 4.2.1 Variance of Confinement ...... 41 4.2.2 Calculation of Longitudinal Steel Reinforcement Ratio ...... 43 4.2.3 Calculation of Plastic Hinge Lengths ...... 44 4.2.4 Shear Strength ...... 45 CHAPTER 5. EXAMINATION OF PRIOR EXPERIMENTAL RESULTS ...... 47 5.1 Introduction ...... 47 5.2 Experimental Tests of Structural Walls ...... 48

[17] 5.2.1 Goodsir ...... 48 [48] 5.2.2 He and Priestley ...... 51 [22] 5.2.3 Ji ...... 54 [23] 5.2.4 Jiang ...... 56 [26][27] 5.2.5 Lefas and Kotsovos ...... 58 [39][40] 5.2.6 Oesterle et al...... 60 [53] 5.2.7 Zhang ...... 62 [54] 5.2.8 Zhou ...... 63 5.2.9 Comparison of Experimental Results with Predictions ...... 65 5.3 Experimental Tests of Prism Specimens...... 69 [3] 5.3.1 Azimikor et al...... 69 [7] 5.3.2 Chai and Elayer ...... 71 [9][10] 5.3.3 Chrysandis and Tegos ...... 72 [11] 5.3.4 Creagh et al...... 74 [17] 5.3.5 Goodsir ...... 76 5.3.6 Comparison of Prism Results...... 78 5.4 Conclusions from Prior Experimental Testing ...... 80 CHAPTER 6. PARAMETRIC STUDIES ...... 82 6.1 Introduction ...... 82 6.2 Parametric Study: Phase I ...... 84 6.2.1 Introduction ...... 84 6.2.2 Phase I Results ...... 85 6.2.3 Comparison with Code Requirements ...... 93 6.3 Parametric Study: Phase II ...... 94 6.3.1 Introduction ...... 94 6.3.2 Phase II Results ...... 95 6.3.3 Comparison with Code Requirements ...... 106 6.4 Parametric Study: Phase III ...... 107 6.4.1 Introduction ...... 107 6.4.2 Generation of Aspect Ratio Limit Curves ...... 107 6.4.3 Comparison with Code Requirements ...... 111 6.5 Parametric Study Conclusions ...... 111 CHAPTER 7. CASE STUDIES...... 113 7.1 Introduction ...... 113 7.2 General Model Information ...... 113 7.2.1 Overview ...... 113 7.2.2 General Material Properties ...... 114 7.2.3 Soil-Structure Interaction ...... 115

7.2.4 Frame Elements ...... 115 7.2.5 Structural Wall Elements ...... 116 7.2.6 Foundation Elements ...... 116 7.2.7 Floor Elements ...... 117 7.2.8 Structural Damping ...... 117 7.2.9 Vertical Loads ...... 118 7.2.10 Ground Motions ...... 119 7.3 Canterbury Television (CTV) ...... 123 7.3.1 Description ...... 123 7.3.2 Observed Damage ...... 126 7.3.3 Model Material Properties ...... 129 7.3.4 Model Results ...... 131 7.4 Pacific Brands House ...... 140 7.4.1 Description ...... 140 7.4.2 Observed Damage ...... 142 7.4.3 Model Material Properties ...... 144 7.4.4 Model Results ...... 145 7.5 Pyne Gould Corporation (PGC) Building ...... 152 7.5.1 Description ...... 152 7.5.2 Observed Damage ...... 154 7.5.3 Model Material Properties ...... 156 7.5.4 Model Response ...... 158 7.6 Case Study Conclusions ...... 169 CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS ...... 170 8.1 Summary ...... 170 8.2 Recommendations for Future Research ...... 171 8.3 Recommendations for Consideration of Wall Designs ...... 172 8.4 Recommendations for Future Field Examinations of Damaged Buildings ...... 173 CHAPTER 9. BIBILIOGRAPHY ...... 174 APPENDICES...... 181 Appendix A. Code for CumbiaWall (and with example input) ...... 182 Appendix B. Parametric Study Reference Data ...... 205 Appendix C. Prior Experimental Test Reference Data ...... 215

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LIST OF TABLES

Table 1 Limit state strains used in CumbiaWall ...... 83 Table 2 Variables and typical ranges considered for Phase I of the parametric study ...... 84 Table 3 Variables and typical ranges considered for Phase II of the parametric study ...... 94 Table 4 Properties assumed for strain-based aspect ratio limits ...... 108 [19] Table 5 Earthquake record station locations ...... 119 Table 6 Adopted start and end record times ...... 123 Table 7 Concrete material properties adopted for modeling the CTV ...... 129 Table 8 Steel material properties adopted for modeling the CTV ...... 130 [20] Table 9 Soil stiffnesses adopted from Hyland's modeling of the CTV ...... 131 Table 10 Material properties adopted for modeling the Pacific Brands House ...... 144 Table 11 Steel material properties adopted for modeling the Pacific Brands House ...... 144 Table 12 Material properties adopted for modeling the Pyne Gould Corporation building . 157 Table 13 Steel material properties adopted for modeling the Pyne Gould Corporation building ...... 157 Table 14 List of wall models and parameters for Phase I parametric study ...... 205 Table 15 List of wall models and parameters for Phase II parametric study ...... 207 Table 16 Prior Experimental Wall Tests - Loading and Geometry ...... 215 Table 17 Prior Experimental Wall Tests - Material Properties ...... 218 Table 18 Prior Experimental Wall Tests - Reinforcing Details ...... 220 Table 19 Prior Experimental Wall Tests - Displacements and Normalized Results ...... 223 Table 20 Prior Experimental Prism Tests - Loading and Geometry ...... 238 Table 21 Prior Experimental Prism Tests - Material Properties ...... 240 Table 22 Prior Experimental Prism Tests - Reinforcing Details ...... 241 Table 23 Prior Experimental Prism Tests - Strains and Normalized Results ...... 244

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LIST OF FIGURES

Figure 1 Example buildings with structural walls (Left[41], Right: Photo courtesy of Sri Sritharan)...... 1 Figure 2 Global out-of-plane instability ...... 2 Figure 3 Local buckling of an L-Wall (Photos courtesy of Sri Sritharan) ...... 3 Figure 4 Typical structural wall geometries and reinforcing patterns. From left to right: a rectangular wall with distributed reinforcement, a barbell wall with concentrated reinforcement and a T-wall with concentrated reinforcement...... 8 [17] Figure 5 Lateral instability of Wall 2 (Left) and Wall 3 (Right) as tested by Goodsir ..... 10 [17] Figure 6 Depiction of buckling damage seen in Wall 2 ...... 11 [17] Figure 7 Goodsir’s proposed buckling failure mechanism ...... 12 [17] Figure 8 Prism elements as tested by Goodsir ...... 13 Figure 9 Geometry of a strip of buckled wall of length ℓo ...... 14 Figure 10 Strain profile for cross section of a buckled wall’s core with a distance z between longitudinal reinforcement ...... 15 Figure 11 Strain profile for a cross section of buckled wall of thickness "b" ...... 17 Figure 12 Cross section of wall compression zone experiencing buckling ...... 18 [46] Figure 13 Internal forces for the buckling end region of a wall ...... 19 [45] Figure 14 Boundary element dimensions ...... 29 [19] Figure 15 Magnitude of seismic events following the Darfield Earthquake (Data: Geonet ) ...... 30 [41] Figure 16 Ground accelerations from Darfield Earthquake ...... 31 [18] Figure 17 Examples of differential settling (Left) and liquefaction (Right) ...... 32 [16] Figure 18 Darfield Earthquake and Subsequent Aftershocks ...... 34 [41] Figure 19 Ground accelerations from Christchurch Earthquake ...... 35 [15] Figure 20 Examples of liquefaction (Left) and differential settling (Right) ...... 37 [15][11] Figure 21 Two different rock falls near Redcliffs, east of Christchurch ...... 37 [20] Figure 22 Post-earthquake collapse of the Canterbury Television building (Left ) and Pyne [4] Gould Corporation building (Right ) ...... 37 Figure 23 Damage incurred by the Pacific Brands House (Courtesy of Sri Sritharan) ...... 38 Figure 24 Example of sub-dividing a rectangular wall (Left) and a barbell wall (Right) for analysis ...... 42 [17] Figure 25 Typical reinforcing layout for Goodsir's rectangular walls ...... 49 [17] Figure 26 Typical reinforcing layout for Goodsir's T-wall ...... 50 [17] Figure 27 Typical local buckling damage seen in Goodsir’s walls ...... 51 [48] Figure 28 Typical reinforcement layout for the T-walls tested by He and Priestley ...... 52

[48] Figure 29 Web out-of-plane curvature in He and Priestley's masonry T-walls ...... 53 [22] Figure 30 Reinforcing details for specimens tested by Ji ...... 54 Figure 31 Buckling of rectangular wall (Left) and cracking of barbell wall from Ji's tests [22] (Right) ...... 55 Figure 32 Details for standard (left) and energy dissipating (right) specimens tested by Jiang [23] ...... 56 [23] Figure 33 Damage seen at the end of Jiang's rectangular wall tests ...... 57 [26][27] Figure 34 Dimensions for wall specimens tested by Lefas and Kostovos ...... 58 Figure 35 Typical damage occurring in Lefas and Kostovos' walls under monotonic loading [27] (Left) and cyclic loading (Right) ...... 59 Figure 36 Typical reinforcing layout for rectangular (Left), barbell (Middle) and double [39] flanged walls (Right) tested by Oesterle et al...... 60 Figure 37 Failure modes in walls tested by Oesterle et al: Local buckling (Left), rebar [39] buckling and fracture (Middle), and concrete crushing (Right) ...... 61 [53] Figure 38 Typical reinforcement layout for walls tested by Zhang ...... 62 [53] Figure 39 Typical damage observed by Zhang ...... 62 [54] Figure 40 Reinforcing layout for walls tested by Zhou ...... 63 Figure 41 Damage seen in walls tested by Zhou: concrete crushing in a standard wall with no axial load (Left) and crushing in toe regions of a diagonally reinforced wall with axial [54] load (Right) ...... 64 Figure 42 Wall experimental displacements normalized to PPBM displacements ...... 65 Figure 43 Wall experimental displacements normalized to CEBM displacements ...... 66 Figure 44 Typical reinforcement and GFRP layout for walls tested by Azimikor et al. (left), [3] and local buckling typical of prisms tested by Azimikor (right) ...... 70 [7] Figure 45 Typical geometry and reinforcing details for prisms tested by Chai and Elayer 71 Figure 46 Typical buckling (Left) and crushing of cover concrete following buckling (right) [7] in prisms tested by Chai and Elayer ...... 72 Figure 47 Typical geometry and reinforcing details for prisms tested by Chrysandis and [3] Tegos ...... 73 [3] Figure 48 Prism specimen failure due to crushing and spalling (left) and buckling (right) 74 [11] Figure 49 Typical geometry and reinforcing details for prisms tested by Creagh ...... 75 [11] Figure 50 Prism specimen failure due to brittle crushing (left) and ductile buckling (right) ...... 75 [17] Figure 51 Reinforcing layout typical of Goodsir's prisms tests ...... 76 Figure 52 Damage seen in Goodsir's tall prism #4 (left), medium prism #6 (center), and squat [17] prism #8 (right) specimens ...... 77

Figure 53 Experimental prism strains normalized to PPBM strains ...... 79 Figure 54 Experimental prism strains normalized to CEBM strains ...... 79 Figure 55 Effect of wall length (Lw) on buckling capacity and strain-displacement response 86 Figure 56 Effect of wall height (Hw) on buckling capacity and strain-displacement response87 Figure 57 Phase I: Effect of wall thickness (tw) on buckling capacity and strain-displacement response...... 87 Figure 58 Phase I: Effect of concrete strength (fc') on buckling capacity and strain- displacement response ...... 88 Figure 59 Phase I: Effect of steel strength (fy) on buckling capacity and strain-displacement response...... 89 Figure 60 Phase I: Effect of longitudinal steel ratio (ρl) on buckling capacity and strain- displacement response ...... 90 Figure 61 Phase I: Effect of transverse steel ratio (ρt) on buckling capacity and strain- displacement response ...... 90 Figure 62 Phase I: Effect of axial load ratio (ALR) on buckling capacities and strain- displacement response ...... 91 Figure 63 Phase I: Effect of plastic hinge length (Lp) on buckling capacities and strain- displacement response ...... 92 Figure 64 Phase II: Effect of varying wall length (Lw) and wall thickness (tw) on buckling capacities and strain-displacement response ...... 96 Figure 65 Phase II: Effect of varying wall height (Hw) and wall thickness (tw) on buckling capacities and strain-displacement response ...... 97 Figure 66 Phase II: Effect of varying steel strength (fy) and wall thickness (tw) on buckling capacities and strain-displacement response ...... 99 Figure 67 Phase II: Effect of varying concrete strength (fc') and wall thickness (tw) on buckling capacities and strain-displacement response ...... 100 Figure 68 Phase II: Effect of varying longitudinal steel ratio (ρl) and wall thickness (tw) on buckling capacities and strain-displacement response ...... 101 Figure 69 Phase II: Effect of varying transverse steel ratio (ρt) and wall thickness (tw) on buckling capacities and strain-displacement response ...... 102 Figure 70 Phase II: Effect of varying axial load ratio (ALR) and wall thickness (tw) on buckling capacities and strain-displacement response ...... 104 Figure 71 Phase II: Effect of varying plastic hinge length factor (Lp) and wall thickness (tw) on buckling capacities and strain-displacement response ...... 105 Figure 72 Wall aspect ratios for walls of varying heights and longitudinal steel ratios calculated using PPBM ...... 109 Figure 73 Wall aspect ratios for walls of varying heights and longitudinal steel ratios calculated using CEBM ...... 110 Figure 74 Location of case study buildings and recording Stations (Image Courtesy of Google Maps)...... 119

Figure 75 Darfield acceleration time histories from station CCCC North (top) and East [19] (bottom) ...... 121 Figure 76 Darfield acceleration time histories from station REHS North (top) and East [19] (bottom) ...... 121 Figure 77 Christchurch acceleration time histories from station CCCC North (top) and East [19] (bottom) ...... 122 Figure 78 Christchurch acceleration time histories from station REHS North (top) and East [19] (bottom) ...... 122 Figure 79 View of the southeast corner of the CTV building prior to the Christchurch [20] earthquake (Photo courtesy of Phillip Pearson, Derivative work from Schwede66) . 124 [20] Figure 80 Location and shape of the CTV's North Core and South Wall ...... 125 [20] Figure 81 The CTV building, immediately following collapse ...... 127 Figure 82 Photo of the base of CTV's South wall, exhibiting slight out-of-plane [20] deformation ...... 128 [20] Figure 83 Location of soil springs on foundation elements ...... 130 Figure 84 CTV model with all elements visible (left), and with slab and roof elements hidden (right) ...... 131 Figure 85 CTV structural model with wall and foundation elements visible ...... 132 Figure 86 Response of CTV's top floor center of mass at roof height to REHS Darfield load history ...... 133 Figure 87 Response of CTV's top floor center of mass to REHS Christchurch and sequential load histories ...... 133 Figure 88 Naming convention for selected wall ends in CTV analysis model ...... 134 Figure 89 Strain demands in extreme longitudinal steel in South Wall end region CTV-1 . 135 Figure 90 Strain demands in extreme longitudinal steel in South Wall end region CTV-2 . 136 Figure 91 Strain demands in extreme longitudinal steel in South Wall end region CTV-3 . 136 Figure 92 Strain demands in extreme longitudinal steel in South Wall end region CTV-4 . 137 Figure 93 Strain demands in extreme longitudinal steel in North Core end region CTV-5 . 138 Figure 94 Strain demands in extreme longitudinal steel in North Core end region CTV-6 . 138 Figure 95 Strain demands in extreme longitudinal steel in North Core end region CTV-7 . 139 Figure 96 Strain demands in extreme longitudinal steel in North Core end region CTV-8 . 139 Figure 97 North-east face of the Pacific Brands House prior to the Christchurch earthquake (left), and the building's location and orientation (Courtesy of Google maps) ...... 141 Figure 98 Floor plan of assumed beam and column locations for modeling the PBH ...... 141 Figure 99 Buckling of PBH's L-wall as seen from the building's interior (courtesy of Sri Sritharan)...... 142 Figure 100 Buckling of PBH's L-wall as seen from the building's exterior (courtesy of Sri Sritharan)...... 143 Figure 101 PBH's southeastern unbuckled L-wall (courtesy of Sri Sritharan) ...... 143

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Figure 102 PBH model with all elements visible (left), and with slab and roof elements hidden (right) ...... 145 Figure 103 PBH structural model with wall and foundation elements visible ...... 145 Figure 104 PBH response to REHS Darfield load history at roof height of center of mass . 146 Figure 105 PBH response to REHS Christchurch and sequential load histories ...... 147 Figure 106 Naming convention for selected wall ends in CTV analysis model ...... 148 Figure 107 Strain demands in extreme longitudinal steel in L-Wall end region PBH-1 ...... 149 Figure 108 Strain demands in extreme longitudinal steel in L-Wall end region PBH-2 ...... 149 Figure 109 Strain demands in extreme longitudinal steel in L-Wall end region PBH-3 ...... 150 Figure 110 Strain demands in extreme longitudinal steel in L-Wall end region PBH-4 ...... 150 Figure 111 View of the southeast corner of the PGC building prior to the Christchurch and [4] Darfield earthquakes ...... 152 [4] Figure 112 Bottom floor plan (left) and typical upper floor plan (right) of the PGC building ...... 153 [4] Figure 113 The PGC building post-event ...... 155 Figure 114 PGC model with all elements visible (left), and with slab and roof elements hidden (right) ...... 158 Figure 115 PGC structural model with wall and foundation elements visible ...... 158 Figure 116 PGC response to REHS Darfield load history at roof height of center of mass . 159 Figure 117 PGC response to REHS Christchurch and sequential load histories ...... 160 Figure 118 Naming convention for selected wall ends in PGC analysis model (first floor) 161 Figure 119 Naming convention for selected wall ends in PGC analysis model (second floor) ...... 162 Figure 120 Strain demands in extreme longitudinal steel in end region PGC-1 ...... 163 Figure 121 Strain demands in extreme longitudinal steel in end region PGC-2 ...... 163 Figure 122 Strain demands in extreme longitudinal steel in end region PGC-3 ...... 164 Figure 123 Strain demands in extreme longitudinal steel in end region PGC-4 ...... 164 Figure 124 Strain demands in extreme longitudinal steel in end region PGC-5 ...... 165 Figure 125 Strain demands in extreme longitudinal steel in end region PGC-6 ...... 165 Figure 126 Strain demands in extreme longitudinal steel in end region PGC-7 ...... 166 Figure 127 Strain demands in extreme longitudinal steel in end region PGC-8 ...... 166 Figure 128 Strain demands in extreme longitudinal steel in end region PGC-9 ...... 167 Figure 129 Strain demands in extreme longitudinal steel in end region PGC-10 ...... 167 Figure 130 Strain demands in extreme longitudinal steel in end region PGC-11 ...... 168 Figure 131 Strain demands in extreme longitudinal steel in end region PGC-12 ...... 168

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CHAPTER 1. INTRODUCTION

1.1 Background

Reinforced concrete and masonry structural walls have been a staple lateral force resisting system in multi-story buildings for the past century. Though they are commonly referred to as shear walls, these walls are typically designed to respond in a ductile and flexural manner through adequate detailing of critical regions to prevent brittle modes of failure. To prevent confusion between the name of the system and the failure mode, this thesis refers to them henceforth as “structural walls” or simply “walls”. Figure 1 displays two examples of structural wall buildings: the hotel on the left has interior walls of various geometries, while the office on the right has two L-walls on either side of its exterior. Both structures have additional columns and beams to distribute floor loads.

[41] Figure 1 Example buildings with structural walls (Left , Right: Photo courtesy of Sri Sritharan)

Wall systems are efficient at supporting both vertical gravity loads as well as in-plane lateral loads such as wind or seismic demands, but are relatively weak in the out-of-plane direction. Structural walls often have boundary elements at their ends, which typically are more heavily reinforced. Thicker boundary elements may be added to further improve lateral stability and load capacity. Nevertheless, due to limitations imposed by either function or aesthetic requirements, thin walls without boundary elements are occasionally designed and implemented. In such cases, many current standards and codes prescribe limits to aspect ratios or suggest varying methods of design for loads that may induce lateral instability. Most of these limits only consider global lateral instability, where the entire length of the wall is expected to buckle in an elastic fashion seen in Figure 2.

Figure 2 Global out-of-plane instability

This global buckling mechanism contrasts with another type of lateral instability emphasized in prior work by Goodsir [17], Paulay and Priestley [46] and Chai and Elayer [7]: locally concentrated, out-of-plane buckling resulting from large inelastic load reversals.

Figure 3 shows one such example of local buckling found in the Pacific Brands House building in Christchurch, New Zealand resulting from the 2011 Christchurch earthquake. It should be noted that only the end region of the wall has buckled, not its entire length and the remainder of the wall was left relatively undamaged.

Figure 3 Local buckling of an L-Wall (Photos courtesy of Sri Sritharan)

While this specific structure experienced local buckling and did not collapse, such a failure leads to a rapid strength loss in the damaged wall. This results in the redistribution of the wall’s demands to other elements in the building, possibly causing additional failures in the surrounding structural elements. If the structure cannot reach equilibrium, this could lead to a complete collapse.

The research presented aims to review existing plastic buckling models in light of the

2010 and 2011 New Zealand earthquakes to support the need for their continued improvement, verification, and implementation to prevent future structural buckling failures.

1.2 Motivation and Objective

The 2010 Darfield and 2011 Christchurch earthquakes and their subsequent aftershocks in New Zealand claimed the lives of 185 people and injured several thousands more while incurring an estimated NZ$19 billion of damage to structures [11][34]. A large number of the injuries and deaths resulted from poor structural performance and partial or total collapse of several multi-story structures supported by structural walls. The Pacific

Brands House building exhibited instability issues where a wall experienced severe out-of- plane displacements and buckled significantly. Both of these structures remained standing, but were later demolished due to the likelihood of future collapse and the impracticality or impossibility of repairs. Two additional buildings – the Pyne Gould Corporation and

Canterbury Television buildings – were found via post-failure analyses to have collapsed due to a progressive failure of concrete crushing in walls and columns [21][20]. It should be noted

that out-of-plane instability of walls was not considered in these investigations and may have still played a part in their collapse.

Previous work has identified out-of-plane stability as an important issue in structural walls and created phenomenological models to relate the steel tensile strain required to induce the onset of buckling; however, there have been few instances of verifying their accuracy using results from actual seismic events. Few codes directly address the inelastic buckling failure mode, and instead provide limits to prevent elastic buckling. Most notably,

New Zealand's current structural standards include an adaptation of the current models, but the general lack of verification using field data brings such methods into question. This thesis evaluates the accuracy of buckling models at predicting the onset of plastic buckling, and assesses current code practices' effectiveness at preventing such a failure mode.

1.3 Research Goals and Scope

An initial literature review was performed regarding the buckling of structural walls, from which two buckling models were selected for comparison with results from a wide range of prior experimental tests. A parametric study was performed on a variety of structural walls to discover key influential material and geometric parameters that affect the displacement at which buckling was expected to occur. Buckling capacities and predicted damage using selected buckling models was compared with design code requirements. Three structural wall buildings damaged by the 2010 Darfield and 2011 Christchurch earthquakes in New Zealand were modeled and subjected to non-linear time history analyses. Results from these analyses were compared with the damage observed in the field, from which each

buckling models' accuracy was assessed, and additional improvements suggested. Each method aims to support either the current models' use in structural wall designs or to justify making modifications to improve accuracy and prevent plastic lateral buckling in future structures.

CHAPTER 2. BACKGROUND

2.1 General Information

2.1.1 Wall Background

Structural walls are commonly used to resist both axial and lateral forces in multi- story buildings. They are efficient at resisting both in-plane elastic horizontal loads such as wind and larger inelastic seismic loadings, provided they are designed to be ductile and are adequately detailed against shear and lateral instability. In a majority of cases where they are expected to resist extreme seismic loads, structural walls are designed to respond in a flexural nature by forming plastic hinges at their base to dissipate energy through the yielding of longitudinal steel and degradation of material in their end regions.

A variety of materials, shapes and geometries can be used to achieve a flexural, ductile response. Walls can be made of masonry units or concrete and are typically reinforced with longitudinal steel. Concrete walls include transverse steel to provide confinement and shear strength, while masonry walls may include steel plates along grout beds to do the same. Longitudinal reinforcing follows one of two patterns: one with reinforcing steel distributed evenly along the full length of the wall and another with reinforcing steel concentrated in the boundary elements and joints of the wall, but a reduced area of steel along their web. Some possible cross-sections and reinforcing pattern combinations are shown in Figure 4.

Figure 4 Typical structural wall geometries and reinforcing patterns. From left to right: a rectangular wall with distributed reinforcement, a barbell wall with concentrated reinforcement and a T-wall with concentrated reinforcement.

For a given material, shape and area of longitudinal reinforcement, structural walls have similar flexural strengths regardless of the reinforcing arrangement. Distributed reinforcing steel such as that in Figure 4above increases shear resistance, while slightly reducing the cross-section's first yield moment, and causes little change in overall strength[47].

Despite this, present day codes typically focus on reinforcing and confinement requirements in wall end regions. Additionally, due to their design requirements, materials, and aesthetic implications, over-sized end regions are generally only implemented when a design's load

necessitates their use. The resulting concentrations of longitudinal reinforcement in thin wall end regions have yet to be fully examined and reconciled against the plastic local buckling failure mode due to a general lack of field information and test data that warrants further research.

2.2 Literature Review

There exists extensive research on structural walls regarding their design, analysis, characteristics and behavior. As such, a full review of literature regarding structural walls is infeasible and will not be done here; instead this section focuses on literature pertaining to inelastic lateral instability. Goodsir[17] proposed an initial mechanism and model for the calculation of wall buckling loads and their resulting out-of-plane displacements based on a number of experimental tests. Paulay and Priestley [46] developed a more refined version of this model and proposed a method for the calculation of a wall stability criterion. This model was then further refined by Chai and Elayer [7] to include additional effects of reinforcing steel hysteretic response and to account for additional experimental data. These works and their proposed models are discussed at length below, followed by a review of current design codes and their standards relating to the design of walls to prevent lateral instability. Outside of this section, the Paulay and Priestley Buckling Model and the Chai and Elayer Buckling

Model are referred to as PPBM and CEBM, respectively.

2.2.1 Design of Coupled Wall-Frame Structures for Seismic Actions, Goodsir [17]

Goodsir proposed a capacity-based seismic design methodology for reinforced concrete frame-wall structures. This method attempts to restrict plastic demands to

adequately detailed regions to ensure sufficient energy dissipates through flexural deformations. In addition to performing non-linear time history analyses, Goodsir constructed and experimentally tested four 1:3 scale structural wall-frame systems using the proposed design method. While the walls displayed both good hysteretic behavior and energy dissipation, two failed due to local out-of-plane instability. Figure 5 displays the buckled end region of these two test specimens. A simplified illustration of the buckling damage experienced by Wall 2 is shown in Figure 6.

[17] Figure 5 Lateral instability of Wall 2 (Left) and Wall 3 (Right) as tested by Goodsir

[17] Figure 6 Depiction of buckling damage seen in Wall 2

Goodsir suggested that this lateral instability was the result of a specific reproducible and predictable mechanism based on the plastic demands experienced by the end regions of the specimens. When subjected to sufficiently large tensile demands shown in Figure 7(a), the reinforcing steel experiences plastic tensile strains and the confined end region concrete begins to crack. Upon a load reversal, the stress in the reinforcing steel progresses from tensile to compressive. Small variances in the compressive forces in the reinforcing bars due to geometric and material differences results in the concrete cracks closing first on one wall face, causing the wall to begin to displace out-of-plane. Assuming the displacement exceeds some critical threshold, the structural wall experiences a local buckling failure as seen in

Figure 7(b). A rapid loss of both axial and horizontal load capacity accompanies this failure.

If the structure cannot redistribute demands, this can lead to an undesirable sudden collapse.

It should be noted that if this displacement threshold is not reached, the concrete cracks close

and additional load cycles are possible. It is likely that some residual out-of-plane displacement still exists as in Figure 7(c) due to the inability of the wall to fully recover to its original undeformed shape.

[17] Figure 7 Goodsir’s proposed buckling failure mechanism

Goodsir continued to investigate the lateral instability issues observed during his large scale tests by constructing a series of nine prism elements and subjecting them to cyclic axial load reversals. These prisms were intended to represent the end regions of walls as shown in Figure 8.

[17] Figure 8 Prism elements as tested by Goodsir

From the resulting prism test data, Goodsir proposed a model based on Euler buckling theory, from which critical buckling loads and their corresponding out-of-plane

displacements can be calculated. The critical buckling load takes the form of (1.1), where Et

is the longitudinal steel’s tangent modulus of elasticity and e is the effective height of the end region of the wall and I is the moment of inertia of the reinforcing steel about the wall’s weak axis.

2  EIt Pcr  2 (1.1) e

When this axial load capacity is exceeded by the axial force applied to the longitudinal steel, the specimen is considered to have failed due to local lateral instability.

The out-of-plane displaced shape was assumed to have constant curvature as seen in Figure

9. Solving for the radius of curvature R, and assuming the structural wall’s total out-of-plane

displacement  is relatively small, gives (1.2). The buckling length o was assumed to be the full length of the prism units.

Figure 9 Geometry of a strip of buckled wall of length ℓo

2 R  o (1.2) 8

Figure 10 displays the assumed distribution of longitudinal steel residual strain across the core of the wall, without the inclusion of the cover. Solving for the radius of curvature in terms of the residual strain in the longitudinal steel results in (1.3), where z is the out-of- plane distance between the two layers of reinforcement.

Figure 10 Strain profile for cross section of a buckled wall’s core with a distance z between longitudinal reinforcement

z R  (1.3)  sm

Solving (1.2) and (1.3) for the out of plane displacement  , results in (1.4), allowing for the calculation of out-of-plane displacements for a given tension residual strain.

 2   s (1.4) 8z

Goodsir calculated the predicted buckling loads using (1.1) and the buckling displacements using (1.4) for the nine prisms tested and compared them to the measured experimental data.

Only one specimen’s behavior was closely approximated with the proposed equations. The large disparity between experimental response and the predicted behavior was attributed to bond degradation between the longitudinal steel and the surrounding concrete. This occurred due to the fact that tensile loads were applied directly to the longitudinal steel reinforcement and compressive loads were applied to the face of the concrete only, rather than to the entire top and bottom face of the specimen. Insufficient length to transfer these stresses from reinforcing steel to the surrounding concrete led to a strength loss under repeated load cycles.

Acknowledging the need for additional experimental data, Goodsir notes a number of additional factors that likely affect the mechanism described: disturbed aggregate, shear forces, cyclic loading, axial load level, cover spall, transverse reinforcement spacing and the scale of the tested walls. None of these effects were explicitly tested or expanded on within the experimental portion of this study, but the possible scope of each is discussed briefly and the importance of further study discussed.

2.2.2 Stability of Ductile Structural Walls, Paulay and Priestley [46]

Paulay and Priestley continued to expand and develop Goodsir’s initial buckling mechanism into a phenomenological model that accounts for geometric and material properties of the wall as well as the arrangement of longitudinal reinforcing steel. The model is based on two primary components: 1) a model relating residual strain in longitudinal steel

to the out-of-plane eccentricity ratio and 2) a stability criterion, the maximum eccentricity ratio sustainable by a given structural wall.

Paulay and Priestley assumed the buckling region corresponded to the plastic hinge region at the base of the wall and took the maximum curvature within the region to be constant, following Goodsir’s approach to solve for the radius of curvature as (1.5).

2 R  o (1.5) 8

Where Goodsir assumed a strain distribution only between longitudinal reinforcement, Paulay and Priestley assumed the strain to be distributed as shown in Figure

11, where b is the thickness of the wall and b is the transverse distance between the wall’s edge and the extreme layer of reinforcing steel. Solving for the radius of curvature in terms of the residual strain in the longitudinal steel results in (1.6).

Figure 11 Strain profile for a cross section of buckled wall of thickness "b"

b R  (1.6)  sm

Solving (1.5) and (1.6) for the eccentricity ratio  , given as  b , results in the final form (1.7), which can be used to calculate the eccentricity ratio imparted by a given residual strain. Conversely, it can be used to calculate the tensile strain associated with a wall’s maximum supported eccentricity ratio.

2  sm o    (1.7) 8 b

Where: b = distance from the interior side of the wall to the extreme longitudinal reinforcement

 sm = residual strains in the extreme longitudinal steel

o = height of the buckled region  = out-of-plane displacement of the wall

To develop an upper limit to the eccentricity ratio, Paulay and Priestley used force and moment equilibrium over an assumed compression zone as seen in Figure 12. This compression zone utilizes an equivalent stress block in both directions, where  is the compression zone’s depth of the in-plane direction and a is the compression zone’s depth in the out-of-plane direction.

Figure 12 Cross section of wall compression zone experiencing buckling

Considering the in-plane direction for a given length of the end region  and using

the notation found in Figure 13, the concrete compressive force, Cc can be shown to be (1.8),

where l is the longitudinal reinforcement ratio in the compression zone undergoing buckling.

[46] Figure 13 Internal forces for the buckling end region of a wall

bf C  ly (1.8) c  1 

Solving for the ratio  in the out-of-plane direction and assuming an equivalent rectangular stress block results in (1.9).

1 C c (1.9)  1 ' 2 0.85 fbc

Substituting (1.8) into (1.9) and rearranging to solve for the maximum eccentricity

ratio,cr gives (1.10), where m is the mechanical reinforcement ratio for the end region. The right hand limit represents the wall failing if the out-of-plane displacement causes the load resisted by the section to extend outside the lateral boundaries of the wall. When this critical value is exceeded, the structural wall fails due to out-of-plane instability and buckles.

2 lyf  0.5 1  2.35m  5.53 m  4.70 m  0.5 ; m  (1.10) cr   ' fc

Paulay and Priestley calculated the expected longitudinal steel residual strains in the plastic hinge region to cause buckling for “Wall 2” tested by Goodsir [16] and compared it to strains measured prior to buckling. The authors also used their buckling model to calculate ultimate displacement ductility for a series of masonry T-section walls previously tested by

Priestley, Seible and Calvi[48]. In both instances, the predictions showed relatively close agreement with results from the experimental tests given the somewhat coarse assumptions made by the model.

To provide design guidelines to prevent lateral instability, Paulay and Priestley developed a requirement for minimum wall thickness based on their stability criterion and buckling strain relationship. Starting with (1.7) - restated as (1.11) - and solving for b while

making assumptions regarding axial load, geometry, and reinforcement detailing results in a simplified design expression for minimum thickness of the form (1.12). For singly reinforced walls, the thickness calculated, b increases by a factor of 1.26.

2  sm o    (1.11) 8 b

 b  0.019  (1.12) p 

2.2.3 Lateral Stability of Reinforced Concrete Columns under Axial Reversed Cyclic

Tension and Compression, Chai and Elayer [7]

Chai and Elayer performed a series of tests on reinforced concrete columns to simulate the end region of a structural wall and adjusted the model proposed by Paulay and

Priestley to account for the new experimental data. Paulay and Priestley[46] previously derived

the tensile strain, sm necessary to induce local lateral instability, repeated as (1.13).

However, this method does not account for the hysteretic behavior for the reinforcing steel.

In order to adjust for this, they proposed another form of shown as (1.14).

2 b (1.13) sm  8   o

* sm  e   r   a (1.14)

Where:

 e = strain elastically recovered during initial reloading

 r = additional reloading strain required to yield the reinforcement in compression *  a = residual tension strain at the first closure of cracks

Both the elastic strain,  e and recovery strain, r were assumed to be proportional to

the yield strain of the longitudinal steel, of the forms ey 1  and ry 2  . During

experimental testing 1 was found to range from 1 to 1.5 and 2 was found to vary from 3 to

* [7] 5. In order to calculate the residual tension strain  a , Chai and Elayer proposed (1.15), a modified version of Paulay and Priestley’s model.

2 *2b a     (1.15) o

Where the previous model assumed a constant curvature throughout the plastic hinge region, Chai and Elayer[7] assumed a more realistic (but less conservative) sinusoidal

curvature distribution. They also recommended slightly conservative values of 1 1 and

2  2 , which results in lower values of strain being supportable prior to buckling. This

simplifies the expression to (1.16), the final form for the maximum tensile strain  sm . For comparison, Paulay and Priestley’s[46] formulation is provided again as (1.17).

2 b 2 (1.16) sm   3  y o

2 b (1.17) sm  8   o

Chai and Elayer[7] adopted the unmodified formulation of Paulay and Priestley’s[46] stability criterion equation, repeated as (1.18).

2 lyf  0.5 1  2.35m  5.53 m  4.70 m  0.5 ; m  (1.18) cr   ' fc

Following Goodsir’s method of testing prisms[17], fourteen 4x8 inch rectangular column prisms were constructed and experimentally tested to help verify the accuracy of the new model. It was observed that the new modified strain equation (1.16) showed better agreement with the test results than Paulay and Priestley’s[46] prior strain equation (1.17).

However, both were conservative, underestimating the maximum tensile strain supportable prior to buckling upon a load reversal.

2.2.4 Minimum thickness of ductile RC structural walls, Chai and Kunnath[8]

Chai and Kunnath calculated minimum thicknesses for walls of varying heights, lengths, wall-to-floor ratios, and steel reinforcement ratios when subjected to ground motions. Acceleration demands with a peak ground acceleration of 0.4g were selected for analysis, corresponding to UBC seismic intensities designed for in "zone IV" seismic areas.

Site effects such as soil variance and differences in attenuation across seismic frequencies were accounted for by varying peak ground acceleration to peak ground velocity (/)av ratios. Required thicknesses were calculated using Chai and Elayer's[7] buckling strain formulation and Paulay and Priestley's[46] stability criterion, which were compared to thicknesses required by New Zealand Standards of 1995[36] (NZS) and the Uniform Building

Code of 1996[51] (UBC).

Examining the author's calculated minimum thickness to prevent buckling, it was found thicker walls are required for lower ratio seismic demands, due to the stiffer

walls tending to result in a longer period and acceleration spectrum demand. Similarly, smaller wall-to-floor area ratios resulted in a general increase in required wall thickness.

Minimum wall thickness was insensitive to changes the tributary floor weight, and instead was overly affected by changes in wall height, with shorter walls resulting in larger required thicknesses. Lower levels of longitudinal reinforcement resulted in thinner required thicknesses, but the effect was less pronounced as low levels of reinforcement ratio.

Comparing the authors' calculated thicknesses to those required by NZS and UBC, the author's calculated required thickness agreed better with those determined by NZS in

most cases. Taller, slender (large Htww ratio), and more square (small HLww ratio) walls showed poorer agreement between minimum thicknesses. Walls with above average (/)av seismic ratios, large reinforcement ratios and large wall-to-floor ratios were also found to show poorer agreement among calculated thickness requirements. Paulay and Priestley suggest that prior to possible code revision, additional experimental research on plastic buckling be performed to assess the accuracy of the existing code requirements and buckling models.

2.3 Design Code Considerations

Although many design codes recognize lateral instability as an issue for structural walls, few prescribe specific requirements to avoid the out-of-plane buckling mechanism previously described. This section reviews code regulations related to lateral instability that may influence a wall's plastic buckling capacity. Requirements typically consist of

limitations on aspect ratios, geometry, and reinforcement ratios, or design methods that indirectly result in limitations on such properties.

2.3.1 Uniform Building Code 1996 Vol. 2 [51]

The Uniform Building Code (UBC) of 1996 attempts to prevent lateral instability in walls resisting both axial and flexural loads by requiring a minimum wall thickness sixteen times the wall's clear (between floors) height, shown as (1.19). While this limitation is likely too simple to accurately design against plastic buckling, it does ensure some level of out-of- plane stiffness. Unfortunately, the limit does not scale with wall heights, resulting in an mistakenly constant minimum thickness as long as story height does not change.

H t  cl (1.19) w 16

Walls are also designed considering out-of-plane failure under seismic loads using second-order analyses, but capacities are calculated without considering plastic buckling directly. Out-of-plane displacements are designed for, but the limits only apply under service loads, which should not cause inelastic steel strains. Excessively loaded walls or those with large compressive strains are required to include boundary elements with higher levels of reinforcing and confinement. The inclusion of boundary elements may prevent buckling of the wall's ends, but if the boundary element is sufficiently long and slender, it too may be prone to instability issues and plastic buckling.

2.3.2 American Concrete Institute Building Code Requirements (ACI 318-08)[1]

Wall detail requirements in ACI are similar to those in the UBC(1996)[51] in that lateral stability is considered for wall design, but plastic buckling is not directly addressed.

Second order analyses based on applied loads, limits to out-of-plane service, minimum wall thicknesses, and specially reinforced boundary element requirements all are similar in function to those found in the UBC and do not provide a reliable and accurate method to prevent local plastic buckling.

Empirically designed walls have a unique design methodology with additional wall thickness requirements similar to the UBC's approach, and are shown as (1.20). It should be noted that these requirements generally result in a smaller thickness being required than those required by (1.19), and despite empirical walls limiting eccentric loading, plastic buckling does not require out-of-plane loads to occur.

HL tuu and t  and t 100 mm (1.20) w25 w 25 w

2.3.3 Chile Structural Code, Normal Chilena Oficial (NCh 430-2008 and 433-1996) [32][33]

Chile's structural design code, the Norma Chilena Oficial, is a lightly modified version of the ACI 318-95[2] seismic provisions and ACI 318-05[1] concrete design codes. Due to this, requirements for structural wall systems are exceedingly similar to those previously listed. Design methods and their associated geometric or reinforcing limitations for walls either do not address lateral instability, or are based on an elastic buckling failure mechanism. One major difference between NCh and ACI's design requirements is that wall

boundary elements are not required by NCh to have additional transverse reinforcement and confinement and instead are simply required to resist applied overturning demands. This allowance was made following the good performance of buildings with poorly confined and unconfined end regions in the Viña del Mar 1985 earthquake. This change is expected to result in poorer performance of structural walls during seismic events where cyclic loads are likely to cause rapid spalling and a loss of poorly confined core concrete.

2.3.4 New Zealand Concrete Structures Standard (NZS 3101:1995,2006) [36][46]

The New Zealand Concrete Structures Standard currently provides guidelines that are meant to prevent plastic lateral instability in thin walls. NZS 3101:1995[36] requires for walls

to meet the minimum required thickness, breq as calculated using (1.21), (1.23) and (1.24).

These equations are simplified approximations from Paulay and Priestley's[46] research and recommendations. NZS 3101:2006[37] changed the calculation of to (1.22) , adding a

factor r to account for whether a wall is reinforced in a single layer or multiple layers and

simplifying the calculation of  to the factor  , which has predetermined constant values.

km 22 A r L w b  (NZS3101 :1995) (1.21) req 1700 

rk m A r 2 L w b  (NZS3101 : 2006) (1.22) req 1700 

Ln km 1.0 (1.23) 0.25 0.055ALrw

lyf  0.3 '  0.1 (1.24) 2.5 fc

Where: breq = Required wall thickness

 = Displacement ductility demand on the wall

r = 1.0 for doubly reinforced walls and 1.25 for singly reinforced walls  = 5 for limited ductile plastic regions, 7 for ductile plastic regions

Ar = Aspect ratio (total wall height divided by wall length)

Lw = Wall length

Ln = Wall clear height between floors

l = Longitudinal reinforcement ratio in the end region of the structural wall f y = Steel yield stress ' fc = Concrete compressive stress

When (1.22) controls the thickness of the wall at the end regions, NZS 3101:1995[36] provides additional requirements suggested in Paulay and Priestley's work [46], shown as

(1.25) and utilizes dimensions depicted in Figure 14. These additional limits aim to prevent the design of boundary elements prone to lateral instability. Though NZS specifically imposes limits to prevent plastic buckling, the codified equations are heavily simplified expressions for generalized wall systems, possibly resulting in a loss of accuracy in cases where wall properties or seismic demands differ from those assumed.

b bA2 1 w (1.25) 1 wb 10

[45] Figure 14 Boundary element dimensions

CHAPTER 3. EARTHQUAKES OF INTEREST

3.1 Introduction

On September 4, 2010, a series of large seismic events began to occur around the area of Christchurch, NZ over a year's time. This section gives a general overview of the two most largest events: the Darfield and Christchurch earthquakes. The Darfield event was followed by several thousand smaller, successive aftershocks. Despite many seismologists considering the Christchurch event to be an aftershock of the Darfield earthquake, it will be discussed as a separate earthquake due to the earthquake's impact on the people and infrastructure of New

Zealand. The general progression and magnitude of these events can be seen in Figure 15.

Size and darkness of the circles corresponds to the relative magnitude of each event.

Additional information regarding each event is provided in the following sections.

Darfield Earthquake Christchurch Earthquake Sept. 4, 2010 Minor Aftershock February 22, 2011 Jan. 20, 2011

Major Aftershock Minor Aftershock Sept. 8, 2010 Dec. 26, 2010

[19] Figure 15 Magnitude of seismic events following the Darfield Earthquake (Data: Geonet )

3.2 Darfield Earthquake (September 4th, 2010)

The Darfield earthquake (also referred to as the Canterbury earthquake) occurred at

4:35 am (NZDT) on September 4, 2010 with a moment magnitude of 7.1 Mw. The epicenter was located 10 km southeast of the city Darfield and was relatively shallow with a focus approximately 10 km deep[11]. Figure 16 contains accelerations measured during the event by seismographs stations in the surrounding area: red arrows display the maximum vertical acceleration while blue arrows indicate the maximum horizontal acceleration recorded, independent of their direction of travel. The relative size of the arrows corresponds to the magnitude of the acceleration. The largest peak ground accelerations near the area were 1.26 g vertically and 0.76 g horizontally. All stations surrounding the surface rupture had vertical- horizontal acceleration ratios greater than 1.5. These large vertical accelerations are thought to have exacerbated damage caused by the already large horizontal motions[11].

Darfield Earthquake Epicenter

Christchurch Building District

[41] Figure 16 Ground accelerations from Darfield Earthquake

The soil in the area is largely variable, but typically consists of a mix of fine particles from glacial silts/clay particles and some volcanic rock. This soft soil resulted in a large amount of seismic attenuation before reaching Christchurch, with demands generally lessening as they approached the city center. The epicenter also occurred in largely rural area away from larger structures. For both of these reasons, the Darfield earthquake resulted in no deaths and only two injuries: one from a falling chimney and another from broken glass [11].

Damage was primarily limited to small residential properties (typically unreinforced masonry buildings) and occurred as a result of differential foundation settling caused by stress-induced lateral spreading and liquefaction. Areas subjected to liquefaction also experienced sand boils, where a fine soil particles and water were forced to the surface, leaving behind deposits[18] as seen in Figure 17.

[18] Figure 17 Examples of differential settling (Left) and liquefaction (Right)

Examining damage occurring beyond that caused by geotechnical issues, damage was concentrated in the many unreinforced masonry (URM) buildings within the Darfield region.

A follow-up investigation surveyed 595 of the 958 URM structures in the area and found 125 to be unsafe for continued use, and 167 required repairs before continued use. Some URM buildings that had undergone partial or full seismic retrofitting withstood the earthquake with only minor damage. Those without retrofits fared poorly due to having limited deformation capacities. Buildings also experienced non-structural damage such as broken windows, minor damage to cosmetic elements, and varying levels of damage to building components and contents [11].

3.3 Darfield Aftershocks

In the weeks following the Darfield earthquake, thousands of aftershocks were reported to have Richter magnitudes of ML > 2 and eleven larger aftershocks of ML > 5.0 occurred around the fault line. Again, these events were characterized by large vertical- horizontal acceleration ratios. A major ML 5.2 aftershock struck September 8, 2010. This event occurred much closer to Christchurch and was only 7km southeast from the city center with a focus 6km deep. As a result, it accounts for a majority of the damage in the days after the Darfield earthquake.[18] Most other aftershocks were significantly smaller in magnitude or further from the city center. Figure 18 displays seismic events occurring in the Christchurch area following the Darfield earthquake on September 4.

Major Darfield Aftershock, Earthquake, September 8th September 4th 2010

[16] Figure 18 Darfield Earthquake and Subsequent Aftershocks

Damage from the Darfield aftershocks was typically less serious in magnitude, but of a similar nature to that of the primary earthquake: damage due to foundation spreading and liquefaction and a concentration of damage in unreinforced masonry buildings. Despite these aftershocks, many businesses reopened within two weeks of the initial event. Buildings too damaged to be repaired and deemed unsafe and unstable were demolished in the following weeks and months [11].

3.4 Christchurch Earthquake

The Christchurch earthquake (also referred to as the Lyttelton aftershock) occurred at

12:51 pm (NZDT) on February 22, 2011 as a continuation of the seismic events following the

Darfield earthquake. The epicenter of the moment magnitude 6.3 Mw earthquake was located

8km southeast of Christchurch, the largest city-center on the South Island. The event was relatively shallow with a focus approximately 4km deep.[15]

Christchurch Earthquake Christchurch Epicenter Building District

[41] Figure 19 Ground accelerations from Christchurch Earthquake

Figure 19 contains accelerations measured during the event: red arrows display the maximum vertical acceleration while blue arrows indicate the maximum horizontal

acceleration recorded, independent of their direction of travel [41]. Again, recorded vertical accelerations were larger than expected, often exceeding the horizontal accelerations by a factor of 2. When compared to the same depiction of the Darfield event in Figure 16 it can be seen that due to the close proximity of the epicenter, peak ground accelerations within

Christchurch are much larger, often exceeding 1g inside the Christchurch Building District

(CBD) and nearing 2g’s in the surrounding areas.

Again, the soil in the area was comprised of fine particles from glacial silts and clay particles and volcanic rock. Despite the soft soil, the proximity of the event to the CBD led to

185 deaths and a large number of injuries as a result of liquefaction, rock falls and structural damage leading to the collapse of many buildings. Liquefaction was much more pronounced during this event, affecting as much as 50% of Christchurch and the surrounding areas [15].

Figure 20 demonstrates the severity of the liquefaction and differential settling, while Figure

21 shows two rock falls near Redcliffs, a town East of Christchurch.

The damage occurring as a result of the liquefaction, differential settling and rock falls led to a significant number of injuries and deaths, however the majority of both resulted from the partial or complete collapse of several structures during the Christchurch earthquake. Most notably, the Canterbury Television building and the Pyne Gould

Corporation building suffered complete collapses. An exterior L-wall of the Pacific Brands

House offices exhibited partial buckling, but remained standing. Figure 22 and Figure 23 contain photos of each of these structural failures.

[15] Figure 20 Examples of liquefaction (Left) and differential settling (Right)

[15][11] Figure 21 Two different rock falls near Redcliffs, east of Christchurch

[20] Figure 22 Post-earthquake collapse of the Canterbury Television building (Left ) and Pyne Gould [4] Corporation building (Right )

Figure 23 Damage incurred by the Pacific Brands House (Courtesy of Sri Sritharan)

Within the three structures discussed above, a variety of structural walls experienced a range of seismic demands known to cause plastic buckling in at least one case. These structures were selected as case studies with the purpose of gathering valuable non- experimental field data from full-scale walls for use with Paulay and Priestley's buckling

model and Chai and Elayer's buckling model to help assess the models' accuracy. These structures, the damage they incurred, the non-linear analyses performed, and their respective results are discussed in CHAPTER 7.

It should be noted that additional buildings (most notably the Smith City Parking

Deck, among others) also experienced excessive damage or collapses. These structures are not discussed here due to having no structural walls or because their walls were formed a

“closed” shear core that lacked significant instability to be considered within the scope of this thesis. Beyond these specific buildings, hundreds of unreinforced masonry (URM) structures collapsed or experienced heavy damage. These collapses caused relatively few deaths or injuries due to local authorities closing or limiting access to many of these buildings after initial damage from the Darfield event. URM buildings retrofitted to more current standards prior to the Christchurch event generally sustained less damage than those left un-renovated.

Steel structures fared particularly well, with only a few structures being red-tagged as unusable[11].

3.5 Christchurch Aftershocks

In the days following the Christchurch earthquake, a series of successively smaller quakes occurred, but little to no damage was observed in comparison to the primary ground motion. Due to the rapid assessment of damaged buildings and the demolishing of structures at-risk for collapse, few injuries and even fewer deaths occurred as a result of the aftershocks following the Christchurch earthquake [11].

CHAPTER 4. RESEARCH METHODS

4.1 General Discussion

As previously mentioned, this project utilized a range of methods and tools to calculate displacements and strains within walls subjected to various axial and horizontal loadings. A modified version of Cumbia[30] was used to estimate force-displacement responses of individual members for comparison with prior experimental results and to conduct a parametric study on the selected buckling models. The three buildings selected as part of a case study were modeled in the analysis program SAP2000v14[13] and were subjected to non-linear time history analyses. Specifics regarding the program Cumbia and its modification for use with wall systems are given below. Due to the range of information conveyed, each of the three analyses performed in SAP2000v14 are discussed at length in

Section 7.2.

4.2 Cumbia

Cumbia is a Matlab program that performs detailed moment-curvature and force- displacement response calculations for reinforced concrete sections. The moment-curvature analysis is performed by calculating the moment and curvature response of a given cross- section for a wide range of feasible strains. To simplify the analysis, the cross-section is divided into a series of discrete layers, each with its own strain and stress level. Both the confined and unconfined concrete material models are those proposed by Mander[28], while the steel material model used was proposed by King[24]. A masonry material model was implemented based on a simplified expression for unconfined concrete with reduced strain

capacities, as suggested by Priestley, Calvi and Kowalsky[47]. Cumbia was originally intended for the sectional analysis of rectangular or circular members rather than walls, leading to a modifications being made to account for this difference. The modified program is referred to as CumbiaWall to prevent confusion between the original program and the adjustments made for analyses performed in this thesis. Additional information regarding the operation of

Cumbia and the methods it uses to calculate values for displacement, strains, and forces can be found in the user manual[31]. Code for the modified version of CumbiaWall and an example of its input is provided in Appendix A. The program was neither created nor tested for general use, and great care should be taken in verifying any output provided.

4.2.1 Variance of Confinement

Cumbia calculates an average level of confinement for an entire cross-section. While this approximation negligibly affects results for smaller, more regular cross-sections, it may lead to inaccurate results when analyzing structural walls. This is because walls often have additional reinforcement in their end regions and significantly less confined web regions. To adjust for this, CumbiaWall divides the cross-section into sub-sections where confinement is calculated individually. Figure 24 shows two examples of the sub-division's implementation.

Each sub-section is composed of an unconfined region and a confined region whose size and approximate location are based on recommendations for confined core size by the aforementioned Mander model. For the barbell wall shown in Figure 24, it can be seen that

“Confined Region 2” extends into the top and bottom sections slightly. This was done to

simplify connecting sub-sections' geometries and was found to have negligible effect on results for the cases considered.

Unconfined Region 1

Confined Region1

Unconfined Region 2

Confined Region 2

Unconfined Region 3

Confined Region 3

Figure 24 Example of sub-dividing a rectangular wall (Left) and a barbell wall (Right) for analysis

It is important to note that this approach can result in high levels of concrete strength in end regions and low levels of concrete strength in webs. While it is likely some interaction between end region confining steel and the web leads to a gradual transition between end region concrete strength and web concrete strength, CumbiaWall models the transition as a discrete change at the end region and web interface. Though such an instantaneous drop in strength is unlikely, crushing of web concrete occurs in walls with thicker boundary elements

and sparse web confinement and is typically concentrated at the ends of the web. This suggests the concrete strength "transition zone" between levels of confinement is relatively short in length, and therefore its effect was assumed to be negligible.

4.2.2 Calculation of Longitudinal Steel Reinforcement Ratio

CumbiaWall was also utilized to calculate the displacement at which the selected buckling models predict failure. As previously discussed, both models’ stability criterion is

' based upon the mechanical reinforcement ratio, m l f y/ f c , which is largely determined by

the local longitudinal steel reinforcement ratio, l found in the compression region undergoing buckling. Since the compression zone changes throughout the loading process,

the longitudinal steel ratio should theoretically vary. Alternatively, an average value of l can be assumed typical of the boundary element considered for buckling. For walls with distributed reinforcement, the difference in results between these two methods is relatively small, however for walls with reinforcement lumped in end regions, can vary by a larger amount if the compression zone extends into the sparsely reinforce web region. It was found that only drastically under-designed sections were observed to be affected by iteratively calculating . Because of this, it was assumed that the stability of most walls considered are likely controlled by their boundary element reinforcement ratio, resulting in the average being calculated and utilized in CumbiaWall.

4.2.3 Calculation of Plastic Hinge Lengths

While Cumbia calculates plastic hinge lengths for a member in single bending following the recommendations from Priestley, Seible and Calvi[48] of the form (1.26),

CumbiaWall uses a conservative modification (1.27) as suggested by Priestley, Calvi and

Kowalsky[47] based upon previous work by Paulay and Priestley[45]. The new method utilizes

the effective height of the wall instead of the contraflexure length. The additional term 0.1Lw is introduced to adjust for additional strain penetration induced by the larger amounts of tension shift commonly found in wall systems. Similarly, (1.28) is a modification suggested for masonry walls by Priestley, Calvi and Kowalsky[47], which takes into account the typical reduced bond capacity and increased strain penetration seen in prior experimental work.

LP k  L c  L sp  2 L sp (1.26)

LP k  H e 0.1 L w  L sp  2 L sp (1.27)

LHLLLP0.04  e  0.1 w  sp  3 sp (1.28)

Where: f k = 0.2u  1 0.08  f y

Lc = distance to point of contraflexure, typically L for members in single bending 2 He = wall effective height, taken as H w , the height of the wall 3

Lsp = 0.022 fdy bl fu = ultimate strength of local longitudinal reinforcement f y = yield strength of local longitudinal reinforcement dbl = diameter of local longitudinal reinforcement

4.2.4 Shear Strength

Cumbia[30] also calculates the shear strength envelope using the revised UCSD shear model[25]. The structural walls examined in this thesis are generally tall and slender, with a response dominated by flexural action. Due to this, CumbiaWall assumes shear strength to be a non-issue. CumbiaWall calculates shear deformations using the same method as Cumbia following the procedure outlined by Montejo[31] and initially provided by Priestley, Calvi and

Kowalsky[47] as outlined below.

AA 5 (1.29) sg6

M ' y (1.30) Ieff  ' Ecy

IAeff s k  (1.31) sg L

Ieff kks, eff sg (1.32) I g

0.25 y 2 ks, cr E s 0.6 D (1.33) 0.25yy 10

For each rectangular sub-section of a wall, the effective shear area is calculated via

(1.29) from which the effective moment of inertia can be found (1.30). The effective inertia is then used to proportion the gross shear stiffness (1.31) to the pre-cracking shear stiffness

(1.32). Post-cracking shear stiffness is calculated using an equivalent strut and tie model

(1.33). Shear displacements for both cases are calculated from the summation of shear stiffness across each sub-section of wall and the calculated shear demands. In general, shear

displacements were found to be small compared to flexural displacements, which was expected for the generally tall and slender walls examined.

CHAPTER 5. EXAMINATION OF PRIOR EXPERIMENTAL RESULTS

5.1 Introduction

Many past researchers have performed experimental tests on structural wall systems and their boundary elements while focusing on other topics such as the effects of seismic detailing, ductility, and nearly countless other topics. Where out-of-plane failures occur during these tests, many authors attribute the failure to a general lateral instability, but few distinguish between a global, elastic failure of an entire specimen and the local, plastic failure of a wall’s end region. Aside from the prior works reviewed, little to no research examines the failure mechanism of plastic out-of-plane buckling in great depth. Additionally, the prior models have been compared with a small selection of experimental wall test results, providing a limited understanding of the models’ accuracy. To address this, this section reviews a large number of prior experimental tests and compares their results with those predicted by models proposed by Paulay and Priestley [46] and Chai and Elayer[7].

The local buckling mechanism relies upon plastic tensile strains to be developed prior to a load reversal. Because of this, short and squat walls were neglected in favor of a walls with a height-to-length ratio of approximately 2:1 to ensure flexural behavior controlled, allowing for the formation of plastic hinges at their base. Both reinforced concrete and masonry walls were included in the review due to lateral instability not being exclusive to reinforced concrete wall systems. The review is split into two groups: experimental tests on partial-scale structural wall systems and axial load tests on prisms representative of boundary elements. A very brief overview of the experimental programs is provided, followed by a

comparison of each specimen’s results with those predicted by the selected buckling models.

In all, sixty-three tests on structural walls and forty-three tests on boundary prism elements are examined. Appendix C contains tables of information for each wall and prism specimen considered: Table 16 through Table 18 contains loading, dimensions, material properties and reinforcement details for wall specimens. The same information for prism specimens is provided in Table 20 through Table 22.

5.2 Experimental Tests of Structural Walls

5.2.1 Goodsir [17]

Goodsir tested four reinforced concrete structural walls constructed to approximately

1:3 to 1:4 scale. Of the four walls, three were rectangular and one was a flanged T-wall. All walls had boundary elements with higher amounts of longitudinal and transverse reinforcing steel, and also had a floor slab approximately 1 meter above their base. Figure 25 and Figure

26 show the typical reinforcing pattern for Goodsir’s rectangular walls and T-wall, respectively. Each wall was subjected to a combination of axial and horizontal loading in a series of increasing load cycles. The axial load varied in magnitude depending on the direction of loading. Significantly less compressive axial load was applied while pushing east than when pushing west to allow larger tension strains to develop in the member prior to each load reversal

[17] Figure 25 Typical reinforcing layout for Goodsir's rectangular walls

[17] Figure 26 Typical reinforcing layout for Goodsir's T-wall

Due to the slenderness of the walls, all four specimens experienced lateral instability in their end regions and buckled locally. The T-section experienced lateral instability only in its web, but continued to sustain loads in the reverse direction until the flange failed in compression under successively larger compressive cycles. Figure 27 displays the typical buckling failure mode. The buckled end region is circled, and the displaced shape’s centerline is traced with a dashed line.

[17] Figure 27 Typical local buckling damage seen in Goodsir’s walls

5.2.2 He and Priestley [48]

He and Priestley constructed and tested nine grouted, reinforced masonry T-walls.

Four were subjected to pseudo-static cyclic loadings and five were tested under a range of

scaled dynamic loadings. In both cases an additional compressive axial load was applied.

Walls were subjected to a pair of load reversals to a pre-determined ductility level, followed by a load cycle to half-yield and another cycle to quarter-yield. If a wall continued to sustain load, the process was repeated by incrementing the loading of the first two cycles to induce a higher level of ductility. The half-yield and quarter-yield cycles remained constant in magnitude throughout the test, only the "ductile" cycles were incremented. All walls had uniform amounts of longitudinal steel along their web and flange, which was placed in a single layer at the center of the masonry blocks and grouted in place. Horizontal reinforcing steel consisting of a single rebar per layer was typically spaced closer together in the web than the flange. Figure 28 shows the typical reinforcing pattern for these T-walls.

[48] Figure 28 Typical reinforcement layout for the T-walls tested by He and Priestley

In a majority of the nine tests, the walls failed with their web in compression. It is important to note again that the specimens only have a single layer of reinforcement and are significantly less stable laterally. Prior to failure, there was little indication of an impending failure beyond small out-of-plane deflections. As a result, it is difficult to determine if walls failed due to local instability or due to bar buckling. Inspection of images preceding collapse led to assuming that four walls buckled due to lateral instability. The walls that buckled all had higher longitudinal reinforcement ratios. A fifth wall with similar reinforcing steel did not buckle, but this was likely due to the wall being loaded at a 45° incline to the web. This would lead to reduced displacement demands in the direction of the web. Similar to

Goodsir’s tests, regardless of their failure mode, the flanged T-walls continued to sustain loads with the flange in compression under successively larger compressive cycles. Figure 29 displays the out-of-plane curvature of a specimen's web prior to its eventual lateral buckling.

[48] Figure 29 Web out-of-plane curvature in He and Priestley's masonry T-walls

[22] 5.2.3 Ji

Ji performed a series of lateral load tests on three reinforced concrete structural wall specimens constructed to 1:4 scale. The specimens had portions of reinforced concrete slab cast at equidistant points along the height of the wall to simulate floor slabs cast into a structural wall. A rectangle, barbell, and I-shaped specimen were tested under axial and cyclic in-plane lateral loading until failure. Large bearing rollers mounted to cross-beams were provided for the rectangular wall at the second and fourth floors to help prevent lateral instability. Reinforcing details for each wall can be found in Figure 30. All specimens had greater levels of longitudinal reinforcement confining steel in their end regions.

[22] Figure 30 Reinforcing details for specimens tested by Ji

Of the three walls tested, only the rectangular wall exhibited local instability and buckled out-of-plane on the second floor between the first and second floor slab stubs. The barbell and double flanged wall displayed mild amounts of flexural cracking and some cover spalling, but did not buckle. The effectiveness of thicker boundary elements at preventing lateral stability is easily observed by comparison with the rectangular wall in Figure 31.

[22] Figure 31 Buckling of rectangular wall (Left) and cracking of barbell wall from Ji's tests (Right)

As seen in Figure 31, the rectangular wall buckled between the representative first and second floors. This is unusual both due to the fact that lateral bracing was provided here, and because buckling should occur where plastic demands are largest: near the base of the wall. Vertical reinforcing irregularity such as splices not specified in the paper, or material imperfections leading to a reduced level of strength could have led to the unexpected failure location.

5.2.4 Jiang [23]

Jiang performed several lateral load tests on reinforced concrete rectangular wall specimens constructed at 1:3 scale to examine the effect of adding a thin layer of energy dissipating neoprene halfway along the length of the wall. Of those tested, two tall and slender walls were selected for their flexural behavior: a standard rectangular reinforced concrete wall, and an “energy dissipating wall” of similar dimensions. The energy dissipating wall consisted of two separate reinforced concrete walls separated halfway along its length by a thin vertical layer of neoprene. Their respective reinforcing layout and location of the neoprene layer are shown in Figure 32. Both specimens had reinforcement and confinement concentrated in the end regions.

[23] Figure 32 Details for standard (left) and energy dissipating (right) specimens tested by Jiang

For the energy-dissipating specimen, edges adjacent to the neoprene layer were not considered boundary elements and were reinforced similar to the web of the wall. To prevent puncturing the neoprene layer, transverse steel was detailed around each individual wall section. Due to this separation, each half of the wall was conservatively assumed to respond independently for the purposes of estimating the top displacement necessary to induce buckling. While this likely overestimated the wall’s displacement, it did not exceed the smaller, more conservative estimate for buckling displacement of the more rigid, single wall.

Both walls exhibited standard crushing at their corners, as seen in Figure 33, but no out-of- plane buckling occurred.

[23] Figure 33 Damage seen at the end of Jiang's rectangular wall tests

5.2.5 Lefas and Kotsovos [26][27]

Lefas and Kostovos tested several 1:3 to 1:4 scale reinforced concrete rectangular wall specimens. Only the ten taller, flexural walls with a height-to-length ratio of 2:1 are examined here. Some walls were subjected to an axial load while others had none. Seven specimens were subjected to a monotonically increasing horizontal load, and the remaining three were placed in a series of increasing cyclic horizontal loads designed to induce post- yield behavior in the extreme longitudinal reinforcement. The specimen’s geometries and reinforcing details are shown in Figure 34. All specimens had concentrations of longitudinal and transverse reinforcement in their end regions.

[26][27] Figure 34 Dimensions for wall specimens tested by Lefas and Kostovos

Despite varying concrete strengths and reinforcing details, none of the walls experienced local buckling. This outcome is expected for the seven walls monotonically loaded since no load reversals occurred following the development of plastic strains in the longitudinal steel. A majority of the walls cyclically loaded experienced excessive concrete crushing or rebar fracture prior to buckling. The typical damage observed in these walls can be seen in Figure 35.

Figure 35 Typical damage occurring in Lefas and Kostovos' walls under monotonic loading (Left) and [27] cyclic loading (Right)

5.2.6 Oesterle et al. [39][40]

Oesterle et al. performed a series of fourteen experimental tests on reinforced concrete walls. Two rectangular walls, two double flanged walls and ten barbell walls were constructed and subjected to compressive axial loading. One barbell wall was subjected to a monotonic horizontal load, while the remaining specimens were subjected to cyclic horizontal load reversals. All of the walls had the same height, length, and web thickness.

Longitudinal and confining steel was concentrated in the boundary elements for all specimens: the flanges, barbell, and a short length of the end region for the rectangular walls.

Typical steel layout for each of the three walls is shown in Figure 36.

Figure 36 Typical reinforcing layout for rectangular (Left), barbell (Middle) and double flanged walls [39] (Right) tested by Oesterle et al.

Only one wall of the fourteen tested failed due to local out of plane instability, the rectangular wall with a higher longitudinal steel reinforcement ratio in its end region. The other rectangular wall suffered rebar buckling and fracture on subsequent cycles, however no lateral instability occurred. The monotonically loaded barbell wall did not buckle locally due to the lack of load reversals and inability to develop plastic tensile strains necessary to cause local instability. The remaining nine barbell walls and two double flanged walls subjected to cyclic loadings did not buckle locally. Their failure modes included reinforcement bar buckling and fracture in walls with larger amounts of longitudinal steel and crushing of the boundary element’s concrete for those with less longitudinal steel or confinement. Examples of these failures are shown in Figure 37.

Figure 37 Failure modes in walls tested by Oesterle et al: Local buckling (Left), rebar buckling and [39] fracture (Middle), and concrete crushing (Right)

5.2.7 Zhang [53]

Zhang tested seventeen 1:3 to 1:4 scale reinforced concrete rectangular walls and subjected them to compressive axial loads and cyclic load reversals. Axial load, aspect ratio, concrete strength, boundary region length, longitudinal reinforcement ratio, and level of confinement were varied specimen to specimen. Figure 38 shows a typical specimen’s reinforcement layout, though the longitudinal and transverse rebar sizes and spacing vary.

None of the walls tested by Zhang failed due to lateral instability, and most of the walls either failed due to rebar buckling or concrete crushing. Some tests were stopped prior to reaching their maximum feasible displacements, resulting in less observed damage.

[53] Figure 38 Typical reinforcement layout for walls Figure 39 Typical damage observed by Zhang [53] tested by Zhang

5.2.8 Zhou [54]

Zhou tested four 1:3 to 1:4 scale reinforced concrete rectangular walls under cyclic loadings to determine the effect of including diagonal reinforcing across the bottom portion of structural walls. All four had longitudinal reinforcing and confining steel concentrated in their boundary elements. Two of these walls had slightly sparse transverse reinforcing in the web, but included additional diagonal reinforcing bars across this region. Two walls (one standard wall and one diagonally reinforced wall) were subjected to axial loads in addition to their horizontal loadings. No axial load was applied to the remaining two walls. Figure 40 shows reinforcing layouts for both walls and the orientation of the additional diagonal bars.

[54] Figure 40 Reinforcing layout for walls tested by Zhou

Neither the standard walls nor the diagonally reinforced walls displayed signs of lateral instability during testing. As no instability was observed, the effect of diagonal bars on a wall’s propensity for buckling is unknown. The additional diagonal reinforcing appeared to stiffen the walls in-plane response slightly while concentrating damage in the toe regions of the wall. All of the walls had large amounts of crushing at their toe regions. The specimen without axial load and no diagonal reinforcing exhibited greater levels of concrete crushing and spalling across the length of its base. Examples of damage sustained by both types of walls are shown in Figure 41.

Figure 41 Damage seen in walls tested by Zhou: concrete crushing in a standard wall with no axial load [54] (Left) and crushing in toe regions of a diagonally reinforced wall with axial load (Right)

5.2.9 Comparison of Experimental Results with Predictions

For each of the experimental wall tests discussed above, the maximum longitudinal tension strain supportable by a given wall prior to buckling on a load reversal was calculated using Paulay and Priestley’s buckling model (PPBM) and Chai and Elayer’s buckling model

(CEBM) as discussed in Sections 2.2.2 and 2.2.3. Steel strains were considered potential failure limit states for the purpose of calculating their corresponding displacement limits (or capacities) using a moment curvature sectional analysis program, CumbiaWall, as discussed previously in Section 4.2, which are then compared with the experimental displacements recorded. The experimental displacements, Δexp are normalized to PPBM and CEBM predicted buckling displacements, Δppbm and Δcebm and plotted in Figure 42 and Figure 43 respectively. The raw data for these plots can be found in Table 19 in Appendix C.

Goodsir He & Priestley Ji Jiang Lefas & Kostovos Oesterle Zhang Zhou

Figure 42 Wall experimental displacements normalized to PPBM displacements

Goodsir He & Priestley Ji Jiang Lefas & Kostovos Oesterle Zhang Zhou

Figure 43 Wall experimental displacements normalized to CEBM displacements

In the Figure 42 and Figure 43, each data point represents a displacement resulting in tension strain preceded and followed by a compressive strain loading. For dynamic loadings with many partial reversals, only the largest displacement corresponding to the tension strain peak between a full compression-tension-compression reversal is shown. For symmetric walls subjected to multiple cyclic loads, displacement data for each tension cycle for a single edge of the wall is shown. Monotonic load cases that did not apply any tension demands are recorded as a "zero".

It should be noted that whether a wall was observed to have "buckled" is subjective in nature, and may vary from the each original author's observations. Some considered a loss of load with any out-of-plane motion buckling, while others required only lateral motion and

cover spalling. For the previously discussed cases, a wall was assumed to have buckled if it displayed a significant out-of-plane displacement and underwent a noticeable loss of strength

(on the margin of 10-30%) not attributable to another failure mechanism such as bar buckling, bar fracture, or concrete crushing. Additionally, while values closer to a value of unity indicate a model’s predicted buckling displacement is close to the experimental displacement, this does not necessarily mean the model is accurate, due to the discrete nature of cyclic loading. Instead, data points with less scatter, corresponding to specimens with smaller load or displacement increments provide a greater level of confidence than those that rapidly jump orders of magnitude in size.

Observing Figure 42 and Figure 43, it can be seen that CEBM typically shows a better agreement than PPBM, as the former has less false positives (predicts buckling, but no buckling occurred) and false negatives (predicts no buckling, but buckling occurred). Four walls appear inconsistent regardless of the model used and are discussed in greater detail below: Wall #2 tested by Goodsir[17], Wall #14 tested by Ji[22], and Walls #41 and #42 tested by Oesterle[39][40].

Wall #2 was loaded beyond its predicted buckling capacity, but did not buckle until subjected to a second load cycle nearly identical to the first. This suggests that load history and cyclic fatigue likely plays some role in determining the likelihood of out-of-plane buckling. It is expected that the additional concrete spalling and crushing, as well as cyclic fatigue between cycles contributes to the overall softening of a wall, but the magnitude of such an effect is unable to be estimated from a single test.

Wall #14 buckled well before its capacity estimated by both models, failing at 72% of its buckling displacement calculated using PPBM and 46% of its buckling displacement calculated using CEBM. Additionally, despite being laterally supported at the second and fourth floor levels, it buckled at the second floor level between floor slab stubs, rather than near its base, where the greatest demands should have occurred. As previously mentioned, it is possible reinforcement or material imperfections could have led to this unusual failure. It is also feasible that the wall attempted to deflect laterally but pivoted about its bracing, resulting in increased lateral displacements and a reduced buckling capacity.

Wall #41 displaced laterally slightly, but was never accompanied by a drop in load capacity. Instead of plastically buckling in a global fashion as predicted, Wall #41 experienced local rebar buckling and eventually rebar fracture. Paulay and Priestley's buckling mechanism relies on longitudinal steel preventing crack closure long enough to develop instability, and as a result it is likely that rebar buckling lessens the possibility of buckling due to its inability to sustain large enough compressive loads to result in out-of- plane displacements large enough to cause plastic buckling.

Wall #42 tested by Oesterle[39][40] exhibited out-of-plane displacement during the two cycles prior to its eventual buckling, but lateral bracing was applied to both ends of the wall at a height corresponding to a scaled first story height. Though this was likely meant to simulate lateral resistance provided by beams or slabs, its effectiveness in reproducing similar effects is unknown. Other wall specimens tested by Goodsir included laterally unbraced floor diaphragm stubs and still experienced buckling, suggesting the lateral bracing

provided by Oesterle may have prevented an earlier onset of buckling by simulating an exceptionally rigid floor diaphragm incapable of out of plane displacements.

Neglecting these four irregularities, the Chai and Elayer method of calculating buckling displacements appears more consistent, and has fewer false positives, as well as a narrower range of values. Despite this visually observable improvement, wall #3 by

Goodsir[17] buckled at 85% of its CEBM estimated displacement. Due to the coarseness of the method, this level of error is relatively small and within acceptable tolerances.

5.3 Experimental Tests of Prism Specimens

5.3.1 Azimikor et al. [3]

Azimikor et al. constructed and tested five reinforced, grouted masonry prisms with varying levels of longitudinal steel. A single specimen was subjected to a monotonic axial compression load while the remaining four prisms were tested under a series of cyclic axial loads. Three of the four cyclic prisms were strengthened with Glass Fiber Reinforced

Polymer (GFRP) through applying a varying number of vertical strips to the upper half of each face of the specimen. Additional horizontal GFRP layers were added at the top and bottom ends of the vertical GFRP strips. Figure 44 shows a typical arrangement of GFRP and reinforcing layout for the prisms.

Three of the prisms subjected to cyclic loading buckled: the plain prism without

GFRP and two prisms with less GFRP and a greater level of longitudinal reinforcement. The fifth prism with the least longitudinal reinforcement and a large amount of GFRP failed due to bar buckling and fracture followed by concrete crushing. These results suggest that the

additional confinement provided by this GFRP wrap effectively limited the spread of plasticity up the member. This reduced the plastic hinge length and improved the section’s resistance to lateral instability, leading to a higher level of strain being needed to induce buckling. Figure 44 shows typical buckling seen in Azimikor’s GFRP specimens where the wall buckles only in the section without the confining wrap.

Figure 44 Typical reinforcement and GFRP layout for walls tested by Azimikor et al. (left), and local [3] buckling typical of prisms tested by Azimikor (right)

5.3.2 Chai and Elayer [7]

Chai and Elayer tested fourteen reinforced concrete prisms while revising Paulay and

Priestley’s buckling model. Prisms were constructed with varying lengths and longitudinal reinforcing ratios. Specimens with a higher longitudinal reinforcement ratio had transverse confining steel spaced slightly farther apart as shown in Figure 45. All specimens were subjected to increasing cyclic axial loads until buckling occurred. Prisms with larger height- length ratios consistently buckled at lower strains. Similarly, specimens with higher levels of longitudinal reinforcement and lower levels of transverse reinforcement buckled at lower strains. Some specimens also experienced crushing of cover concrete after buckling due to the increasing compressive demands imparted upon the concave side of the wall. Figure 46 displays both the buckling typically observed and the crushing that occurred during testing.

[7] Figure 45 Typical geometry and reinforcing details for prisms tested by Chai and Elayer

Figure 46 Typical buckling (Left) and crushing of cover concrete following buckling (right) in prisms [7] tested by Chai and Elayer

5.3.3 Chrysandis and Tegos [9][10]

Chrysandis and Tegos tested a series of sixteen reinforced concrete prisms to examine the effect of reinforcing steel tension demands and longitudinal reinforcement ratio on lateral instability. All of the prisms were of similar geometry, with eleven having varying levels of longitudinal reinforcement and the remaining five having a constant amount of longitudinal steel. Reinforcing details and dimensions typical of these specimens are shown in Figure 47.

[3] Figure 47 Typical geometry and reinforcing details for prisms tested by Chrysandis and Tegos

Of the five similarly reinforced specimens, four were loaded in tension to reach a predetermined level of strain in the specimen, and then specimen was compressed until failure.

The remaining similarly reinforced specimen underwent a single cycle of compression until failure. The two specimens with the largest tensile demands buckled out-of-plane while the other three failed due to degradation of the compressive zone typically associated with high axial demands. The other eleven specimens were loaded to a tension strain of approximately

0.03, and then compressed until failure. Despite varying levels of reinforcement, all eleven specimens failed due to out-of-plane buckling. Both failure modes are depicted in Figure 48.

[3] Figure 48 Prism specimen failure due to crushing and spalling (left) and buckling (right)

5.3.4 Creagh et al. [11]

Creagh et al. tested two rectangular reinforced concrete prisms of similar geometry and reinforcement content to examine lateral instability in walls. One specimen was subjected to a tensile strain of approximately 0.04 followed by compression until it experienced a ductile buckling failure, while the other was simply compressed until a brittle crushing failure occurred. The reinforcing details are shown in Figure 49, while Figure 50 contains images of both specimens' failures. Again, a large tension strain is necessary to induce plastic buckling within the specimen. Applying only a compressive load leads to a brittle crushing failure, which is consistent with Paulay and Priestley's proposed buckling mechanism.

[11] Figure 49 Typical geometry and reinforcing details for prisms tested by Creagh

[11] Figure 50 Prism specimen failure due to brittle crushing (left) and ductile buckling (right)

5.3.5 Goodsir [17]

In addition to the afore mentioned reinforced concrete walls, Goodsir also tested a series of nine reinforced concrete prisms. All prisms shared similar cross section sizes, but varied in length and reinforcing layout. Five tall, two medium, and two squat specimens were built and tested. The specimens' lengths and reinforcing details are shown in Figure 51.

[17] Figure 51 Reinforcing layout typical of Goodsir's prisms tests

One of the tall specimens was loaded in a single compressive cycle until failure, while the remaining eight specimens were loaded in a series of tension-compression cycles of increasing magnitude. The tension force was applied directly to the longitudinal steel and the compressive force was applied to the ends of each specimen, which were capped with a steel plate to help distribute stresses. Transferring stresses in this fashion led to a ratcheting effect between the longitudinal steel and the surrounding concrete, causing rapid degradation of the rebar's bond to the concrete. Due to this, Goodsir comments that the behavior of the specimens' end regions may be affected. This concentration of damage at the specimen's ends can be seen in Figure 52 for some of the specimens.

Figure 52 Damage seen in Goodsir's tall prism #4 (left), medium prism #6 (center), and squat prism #8 [17] (right) specimens

Despite the loss of bond strength at the specimens' ends, each of the four tall prisms subjected to cyclic loading buckled in an out-of-plane fashion. The two medium specimens displaced laterally at higher load cycles, but did not display the loss of strength associated with a buckling failure until concrete crushing occurred. The two squat specimens exhibited little to no out-of-plane displacements and eventually failed due to concrete crushing. The tall prism loaded only in compression failed due to concrete crushing and did not show any lateral buckling.

5.3.6 Comparison of Prism Results

For each of the experimental prism tests discussed above, the maximum longitudinal tension strain supportable by a given specimen prior to buckling on a load reversal was calculated using Paulay and Priestley’s buckling model (PPBM) and Chai and Elayer’s buckling model (CEBM) as laid out in Sections 2.2.2 and 2.2.3. These steel strains were considered as failure limit states and are compared with the experimental strains recorded.

The experimental strains, ɛexp are normalized to the PPBM and CEBM predicted buckling strains, ɛppbm and ɛcebm, and plotted in Figure 53 and Figure 54 respectively. The raw data for these plots can be found in Table 23 in Appendix C. In the figures below, each data point represents a tension strain followed by a compressive strain loading. For cyclic load patterns with many partial reversals, only the largest strain demand corresponding to the peak tension strain between a full compression-tension-compression reversal is shown. Monotonic load cases that did not apply any tension demands are recorded as a "zero".

Azimikor Chai & Elayer Creagh Crysandis Goodsir

Figure 53 Experimental prism strains normalized to PPBM strains

Azimikor Chai & Elayer Creagh Crysandis Goodsir

Figure 54 Experimental prism strains normalized to CEBM strains

As previously discussed in Section 5.2.9, it was noted that whether a wall was observed to have "buckled" is subjective. This holds particularly true for prisms due to their greater instability compared to the end-regions of a wall, which is partial restrained by a portion of wall not yet buckling. Similar to the experimental wall tests, a prism was assumed to have buckled if it displayed a significant out-of-plane displacement and underwent a noticeable loss of strength (on the margin of 10-30%) not attributable to another failure mechanism such as bar buckling, bar fracture, or concrete crushing.

From Figure 53 and Figure 54, it can be seen that CEBM again shows typically a better agreement than when using PPBM, however the improvement in relative accuracy is much greater for prisms. In particular, singly reinforced masonry prisms such as those tested by Azimikor[3] have their buckling capacities severely underestimated. Data from prisms tested by Goodsir[17] and Chai and Elayer[7] indicates that PPBM more accurately predicts buckling capacities for shorter, squat reinforced concrete prisms, with increasing levels of inaccuracy for more slender members. This trend is not noticeable when using CEBM, but the method's accuracy for prisms is lower than that for walls. In the most extreme cases, it over-estimates buckling strain capacities by at least 15% (for Prism #42) and underestimates buckling strain capacities by at least 32% (for Prism #6). Again, these bounds represent the minimum levels of inaccuracy observed in the selected prisms.

5.4 Conclusions from Prior Experimental Testing

Within this chapter, results from prior experimental tests on wall and prism specimens were examined. Buckling capacities were calculated using two buckling models to

predict the occurrence of plastic buckling, which was compared with the experimental damage. In general, the less conservative CEBM was found to predict buckling capacities more accurately for a wider range of wall and prism specimens. Unfortunately, many of the wall and prisms considered were loaded under cyclic loads whose intervals were too large to provide the desired level of resolution. Many wall specimens were controlled by alternative limit states, and never approached demands large enough to buckle. Finally, issues of scale were not considered, as no full-scale wall tests were considered. For these reasons, additional tests on walls designed to buckle are warranted. While additional prism tests may provide information regarding buckling capacity, walls are inherently more stable than prisms due to their unbuckled length providing additional lateral resistance. Using prism data for developing buckling models may result in the under-estimation of buckling capacities.

CHAPTER 6. PARAMETRIC STUDIES

6.1 Introduction

Given the nature of plastic out-of-plane wall buckling, which results in a sudden strength loss and a high probability of immediate failure, it is desirable to prevent such a failure mode where feasible. Member designs that lead to failure prior to reaching ultimate strength are not only costly and inefficient, but often dangerous, especially in this instance.

CHAPTER 5 provided some prior experimental evidence towards verifying the accuracy of the buckling models, but the implications of such models needs further investigation. To improve understanding of both the Paulay and Priestley buckling model (PPBM)[45] and the

Chai and Elayer buckling model (CEBM)[7], a series of parametric studies were performed to identify the parameters most influential in determining a given wall's buckling capacity and whether it will buckle prior to reaching ultimate strength. The study was performed in three phases:

 Phase 1: An initial small-scale parametric study to determine influential variables.

 Phase 2: Expanding on Phase 1, additional models were created simultaneously

varying wall thickness and several selected parameters.

 Phase 3: Buckling models were reparameterized in terms of key parameters to

provide aspect ratio curve limits that prevent buckling from occurring prior to

longitudinal steel reaching its ultimate tension strain.

In each phase, models of structural walls were created and analyzed using a modified version of the Matlab[20] moment-curvature sectional analysis program, Cumbia.[30] Strain

values corresponding to serviceability, damage control and ultimate design limit states as suggested by Priestley, Calvi and Kowalsky [47] are shown in Table 1. The serviceability limit state corresponds to a compression strain prior to the occurrence of concrete cover spalling and a tension strain resulting in crack widths less than 1 mm in thickness. The damage control limit state corresponds to strains likely to result in repairable concrete spalling, large residual crack widths. Finally, the ultimate limit state strains correspond to excessive, likely un-repairable damage to concrete and reinforcement prior to collapse. Similarly, buckling strain limits were calculated using PPBM and CEBM. Displacements were calculated for each of these strains using strain-displacement relationships obtained for each analytical wall model using CumbiaWall.

Table 1 Limit state strains used in CumbiaWall

Steel Concrete Serviceability 0.015 0.004 Damage Control 0.06 * Ultimate 0.09 * * : Calculated using Mander concrete model.

Walls predicted to buckle prior to reaching their ultimate limit state are assumed to represent walls that will buckle if subjected to sufficiently large demands and never reach their ultimate strengths. From a design perspective, this is a highly undesirable outcome, and should be avoided by ensuring buckling does not control over more ductile failure modes.

Instances where buckling limit states control are checked against code requirements in the

UBC, ACI, and NZS to determine their effectiveness at preventing buckling.

6.2 Parametric Study: Phase I

6.2.1 Introduction

Phase I of the parametric study consisted of creating a series of structural wall models with a range of geometric and material properties to determine the most influential parameters. Variables considered included: wall thickness, length and height, steel and concrete strength, axial load ratio, longitudinal and transverse steel reinforcement ratios, plastic hinge length (Lp) and aspect ratio (H/L). A standard wall with typical properties was assumed, from which additional models were created by varying each parameter individually across a range of values to be compared on a variable-by-variable basis. While some correlation between parameters is likely, this method reduces sample size while still providing information on each property's influence. Table 2 summarizes the variables and their respective ranges utilized for the initial parametric study. Values under the column labeled “Typical” were held constant while varying a single parameter along the selected range of values, resulting in the creation of a total of forty walls for Phase I.

Table 2 Variables and typical ranges considered for Phase I of the parametric study

Variable Typical Wall length, L (m) 2 3 4 6 10 Wall height, H (m) 6 10 15 20 25 Wall thickness, t (m) 0.1 0.2 0.3 0.4 0.5 Steel Strength (MPa) 350 400 450 500 550 Concrete strength (MPa) 20 25 30 35 40 ' Axial load ratio ( P/ fcg A ) -0.05 0 0.1 0.2 0.3 Longitudinal steel ratio (%) 0.5 % 1.0 % 2.0 % 3 % 4 % Transverse steel ratio (%) 0.25 % 0.5 % 0.75 % 1.25 % 1.5 % Plastic hinge length, LP (%) 50 % 75 % 100 % 125 % 150 %

Each parameter's range was selected to account for normal variances resulting from construction practices and design codes changing over time, as well as geographically. It should be noted that exceptionally thin walls, while not prevalently used in constructed buildings, were included in the study due to the common practice of experimental studies

utilizing scaled models of walls. Where Lp varied, it was calculated as a percentage of the value found using (1.27). For further reference, Table 14 in Appendix B contains a full list of models created for Phase I and their relevant properties.

Each of the plots in the following sections display strain-displacement responses for a group of walls that vary a single parameter at a time. A solid line indicates the predicted response for an individual wall, while the fine dotted line represents the extrapolated behavior that would have occurred if other failure modes had controlled. Values corresponding to predicted buckling capacities calculated using PPBM and CEBM are plotted along each wall's response curve. Horizontal dashed lines indicate steel limit state strains, which typically controlled the failure response of the walls. Instances where concrete limit states controlled are indicated by the early termination of the wall's strain-displacement section response indicated by a solid line.

6.2.2 Phase I Results

Figure 55, Figure 56, and Figure 57 display the effect of varying length, height, and thickness, respectively. As wall length increases, buckling strain and displacement capacities decrease due to the increase in calculated plastic hinge length using (1.27). As wall height

and flexural demands increase, estimated buckling strain capacities decrease, but the walls buckle at a higher level of displacement. Wall thickness had little impact on strain- displacement relationship, but greatly influenced buckling capacities. Despite the relatively large variance in geometry, few walls were predicted to reach their ultimate strain (and strength) prior to buckling, with wall thickness having the largest effect.

0.2

0.15

0.1 Ult.

PPBM Limit

D.C. CCBM Limit Steel Tension Strain Tension Steel 0.05 Section Response

Extrapolated Response

Ser. Design Limit States 0 0 0.5 1 1.5 2 2.5 3 3.5 Displacement (m)

Figure 55 Effect of wall length (Lw) on buckling capacity and strain-displacement response

0.2

0.15

0.1 Ult.

PPBM Limit D.C. 0.05 CCBM Limit

Steel Tension Strain Tension Steel Section Response Extrapolated Response Ser. Design Limit States 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Displacement (m)

Figure 56 Effect of wall height (Hw) on buckling capacity and strain-displacement response

0.5m

0.4m

0.3m

0.2m

0.1m

Figure 57 Phase I: Effect of wall thickness (tw) on buckling capacity and strain-displacement response

Figure 58 and Figure 59 show the effect of varying concrete strength and steel strength on buckling capacity and wall response. Higher strength concrete and lower strength steel resulted in an increased buckling strain and displacement capacity. This suggests high strength concrete walls are less likely to experience plastic buckling, but high-strength reinforcement causes walls to be more prone to buckling failure. Though both concrete and steel strength affected buckling, the change was relatively small when compared to the effect of other variables such as wall geometry.

Increasing fc'

Figure 58 Phase I: Effect of concrete strength (fc') on buckling capacity and strain-displacement response

0.2

0.15

0.1 Ult.

PPBM Limit D.C. 0.05 CCBM Limit Steel Tension Strain Tension Steel Section Response Extrapolated Response Ser. Design Limit States 0 0 0.5 1 1.5 2 2.5 Displacement (m)

Figure 59 Phase I: Effect of steel strength (fy) on buckling capacity and strain-displacement response

Figure 60 and Figure 61 show the effect of varying longitudinal and transverse steel ratios on buckling strain and displacement capacities. As longitudinal steel content increase, buckling strain and displacement capacities sharply drops. This particularly concern because many codes provide boundary elements requirements that often result in wall designs with high volumes of longitudinal reinforcement lumped at their end regions. Such walls have a greatly reduced buckling capacity compared to wall of similar geometry and material properties, but calls for longitudinal reinforcement to be distributed along its length. While neither PPBM nor CEBM include the effect of transverse reinforcement in the calculation of buckling capacity, more transverse steel would likely increase strains and displacements sustainable before buckling due to an increase in concrete confinement and strength levels.

0.2

0.15

0.1 Ult. PPBM Limit D.C. 0.05 CCBM Limit

Section Response Steel Tension Strain Tension Steel Extrapolated Response Ser. Design Limit States 0 0 0.5 1 1.5 2 2.5 Displacement (m)

Figure 60 Phase I: Effect of longitudinal steel ratio (ρl) on buckling capacity and strain-displacement response

0.2

0.15

0.1 Ult. PPBM Limit D.C. 0.05 CCBM Limit

Section Response Steel Tension Strain Tension Steel Extrapolated Response Ser. Design Limit States 0 0 0.5 1 1.5 2 2.5 Displacement (m)

Figure 61 Phase I: Effect of transverse steel ratio (ρt) on buckling capacity and strain-displacement response

Figure 62 shows the effect of varying axial load ratio on wall buckling capacity and strain-displacement response. Similar to transverse steel ratio, neither PPBM nor CEBM directly accounts for axial load ratio in determining whether a wall is likely to buckle. Large compression or tension demands can cause limit states other than buckling to control. In the case of tension loads large enough to prevent the wall from reversing and achieving compressive yielding of longitudinal steel, plastic buckling cannot occur. Similarly, if compression loads are large enough to prevent the wall from experiencing plastic tension strains, the wall cannot buckle in a plastic manner. It should be noted that in the second case, elastic Euler buckling or concrete crushing can still occur.

0.2

0.15

0.1 Ult.

PPBM Limit D.C. CCBM Limit 0.05 Steel Tension Strain Tension Steel Section Response Extrapolated Response Ser. Design Limit States 0 0 0.5 1 1.5 2 2.5 Displacement (m)

Figure 62 Phase I: Effect of axial load ratio (ALR) on buckling capacities and strain-displacement response

Figure 63 shows the effect of changing plastic hinge length on buckling capacity.

Many variations on plastic hinge models exist, leading to some uncertainty surrounding the values calculated. Longer plastic hinges drastically reduce a wall's predicted buckling capacity, while shorter plastic hinges stiffen the section. As discussing in Section 5.3, experimental prisms tested by Azimikor et al. [3] were wrapped with GFRP for various distances along their length. This stiffened the specimens against lateral failure and reduced the length of specimen that exhibited plastic behavior, effectively reducing the plastic hinge length and successfully increasing buckling capacity.

0.3

0.25

0.2

0.15

0.1 PPBM Limit

Ult. CCBM Limit Steel Tension Strain Tension Steel D.C. Section Response 0.05 Extrapolated Response Ser. Design Limit States 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Displacement (m)

Figure 63 Phase I: Effect of plastic hinge length (Lp) on buckling capacities and strain-displacement response

6.2.3 Comparison with Code Requirements

Of the forty wall models tested and analyzed in phase one, UBC's requirements for wall thickness (1.19) were met by all but the thinnest wall, and ACI's requirements for wall thickness (1.20) were met by all but the thinnest wall and the longest wall. The thinnest two walls, the tallest wall, and the longest wall did not meet NZS 3101:2006 requirements (1.22).

Despite the overwhelming appearance of meeting standards designed to prevent lateral instability, all but three walls were predicted by PPBM to buckle prior to their ultimate limit state. The less conservative CEBM predicted all but six walls buckling at strains under their ultimate limit state. A full listing of model agreement with selected codes and whether predicted buckling strains exceed the ultimate limit state strain is provided in Table 14 in

Appendix B.

In many instances of exceedingly thin, tall, or long walls, or walls with high levels of longitudinal reinforcement, buckling was predicted to occur before the damage control limit state was reached. In the most severe case, both PPBM and CEBM predicted the thinnest wall (with a thickness of 100 mm) to buckle prior to ever reaching displacements or strains corresponding to the serviceability limit state, which is unacceptable. Due to wall thickness having a high level of influence over buckling capacities, and the likelihood of it having a confounding effect on other variable's trends, a large number of additional parametric models were created in the second phase of the parametric study.

6.3 Parametric Study: Phase II

6.3.1 Introduction

Phase II of the parametric study consisted of creating structural wall models similar to those in Phase I, but in greater numbers to allow for the variation of wall thickness and another parameter at the same time. The variables considered included: length and height, steel and concrete strength, axial load ratio, longitudinal and transverse steel reinforcement ratios, and plastic hinge length (Lp). Models were created using a method similar to Phase I, but individual parameters were varied for a range of wall thicknesses rather than assuming a typical wall thickness. Values under the column labeled “Typical” were held constant while varying a single parameter along its selected range of values for each wall thickness, resulting in the creation of a total of 240 walls. Table 3 summarizes the variables and their respective ranges utilized for the initial parametric study. For further reference, Table 15 in

Appendix B contains a full list of models created for Phase II and their properties.

Table 3 Variables and typical ranges considered for Phase II of the parametric study

Variable Typical Wall thickness, t (m) 0.15 0.2 .25 0.3 0.35 0.4 Wall length, L (m) 2 3 4 6 10 Wall height, H (m) 6 10 15 20 25 Steel Strength (MPa) 350 400 450 500 550 Concrete strength (MPa) 20 25 30 35 40 ' Axial load ratio ( P/ fcg A ) -0.05 0 0.1 0.2 0.3 Longitudinal steel ratio (%) 0.5 % 1.0 % 2.0 % 3 % 4 % Transverse steel ratio (%) 0.25 % 0.5 % 0.75 % 1.25 % 1.5 % Plastic hinge length, LP (%) 50 % 75 % 100 % 125 % 150 %

Plots in the following sections are grouped by the parameter varied, with each individual plot containing data from walls of a given thickness. A solid line indicates the predicted response for an individual wall, while the fine dotted line represents the extrapolated behavior that would have occurred if other failure modes had controlled. Values corresponding to the predicted buckling capacities calculated using PPBM and CEBM are plotted along each wall's response curve. Horizontal dashed lines indicate steel limit state strains, which typically controlled the failure response of the walls. Instances where concrete limit states controlled are indicated by the early termination of the wall's strain-displacement section response indicated by a solid line.

6.3.2 Phase II Results

Figure 64 and Figure 65 (and sub-figures "a" through "f") display the effect of varying length or height for a range of wall thicknesses. Similar to Phase I, as wall length increases, buckling strain and displacement capacities decrease due to the increase in plastic hinge length and inelastic reinforcing strains. As wall heights increase, estimated buckling strain capacities decrease, but the walls also buckle at a higher level of displacement.

Increasing wall thickness amplifies the magnitude of these changes to buckling capacities.

Despite the relatively large variance in geometry, few walls were predicted to reach ultimate strain (and strength) prior to buckling, with wall thickness having the largest effect, followed by wall heights.

0.2 0.2 tw = 0.15m tw = 0.2m

Lw = Various Lw = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. PPBM D.C.

0.05 CCBM 0.05

Steel Tension Strain Tension Steel Strain Tension Steel Section Response Extrapolated Response Ser. Design Limit States Ser. 0 0 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Displacement (m) Displacement (m)

(a) Varying wall length with tw = 0.15m (b) Varying wall length with tw = 0.2m

0.2 0.2 tw = 0.25m tw = 0.3m

Lw = Various Lw = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C.

0.05 0.05

Steel Tension Strain Tension Steel Strain Tension Steel

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Displacement (m) Displacement (m)

0.2 0.2 tw = 0.35m tw = 0.4m

Lw = Various Lw = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C.

0.05 0.05

Steel Tension Strain Tension Steel Strain Tension Steel

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Displacement (m) Displacement (m)

(e) Varying wall length with tw = 0.35m (f) Varying wall length with tw = 0.4m

Figure 64 Phase II: Effect of varying wall length (Lw) and wall thickness (tw) on buckling capacities and strain-displacement response

0.25 0.25

tw = 0.15m, Hw = Various tw = 0.2m, Hw = Various

0.2 0.2

0.15 0.15

0.1 0.1 Ult. Ult. PPBM

D.C. CCBM D.C.

Steel Tension Strain Tension Steel Strain Tension Steel 0.05 Section Response 0.05 Extrapolated Response Ser. Design Limit States Ser. 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Displacement (m) Displacement (m)

(a) Varying wall height with tw = 0.15m (b) Varying wall height with tw = 0.2m

0.25 0.25

tw = 0.25m, Hw = Various tw = 0.3m, Hw = Various

0.2 0.2

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C. Steel Tension Strain Tension Steel Steel Tension Strain Tension Steel 0.05 0.05

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Displacement (m) Displacement (m)

0.25 0.25

tw = 0.35m, Hw = Various tw = 0.4m, Hw = Various

0.2 0.2

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C. Steel Tension Strain Tension Steel Steel Tension Strain Tension Steel 0.05 0.05

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Displacement (m) Displacement (m)

(e) Varying wall height with tw = 0.35m (f) Varying wall height with tw = 0.4m

Figure 65 Phase II: Effect of varying wall height (Hw) and wall thickness (tw) on buckling capacities and strain-displacement response

Figure 66 and Figure 67 display the effect of varying concrete compressive strength or steel yield stress for a range of wall thicknesses. Larger values of steel yield strengths reduced buckling capacities, however this change was significantly impacted by wall thickness. Exceedingly thin walls (with a thickness of 0.15 or 0.2 m) saw little to no change in buckling capacity, while thicker walls saw a much larger change. This suggests that for overly slender walls, geometries limit the buckling capacities in both PPBM and CEBM, rather than the material properties. For thicker walls that are less affected by geometrically induced instability, an increased steel strength leads to more significant impacts. This trend is also observable when varying concrete strengths, but is less pronounced. Holding all other variables constant, this means older buildings are less likely to buckle due to their concrete continuing to cure and gain strength as it ages.

Figure 68 and Figure 69 show the effect of varying steel longitudinal and transverse reinforcement ratios for a range of wall thicknesses. Lower levels of longitudinal steel significantly increase buckling strain and displacement capacities. Again, this effect is more pronounced for thicker walls. The observed behavior supports distributing longitudinal reinforcement along the length of seismically designed walls rather than lumped at end regions. The effect of transverse reinforcement was not captured by the analyses, nor the buckling models due to neglecting the effect of additional confinement on concrete strength.

Increasing transverse steel ratio (and confinement levels) should increase concrete strengths, increasing buckling capacities similar to those previously discussed and seen in Figure 67.

0.2 0.2 tw = 0.15m tw = 0.2m

fy = Various fy = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. PPBM D.C.

0.05 CCBM 0.05

Steel Tension Strain Tension Steel Strain Tension Steel Section Response Extrapolated Response Ser. Design Limit States Ser. 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Displacement (m) Displacement (m)

(a) Varying steel strength with tw = 0.15m (b) Varying steel strength with tw = 0.2m

0.2 0.2 tw = 0.25m tw = 0.3m

fy = Various fy = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C.

0.05 0.05

Steel Tension Strain Tension Steel Strain Tension Steel

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Displacement (m) Displacement (m)

0.2 0.2 tw = 0.35m tw = 0.4m

fy = Various fy = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C.

0.05 0.05

Steel Tension Strain Tension Steel Strain Tension Steel

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Displacement (m) Displacement (m)

(e) Varying steel strength with tw = 0.35m (f) Varying steel strength with tw = 0.4m

Figure 66 Phase II: Effect of varying steel strength (fy) and wall thickness (tw) on buckling capacities and strain-displacement response

0.2 0.2 tw = 0.15m tw = 0.2m

fc' = Various fc' = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. PPBM D.C.

0.05 CCBM 0.05

Steel Tension Strain Tension Steel Strain Tension Steel Section Response Extrapolated Response Ser. Design Limit States Ser. 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Displacement (m) Displacement (m)

(a) Varying concrete strength with tw = 0.15m (b) Varying concrete strength with tw = 0.2m

0.2 0.2 tw = 0.25m tw = 0.3m

fc' = Various fc' = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C.

0.05 0.05

Steel Tension Strain Tension Steel Strain Tension Steel

Ser. Ser. 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Displacement (m) Displacement (m)

0.2 0.2 tw = 0.35m tw = 0.4m

fc' = Various fc' = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C.

0.05 0.05

Steel Tension Strain Tension Steel Strain Tension Steel

Ser. Ser. 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Displacement (m) Displacement (m)

(e) Varying concrete strength with tw = 0.35m (f) Varying concrete strength with tw = 0.4m

Figure 67 Phase II: Effect of varying concrete strength (fc') and wall thickness (tw) on buckling capacities and strain-displacement response

100

0.3 0.3 tw = 0.15m tw = 0.2m ρl = Various ρl = Various

0.25 0.25

0.2 0.2

0.15 0.15

0.1 0.1 Ult. PPBM Ult.

CCBM Steel Tension Strain Tension Steel Steel Tension Strain Tension Steel D.C. D.C. 0.05 Section Response 0.05 Extrapolated Response Ser. Design Limit States Ser. 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 Displacement (m) Displacement (m)

(a) Varying longitudinal steel ratio with tw = 0.15m (b) Varying longitudinal steel ratio with tw = 0.2m

0.3 0.3 tw = 0.25m tw = 0.3m ρl = Various ρl = Various

0.25 0.25

0.2 0.2

0.15 0.15

0.1 0.1

Ult. Ult. Steel Tension Strain Tension Steel Steel Tension Strain Tension Steel D.C. D.C. 0.05 0.05

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 Displacement (m) Displacement (m)

0.3 0.3 tw = 0.35m tw = 0.4m ρl = Various ρl = Various

0.25 0.25

0.2 0.2

0.15 0.15

0.1 0.1

Ult. Ult. Steel Tension Strain Tension Steel Steel Tension Strain Tension Steel D.C. D.C. 0.05 0.05

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 Displacement (m) Displacement (m)

(e) Varying longitudinal steel ratio with tw = 0.35m (f) Varying longitudinal steel ratio with tw = 0.4m

Figure 68 Phase II: Effect of varying longitudinal steel ratio (ρl) and wall thickness (tw) on buckling capacities and strain-displacement response

101

0.2 0.2 tw = 0.15m tw = 0.2m

ρt = Various ρt = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. PPBM D.C.

0.05 CCBM 0.05

Steel Tension Strain Tension Steel Strain Tension Steel Section Response Extrapolated Response Ser. Design Limit States Ser. 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Displacement (m) Displacement (m)

(a) Varying transverse steel ratio with tw = 0.15m (b) Varying transverse steel ratio with tw = 0.2m

0.2 0.2 tw = 0.25m tw = 0.3m

ρt = Various ρt = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C.

0.05 0.05

Steel Tension Strain Tension Steel Strain Tension Steel

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Displacement (m) Displacement (m)

0.2 0.2 tw = 0.35m tw = 0.4m

ρt = Various ρt = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C.

0.05 0.05

Steel Tension Strain Tension Steel Strain Tension Steel

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Displacement (m) Displacement (m)

(e) Varying transverse steel ratio with tw = 0.35m (f) Varying transverse steel ratio with tw = 0.4m

Figure 69 Phase II: Effect of varying transverse steel ratio (ρt) and wall thickness (tw) on buckling capacities and strain-displacement response

102

Figure 70 and Figure 71 contain results from varying axial load ratios or plastic hinge lengths for a range of wall thicknesses. As before in Phase I, axial load ratio is not accounted for in either buckling model. Instead the axial load affects wall strain-displacement behavior.

Walls with a large compressive axial load ratio of 20% or 30% resulted in walls failing prior to reaching their ultimate steel strain limit state, and instead were controlled by the concrete degradation and crushing. If the compressive loads are large enough to prevent walls from achieving significant inelastic steel strains, inelastic buckling cannot occur, but this comes at the cost of designing to prevent or mitigate the brittle compressive failure likely to control.

Smaller values of plastic hinge length leads to higher estimated buckling strain and displacement capacities due to reducing the length of wall made unstable by inelastic demands. The effect is amplified by increases in wall thickness, which also reduces plastic hinge length due to the increase in stiffness provided to both the in-plane and out-of-plane directions. As previously discussed, GFRP wraps to limit regions of inelastic demands at the base of walls were found to be effective at increasing buckling capacities. This comes at the cost of stiffening the wall significantly, which reduces ductility and could lead to brittle failures when the strength of GFRP is exceeded. GFRP could also be used to repair and restrain walls prone to plastic buckling due to prior inelastic tensile steel strains.

103

0.2 0.2 tw = 0.15m tw = 0.2m

ALR = Various ALR = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. PPBM D.C.

0.05 CCBM 0.05

Steel Tension Strain Tension Steel Strain Tension Steel Section Response Extrapolated Response Ser. Design Limit States Ser. 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Displacement (m) Displacement (m)

(a) Varying axial load ratio with tw = 0.15m (b) Varying axial load ratio with tw = 0.2m

0.2 0.2 tw = 0.25m tw = 0.3m

ALR = Various ALR = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C.

0.05 0.05

Steel Tension Strain Tension Steel Strain Tension Steel

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Displacement (m) Displacement (m)

0.2 0.2 tw = 0.35m tw = 0.4m

ALR = Various ALR = Various

0.15 0.15

0.1 0.1 Ult. Ult.

D.C. D.C.

0.05 0.05

Steel Tension Strain Tension Steel Strain Tension Steel

Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Displacement (m) Displacement (m)

(e) Varying axial load ratio with tw = 0.35m (f) Varying axial load ratio with tw = 0.4m

Figure 70 Phase II: Effect of varying axial load ratio (ALR) and wall thickness (tw) on buckling capacities and strain-displacement response

104

0.4 0.4 tw = 0.15m tw = 0.2m

0.35 Lp Factor = Various 0.35 Lp Factor = Various

0.3 0.3

0.25 0.25

0.2 0.2

0.15 0.15 PPBM

0.1 CCBM 0.1 Steel Tension Strain Tension Steel Steel Tension Strain Tension Steel Ult. Ult. Section Response D.C. D.C. 0.05 Extrapolated Response 0.05 Ser. Design Limit States Ser. 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Displacement (m) Displacement (m)

(a) Varying plastic hinge length with tw = 0.15m (b) Varying plastic hinge length with tw = 0.2m

0.4 0.4 tw = 0.25m tw = 0.3m

0.35 Lp Factor = Various 0.35 Lp Factor = Various

0.3 0.3

0.25 0.25

0.2 0.2

0.15 0.15

0.1 0.1 Steel Tension Strain Tension Steel Steel Tension Strain Tension Steel Ult. Ult. 0.05 D.C. 0.05 D.C. Ser. Ser. 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Displacement (m) Displacement (m)

0.4 0.4 tw = 0.35m tw = 0.4m

0.35 Lp Factor = Various 0.35 Lp Factor = Various

0.3 0.3

0.25 0.25

0.2 0.2

0.15 0.15

(e) Varying plastic hinge length with tw = 0.35m (f) Varying plastic hinge length with tw = 0.4m

Figure 71 Phase II: Effect of varying plastic hinge length factor (Lp) and wall thickness (tw) on buckling capacities and strain-displacement response

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6.3.3 Comparison with Code Requirements

Of the 240 wall models tested and analyzed in phase II, UBC's requirements for wall thickness (1.19) were met by all walls. ACI's requirements for wall thickness (1.20) were met by 196 walls. Walls with a thickness of 0.15m or 0.2m failed, as did long, thin walls. Only

157 walls met NZS 3101:2006 requirements (1.22). Thin walls and tall walls, as well as walls with higher strength steel failed to meet the minimum thickness calculated. All but fourteen walls were predicted by PPBM to buckle prior to their ultimate limit state. The less conservative CEBM predicted all but 42 walls buckling at strains under their ultimate limit state. In general, for the wall parameters considered, current code standings do not accurately provide requirements resulting in the successful prevention of buckling according to capacities calculated by PPBM and CEBM. A full listing of model agreement with selected codes and whether predicted buckling strains exceed the ultimate limit state strain is provided in Table 15 in Appendix B.

As with Phase I, many instances of exceedingly thin, tall, or long walls, or walls with high levels of longitudinal reinforcement, were predicted to buckle before the damage control limit state was reached. PPBM and CEBM predicted many of the thinnest walls (with a thickness of 150mm) to buckle prior to ever reaching displacements or strains corresponding to the serviceability limit state, which is unacceptable. The determination of a minimum required wall thickness in design codes is likely of the greatest importance to prevent plastic buckling.

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6.4 Parametric Study: Phase III

6.4.1 Introduction

The prior parametric studies showed that simplified structural code limitations poorly capture the complex nature of plastic wall buckling. Even in the case of NZS requirements

(based loosely on PPBM), simplifying assumptions of wall properties and ductility demands led to multiple instances where buckling controlled over a given wall's ultimate limit state.

Rather than attempting to over-simplify the phenomenon, or designing for assumed demands, it is suggested that the buckling limit state be designed at a strain level. Requiring buckling strains to be larger than the steel's ultimate strain level prevents buckling from occurring, and allows for the creation of relatively simple plots for design purposes.

6.4.2 Generation of Aspect Ratio Limit Curves

Substituting equations for plastic hinge length (1.34) and stability criterion (1.35) into

Paulay and Priestley's[46] buckling strain formulation (1.36) yields a function (1.37) that solves for buckling strain in terms only of wall dimensions (height, length, and thickness) and longitudinal steel ratio. A similar equation can be created for Chai and Elayer's buckling

strain can be created as well. By setting  sm equal to an assumed steel ultimate tension strain of 0.09 as suggested by Priestley, Calvi and Kowalsky[47], it is possible to generate curves representing walls of with a given reinforcement ratio and across a range of heights that reach their ultimate steel strains prior to buckling. Due to many code limitations commonly prescribing aspect ratio limitations, the functions can be reparameterized in terms of aspect

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H H ratios and wall height by setting L  w and t  w . Figure 72 and Figure 73 w H w H w w Lw tw display buckling curves based on aspect ratio limitations for PPBM and CEBM. Each sub- plot assumes a given longitudinal steel ratio and displays aspect ratio limits for walls of varying heights. Basic material properties assumed for the generation of these plots are shown in Table 4.

f L k  H 0.1 L  L  2 L ; k  0.2u  1  0.08 , L  0.022 f d (1.34) P e w sp sp sp y bl f y

2 lyf  0.5 1  2.35m  5.53 m  4.70 m  0.5 ; m  (1.35) cr   ' fc

2 b (1.36) sm  8   o

2 6.8*tw * Min 0.5,0.5 17.625  0.5 70.5   1244.25   (1.37)  sm  Max0.4752,0.2376 0.5333* Hww 0.1* L 

Table 4 Properties assumed for strain-based aspect ratio limits

fy fu Es fc' dbl β (MPa) (MPa) (GPa) (MPa) (mm) (N/A) 450 675 200 30 24 0.85

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Hw tw Hw tw Hw tw 100 100 100

80 H 48m 80 80 H 24m H 48m H 24m H 48m H 12m H 24m 60 60 H 12m 60 H 12m H 6m H 6m H 6m 40 40 40 H 3m H 3m H 3m 20 20 20

0 H L 0 H L 0 H L 0 5 10 15 20 w w 0 5 10 15 20 w w 0 5 10 15 20 w w Figure 72 (a) Walls with ρl = Figure 72 (b) Walls with ρl = 0.005 Figure 72 (c) Walls with ρl = 0.0025 0.0075

Hw tw Hw tw Hw tw 100 100 100

80 80 80

H 48m 60 60 H 48m 60 H 48m H 24m H 24m H 12m H 24m H 12m H 12m H 6m 40 40 H 6m 40 H 6m

H 3m H 3m H 3m 20 20 20

0 H L 0 H L 0 H L 0 5 10 15 20 w w 0 5 10 15 20 w w 0 5 10 15 20 w w Figure 72 (d) Walls with ρl = 0.01 Figure 72 (e) Walls with ρl = Figure 72 (f) Walls with ρl = 0.015 0.0125

Hw tw Hw tw Hw tw 100 100 100

80 80 80

60 60 60 H 48m H 24m H 48m H 12m H 24m H 48m 40 40 H 12m 40 H 24m H 6m H 12m H 6m H 6m H 3m 20 20 H 3m 20 H 3m

0 H L 0 H L 0 H L 0 5 10 15 20 w w 0 5 10 15 20 w w 0 5 10 15 20 w w Figure 72 (g) Walls with ρl = 0.02 Figure 72 (h) Walls with ρl = 0.03 Figure 72 (i) Walls with ρl = 0.04

Figure 72 Wall aspect ratios for walls of varying heights and longitudinal steel ratios calculated using PPBM

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Hw tw Hw tw Hw tw 100 100 100 H 48m H 48m 80 H 24m 80 80 H 24m H 48m H 12m H 24m H 12m H 12m 60 H 6m 60 60 H 6m H 6m 40 40 40 H 3m H 3m H 3m 20 20 20

0 H L 0 H L 0 H L 0 5 10 15 20 w w 0 5 10 15 20 w w 0 5 10 15 20 w w Figure 73 (a) Walls with ρl = Figure 73 (b) Walls with ρl = 0.005 Figure 73 (c) Walls with ρl = 0.0025 0.0075

Hw tw Hw tw Hw tw 100 100 100

80 80 80 H 48m H 48m H 24m H 48m H 24m H 24m 60 H 12m 60 60 H 12m H 12m H 6m H 6m H 6m 40 40 40

H 3m H 3m H 3m 20 20 20

0 H L 0 H L 0 H L 0 5 10 15 20 w w 0 5 10 15 20 w w 0 5 10 15 20 w w Figure 73 (d) Walls with ρl = 0.01 Figure 73 (e) Walls with ρl = Figure 73 (f) Walls with ρl = 0.015 0.0125

Hw tw Hw tw Hw tw 100 100 100

80 80 80

H 48m 60 H 24m 60 H 48m 60 H 12m H 24m H 48m H 12m H 24m 40 H 6m 40 40 H 12m H 6m H 6m H 3m 20 20 H 3m 20 H 3m

0 H L 0 H L 0 H L 0 5 10 15 20 w w 0 5 10 15 20 w w 0 5 10 15 20 w w Figure 73 (g) Walls with ρl = 0.02 Figure 73 (h) Walls with ρl = 0.03 Figure 73 (i) Walls with ρl = 0.04

Figure 73 Wall aspect ratios for walls of varying heights and longitudinal steel ratios calculated using CEBM

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Given a specific plot, such as Figure 73(g), which contains walls with a longitudinal steel ratio of 2%, each curve depicts the upper limit to wall aspect ratios that reach ultimate steel strain prior to buckling. For example, walls 12 meters tall with aspect ratios below the curve labeled "H = 12 m" will be controlled by buckling as predicted by CEBM. Such plots are visually easy to apply, and provide a greater level of confidence by limiting plastic buckling at a material level rather than attempting to tailor designs to individual loads. While differences in material strengths, longitudinal bar size, and cover to longitudinal steel will affect the buckling capacities shown in the figures above, the variance is relatively small when compared to the parameters considered.

6.4.3 Comparison with Code Requirements

Code restrictions that limit HLww aspect ratios do little in terms of preventing

plastic buckling above values of 15 or 16. Instead, Htww plays a larger role in determining

whether a given wall will buckle. However, a flat restriction such as requiring Htww 20 is overly conservative in most cases, due to taller walls allowing larger ratios. While it may be possible to generate simplified aspect ratio limits, until a greater level of understanding of the plastic buckling mechanism and confidence in the buckling models, assumptions inherent in such contrived equations may lead to inaccurate limits.

6.5 Parametric Study Conclusions

In this chapter, wall geometric and material properties were varied to determine each parameter's effect on wall strain-displacement response and buckling capacity. In general,

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wall geometric properties such as height, length, and thickness greatly affected a wall's propensity to buckle. As height and length increase, and thickness decreases, buckling capacities were found to generally decrease. Additionally, increasing longitudinal strain ratios resulted in significantly reduced buckling capacities, supporting the use of distributing longitudinal reinforcement along the length of a wall rather than lumping the steel in end regions. While other parameters affected strain-displacement response and buckling capacities, the associated changes were smaller in comparison to those previously discussed.

Plots provided as part of phase III impose limitations on aspect ratios for walls of a given height and longitudinal steel to prevent buckling from occurring prior to the longitudinal steel reaching its ultimate tension strain. These plots are meant as visual tools for general design purposes, but warrant further examination once additional data from experimental tests and field data is available for verification of the buckling models.

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CHAPTER 7. CASE STUDIES

7.1 Introduction

As a result of the damage occurring primarily during the Christchurch earthquake, many buildings suffered damage ranging from light non-structural damage to full collapse.

Prior chapters focused on first assessing the accuracy of existing models at predicting buckling strains, and their dependency on relevant design parameters. This chapter describes analyses performed on three structural wall buildings that experienced both the Darfield and

Christchurch earthquakes: the Canterbury Television building, Pyne Gould Corporation and

Pacific Brands House. General modeling procedure and analysis methods are provided, followed by a building-by-building breakdown describing the structure and damage observed during the Darfield and Christchurch earthquakes. For each building, analysis model response, wall demands, and buckling capacities are provided. Coarse, order-of-magnitude agreement between observed damage and outcomes predicted by buckling models is discussed, to provide field verification for the buckling models.

7.2 General Model Information

7.2.1 Overview

Three-dimensional models were created for selected buildings using the structural analysis program, SAP2000[13] and subjected to acceleration time histories representative of the 2010 Darfield and 2011 Christchurch earthquakes. Demands on wall systems were calculated using non-linear time history analyses via direct integration of each model's dynamic equilibrium equations. These demands were compared with buckling strain

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capacities estimated by selected buckling models. The predicted outcomes are compared with observed damage to determine each model’s agreement and relative accuracy.

SAP2000 was selected to model each of the structures using a combination of frame and shell elements. This simplified the analysis process, but prevented the inclusion of more complicated effects on wall stability such as: eccentric geometries, eccentric loading, out-of- plane loading, among many others likely to influence plastic buckling capacities or demands.

While more complex models such as a fiber or finite element model can account for such issues, the additional accuracy gained was found to be far outweighed by the increased computational cost associated with such methods.

7.2.2 General Material Properties

In situ concrete strengths were estimated using material testing on samples taken from each building. Where concrete cores were unavailable in sufficient numbers to provide reliable information, in situ concrete strength was estimated to be 1.5 times the specified strength as allowed by NZSEE[38]. A similar factor was used in a similar post-event assessment analysis of the CTV building performed by Hyland[20], following recommendations by Priestley and Sieble[48]. The increase in strength accounts for both strength gained as concrete ages, as well as the typical practice of concrete producers to deliver higher strength mixes than specified[20]. Where unavailable from testing, concrete modulus was calculated using (1.38), following provisions from NZS 3101[37]. Where no material information was available, values were assumed to be similar to those used for the

CVT building model. Confined and unconfined concrete stress-strain relationships were

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based on Mander's model[28]. Hysteretic behavior was included via applying SAP2000's built- in Takeda model[13].

' Ecc3320 f 6900 ( MPa ) (1.38)

Steel material properties were estimated using material testing on samples taken from each building. Where unavailable, an ultimate steel stress was estimated to be 1.5 times the steel's yield stress[47], and other properties were assumed to be similar to those used in modeling the CTV building. SAP2000's Park[13] stress-strain model was adopted, with hysteretic behavior included via SAP2000's built-in Takeda model[13].

7.2.3 Soil-Structure Interaction

Due to none of the selected case study buildings exhibiting damage resulting from liquefaction, its effect was not included in the analyses performed. Soil stiffness was modeled using non-linear "gap" springs, that are compression only, resulting in neglecting soil suction. If soil test data or properties were unavailable, an average stiffness weighted by area was calculated from values utilized by Hyland[20] and applied to each foundation element. Springs were applied to the areas of foundation shells and distributed along the length of thin foundation beams.

7.2.4 Frame Elements

Beams and columns were modeled as elastic frame members, whose non-linear behavior is captured by fiber hinges at their ends. These fiber hinges account for the coupled behavior of the elements under axial loads and biaxial moment demands. Frame elements

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expected to undergo inelastic demands had their bending stiffness reduced to levels corresponding to their strength after cracking. This value was typically 30% of the gross stiffness for beams and 40% for columns. Frame elements expected to remain elastic, such as foundation elements, were left unmodified.

7.2.5 Structural Wall Elements

Walls were modeled using multi-layered nonlinear shell elements, with individual layers modeling nonlinear material behavior. Out-of-plane stiffness was reduced to approximately 30%, corresponding to each wall's cracked stiffness. Longitudinal and transverse reinforcing bars were modeled as individual layers within the plane of the wall.

Transverse steel running in the out-of-plane direction was not explicitly modeled, but its contribution to concrete confinement was included within each wall's concrete material model. Transfer beams between structural walls were modeled using a combination of multi- layered non-linear shell elements and elastic frame elements with non-linear fiber hinges at their ends. This method was implemented by Hyland[20] to analyze the CTV building's collapse and was reasonably successful in approximating transfer beam load bearing and deformation behavior.

7.2.6 Foundation Elements

Foundation pads, mats, and beams with exceptionally large footprints were modeled as elastic shell elements. Smaller foundation beams were modeled as elastic frame elements.

In both cases, reinforcing details were explicitly modeled. Soil-structure interaction was

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included via non-linear, compression-only springs representative of soil stiffness applied to either a shell's surface or along a frame element's length. Where necessary, piles were modeled as elastic frame elements with non-linear fiber hinges at their tops. Pile rotations were restrained at their depth-of-fixity as estimated from the building's soil profile to be at a depth corresponding to a thick level of gravel. Additional springs modeled the pile's toe bearing capacity and skin friction capacity. None of the buildings examined case study were observed to have experienced enough liquefaction to impact building behavior or strength, so its effects have not been considered in any of the analytical models.

7.2.7 Floor Elements

Floor slabs were modeled with multi-layered inelastic shell elements with out-of- plane stiffness reduced to approximately 30%, corresponding to expected cracked stiffness.

Longitudinal and transverse reinforcing bars were modeled as individual layers within the plane of the slab. Where present, steel decking was also modeled with appropriate material and physical properties.

7.2.8 Structural Damping

A modal analysis was used to determine mass-participation ratios and modal periods for each structural analysis model. All structures were found to be significantly dominated by their first two modes, with additional modes having little or no participating mass. The first two modal periods were used to calculate mass and stiffness proportional constants corresponding to a selected Rayleigh damping ratio. While a lower damping coefficient of 2-

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3% is typical for structures undergoing inelastic actions due to softening of structural elements, a slightly higher value was selected to represent structural behavior prior to drastic inelastic actions, for reasons discussed further below.

While the Pyne Gould Corporation and Canterbury Television building' debris were suggestive of extreme inelastic demands, such failures redistribute load in a manner unlikely to cause high tension strains in wall end regions that will be subjected to a load reversal.

Instead, such rapid, progressive collapses tend towards failing in a single direction, without sustaining the additional reversals necessary to induce wall buckling. To account for this possibility, that wall buckling may have occurred prior to sufficient plastic softening of structural elements, a damping value of 5% was adopted for the calculation of mass and stiffness damping coefficients in each model. In the case of the Pacific Brands House building, while buckling was observed in a structural wall, little other inelastic structural damage was observed. Both structures' behaviors were taken to also support the usage of a higher damping ratio.

7.2.9 Vertical Loads

Vertical gravity loads were adopted from Hyland's seismic analyses on the CTV building[20], which were developed following NZS 1170[35] requirements. Superimposed dead and live loads designated by occupancy use were applied as a single combined loading of approximately 3.7 kPa. Seismic masses were calculated based on modal load participation by

SAP2000 for use in non-linear time history analyses.

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7.2.10 Ground Motions

Ground motion records for the Darfield and Christchurch earthquakes were available from four seismic stations near the Christchurch CBD. Each station's location is provided in

Table 5 and their position relative to the selected case study buildings is shown in Figure 74.

[19] Table 5 Earthquake record station locations

Station ID Location Christchurch Botanic Gardens CBGS 43.53°S, 172.62°E Christchurch Cathedral College CCCC 43.54°S, 172.65 °E Christchurch Hospital CHHC 43.54°S, 172.63°E Resthaven REHS 43.52 S, 172.64 E

REHS Building

Station

PBH PGC

CTV CBGS

Darfield EQ Epicenter Christchurch EQ CHHC Epicenter

CCCC

Figure 74 Location of case study buildings and recording Stations (Image Courtesy of Google Maps)

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Time histories from the stations at Resthaven (REHS) and Christchurch Cathedral

College (CCCC) were adopted for analysis due to their closer proximity to all of the case structures than the stations at Christchurch Botanical Gardens (CBGS) or Christchurch

Hospital (CHCC). Analytical models for the Pacific Brands House and Pyne Gould

Corporation buildings were subjected to time histories from the REHS station while the

Canterbury Television building was loaded using time histories from the CCCC station.

Acceleration time histories from the REHS and CCCC stations were obtained from

Geonet[19] for both the Darfield and Christchurch events and realigned to orient North and

East. While abnormally large vertical accelerations were observed during both earthquakes, initial sensitivity analyses showed that their inclusion significantly increased computation time and negligible changes in demands. As such, vertical accelerations were neglected, and only horizontal accelerations were selected for use as load histories. Darfield and

Christchurch records for CCCC and REHS stations are displayed in Figure 75 through Figure

78. To reduce analysis time, records were reduced in length to contain only times with significant ground motions. Adopted record beginning and ending times selected for use in analyses are provided in Table 6. Each building model was subjected to the Darfield and

Christchurch events individually. To determine if the Darfield event's damage significantly impacted seismic responses, an additional load case comprised of the Darfield record followed by the Christchurch record was included.

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[19] Figure 75 Darfield acceleration time histories from station CCCC North (top) and East (bottom)

0.3 0.2 0.1 0 -0.1 -0.2 Acceleration(g) -0.3 0 5 10 15 20 25 30 35 40 Time (s)

[19] Figure 76 Darfield acceleration time histories from station REHS North (top) and East (bottom)

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[19] Figure 77 Christchurch acceleration time histories from station CCCC North (top) and East (bottom)

[19] Figure 78 Christchurch acceleration time histories from station REHS North (top) and East (bottom)

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Table 6 Adopted start and end record times

Adopted Start Adopted End Record Time (s) Time (s) Darfield 5 20 Christchurch 0 10

It should be noted that due to this section's focus on gathering data for assessing the accuracy of buckling models, a detailed analysis of each structure's collapse mechanism is not performed. In addition to causing increasingly complex loads, progressive building failures are unlikely to cause large strain reversals in wall elements. Due to this, while the structural models created account for non-linear behavior, little attempt was made to verify results beyond the model's initial collapse.

7.3 Canterbury Television (CTV)

7.3.1 Description

The Canterbury Television building (CTV) was a six-story reinforced concrete structure located at 249 Madras Street within the Christchurch central business district. The

CTV structure was approximately 21x27m and about 20 meters tall. Figure 79 shows the undamaged structure prior to damage occurring during the Darfield earthquake and before its collapse during the Christchurch earthquake[20].

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Figure 79 View of the southeast corner of the CTV building prior to the Christchurch earthquake (Photo [20] courtesy of Phillip Pearson, Derivative work from Schwede66)

The CTV building was designed in 1986 according to NZS 3104:1983 requirements to resist seismic loads using a pair of reinforced concrete walls. A narrow wall along the south edge housed a fire escape, and a large C-shape wall core contained two elevators, a stairway, and the structure's bathrooms. These walls are referred to henceforth as the South

Wall and North Core, respectively. The South Wall was comprised of two walls connected by seismically designed transfer beams and the North Core was made of a multi-flanged thicker wall system shown in Figure 80. [20]

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North Core

South Wall

[20] Figure 80 Location and shape of the CTV's North Core and South Wall

An infill reinforced concrete masonry wall was built along the bottom three floors of the building's West edge, however it was found to have little effect on the seismic response of the structure. The foundation was composed of strip and pad footings with interlinking foundation grade beams, and the roof was made of a lightweight steel sheeting supported by steel rafters[20]. The interior structural frame system was composed of reinforced concrete

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columns with precast reinforce concrete beams running East-West along the length of building, which can be seen in the typical floor plan also shown in Figure 80.

The CTV was originally intended and designed as an office building but throughout its lifetime changed to a school, a medical facility, a radio station, and finally a television studio. Post-construction, several structural modifications were made to the building to either improve its seismic behavior or to account for its change of usage. Drag bars were installed on the fourth, fifth, and sixth floors to strengthen each slab's connections to the North Core.

A stairway door was cut into the South Wall's second floor, and various cores were drilled through floor slabs to allow for the installation of additional electrical conduit and piping[20].

7.3.2 Observed Damage

Hyland Consultants and Structure Smith Ltd. [20] performed a collapse assessment for the CTV building for both the Darfield and Christchurch events and their aftershocks. During the Darfield earthquake, minor damage was reported including minor shear cracking in the

North Core, South Wall, and several columns along the building's exterior edges. Additional non-structural failures occurred in stair landings that lacked displacement capacity, resulting in some crushing of cover concrete. Damage to non-structural elements such as room partitions and windows also occurred. The building was deemed safe enough for continue occupancy after superficial repairs were made. Following the Darfield earthquake, an adjacent building was demolished, but was found to cause negligible distress to the CTV building. During another major aftershock on December 26, 2010, additional minor non- structural damage occurred, but no major damage to primary components was observed[20].

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During the Christchurch earthquake, the building suffered a nearly complete collapse, with only the North Core remaining standing. Eyewitnesses interviewed generally agreed that the building's failure occurred quickly and violently, with initial shaking movements leading up to a nearly vertical failure[20]. The collapsed building and still-standing North Core can be seen in Figure 81. Concrete slab floors detached from higher levels of the North Core, while lower levels hinged about their connecting reinforcement.

North

[20] Figure 81 The CTV building, immediately following collapse

A post-collapse analysis performed by Hyland[20]suggested several possible collapse scenarios. The most likely involved the eastern external columns failing at their bases, leading to the building tilting east and the progressive collapse of additional interior columns.

This failure mechanism shows the greatest agreement with the fact that many eyewitnesses

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agreed upon the building tilting eastward as it failed. Additionally, as observed in Figure 81, more debris fell eastward than in other directions. Other less feasible collapse initiators included internal column failures leading to an inwards collapse, and floor slab detachment from the north core at the lower or upper levels leading to a loss of strength and increased displacement demands. Lateral instability of the South Wall was not discussed as a possible option, but examination of a photo taken of the wall's base as in Figure 82 reveals slight out- of-plane deformations. This may be indicative of plastic buckling, but could have also been caused by the structure's collapse pulling the wall northward.

[20] Figure 82 Photo of the base of CTV's South wall, exhibiting slight out-of-plane deformation

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7.3.3 Model Material Properties

Material properties for modeling concrete and steel behavior were adopted from an analysis performed by Hyland[20] in which sample cores taken from various structural elements in the CTV building. Concrete strengths were found to be highly variable between samples due to degradation and damage experienced during the Darfield and Christchurch earthquakes, resulting in adopted strengths calculated as 1.5 times the specified concrete strength as detailed by NZSEE[38] standards. Ultimate concrete strains from the material tests were more consistent, and were adopted. A lack of samples from foundation concrete led to assuming an ultimate concrete strain similar to that adopted for slabs, beams, and walls due to a similarity in strength. Modulus of elasticity for each concrete strength was calculated using the previously discussed equation (1.38) from NZS 3101[37]. Table 7 and Table 8 list selected concrete and steel properties, respectively. Soil stiffnesses adopted from Hyland's[20] analysis of the building, have their locations and values provided in Figure 83 and Table 9, respectively.

Table 7 Concrete material properties adopted for modeling the CTV

Adopted Adopted Strength, fc' Modulus, Ec Element (MPa) (GPa) Walls 37.5 27.2 Columns (up to 2nd floor) 52.5 31 Columns (2nd to 3rd floor) 45 29.2 Columns (above 3rd floor) 37.5 27.2 Beams 37.5 27.2 Slabs 37.5 27.2 Foundation Pads / Beams 30 25.1

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Table 8 Steel material properties adopted for modeling the CTV

Yield Ultimate Ultimate Adopted stress, fy stress, fu Strain, εult Modulus, Es Steel (Use) (MPa) (MPa) (N/A) (GPa) G275 (transverse) 321 451 .202 200 G380 (longitudinal) 448 603 .168 200 664 Mesh (slab) 615 665 .042 200

[20] Figure 83 Location of soil springs on foundation elements

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[20] Table 9 Soil stiffnesses adopted from Hyland's modeling of the CTV

Foundation Stiffness (#) (MN/m3) 1 123 1a 131 1b 66 2 85 2a 53 3 117 3a 79 4 160 4a 74 5 104 6 185

7.3.4 Model Results

Figure 84 and Figure 85 contain images of the analysis model created for analysis of the CTV building as viewed from the southeast. The model was subjected to the adopted

REHS record from the Darfield and Christchurch events individually and then sequentially.

Figure 84 CTV model with all elements visible (left), and with slab and roof elements hidden (right)

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Figure 85 CTV structural model with wall and foundation elements visible

Figure 86 and Figure 87 show CTV's center-of-mass displacement at roof height caused by the Darfield (DF), Christchurch (CC) and combined loading (DF → CC) load history. Displacements during the Christchurch event are significantly larger than those from the more distant Darfield earthquake. The CTV responded to the Darfield earthquake in a diagonal northeast-southwest fashion, but experienced larger displacements to the southeast during the Christchurch event. This is consistent with the failure mechanism suggested by

Hyland[20] initiating from column failure along the southern or eastern building edge at a drift of approximately ~1%. This corresponds to a top displacement of about 200mm, which was exceeded during the analysis. Large displacements east-west would impart large displacement demands on the South Wall, further contributing to the CTV's collapse. While some difference in building response is observable between the individual Christchurch and the sequential Darfield-Christchurch loading, displacements are of similar magnitudes.

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300 DF: REHS 200

100

0 (mm) -100

-200 Displacement South/North South/North Displacement -300 -300 -200 -100 0 100 200 300 Displacement West/East (mm)

Figure 86 Response of CTV's top floor center of mass at roof height to REHS Darfield load history

300 CC: REHS 200 DF→CC: REHS

100

0 (mm) -100

-200 Displacement South/North South/North Displacement -300 -300 -200 -100 0 100 200 300 Displacement West/East (mm)

Figure 87 Response of CTV's top floor center of mass to REHS Christchurch and sequential load histories

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Name designations for each wall end region are provided in Figure 88. Longitudinal strains are presented in Figure 89 through Figure 96 for South Wall and North Core end regions for each of the three seismic load cases applied. The steel demands are compared to buckling capacities calculated using PPBM and CEBM, and agreement between the predicted and observed damage is discussed.

CTV-5

CTV-1 CTV-6

CTV-2 CTV-7

CTV-3

CTV-4 CTV-8

Figure 88 Naming convention for selected wall ends in CTV analysis model

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Strain demands for the South Wall's wall ends (CTV-1 through CTV-4) are displayed in Figure 89 through Figure 92. Despite the lateral deformation observed post-failure in the

South Wall, strain demands do not even exceed half of the buckling capacities estimated by either model. This suggests that the curvature seen was caused by the structure's collapse, rather than plastic buckling. Due to the magnitude of this difference, however, it is assumed that the damage (or lack of buckling) observed and buckling model predictions agrees.

0.16000 DF: REHS 0.14000 CC: REHS 0.12000 DF→CC: REHS PPBM Limit 0.10000 CEBM Limit 0.08000

0.06000

0.04000

0.02000 Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.02000 0 5 10 15 Time (s)

Figure 89 Strain demands in extreme longitudinal steel in South Wall end region CTV-1

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0.16000 DF: REHS 0.14000 CC: REHS 0.12000 DF→CC: REHS PPBM Limit 0.10000 CEBM Limit 0.08000

0.06000

0.04000

0.02000

Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.02000 0 5 10 15 Time (s)

Figure 90 Strain demands in extreme longitudinal steel in South Wall end region CTV-2

0.16000

0.14000 DF: REHS CC: REHS 0.12000 DF→CC: REHS 0.10000 PPBM Limit CEBM Limit 0.08000

0.06000

0.04000

0.02000

Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.02000 0 5 10 15 Time (s)

Figure 91 Strain demands in extreme longitudinal steel in South Wall end region CTV-3

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0.16000 DF: REHS 0.14000 CC: REHS 0.12000 DF→CC: REHS PPBM Limit 0.10000 CEBM Limit 0.08000

0.06000

0.04000

0.02000

Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.02000 0 5 10 15 Time (s)

Figure 92 Strain demands in extreme longitudinal steel in South Wall end region CTV-4

Strain demands for each of the North Core's protruding wall ends (CTV-5 through

CTV-8) are displayed in Figure 93 through Figure 96. Significantly smaller buckling strain capacities were calculated for the North Core's longer, thinner walls than the thicker, shorter

(lengthwise) walls. Despite this difference, strain demands from the analysis model rarely exceeded one third of each wall's buckling strain capacities. The low steel strain demands agree with the North Core's relatively undamaged state (aside from flexural and shear cracking at its base) following the Darfield and Christchurch earthquakes. Even with the coarse assumptions made in the analysis model and the phenomenological buckling models, the results show coarse agreement for both PPBM and CEBM in predicting a lack of buckling in the North Core Walls.

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0.05000

DF: REHS

0.04000 CC: REHS DF→CC: REHS PPBM Limit 0.03000 CEBM Limit

0.02000

0.01000

0.00000 Longitudinal Steel Strain Steel Longitudinal

-0.01000 0 5 10 15 Time (s)

Figure 93 Strain demands in extreme longitudinal steel in North Core end region CTV-5

0.06000 DF: REHS 0.05000 CC: REHS DF→CC: REHS 0.04000 PPBM Limit CEBM Limit 0.03000

0.02000

0.01000

Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.01000 0 5 10 15 Time (s)

Figure 94 Strain demands in extreme longitudinal steel in North Core end region CTV-6

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0.06000 DF: REHS 0.05000 CC: REHS 0.04000 DF→CC: REHS PPBM Limit 0.03000 CEBM Limit

0.02000

0.01000

0.00000

Longitudinal Steel Strain Steel Longitudinal -0.01000

-0.02000 0 5 10 15 Time (s)

Figure 95 Strain demands in extreme longitudinal steel in North Core end region CTV-7

0.06000 DF: REHS 0.05000 CC: REHS DF→CC: REHS 0.04000 PPBM Limit CEBM Limit 0.03000

0.02000

0.01000

Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.01000 0 5 10 15 Time (s)

Figure 96 Strain demands in extreme longitudinal steel in North Core end region CTV-8

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7.4 Pacific Brands House

7.4.1 Description

The Pacific Brands House (PBH) was a multi-story reinforced concrete structure located at 123 Victoria Street within the Christchurch central business district. The structure consisted of seven floors with mixed usage as office and retail space. The main structure was approximately 21x29m and was 21 meters tall. Figure 97 shows the undamaged building prior to its demolition in the months following the Christchurch earthquake, and its location and orientation.

The PBH was designed as a dual wall-frame system for the purpose of resisting seismic loads. While the building's age, or the code to which it was designed was unknown, the building appears to be of a newer construction than the CTV or PBH. The foundation was assumed to be composed of strip and pad footings with interlinking foundation grade beams, similar to the CTV. The concrete frame system was made up on interior round columns and exterior deep columns that gradually tapered at higher floors. Reinforced concrete beams ran around the building's perimeter and across the short dimension of the building. On either long side of the frame system, two large L-walls provided seismic resistance to the building. A typical floor plan for the PBH building can be seen in Figure 98. Due to design information being unavailable, geometric and reinforcement properties were estimated and calculated from photos taken immediately following the Christchurch earthquake (courtesy of Sri

Sritharan), or during the structure's demolition[50]. Material properties were assumed similar to those used in the CTV analysis model, disregarding differences in concrete age.

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Figure 97 North-east face of the Pacific Brands House prior to the Christchurch earthquake (left), and the building's location and orientation (Courtesy of Google maps)

Columns

L-Wall

Figure 98 Floor plan of assumed beam and column locations for modeling the PBH

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7.4.2 Observed Damage

The Pacific Brand House's behavior during the Darfield earthquake and its aftershocks were also unavailable, but the structure likely incurred moderate non-structural damage to contents and elements such as windows. During the Christchurch earthquake, the building experienced plastic buckling in the end region of one of its L-walls. The resulting out-of-plane deformation and concrete damage can be seen in Figure 99 and Figure 100.

Other structural elements experienced mild, if any damage. Both the flange of the buckled L- wall and the second L-wall displayed some concrete cracking, but did not exhibit any out-of- plane instability, as shown in Figure 101. Despite the building undergoing buckling, no collapse occurred, which was assumed to be a result of the PBH's high level of structural regularity and design redundancy. This would have allowed the redistribution of loads previously supported by the buckled wall without causing further failures due to overloading.

Figure 99 Buckling of PBH's L-wall as seen from the building's interior (courtesy of Sri Sritharan)

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Figure 100 Buckling of PBH's L-wall as seen from the building's exterior (courtesy of Sri Sritharan)

Figure 101 PBH's southeastern unbuckled L-wall (courtesy of Sri Sritharan)

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7.4.3 Model Material Properties

As previously mentioned, little to no information of material strengths was available, leading to the adoption of concrete and steel strengths similar to those implemented in the

CTV model, with some modifications to account for the building's newer design and younger age. Due to their extremely deep cross-section and vertical taper, columns at lower levels of the PBH were assumed to not require higher strength concrete as the CTV specified. Instead, concrete strengths were assumed to be constant across all floors. Modulus of elasticity for each concrete strength was calculated using the previously discussed equation (1.38) from

NZS 3101[37]. Due to steel reinforcement testing results being unavailable, properties were assumed similar to those used in modeling the CTV building. Table 10 and Table 11 contain the adopted concrete and steel properties, respectively.

Table 10 Material properties adopted for modeling the Pacific Brands House

Adopted Adopted Strength, fc' Modulus, Ec Element (MPa) (GPa) Walls 37.5 27.2 Columns 45 29.2 Beams 37.5 27.2 Slabs 32 27.2 Foundation Pads / Beams 32 25.1

Table 11 Steel material properties adopted for modeling the Pacific Brands House

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7.4.4 Model Results

Figure 102 and Figure 103 contain images of the analysis model created for analysis of the PBH building viewed from the southeast. The model was subjected to adopted REHS records from the Darfield and Christchurch earthquakes individually and then sequentially.

Figure 102 PBH model with all elements visible (left), and with slab and roof elements hidden (right)

Figure 103 PBH structural model with wall and foundation elements visible

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The PBH structural model's center-of-mass responses to the three selected time histories are shown in Figure 104 and Figure 105. Again, displacements from the

Christchurch load history are significantly larger than those from the more distant Darfield earthquake. The PBH model responded to the Darfield loading in a diagonal northwest- southeast direction, but displaced east-northeast and west-southwest during the Christchurch event. This response is likely to have imparted large tensile demands followed by compressive demands to the wall end region observed to have buckled. This response is also consistent with the lack of damage observed at other wall end regions, as no other regions would have experienced as large of tension demands.

600.00

DF: REHS 400.00

200.00

0.00

-200.00

-400.00 Displacement South/North (mm) South/North Displacement -600.00 -600.00 -400.00 -200.00 0.00 200.00 400.00 600.00 Displacement East/West(mm)

Figure 104 PBH response to REHS Darfield load history at roof height of center of mass

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PBH-3 experiences high inelastic tension strains PBH-1 undergoes even larger inelastic tension strains

Figure 105 PBH response to REHS Christchurch and sequential load histories

Longitudinal strains are presented for wall end regions of the two L-Walls for each of the three seismic load cases applied. Name designations for each wall end region are provided in Figure 106. The steel demands are compared to buckling capacities calculated using PPBM and CEBM. Strain demands for the Pacific Brand House walls (PBH-1 through

PBH-4) are displayed in Figure 107 through Figure 110. Strain demands for PBH-1 exceed its estimated buckling strain capacity, matching the damage observed to occur. Wall PBH-4 nearly exceeded its strain capacity calculated using the more conservative PPBM, but did not actually buckle. Additionally, this strain likely occurred following the buckling of PBH-1, leading to possible inaccuracies in the reported analysis demands. Strain demands on the

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remaining two wall ends (PBH-2 and PBH-3) were significantly smaller than their capacities.

Again, all four wall ends examined agree to some extent with the PPBM and CEBM models.

PBH-1

North PBH-2

PBH-4

PBH-3

Figure 106 Naming convention for selected wall ends in CTV analysis model

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0.06000

DF: REHS

0.05000 CC: REHS 0.04000 DF→CC: REHS PPBM Limit 0.03000 CEBM Limit 0.02000

0.01000

0.00000

-0.01000

Longitudinal Steel Strain Steel Longitudinal -0.02000

-0.03000 0 5 10 15 20 Time (s)

Figure 107 Strain demands in extreme longitudinal steel in L-Wall end region PBH-1

0.06000 DF: REHS 0.05000 CC: REHS DF→CC: REHS 0.04000 PPBM Limit 0.03000 CEBM Limit

0.02000

0.01000

0.00000

Longitudinal Steel Strain Steel Longitudinal -0.01000

-0.02000 0 5 10 15 20 Time (s)

Figure 108 Strain demands in extreme longitudinal steel in L-Wall end region PBH-2

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0.06000

DF: REHS

0.05000 CC: REHS 0.04000 DF→CC: REHS PPBM Limit 0.03000 CEBM Limit 0.02000

0.01000

0.00000

-0.01000

Longitudinal Steel Strain Steel Longitudinal -0.02000

-0.03000 0 5 10 15 20 Time (s)

Figure 109 Strain demands in extreme longitudinal steel in L-Wall end region PBH-3

0.06000 DF: REHS

0.05000 CC: REHS 0.04000

0.03000 DF→CC: REHS 0.02000

0.01000

0.00000

Longitudinal Steel Strain Steel Longitudinal -0.01000

-0.02000 0 5 10 15 20 Time (s)

Figure 110 Strain demands in extreme longitudinal steel in L-Wall end region PBH-4

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It is important to note that while it can be said that PBH-1 was successfully predicted to buckle and PBH-4 was successfully predicted to not buckle by PPBM, as previously stated, the many assumptions made in modeling the structure as well as the assumptions made in both the PPBM and CEBM make such statements far from likely. The actual numbers may be off a non-significant amount, however for a general order-of-magnitude assessment, a general sense of agreement can be observed.

Additionally, while the difference in including a sequential load history was negligible in the case of the CTV analysis, it can be seen that the load history of both earthquakes "DF→CC" generally results in larger demands. In the instance of end regions

PBH-1 and PBH-4, the sequential load history led to demands likely exceeding each wall's estimated buckling capacity, while the Christchurch time history led to reduced demands significantly less likely to have induced buckling. Despite the many assumptions made in modeling the PBH structure and the inaccuracy likely within the results presented here, it is feasible to assume load history had some impact in the building's buckling failure due to the magnitude of difference between demands. This behavior was common to the other two buildings modeled and discussed below, and despite the results being subjected to the same degrees of inaccuracy, they show reasonable agreement with the behaviors observed to have occurred in the field.

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7.5 Pyne Gould Corporation (PGC) Building

7.5.1 Description

The Pyne Gould Corporation building (PGC) was a five-story reinforced concrete structure that located at 231-233 Cambridge Terrace within the Christchurch central business district. The PGC structure was approximately 25.5m square and about 20 meters tall. Figure

111 shows the undamaged building prior to its collapse during the Christchurch event.

Figure 111 View of the southeast corner of the PGC building prior to the Christchurch and Darfield [4] earthquakes

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The PGC building sat on shallow pad foundations interconnected by foundation beams and relied on a large rectangular core of reinforced concrete walls to resist seismic loads. A series of reinforced concrete beams and reinforced jacketed columns distributed loads to the central core. The shear core grew significantly smaller in size and the walls progress from 200mm thick to 150mm thick between the first and second floors, creating a large irregularity in horizontal stiffness in the East-West direction. Figure 112 shows the orientation of the core walls and their disparity in size between the first and second floors.

Shear Cores

[4] Figure 112 Bottom floor plan (left) and typical upper floor plan (right) of the PGC building

Designed in 1963 to NZS 3401:1963 requirements as an office space, the PGC changed hands several times prior to reaching its current day owners. At the time of the

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earthquake, only the first floor was lightly occupied. Post-construction, many structural modifications were made to the building to either improve its seismic behavior or to suit its current owner's planned usage requirements. Following a structural review in 1997, a series of 72 steel prop columns were added adjacent to external columns to provide additional load capacity to meet newer standards. At the same time, several door openings were cut into walls while others were infilled. In 2008, additional door openings were added and removed, and a large cell-phone tower was added to the roof level. In 2009, repairs were made on minor cracks found in some exterior columns, and access holes were drilled through several floor slabs to install air-condition ducts.[4]

7.5.2 Observed Damage

At the request of the Department of Building and Housing, Beca[4] performed a collapse assessment for the PGC building for the earthquakes leading up to its collapse.

During the Darfield earthquake, minor cracking (typically less than 0.5mm) and shallow spalling in the central core wall between the first and second floor was observed. Some external spandrel beams spalled slightly due to reinforcing corrosion, and some of the fourth floor's ceiling tile framing failed. The building was deemed safe enough for continue occupancy after superficial repairs were made. During another major aftershock on

December 26, 2010, additional minor non-structural damage occurred, but no major damage to the primary building components was apparent. Some cracks in the central core and stairways enlarged and spalled slightly, but were deemed safe following repair using epoxy injection[4].

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During the Christchurch earthquake, the building suffered a nearly complete collapse.

The shear core wall was primarily intact, but tilted eastward by about 22 degrees, with floor slabs collapsing on top of one another providing support against further collapse. Beams and columns were seen to have experienced significant damage, and were often detached from one or both of their adjoining joints. Some eyewitnesses stated the building rocked laterally in the East-West direction before collapsing, while others thought the building twisted in a torsional manner. The collapsed building and still-standing North Core can be seen in Figure

113. Despite the many structural alterations discussed previously, none were found to be the cause of collapse during the Christchurch event. [4]

[4] Figure 113 The PGC building post-event

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Post-collapse analysis by Beca[4] suggests the building's failure initiated due to reinforcement yield and fracture in the central core's east side between the first and second level. Non-linear time history analyses performed predicted the building to experience a large acceleration pulse East, followed by an immediate reversal loading West. This would have caused the building to first rock westward, yielding and fracturing longitudinal steel along the central core's eastern face at the second floor. Demands were likely concentrated at the second floor due to the previously discussed stiffness irregularity caused by the difference in wall thickness and number of walls running East-West. The resulting loss of strength from reinforcement failure was assumed to have caused the wall to fail in compression on subsequent load reversal due to concrete crushing. Load redistribution led to the overloading of columns, which failed at their bases along the second floor. This failure mechanism agrees with what how many eyewitnesses described the tilting motion of the building prior to collapse. Lateral instability of core walls or first floor rectangular support walls radiating out from the core was not considered as a possible failure mode.

7.5.3 Model Material Properties

Material properties for modeling concrete and steel behavior were partially adopted from an analysis performed by Beca[20] that contained material test results on the PGC building. Similar to the concrete cores tested from the CVT building, concrete strengths were found to vary sample-to-sample due to degradation and damage. Where material data was unavailable, strengths were calculated as 1.5 times the specified concrete strength, following

NZSEE[38] provisions. Ultimate concrete strains from the material tests were more consistent,

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and were adopted. A lack of samples from foundation concrete led to assuming an ultimate concrete strain similar to that adopted for slabs, beams, and walls due to a similarity in strength. Modulus of elasticity for each concrete strength was calculated using the previously discussed equation (1.38) from NZS 3101[37]. Table 12 and Table 13 list selected concrete and steel properties, respectively. Due to insufficient soil testing, soil spring stiffness for foundation elements was taken as the average of spring strengths used in modeling the CTV building, weighted by their area of occurrence.

Table 12 Material properties adopted for modeling the Pyne Gould Corporation building

Adopted Adopted Strength, fc' Modulus, Ec Element (MPa) (GPa) Walls 37.5 27.2 Columns 47.3 29.2 Beams 40.7 27.2 Slabs 37.5 27.2 Foundation Pads 30 25.1 Foundation Beams 30 25.1

Table 13 Steel material properties adopted for modeling the Pyne Gould Corporation building

Yield Ultimate Ultimate Adopted stress, fy stress, fu Strain, εult Modulus, Es Steel (Use) (MPa) (MPa) (N/A) (GPa) Light Steel (#4,5,7) 316 452 .218 200 Light Steel (D24) 316 452 .218 200 Heavy Steel (#10) 424 696 .125 200

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7.5.4 Model Response

Figure 114 and Figure 115 contain images of the analysis model created for analysis of the PBH building viewed from the southeast. The model was subjected to adopted REHS recordings from the Darfield and Christchurch earthquakes individually and sequentially.

Figure 114 PGC model with all elements visible (left), and with slab and roof elements hidden (right)

Figure 115 PGC structural model with wall and foundation elements visible

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The structure's displacement at a point corresponding to the building's center-of-mass at roof height due to the Darfield (DF) is shown in Figure 116, while its response to the

Christchurch (CC) and combined loading (DF → CC) is shown in Figure 117. Displacements during the closer Christchurch event are significantly larger than those from the more distant

Darfield earthquake. The PGC generally responded to the Darfield loading in a diagonal northwest-southeast fashion, but displaced east-west during the Christchurch event. This behavior is consistent with the failure mechanism suggested by Beca[4] initiating wall failure along the eastern central core. The large displacement east would impart large tension strains on the eastern core wall, which would likely collapse upon reversal. While some difference in building response is observable between the individual Christchurch and the sequential

Darfield-Christchurch loading, the amount is most nearly negligible.

300 DF: REHS 200

100

0 (mm) -100

-200

Displacement South/North Displacement -300 -600 -400 -200 0 200 400 600 Displacement West/East (mm)

Figure 116 PGC response to REHS Darfield load history at roof height of center of mass

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300 CC: REHS 200 DF→CC: REHS

100

-100

-200

Displacement South/North (mm) South/North Displacement -300 -600 -400 -200 0 200 400 600 Displacement West/East (mm)

Figure 117 PGC response to REHS Christchurch and sequential load histories

Longitudinal strains are presented for selected wall end regions in the PGC building for each of the three seismic load cases applied. Name designations for each wall end region are provided in Figure 118 and Figure 119. Wall ends on either side of large core openings on the second floor were monitored in case the thinner second floor walls had significantly less capacity than the first floor walls. Strain demands for the walls (PGC-1 through PBH-12) are displayed in Figure 120 through Figure 131. Strain demands for PBH-1 exceed its estimated buckling strain capacity, matching the damage observed to occur. Most wall ends experience negligible levels of strain, or experience significant levels of strain after the analysis model becomes unstable and begins to collapse. Strains at PGC-4 exceeded the

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section's strain capacity calculated using PPBM, but whether the wall actually buckled cannot be assessed from the structure's remains. The possibility of PGC-4 actually having buckled is low, since the loss of support would have likely caused the building to collapse westward, rather than eastward. Strain demands at PGC-3 and PGC-11 also sustained significant tension demands, but only after experienced compressive strains exceeding the member's capacity.

PGC-1

PGC-2

PGC-8 PGC-3

PGC-4 PGC-7

PGC-6

PGC-5

Figure 118 Naming convention for selected wall ends in PGC analysis model (first floor)

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PGC-12 PGC-10

PGC-9 PGC-11

Figure 119 Naming convention for selected wall ends in PGC analysis model (second floor)

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0.06000

DF: REHS

0.05000 CC: REHS DF→CC: REHS 0.04000 PPBM Limit CEBM Limit 0.03000

0.02000

0.01000

0.00000 Longitudinal Steel Strain Steel Longitudinal

-0.01000 0 5 10 15 20 Time (s)

Figure 120 Strain demands in extreme longitudinal steel in end region PGC-1

0.06000

DF: REHS

0.05000 CC: REHS DF→CC: REHS 0.04000 PPBM Limit CEBM Limit 0.03000

0.02000

0.01000

Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.01000 0 5 10 15 20 Time (s)

Figure 121 Strain demands in extreme longitudinal steel in end region PGC-2

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0.06000

0.04000

0.02000

0.00000

-0.02000

-0.04000 DF: REHS -0.06000 CC: REHS DF→CC: REHS

Longitudinal Steel Strain Steel Longitudinal -0.08000 PPBM Limit CEBM Limit -0.10000 0 5 10 15 20 Time (s)

Figure 122 Strain demands in extreme longitudinal steel in end region PGC-3

0.06000

0.04000

0.02000

0.00000

-0.02000

-0.04000 DF: REHS -0.06000 CC: REHS DF→CC: REHS

Longitudinal Steel Strain Steel Longitudinal -0.08000 PPBM Limit CEBM Limit -0.10000 0 5 10 15 20 Time (s)

Figure 123 Strain demands in extreme longitudinal steel in end region PGC-4

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0.06000 DF: REHS

0.05000 CC: REHS 0.04000 DF→CC: REHS 0.03000 PPBM Limit 0.02000 CEBM Limit 0.01000 0.00000 -0.01000 -0.02000

-0.03000 Longitudinal Steel Strain Steel Longitudinal -0.04000 -0.05000 0 5 10 15 20 Time (s)

Figure 124 Strain demands in extreme longitudinal steel in end region PGC-5

0.06000

DF: REHS

0.05000 CC: REHS DF→CC: REHS 0.04000 PPBM Limit CEBM Limit 0.03000

0.02000

0.01000

Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.01000 0 5 10 15 20 Time (s)

Figure 125 Strain demands in extreme longitudinal steel in end region PGC-6

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0.06000 DF: REHS 0.05000 CC: REHS DF→CC: REHS 0.04000 PPBM Limit CEBM Limit 0.03000

0.02000

0.01000

Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.01000 0 5 10 15 20 Time (s)

Figure 126 Strain demands in extreme longitudinal steel in end region PGC-7

0.06000 DF: REHS 0.05000 CC: REHS DF→CC: REHS 0.04000 PPBM Limit CEBM Limit 0.03000

0.02000

0.01000

Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.01000 0 5 10 15 20 Time (s)

Figure 127 Strain demands in extreme longitudinal steel in end region PGC-8

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0.06000 DF: REHS 0.05000 CC: REHS DF→CC: REHS 0.04000 PPBM Limit CEBM Limit 0.03000

0.02000

0.01000

Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.01000 0 5 10 15 20 Time (s)

Figure 128 Strain demands in extreme longitudinal steel in end region PGC-9

0.06000 DF: REHS 0.05000 CC: REHS DF→CC: REHS 0.04000 PPBM Limit CEBM Limit 0.03000

0.02000

0.01000

Longitudinal Steel Strain Steel Longitudinal 0.00000

-0.01000 0 5 10 15 20 Time (s)

Figure 129 Strain demands in extreme longitudinal steel in end region PGC-10

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0.20000 DF: REHS 0.15000 CC: REHS 0.10000 DF→CC: REHS PPBM Limit 0.05000 CEBM Limit 0.00000

-0.05000

-0.10000

-0.15000

Longitudinal Steel Strain Steel Longitudinal -0.20000

-0.25000 0 5 10 15 20 Time (s)

Figure 130 Strain demands in extreme longitudinal steel in end region PGC-11

0.06000 DF: REHS 0.05000 CC: REHS 0.04000 DF→CC: REHS PPBM Limit 0.03000 CEBM Limit 0.02000

0.01000

0.00000

-0.01000

Longitudinal Steel Strain Steel Longitudinal -0.02000

-0.03000 0 5 10 15 20 Time (s)

Figure 131 Strain demands in extreme longitudinal steel in end region PGC-12

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7.6 Case Study Conclusions

SAP2000 was used to perform Non-linear Time History Analyses (NTHA) on walls within three structures to determine extreme tension fiber strain demands at their base. These demands were compared the buckling capacity strains estimated using two buckling models to predict the occurrence of plastic buckling. In general, the models showed reasonable agreement with the damage observed, but the data set included only one wall (PBH-1) known to have undergone plastic buckling and another (PGC-4) that may have been incorrectly predicted to buckle. As mentioned previously, these inaccuracies result from the assumptions made in modeling the structures, as well assumptions made by the buckling capacity models.

Despite the variance likely existing in the analysis results, a general sense of agreement was observed. Due to this, it is recommended additional field information be reviewed to continue assessing the buckling models' accuracy. Furthermore, the sequential load history was found to have a noticeable effect on the strain demands calculated for some wall end regions. It is possible that the aftershocks following the Darfield earthquake led to additional degradation in damaged walls, leading to increased demands during the following

Christchurch earthquake.

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CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS

8.1 Summary

In response to damage observed resulting from the 2010 and 2011 New Zealand earthquakes, there was a need to assess the accuracy of current wall buckling models. Data from prior experimental tests on walls and prisms was used to compare predicted outcomes from buckling models by Paulay and Priestley and Chai and Elayer with the experimental results. General agreement was found between experimental data and the models, but Chai and Elayer's model was found to be typically less conservative in its calculation of buckling capacities. In general, the experimental data support confidence in the buckling models' results. A parametric study was performed for structural walls with a wide range of geometric and material properties. The effect of varying each parameter on predicted capacities for both buckling models was discussed, and primary influential variables were determined. Wall geometry and longitudinal steel ratio were found to have a significantly larger effect on predicted strain and displacement capacities. Additionally, various code requirements were found to be inadequate to prevent plastic buckling behavior from occurring prior to a specimen reaching its ultimate strength. Results from non-linear time history analyses of three structures damaged by the 2010 and 2011 earthquakes were provided. The occurrence (or non-occurrence) of buckling was found to be relatively well predicted by from the walls examined, however inclusion of additional data available from other earthquakes and structures is suggested.

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8.2 Recommendations for Future Research

More information is necessary to determine the true accuracy of the buckling models discussed. While the buckling models were found to show coarse agreement, additional experimental and field data would provide further opportunities to refine the models, and to more accurately design structural walls to prevent plastic buckling. To this end, it is recommended that:

 Experiments should be performed with much smaller increments of load

cycles to improve data resolution.

 Testing of wall specimens should be prioritized over prisms representative of

end regions, to avoid underestimating buckling capacities.

 Data from full-size tests has not been used to verify the buckling models,

which should be included to prevent possible issues of scale.

 The effect of failure of cover concrete on reducing lateral stiffness could

affect instability, and should be considered.

 Load history effects from preceding earthquakes (and possibly aftershock)

were shown to influence analysis demands, and should be considered in future

analyses.

 Considering the development of future code requirements, simplifications

should focus on limiting the Htww aspect ratio, rather than HLww due to

the latter having little effect on wall buckling capacities.

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8.3 Recommendations for Consideration of Wall Designs

Despite the possibility of error in the existing buckling models, general agreement was found in their ability to coarsely predict the phenomenon's occurrence. In contrast, code provisions meant to limit lateral stability were found to be ineffective at preventing local plastic buckling from limiting a given wall's ultimate strength. Even where NZS[37] recommendations were based on Paulay and Priestley's buckling model, minimum wall thicknesses were grossly underestimated due to the many simplifying assumptions made.

Despite this, walls designed with enlarged boundary elements meeting code requirements were found to effectively prevent buckling from occurring. To prevent the possibility of lateral instability in future designs of structural walls it is recommended that:

 In addition to meeting relevant code requirements on lateral stability,

structural walls should be designed implementing either Paulay and Priestley

or Chai and Elayer's full, un-simplified buckling models.

 Where possible, longitudinal steel should be distributed along the length of the

wall, rather than concentrated at end regions to increasing buckling capacities.

 Preference should be given to avoiding the use of exceedingly thin walls to

resist structural loads. While over-conservative in most instances, requiring

Htww 20 prevents buckling according to CEBM for a range of cases as

discussed in Section 6.4.

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 Where thin walls are required to resist in-plane loads, and lateral stability is a

concern, enlarged end regions should be included using current code

requirements.

8.4 Recommendations for Future Field Examinations of Damaged Buildings

It is possible that many more structural walls have failed due to plastic buckling during earthquakes, and the resulting damage has simply been missed. Perhaps a building's collapse concealed the occurrence of plastic buckling, or maybe the failure was misattributed to another collapse mechanism. Only the New Zealand Standards directly address the plastic buckling failure mode, and its requirements give little in the way of suggestions as to what the failure looks like or how to assess whether or not such a failure has occurred in the field.

To help prevent the possibility of such failures occurring in the future, it is suggested that additional effort be made to:

 Improve general awareness of the plastic buckling mechanism and how to

design to prevent it.

 Encourage practicing field engineers to look for and consider plastic buckling

as a possible cause when assessing a building's collapse mechanism.

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CHAPTER 9. BIBILIOGRAPHY

[1] ACI 318. Building code requirements for structural concrete and commentary. Detroit:

American Concrete Institute; 2008.

[2] ACI 318. Building code requirements for structural concrete and commentary. Detroit:

American Concrete Institute; 1996.

[3] Azimikor, N. “Out-of-Plane Stability of Reinforced Masonry Shear Walls under Seismic

Loading: Cyclic Uniaxial Tests” Master’s Thesis, April 2012. 196 pp.

[4] Beca. Beca Carter Hollings & Ferner Ltd. 26th September 2011. Technical Report

BUI.CAM233.0195. Canterbury: Royal Commission, 2011.

[5] Buhayar, Noah, Jacob Greber and Nichola Saminather. Bloomberg. 23 February 2011. 2

1 2013.

become-costliest-insured-disaster-since-2008.html>.

[6] Canterbury Earthquakes Royal Commission. Department of Internal Affairs. Updated: 14

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180

APPENDICES

181

Appendix A. Code for CumbiaWall (and with example input)

%======% CUMBIAWALL % SECTION AND MEMBER RESPONSE OF RC MEMBERS OF RECTANGULAR SECTION % LUIS A. MONTEJO & C. KELLY HERRICK % DEPARTMENT OF CIVIL, CONSTRUCTION AND ENVIROMENTAL ENGINEERING % NORTH CAROLINA STATE UNIVERSITY % last updated: 10/01/2013 %======

% ======% ======% ======ENTER INPUT DATA BELOW ======% ======% ======

% input data: name = 'Example'; %identifies actual work, the output file will be name.xls

% section properties: sectiontype = 'L6'; % section shape type: line, c, n, u, L5, invL5, L6, invL6, T, invT, leftT, rightT, H, I ConcMat = [800 300 % Concrete matrix (mm) - Height | Width 2300 300 600 300 300 900 300 2200 300 900];

% vector of # legs transv. steel x_dir (confinement) ncx = [4 2 3 2 2 2];

% vector of # legs transv. steel y_dir (shear) ncy = [2 2 2 4 2 4];

% vector of cover to longitudinal bars (mm) clb = [50 50 50 50 50 50];

% member properties L = 3000*7; % member clear length (mm) bending = 'wall'; % single or double or wall ductilitymode = 'uniaxial'; % biaxial or uniaxial

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% longitudinal reinforcement details, MLR is a matrix composed by % [distance from the top (mm) - # of bars - bar diameter (mm)] each row % corresponds to a layer of reinforcement. The top is in tension. MLR=[62 2 24 238 2 24 338 2 24 438 2 24 538 2 24 638 2 24 738 2 24 838 2 24 1438 2 8 2038 2 8 2638 2 8 3238 2 24 3338 2 24 3438 2 24 3538 2 24 3638 2 24 3738 2 24 3838 16 24 3938 16 24];

% transverse reinforcement details % diameter of transverse reinf. by section x-dir, y-dir(mm) Dh = [10, 10 10, 10 10, 10 10, 10 10, 10 10, 10];

% vector of spacing of transverse steel (mm)* s = [60 120 60 60 120 60];

% applied loads:

P = 4254*1000; % axial load N (-) tension (+)compression

% material models (input the 'name' of the file with the stress-strain relationship % to use the default models: Mander model for confined or unconfined concrete type 'mc' or 'mu'. % For lightweight confined concrete type 'mclw' % King model for the steel 'ks', Raynor model for steel 'ra': confined = 'mc'; unconfined = 'mu'; rebar = 'ra';

% vectors of clear distance between longitudinal bars properly restrained (used only if the mander model is selected) % ** pad rows with 0's as necessary to fill each row to the longest row's length. wi = [152 76. 76. 76. 76. 76. 76. 76. 152 76. 76. 76. 76. 76. 76. 76. 152 582+588 588+582 152 582+588 588+582 0 0 0 0 0 0 0 0 0 0 152 76. 76. 76. 76. 76. 76. 152 76. 76. 76. 76. 76. 76. 0 0 152 76. 76. 76. 76. 76. 76. 76. 152 76. 76. 76. 76. 76. 76. 76. 152 582+588 588+582 152 582+588 588+582 0 0 0 0 0 0 0 0 0 0 152 76. 76. 76. 76. 76. 76. 76. 152 76. 76. 76. 76. 76. 76. 76.];

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% material properties fpc = 37.5; % concrete compressive strength (MPa) Ec = 3320*sqrt(fpc) + 6900; % concrete modulus of elasticity (MPa) or % input 0 for automatic calculation using % 5000(fpc)^0.5 eco = 0.002; % unconfined strain (usually 0.002 for normal weight or 0.004 for lightweight)* esm = 0.12; % max transv. steel strain (usually ~0.10-0.15)* espall = 0.0064; % max uncon. conc. strain (usually 0.0064) fy = 448; % long steel yielding stress (MPa) fyh = 448; % transverse steel yielding stress (MPa) Es = 200000; % steel modulus of elasticity fsu = 603; % long steel max stress (MPa)* esh = 0.008; % long steel strain for strain hardening (usually 0.008)* esu = 0.09; % long. steel maximum strain (usually ~0.10-0.15)*

Ey = 700; % slope of the yield plateau (MPa) C1 = 3.3; % defines strain hardening curve in the Raynor model [2-6]

% *this information is used only if the default matrial models are selected

% Data on Chai and Priestly Buckling Models beta = 0.8333; % ratio of distance to extreme longitudinal steel to section width in buckling region rhostar = 0.0302; % ratio of steel to area for the buckling region bbuckle = 300; % width of extreme section to buckle (mm)

% ======% ======% ======END OF INPUT DATA ======% ======% ======addpath(fullfile(pwd,'models')) % directory with the user specified material models graph = 'n'; % if you want to display graphs, type 'y', otherwise type 'n'

% Deformation Limit States: ecser = 0.004; esser = 0.015; % concrete (ecser) and steel (esser) serviceability strain ecdam = 'twth'; esdam = 0.060; % concrete (ecser) and steel (esser) damage control strain % (to use the 2/3 of the ultimate % concrete strain just tipe 'twth') if strcmp(confined,'macb') || strcmp(unconfined,'macb') ecser = 0.0015; end

% control parameters: itermax = 100; % max number of iterations (100) ncl = 100; % # of concrete layers (100) tolerance = 0.001; % x fpc x Ag (0.001) dels = 0.0001; % delta strain for default material models (0.0001)

% ======

% Assign Initial Vars: if Ec==0 Ec = 5000*(fpc^(1/2)); end

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Ast = sum(MLR(:,2).*0.25.*pi.*MLR(:,3).^2); % Total area of long steel switch lower(sectiontype) case 'rect' ConcCore = ConcMat; % Calculations of core height and core width ConcCore(:,1) = ConcMat(:,1) - (2*clb(1) - Dh(1,1)); % Adjust core height (top) ConcCore(:,2) = ConcMat(:,2) - 2*clb + Dh(:,2); % Adjust core width (all) % distance to the top of the core from the top dcore = [clb(1)-Dh(1,1)/2]; dcore_extr_comp = dcore + ConcCore(:,1); dtopsec = 0; % Distance to top of each section from top (mm) dbotsec = dtopsec + ConcMat(:,1); % Distance to bottom of each section from top (mm) SecHeight = sum(ConcMat(:,1)); % Total Height of Cross Secion (mm) IndivCentroid = cumsum(ConcMat(:,1)) - ConcMat(:,1)/2; % Distance to centroid of each section from the top of the section (mm) IndivArea = ConcMat(:,1).*ConcMat(:,2); % Area of each section (mm) ybar = sum(IndivArea.*IndivCentroid)/sum(IndivArea); % Distance to overall centroid from the top of the section (mm) Ig = sum((ConcMat(:,2).*ConcMat(:,1).^3)/12) + sum(IndivArea.*(IndivCentroid-ybar).^2); controlsec = 1; % Number of the section farthest from the tensile face, deciding strain limit for core failure case 'double' ConcCore = ConcMat; % Calculations of core height and core width ConcCore(1,1) = ConcMat(1,1) - (2*clb(1) - Dh(1,1))/2; % Adjust core height (top) ConcCore(2,1) = ConcMat(2,1) - (2*clb(2) - Dh(2,1))/2; % Adjust core height (bottom) ConcCore(:,2) = ConcMat(:,2) - 2*clb + Dh(:,2); % Adjust core width (all) % distance to the top of the core from the top dcore = [clb(1)-Dh(1,1)/2 ConcMat(1,1)]; dcore_extr_comp = dcore + ConcCore(:,1); dtopsec = cumsum(ConcMat(:,1),1)-ConcMat(:,1); % Distance to top of each section from top (mm) dbotsec = dtopsec + ConcMat(:,1); % Distance to bottom of each section from top (mm) SecHeight = sum(ConcMat(:,1)); % Total Height of Cross Secion (mm) IndivCentroid = cumsum(ConcMat(:,1)) - ConcMat(:,1)/2; % Distance to centroid of each section from the top of the section (mm) IndivArea = ConcMat(:,1).*ConcMat(:,2); % Area of each section (mm) ybar = sum(IndivArea.*IndivCentroid)/sum(IndivArea); % Distance to overall centroid from the top of the section (mm) Ig = sum((ConcMat(:,2).*ConcMat(:,1).^3)/12) + sum(IndivArea.*(IndivCentroid-ybar).^2); controlsec = 2; % Number of the section farthest from the tensile face, deciding strain limit for core failure case 'line' ConcCore = ConcMat; % Calculations of core height and core width ConcCore(1,1) = ConcMat(1,1) - (2*clb(1) - Dh(1,1))/2; % Adjust core height (top) ConcCore(2,1) = ConcMat(2,1); % Adjust core height (middle) ConcCore(3,1) = ConcMat(3,1) - (2*clb(3) - Dh(3,1))/2; % Adjust core height (bottom) ConcCore(:,2) = ConcMat(:,2) - 2*clb + Dh(:,2); % Adjust core width (all) % distance to the top of the core from the top dcore = [clb(1)-Dh(1,1)/2 ConcMat(1,1) ConcMat(1,1) + ConcMat(2,1)]; dcore_extr_comp = dcore + ConcCore(:,1); dtopsec = cumsum(ConcMat(:,1),1)-ConcMat(:,1); % Distance to top of each section from top (mm) dbotsec = dtopsec + ConcMat(:,1); % Distance to bottom of each section from top (mm) SecHeight = sum(ConcMat(:,1)); % Total Height of Cross Secion (mm) IndivCentroid = cumsum(ConcMat(:,1)) - ConcMat(:,1)/2; % Distance to centroid of each section from the top of the section (mm) IndivArea = ConcMat(:,1).*ConcMat(:,2); % Area of each section (mm) ybar = sum(IndivArea.*IndivCentroid)/sum(IndivArea); % Distance to overall centroid from the top of the section (mm) Ig = sum((ConcMat(:,2).*ConcMat(:,1).^3)/12) + sum(IndivArea.*(IndivCentroid-ybar).^2); controlsec = 3; % Number of the section farthest from the tensile face, deciding strain limit for core failure case 'c' ConcCore = ConcMat; % Calculations of core height and core width ConcCore(1,1) = ConcMat(1,1) - 2*clb(1) + Dh(1,1); % Adjust core height (top)

185

ConcCore(2,1) = ConcMat(2,1) + 2*clb(2) - Dh(2,1); % Adjust core height (middle) ConcCore(3,1) = ConcMat(3,1) - 2*clb(3) + Dh(3,1); % Adjust core height (bottom) ConcCore(:,2) = ConcMat(:,2) - 2*clb + Dh(:,2); % Adjust core width (all) % distance to the top of the core from the top dcore = [clb(1)-Dh(1,1)/2 ConcMat(1,1) - clb(2) + Dh(2,1)/2 ConcMat(1,1) + ConcMat(2,1) + clb(3) - Dh(3,1)/2]; dcore_extr_comp = dcore + ConcCore(:,1); dtopsec = cumsum(ConcMat(:,1),1)-ConcMat(:,1); % Distance to top of each section from top (mm) dbotsec = dtopsec + ConcMat(:,1); % Distance to bottom of each section from top (mm) SecHeight = sum(ConcMat(:,1)); % Total Height of Cross Secion (mm) IndivCentroid = cumsum(ConcMat(:,1)) - ConcMat(:,1)/2; % Distance to centroid of each section from the top of the section (mm) IndivArea = ConcMat(:,1).*ConcMat(:,2); % Area of each section (mm) ybar = sum(IndivArea.*IndivCentroid)/sum(IndivArea); % Distance to overall centroid from the top of the section (mm) Ig = sum((ConcMat(:,2).*ConcMat(:,1).^3)/12) + sum(IndivArea.*(IndivCentroid-ybar).^2); controlsec = 3; % Number of the section farthest from the tensile face, deciding strain limit for core failure case 'l4' ConcCore = ConcMat; % Calculations of core height and core width ConcCore(1,1) = ConcMat(1,1) - (2*clb(1) - Dh(1,1))/2; % Adjust core height (top) ConcCore(2,1) = ConcMat(2,1); % Adjust core height (middle1) ConcCore(3,1) = ConcMat(3,1) + (2*clb(3))/2 - (Dh(3,1))/2; % Adjust core height (middle2) ConcCore(4,1) = ConcMat(4,1) - (2*clb(4) - Dh(4,1)); % Adjust core height (bottom) ConcCore(:,2) = ConcMat(:,2) - 2*clb + Dh(:,2); % Adjust core width (all) % distance to the top of the core from the top dcore = [clb(1)-Dh(1,1)/2 ConcMat(1,1) ConcMat(1,1) + ConcMat(2,1) ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1) + clb(4) - Dh(4,1)/2]; dcore_extr_comp = dcore + ConcCore(:,1); dtopsec = cumsum(ConcMat(:,1),1)-ConcMat(:,1); % Distance to top of each section from top (mm) dbotsec = dtopsec + ConcMat(:,1); % Distance to bottom of each section from top (mm) SecHeight = sum(ConcMat(:,1)); % Total Height of Cross Secion (mm) IndivCentroid = cumsum(ConcMat(:,1)) - ConcMat(:,1)/2; % Distance to centroid of each section from the top of the section (mm) IndivArea = ConcMat(:,1).*ConcMat(:,2); % Area of each section (mm) ybar = sum(IndivArea.*IndivCentroid)/sum(IndivArea); % Distance to overall centroid from the top of the section (mm) Ig = sum((ConcMat(:,2).*ConcMat(:,1).^3)/12) + sum(IndivArea.*(IndivCentroid-ybar).^2); controlsec = 4; % Number of the section farthest from the tensile face, deciding strain limit for core failure case 'invl4' ConcCore = ConcMat; % Calculations of core height and core width ConcCore(1,1) = ConcMat(1,1) - 2*clb(1) + Dh(1,1); % Adjust core height (top) ConcCore(2,1) = ConcMat(2,1) + (2*clb(2))/2 - (Dh(2,1))/2; % Adjust core height (middle1) ConcCore(3,1) = ConcMat(3,1); % Adjust core height (middle2) ConcCore(4,1) = ConcMat(4,1) - (2*clb(4) - Dh(4,1))/2; % Adjust core height (bottom) ConcCore(:,2) = ConcMat(:,2) - 2*clb + Dh(:,2); % Adjust core width (all) % distance to the top of the core from the top dcore = [clb(1)-Dh(1,1)/2 ConcMat(1,1) - clb(1) + Dh(1,1)/2 ConcMat(1,1) + ConcMat(2,1) ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1)]; dcore_extr_comp = dcore + ConcCore(:,1); dtopsec = cumsum(ConcMat(:,1),1)-ConcMat(:,1); % Distance to top of each section from top (mm) dbotsec = dtopsec + ConcMat(:,1); % Distance to bottom of each section from top (mm) SecHeight = sum(ConcMat(:,1)); % Total Height of Cross Secion (mm) IndivCentroid = cumsum(ConcMat(:,1)) - ConcMat(:,1)/2; % Distance to centroid of each section from the top of the section (mm) IndivArea = ConcMat(:,1).*ConcMat(:,2); % Area of each section (mm) ybar = sum(IndivArea.*IndivCentroid)/sum(IndivArea); % Distance to overall centroid from the top of the section (mm) Ig = sum((ConcMat(:,2).*ConcMat(:,1).^3)/12) + sum(IndivArea.*(IndivCentroid-ybar).^2); controlsec = 4; % Number of the section farthest from the tensile face, deciding strain limit for core failure

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case 'l5' ConcCore = ConcMat; % Calculations of core height and core width ConcCore(1,1) = ConcMat(1,1) - (2*clb(1) - Dh(1,1))/2; % Adjust core height (top) ConcCore(2,1) = ConcMat(2,1) + (2*clb(2))/2 - (Dh(2,1))/2; % Adjust core height (middle) ConcCore(3:5,1) = ConcMat(3:5,1) - (2*clb(3:5) - Dh(3:5,1));% Adjust core height (bottom sections) ConcCore(1:2,2) = ConcMat(1:2,2) - 2*clb(1:2) + Dh(1:2,2); % Adjust core width (top) ConcCore(3,2) = ConcMat(3,2) - (2*clb(3) - Dh(3,2))/2; % Adjust core width (bottom left) ConcCore(4,2) = ConcMat(4,2); % Adjust core width (bottom middle) ConcCore(5,2) = ConcMat(5,2) - (2*clb(5) - Dh(5,2))/2; % Adjust core width (bottom right) % distance to the top of the core from the top dcore = [clb(1)-Dh(1,1)/2 ConcMat(1,1) ConcMat(1,1) + ConcMat(2,1) + clb(3) - Dh(3,1)/2 ConcMat(1,1) + ConcMat(2,1) + clb(4) - Dh(4,1)/2 ConcMat(1,1) + ConcMat(2,1) + clb(5) - Dh(5,1)/2]; dcore_extr_comp = dcore + ConcCore(:,1); dtopsec = [0 ConcMat(1,1) ConcMat(1,1) + ConcMat(2,1) ConcMat(1,1) + ConcMat(2,1) ConcMat(1,1) + ConcMat(2,1)]; % Distance to top of each section from top (mm) dbotsec = dtopsec + ConcMat(:,1); % Distance to bottom of each section from top (mm) SecHeight = sum(ConcMat(1:3,1)); % Total Height of Cross Secion (mm) IndivCentroid = dtopsec + ConcMat(:,1)/2; % Distance to centroid of each section from the top (mm) IndivArea = ConcMat(:,1).*ConcMat(:,2); % Area of each section (mm) ybar = sum(IndivArea.*IndivCentroid)/sum(IndivArea); % Distance to overall centroid from the top of the section (mm) Ig = sum((ConcMat(:,2).*ConcMat(:,1).^3)/12) + sum(IndivArea.*(IndivCentroid-ybar).^2); controlsec = 3; % Number of the section farthest from the tensile face, deciding strain limit for core failure case 'invl5' ConcCore = ConcMat; % Calculations of core height and core width ConcCore(1:3,1) = ConcMat(1:3,1) - (2*clb(1:3))/2 - (Dh(1:3,1))/2; % Adjust core height (top sections) ConcCore(4,1) = ConcMat(1,1) + (2*clb(4))/2 - (Dh(4,1))/2; % Adjust core height (middle) ConcCore(5,1) = ConcMat(5,1) - (2*clb(5) - Dh(5,1))/2; % Adjust core height (bottom) ConcCore(1,2) = ConcMat(1,2) - (2*clb(1) - Dh(1,2))/2; % Adjust core width (top left) ConcCore(2,2) = ConcMat(2,2); % Adjust core width (top middle) ConcCore(3,2) = ConcMat(3,2) - (2*clb(3) - Dh(3,2))/2; % Adjust core width (top right) ConcCore(4:5,2) = ConcMat(4:5,2) - 2*clb(4:5) + Dh(4:5,2); % Adjust core width (bottom) % distance to the top of the core from the top dcore = [clb(1)-Dh(1,1)/2 clb(2)-Dh(2,1)/2 clb(3)-Dh(3,1)/2 ConcMat(1,1) + (2*clb(4))/2 - (Dh(4,1))/2 ConcMat(1,1) + ConcMat(4,1)]; dcore_extr_comp = dcore + ConcCore(:,1); dtopsec = [0 0 0 ConcMat(1,1) ConcMat(1,1) + ConcMat(4,1)]; % Distance to top of each section from top (mm) dbotsec = dtopsec + ConcMat(:,1); % Distance to bottom of each section from top (mm) SecHeight = ConcMat(1,1) + sum(ConcMat(4:5,1)); % Total Height of Cross Secion (mm) IndivCentroid = dtopsec + ConcMat(:,1)/2; % Distance to centroid of each section from the top of the section (mm) IndivArea = ConcMat(:,1).*ConcMat(:,2); % Area of each section (mm) ybar = sum(IndivArea.*IndivCentroid)/sum(IndivArea); % Distance to overall centroid from the top of the section (mm) Ig = sum((ConcMat(:,2).*ConcMat(:,1).^3)/12) + sum(IndivArea.*(IndivCentroid-ybar).^2); controlsec = 5; % Number of the section farthest from the tensile face, deciding strain limit for core failure case 'l6' ConcCore = ConcMat; % Calculations of core height and core width ConcCore(1,1) = ConcMat(1,1) - (2*clb(1) - Dh(1,1))/2; % Adjust core height (top) ConcCore(2,1) = ConcMat(2,1); % Adjust core height (middle1) ConcCore(3,1) = ConcMat(3,1) + (2*clb(3))/2 - (Dh(3,1))/2; % Adjust core height (middle2)

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ConcCore(4:6,1) = ConcMat(4:6,1) - (2*clb(4:6) - Dh(4:6,1));% Adjust core height (bottom sections) ConcCore(1:3,2) = ConcMat(1:3,2) - 2*clb(1:3) + Dh(1:3,2); % Adjust core width (top) ConcCore(4,2) = ConcMat(4,2) - (2*clb(4) - Dh(4,2))/2; % Adjust core width (bottom left) ConcCore(5,2) = ConcMat(5,2); % Adjust core width (bottom middle) ConcCore(6,2) = ConcMat(6,2) - (2*clb(6) - Dh(6,2))/2; % Adjust core width (bottom right) % distance to the top of the core from the top dcore = [clb(1)-Dh(1,1)/2 ConcMat(1,1) ConcMat(1,1) + ConcMat(2,1) ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1) + clb(4) - Dh(4,1)/2 ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1) + clb(5) - Dh(5,1)/2 ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1) + clb(6) - Dh(6,1)/2]; dcore_extr_comp = dcore + ConcCore(:,1); dtopsec = [0 ConcMat(1,1) ConcMat(1,1) + ConcMat(2,1) ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1) ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1) ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1)]; % Distance to top of each section from top (mm) dbotsec = dtopsec + ConcMat(:,1); % Distance to bottom of each section from top (mm) SecHeight = sum(ConcMat(1:4,1)); % Total Height of Cross Secion (mm) IndivCentroid = dtopsec + ConcMat(:,1)/2; % Distance to centroid of each section from the top (mm) IndivArea = ConcMat(:,1).*ConcMat(:,2); % Area of each section (mm) ybar = sum(IndivArea.*IndivCentroid)/sum(IndivArea); % Distance to overall centroid from the top of the section (mm) Ig = sum((ConcMat(:,2).*ConcMat(:,1).^3)/12) + sum(IndivArea.*(IndivCentroid-ybar).^2); controlsec = 4; % Number of the section farthest from the tensile face, deciding strain limit for core failure case 'invl6' ConcCore = ConcMat; % Calculations of core height and core width ConcCore(1:3,1) = ConcMat(1:3,1) - (2*clb(1:3))/2 - (Dh(1:3,1))/2; % Adjust core height (top sections) ConcCore(4,1) = ConcMat(1,1) + (2*clb(4))/2 - (Dh(4,1))/2; % Adjust core height (middle1) ConcCore(5,1) = ConcMat(5,1); % Adjust core height (middle2) ConcCore(6,1) = ConcMat(4,1) - (2*clb(6) - Dh(6,1))/2; % Adjust core height (bottom) ConcCore(1,2) = ConcMat(1,2) - (2*clb(1) - Dh(1,2))/2; % Adjust core width (top left) ConcCore(2,2) = ConcMat(2,2); % Adjust core width (top middle) ConcCore(3,2) = ConcMat(3,2) - (2*clb(3) - Dh(3,2))/2; % Adjust core width (top right) ConcCore(4:6,2) = ConcMat(4:6,2) - 2*clb(4:6) + Dh(4:6,2); % Adjust core width (bottom) % distance to the top of the core from the top dcore = [clb(1)-Dh(1,1)/2 clb(2)-Dh(2,1)/2 clb(3)-Dh(3,1)/2 ConcMat(1,1) + (2*clb(4))/2 - (Dh(4,1))/2 ConcMat(1,1) + ConcMat(4,1) ConcMat(1,1) + ConcMat(4,1) + ConcMat(5,1)]; dcore_extr_comp = dcore + ConcCore(:,1); dtopsec = [0 0 0 ConcMat(1,1) ConcMat(1,1) + ConcMat(4,1) ConcMat(1,1) + ConcMat(4,1) + ConcMat(5,1)]; % Distance to top of each section from top (mm) dbotsec = dtopsec + ConcMat(:,1); % Distance to bottom of each section from top (mm) SecHeight = ConcMat(1,1) + sum(ConcMat(4:6,1)); % Total Height of Cross Secion (mm) IndivCentroid = dtopsec + ConcMat(:,1)/2; % Distance to centroid of each section from the top of the section (mm) IndivArea = ConcMat(:,1).*ConcMat(:,2); % Area of each section (mm) ybar = sum(IndivArea.*IndivCentroid)/sum(IndivArea); % Distance to overall centroid from the top of the section (mm) Ig = sum((ConcMat(:,2).*ConcMat(:,1).^3)/12) + sum(IndivArea.*(IndivCentroid-ybar).^2); controlsec = 6; % Number of the section farthest from the tensile face, deciding strain limit for core failure case 'symi' ConcCore = ConcMat; % Calculations of core height and core width ConcCore(1,1) = ConcMat(1,1) - (2*clb(1) - Dh(1,1))/2; % Adjust core height (1L) ConcCore(2,1) = ConcMat(2,1); % Adjust core height (2L)

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ConcCore(3,1) = ConcMat(3,1); % Adjust core height (3L) ConcCore(4,1) = ConcMat(4,1); % Adjust core height (4L) ConcCore(5,1) = ConcMat(5,1) - (2*clb(1) - Dh(1,1))/2; % Adjust core height (5L) ConcCore(6,1) = ConcMat(6,1) - (2*clb(1) - Dh(1,1)); % Adjust core height (5L) ConcCore(7,1) = ConcCore(1,1); % Adjust core height (7R) ConcCore(8,1) = ConcCore(2,1); % Adjust core height (8R) ConcCore(9,1) = ConcCore(3,1); % Adjust core height (9R) ConcCore(10,1) = ConcCore(4,1); % Adjust core height (10R) ConcCore(11,1) = ConcCore(5,1); % Adjust core height (11R) ConcCore(:,2) = ConcMat(:,2) - 2*clb(:) + Dh(:,2); % Adjust core width (all, except #6) ConcCore(6,2) = ConcMat(6,2) + (2*clb(6) - Dh(6,2))*2; % Adjust core width (#6) % distance to the top of the core from the top dcore = [clb(1) - Dh(1,1)/2 ConcMat(1,1) ConcMat(1,1) + ConcMat(2,1) ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1) ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1) + ConcMat(4,1) ConcMat(1,1) + ConcMat(2,1) + clb(6) - Dh(6,1)/2 clb(7) - Dh(7,1)/2 ConcMat(7,1) ConcMat(7,1) + ConcMat(8,1) ConcMat(7,1) + ConcMat(8,1) + ConcMat(9,1) ConcMat(7,1) + ConcMat(8,1) + ConcMat(9,1) + ConcMat(10,1)]; dcore_extr_comp = dcore(:) + ConcCore(:,1); dtopsec = [0 ConcMat(1,1) ConcMat(1,1) + ConcMat(2,1) ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1) ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1) + ConcMat(4,1) ConcMat(1,1) + ConcMat(2,1) 0 ConcMat(7,1) ConcMat(7,1) + ConcMat(8,1) ConcMat(7,1) + ConcMat(8,1) + ConcMat(9,1) ConcMat(7,1) + ConcMat(8,1) + ConcMat(9,1) + ConcMat(10,1)]; % Distance to top of each section from top (mm) dbotsec = dtopsec + ConcMat(:,1); % Distance to bottom of each section from top (mm) SecHeight = sum(ConcMat(1:5,1)); % Total Height of Cross Secion (mm) IndivCentroid(1:5) = cumsum(ConcMat(1:5,1)) - ConcMat(1:5,1)/2; % Distance to centroid of each section from the top of the section (mm) IndivCentroid(6) = ConcMat(1,1) + ConcMat(2,1) + ConcMat(3,1)/2; IndivCentroid(7:11) = cumsum(ConcMat(7:11,1)) - ConcMat(7:11,1)/2; IndivCentroid = IndivCentroid'; IndivArea = ConcMat(:,1).*ConcMat(:,2); % Area of each section (mm) ybar = sum(IndivArea.*IndivCentroid)/sum(IndivArea); % Distance to overall centroid from the top of the section (mm) Ig = sum((ConcMat(:,2).*ConcMat(:,1).^3)/12) + sum(IndivArea.*(IndivCentroid-ybar).^2); controlsec = 5; % Number of the section farthest from the tensile face, deciding strain limit for core failure end tcl = SecHeight / ncl; % thickness of concrete layers yl = tcl*(1:ncl); % disance to bottom edge of concrete layers esser = -esser; esdam = -esdam; switch lower(unconfined) case 'mu' [ecun,fcun] = manderun(Ec,fpc,eco,espall,dels); case 'mc' [ecun,fcun] = manderconf(Ec,Ast,Dh,clb,s,fpc,fyh,eco,esm,'rectangular',0,dcore_extr_comp,ConcCore,ncx,ncy,wi,dels,'hoops'); case 'mclw' [ecun,fcun] = manderconflw(Ec,Ast,Dh,clb,s,fpc,fyh,eco,esm,'rectangular',0,dcore_extr_comp,ConcCore,ncx,ncy,wi,dels,'hoops');

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case 'macb' [ecun,fcun] = masonryconc(fpc,eco,dels); otherwise AUX = load ([unconfined,'.txt']); % read the material model data file ecun = AUX(:,1)'; fcun = AUX(:,2)'; end switch lower(confined) case 'mu' [ec,fc] = manderun(Ec,fpc,eco,espall,dels); case 'mc' [ec,fc] = manderconf(Ec,Ast,Dh,clb,s,fpc,fyh,eco,esm,'rectangular',0,dcore_extr_comp,ConcCore,ncx,ncy,wi,dels,'hoops'); case 'mclw' [ec,fc] = manderconflw(Ec,Ast,Dh,clb,s,fpc,fyh,eco,esm,'rectangular',0,dcore_extr_comp,ConcCore,ncx,ncy,wi,dels,'hoops'); case 'macb' [ec,fc] = masonryconc(fpc,eco,dels); ec = ones(size(ConcMat,1),1)*ec; fc = ones(size(ConcMat,1),1)*fc; otherwise AUX = load ([confined,'.txt']); % read the material model data file ec = AUX(:,1)'; fc = AUX(:,2)'; end switch lower(rebar) case 'ks' [es,fs] = steelking(Es,fy,fsu,esh,esu,dels); case 'ra' [es,fs] = Raynor(Es,fy,fsu,esh,esu,dels,C1,Ey); otherwise AUX = load ([rebar,'.txt']); % read the material model data file es = AUX(:,1)'; fs = AUX(:,2)'; end

% maximum strain confined concrete at extreme comp fiber of core if strcmp(confined,'mc') || strcmp(confined,'macb') boundless_ec = zeros(size(ec,1),size(ec,2)-1); for i = 1:size(ec,1) temp = ec(i,:); boundless_ec(i,:) = temp(temp<10); end else ec = [-1e10 ec 1e10]; boundless_ec = ec(2:end-1); fc = [0 fc 0];

ec = ones(size(ConcCore,1),1)*ec; boundless_ec = ones(size(ConcCore,1),1)*boundless_ec; fc = ones(size(ConcCore,1),1)*fc; end ecu = max(boundless_ec(controlsec,:)); % ultimate strain predicted by the original mander model ecumander = ecu/1.5; if strcmp(lower(ecdam), 'twth') ecdam = ecumander; end

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[~,maxsec] = max(max(boundless_ec,[],2)); ec = bsxfun(@times,ec(maxsec,:),ones(size(ConcMat,1),1)); % re-define ec to match full range of strains possible if strcmp(confined,'macb') else ecun = [-1e10 ecun ecun(end)+ecun(:,2)-ecun(:,1) 1e10]; % vector with strains of unconfined concrete fcun = [0 fcun 0 0]; % vector with stresses of unconfined concrete end esu = es(end); % maximum strain steel es = [es es(length(es))+es(:,2)-es(:,1) 1e10]; % vector with strains of the steel fs = [fs 0 0]; % vector with stresses of the steel esaux = zeros(1,length(es)); fsaux = zeros(1,length(es)); for i=1:size(es,2) esaux(i) = es(length(es)-i+1); fsaux(i) = fs(length(fs)-i+1); end es = [-esaux es(2:length(es))]; % vector with strains of the steel fs = [-fsaux fs(2:length(fs))]; % vector with stresses of the steel if strcmp(graph,'y') figure;area(ec,fc,'FaceColor','c') hold on area(ecun,fcun,'FaceColor','b') hold off;ylabel('Stress [MPa]','FontSize',16); xlabel('Strain','FontSize',16); legend(': Confined Concrete',': Unconfined Concrete');grid on;set(gca,'Layer','top'); title('Stress-Strain Relation for Confined and Unconfined Concrete','FontSize',16) axis([ec(2) ec(length(ec)-2) fc(1) 1.05*max(fc)]);

figure; area(es,fs,'faceColor',[0.8 0.8 0.4]); grid on ylabel('Stress [MPa]','FontSize',16); xlabel('Strain','FontSize',16); axis([es(3) es(length(es)-2) 1.05*fs(3) 1.05*max(fs)]);set(gca,'Layer','top'); title('Stress-Strain Relation for Reinforcing Steel','FontSize',16) end

%======CONCRETE LAYERS ======yl = sort([yl dtopsec' dbotsec' dcore' dcore_extr_comp'],2); % add layer divisions at top/bottom of each section yl = unique(yl,'first'); yl(yl==0) = []; yl = bsxfun(@times,yl,ones(size(ConcCore,1),1)); % assemble matrix of concrete layer distances yc = yl.*and(yl > dcore(1), yl < dcore_extr_comp(end)); % assemble matrix of confined concrete layer distances

% Assign 0 to values outside of each section for i = 1:size(yl,1) for j = 1:size(yl,2) if yl(i,j) <= dtopsec(i) || yl(i,j) > dbotsec(i) yl(i,j) = 0; end end end for i = 1:size(yc,1) for j = 1:size(yc,2) if yc(i,j) <= dcore(i) || yc(i,j) > dcore_extr_comp(i) yc(i,j) = 0; end end end

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yltemp = [zeros(size(yl,1),1) yl]; yctemp = [zeros(size(yc,1),1) yc]; yltemp = [zeros(1,size(yltemp,2));yltemp]; yctemp = [zeros(1,size(yctemp,2));yctemp]; for i = 2:size(yltemp,1) for j = 2:size(yltemp,2) if yltemp(i,j) == 0 Atc(i-1,j-1) = 0; elseif yltemp(i,j-1) == 0 && yltemp(i-1,j) == 0 && yltemp(i-1,j-1) == 0 % initial boundary Atc(i-1,j-1) = yltemp(i,j)*ConcMat(i-1,2); elseif yltemp(i,j-1) == 0 && yltemp(i-1,j) == 0 && yltemp(i-1,j-1) ~= 0 Atc(i-1,j-1) = (yltemp(i,j)-yltemp(i-1,j-1))*ConcMat(i-1,2); elseif yltemp(i,j-1) == 0 && yltemp(i-1,j-1) == 0 Atc(i-1,j-1) = (yltemp(i,j)-max(yltemp(:,j-1)))*ConcMat(i-1,2); elseif yltemp(i,j-1) == 0 && yltemp(i,j-2) == 0 Atc(i-1,j-1) = (yltemp(i,j)-max(yltemp(:,j-1)))*ConcMat(i-1,2); else Atc(i-1,j-1) = (yltemp(i,j)-yltemp(i,j-1))*ConcMat(i-1,2); end if yctemp(i,j) == 0 Atcc(i-1,j-1) = 0; elseif yctemp(i,j-1) == 0 && yctemp(i-1,j) == 0 && yctemp(i-1,j-1) == 0 % initial boundary Atcc(i-1,j-1) = (yctemp(i,j)-yltemp(i,j-1))*ConcCore(i-1,2); elseif yctemp(i,j-1) == 0 && yctemp(i-1,j) == 0 && yctemp(i-1,j-1) ~= 0 Atcc(i-1,j-1) = (yctemp(i,j)-yctemp(i-1,j-1))*ConcCore(i-1,2); elseif yctemp(i,j-1) == 0 && yctemp(i-1,j-1) == 0 Atcc(i-1,j-1) = (yctemp(i,j)-max(yctemp(:,j-1)))*ConcCore(i-1,2); elseif yctemp(i,j-1) == 0 && yctemp(i,j-2) == 0 Atcc(i-1,j-1) = (yctemp(i,j)-max(yctemp(:,j-1)))*ConcCore(i-1,2); else Atcc(i-1,j-1) = (yctemp(i,j)-yctemp(i,j-1))*ConcCore(i-1,2); end end end

% Collapse yl, yc, Atc, and Atcc into singular vectors based on geometries if strcmpi(sectiontype,'rect') elseif strcmpi(sectiontype,'l5') yl = sum(yl(1:3,:)); yc = sum(yc(1:3,:)); elseif strcmpi(sectiontype,'invl5') yl = sum(yl([1,4:5],:)); yc = sum(yc([1,4:5],:)); elseif strcmpi(sectiontype,'l6') yl = sum(yl([1,2:4],:)); yc = sum(yc([1,2:4],:)); elseif strcmpi(sectiontype,'invl6') yl = sum(yl([1,4:6],:)); yc = sum(yc([1,4:6],:)); else yl = sum(yl); yc = sum(yc); end if strcmpi(sectiontype,'rect') else Atc = sum(Atc); Atcc = sum(Atcc); end

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% Create concrete layer area array [A uncon|A conf] conclay = [Atc-Atcc; Atcc];

% [center layer|A uncon|A conf|d to top of layer] conclay = [[yl(1)/2 0.5*(yl(1:length(yl)-1)+yl(2:length(yl)))];conclay;yl]';

%======REBARS ======distld = zeros(1,size(MLR,1)); Asbs = zeros(1,size(MLR,1)); diam = zeros(1,size(MLR,1)); distld(:) = MLR(:,1); Asbs(:) = 0.25*pi*(MLR(:,3).^2).*MLR(:,2); diam(:) = MLR(:,3); auxqp = sortrows([distld' Asbs' diam'],1); distld = auxqp(:,1); % y coordinate of each bar Asbs = auxqp(:,2); % area of each bar diam = auxqp(:,3); % diameter of each bar

%======CORRECTED AREAS ======err=0; for i=1:size(distld,2) aux = find(yl>distld(i)); if strcmp(confined,'macb') || strcmp(unconfined,'macb') conclay(aux(1),3) = conclay(aux(1),2)-Asbs(i); else conclay(aux(1),3) = conclay(aux(1),2)-Asbs(i); end % if conclay(aux(1),3)<0 % err = err+1; % end end if err >0 disp('decrease # of layers') return end

%======Define vector (def) with the deformations in the top concrete ======if ecu<=0.0018 def = (0.0001:0.0001:20*ecu); end if ecu>0.0018 && ecu<=0.0025 def = [0.0001:0.0001:0.0016 0.0018:0.0002:20*ecu]; end if ecu>0.0025 && ecu<=0.006 def = [0.0001:0.0001:0.0016 0.0018:0.0002:0.002 0.0025:0.0005:20*ecu]; end if ecu>0.006 && ecu<=0.012 def = [0.0001:0.0001:0.0016 0.0018:0.0002:0.002 0.0025:0.0005:0.005 0.006:0.001:20*ecu]; end if ecu>0.012 def = [0.0001:0.0001:0.0016 0.0018:0.0002:0.002 0.0025:0.0005:0.005 0.006:0.001:0.01 0.012:0.002:20*ecu]; end

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np = length(def); if P>0 for i=1:np temp_sum = 0; for j = 1:length(dcore) for k = 1:size(yl,2) if yl(k) >= dcore(j) && yl(k) <= dcore_extr_comp(j); temp_sum = temp_sum + interp1(ec(j,:),fc(j,:),def(1))*conclay(k,3); end end end compch = sum(interp1(ecun,fcun,def(1)*ones(1,length(yl))).*conclay(:,2)') + ... temp_sum + sum(Ast*interp1(es,fs,def(1))); if compch

% ======Calculation of Axial T or C force to yield ======

Agross = sum(ConcMat(:,1).*ConcMat(:,2)); Acore = sum(ConcCore(:,1).*ConcCore(:,2)); pcy = zeros(size(ConcMat,1),1); steelsec = zeros(size(ConcMat,1),1); for i = 1:size(ConcMat,1) for j = 1:size(MLR,1) if MLR(j,1) > dtopsec(i) && MLR(j,1) > dbotsec(i) steelsec(i) = steelsec(i) + MLR(j,2)*pi*MLR(j,3)^2/4; end end pcy(i) = interp1(ec(i,:),fc(i,:),(fy/Es))*(ConcCore(i,1)*ConcCore(i,2) - steelsec(i)); end % axial forces for yielding PCY = sum(pcy)+interp1(ecun,fcun,(fy/Es))*(Agross-Acore)+Ast*fy; PTY = Ast*fy; % axial force for yielding

% ======ITERATIVE PROCESS TO FIND THE MOMENT - CURVATURE RELATION: ======

message = 0; % stop conditions alt_message = 0; curv(1) = 0; % curvatures mom(1) = 0; % moments ejen(1) = 0; % neutral axis DF(1) = 0; % force eqilibrium vniter(1) = 0; % iterations coverstrain(1) = 0; corestrain(1) = 0; steelstrain(1) = 0; ecu_alt = 0; tol = tolerance*sum(ConcMat(:,1).*ConcMat(:,2))*fpc; % tolerance allowed x(1) = SecHeight/2; momloss = 0; for i=1:np lostmomcontrol = max(mom); if mom(i)<(0.8*lostmomcontrol) && momloss == 0

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momloss = mom(end); message = 4; break end

F(1) = 10*tol; niter = 1; flip = 0; while abs(F(niter)) > tol if niter == 1 && flip == 0 flip = 1; else niter = niter + 1; end

if x(niter) <= SecHeight eec = (def(i)/x(niter))*(conclay(:,1)-(SecHeight-x(niter))); % vector with the strains in the concrete layers ees = (def(i)/x(niter))*(distld-(SecHeight-x(niter))); % vector with the strains in the steel layers end if x(niter) > SecHeight eec = (def(i)/x(niter))*(x(niter)-SecHeight+conclay(:,1)); % vector with the strains in the concrete layers ees = (def(i)/x(niter))*(x(niter)-SecHeight+distld); % vector with the strains in the steel layers end

fcunconf = interp1(ecun,fcun,eec); % vector with stresses in the unconfined concrete layers fcconf = zeros(length(eec),1); % vector with stresses in the confined concrete layers

for j = 1:length(dcore) % iterate through each section of the wall for k = 1:length(eec) % iterate through the layers of confined concrete temp = yl(k); if yl(k) >= dcore(j) && yl(k) <= dcore_extr_comp(j); fcconf(k) = interp1(ec(j,:),fc(j,:),eec(k)); end end end

fsteel = interp1(es,fs,ees); % vector with stresses in the steel layers FUNCON = fcunconf.*conclay(:,2); FCONF = fcconf.*conclay(:,3); FST = Asbs.*fsteel; F(niter) = sum(FUNCON) + sum(FCONF) + sum(FST) - P; if F(niter) > 0 x(niter+1) = x(niter) - 0.05*x(niter); end

if F(niter)<0 x(niter+1) = x(niter) + 0.05*x(niter); end if niter>itermax message = 3; break end end cores = (def(i)/x(niter))*abs(x(niter)-(SecHeight - dcore_extr_comp(controlsec)));

% maximum strain at any location in the specimen (could be in the middle) if alt_message == 0 nacore = x(niter)-(SecHeight-dcore_extr_comp); % Distances from N.A. to extr. comp. fiber of each section eclims = max(boundless_ec,[],2); % Max ec for each section

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nacore( nacore <= 0 ) = 0; % Remove strains in the tension region ( where nacore < 0) ecu_alt(1,i+1) = min(eclims./nacore*max(nacore)); if cores > ecu_alt(1,i+1) && ecu_alt(1,i+1) < ecu alt_message = 1; end end TF = strcmp(confined, unconfined); if message == 0 if TF == 0 if cores>=ecu message = 1; end end if TF == 1 if def(i)>=ecu message = 1; end end if abs(ees(1))>esu message = 2; end end

if TF == 0 if cores>=2*ecu break end end if TF == 1 if def(i)>=2*ecu break end end if ees(1) < -2*esu break end

ejen(1,i+1) = x(niter); DF(1,i+1) = F(niter); vniter(1,i+1) = niter; mom(1,i+1) = (sum(FUNCON.*conclay(:,1)) + sum(FCONF.*conclay(:,1)) + sum(FST.*distld) - P*ybar)/(10^6); if mom(1,i+1)<0 mom(1,i+1) = -0.01*mom(1,i+1); end curv(1,i+1) = 1000*def(i)/x(niter); coverstrain(1,i+1) = def(i); corestrain(:,i+1) = cores; steelstrain(1,i+1) = ees(1); x(1) = x(niter); x(2:length(x))=0; end

LongSteelRatio = (Ast/(Agross)); TransvSteelRatioX = ncx*0.25*pi.*(Dh(:,1).^2)./(s.*ConcCore(:,1)); TransvSteelRatioY = ncy*0.25*pi.*(Dh(:,2).^2)./(s.*ConcCore(:,2)); TransvSteelRatioAverage = sum(TransvSteelRatioX+TransvSteelRatioY)/(length(TransvSteelRatioX)+length(TransvSteelRatioY)); AxialRatio = (P/(fpc*Agross)); if length(coverstrain) == 1 || length(mom) == 1 chaidisp = NaN; priestleydisp = NaN;

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return end

Mn = interp1(coverstrain,mom,0.004); esaux = interp1(mom,steelstrain,Mn); if esaux<-0.015 || isnan(esaux) Mn = interp1(steelstrain,mom,-0.015); end cMn = interp1(mom,ejen,Mn); fycurv = interp1(steelstrain,curv,-fy/Es); % curvature for first yield fyM = interp1(curv,mom,fycurv); % moment for first yield eqcurv = max((Mn/fyM)*fycurv,fycurv); curvbilin = [0 eqcurv curv(length(curv))]; mombilin = [0 Mn mom(length(mom))];

SectionCurvatureDuctility = curv(length(curv))/eqcurv; if strcmp(graph,'y') figure; plot(curvbilin,mombilin,'r',curv,mom,'b--','LineWidth',2); grid on; xlabel('Curvature(1/m)','FontSize',16); ylabel('Moment (kN-m)','FontSize',16); title('Moment - Curvature Relation','FontSize',16); end kkk = min(0.2*(fsu/fy-1),0.08); Lsp = 0.022*fy*max(MLR(:,3)); % Strain penetration length switch lower(bending) case 'single' Lp = max(kkk*L + Lsp,2*Lsp); % Plastic hinge length case 'double' Lp = max(kkk*L/2 + Lsp,2*Lsp); % Plastic hinge length case 'wall' if strcmp(confined,'macb') || strcmp(unconfined,'macb') Lp = max(0.04*L*(2/3) + 0.1*SecHeight + Lsp,3*Lsp); else Lp = max(kkk*L*(2/3) + 0.1*SecHeight + Lsp,2*Lsp); % Plastic hinge length end otherwise disp('bending should be specified as single or double or wall'); return end

% Flexurual deflection: displf = zeros(1,length(curv)); switch lower(bending) case 'single' Leff = L + Lsp; fydisplf = fycurv*((Leff/1000)^2)/3; for i=1:length(curv) if curv(i)<=fycurv displf(i) = curv(i) * ((Leff/1000)^2)/3; else displf(i) = (curv(i)-fycurv*(mom(i)/fyM))*(Lp/1000)*((L+Lsp-0.5*Lp)/1000) + fydisplf*(mom(i)/fyM); end end Force = mom/(L/1000);

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case 'double' Leff = L + 2*Lsp; fydisplf = fycurv*((Leff/1000)^2)/6; for i=1:length(curv) if curv(i)<=fycurv displf(i) = curv(i) * ((Leff/1000)^2)/6; else displf(i) = (curv(i)-fycurv*(mom(i)/fyM))*(Lp/1000)*((L+2*(Lsp-0.5*Lp))/1000) + fydisplf*(mom(i)/fyM); end end Force = 2*mom/(L/1000); case 'wall' Leff = L + Lsp; fydisplf = fycurv*((Leff/1000)^2)/3; for i=1:length(curv) if curv(i)<=fycurv displf(i) = curv(i) * ((Leff/1000)^2)/3; else displf(i) = (curv(i)-fycurv*(mom(i)/fyM))*(Lp/1000)*((L+Lsp-0.5*Lp)/1000) + fydisplf*(mom(i)/fyM); end end Force = mom/(L/1000); otherwise disp('bending should be specified as single or double or wall'); return; end switch lower(bending) case 'single' forcebilin = mombilin/(L/1000); case 'double' forcebilin = 2*mombilin/(L/1000); case 'wall' forcebilin = mombilin/(L/1000); end % Chai & Priestley Wall Buckling Models

Lo = Lp; mstar = rhostar * fy/fpc; xicrit = 0.5*(1 + 2.35*mstar - (5.53*mstar.^2 + 4.7*mstar).^0.5); xicontrol = min(xicrit,0.5); chaistr = beta*pi^2*(bbuckle/Lo)^2*xicontrol + 3*(fy/Es); priestleystr = beta*8*(bbuckle/Lo)^2*xicontrol; fydisp = interp1(steelstrain,displf,-fy/Es); % displacement for first yield eqdisp = max((Mn/fyM)*fydisp,fydisp); if chaistr >= max(-steelstrain) chaidisp = NaN; chaiCurv = NaN; chaiCuDu = NaN; chaiDispDu = NaN; else chaidisp = interp1(steelstrain,displf,-chaistr); chaiCurv = interp1(steelstrain,curv,-chaistr); chaiCuDu = chaiCurv/eqcurv; chaiDispDu = chaidisp/eqdisp; end if priestleystr >= max(-steelstrain)

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priestleydisp = NaN; priestleyCurv = NaN; priestleyCuDu = NaN; priestleyDispDu = NaN; else priestleydisp = interp1(steelstrain,displf,-priestleystr); priestleyCurv = interp1(steelstrain,curv,-priestleystr); priestleyCuDu = priestleyCurv/eqcurv; priestleyDispDu = priestleydisp/eqdisp; end Ieff = (Mn*1000/(Ec*(10^6)*eqcurv))*(10^12); Iratio = (Mn*1000/(Ec*(10^6)*eqcurv))*(10^12)/Ig;

% Shear deflection: G = 0.43*Ec; As = (5/6)*Agross; % Ieff = (Mn*1000/(Ec*(10^6)*eqcurv))*(10^12); beta = min(0.5+20*LongSteelRatio,1); switch lower(bending) case 'single' alpha = min(max(1,3-L/SecHeight),1.5); case 'double' alpha = min(max(1,3-L/(2*SecHeight)),1.5); case 'wall' alpha = min(max(1,3-L/SecHeight),1.5); end

Vc1 = 0.29*alpha*beta*0.8*(fpc^(1/2))*Agross/1000; deff = ConcCore(:,1); kscr = sum((0.25*TransvSteelRatioY*Es.*(ConcCore(:,2)/1000).*(deff)/1000)./(0.25+10*TransvSteelRatioY)*1000); switch lower(bending) case 'single' ksg = (G*As/L)/1000; kscr = (kscr/L); forcebilin = mombilin/(L/1000); case 'double' ksg = (G*As/(L/2))/1000; kscr = (kscr/(L/2)); forcebilin = 2*mombilin/(L/1000); case 'wall' ksg = (G*As/L)/1000; kscr = (kscr/L); forcebilin = mombilin/(L/1000); end kseff = ksg*(Ieff/Ig); aux = (Vc1/kseff)/1000; aux2 = 0; momaux = mom; displsh = zeros(1,length(curv)); for i=1:length(curv) if momaux(i)<=Mn && Force(i)=Vc1 displsh(i) = ((Force(i)-Vc1)/kscr)/1000+aux; end if momaux(i)>Mn momaux = 4*momaux; aux3=i-aux2; aux2 = aux2 + 1;

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displsh(i) = (displf(i)/displf(i-1))*displsh(i-1); end end displ = displsh + displf;

% bilinear approx: dy1 = interp1(curv,displ,fycurv); dy = (Mn/fyM)*dy1; du = displ(length(displ)); displbilin = [0 dy du]; Dduct = displ/dy; DisplDuct = max(Dduct); dy1f = interp1(curv,displf,fycurv); dyf = (Mn/fyM)*dy1f;

Ieq = (Mn/(eqcurv*Ec))/1000; % equivalent I for NL THA Bi = 1/(((mombilin(2))/(curvbilin(2)))/((mombilin(3)-mombilin(2))/(curvbilin(3)-curvbilin(2)))); % Bilinear factor

% Limit States: displdam = 0; displser = 0; displalt = 0; displult = 0; displmax = 0; Dductdam = 0; Dductser = 0; Dductalt = 0; Dductult = 0; Dductmax = 0; curvdam = 0; curvser = 0; curvalt = 0; curvult = 0; curvmax = 0; CuDudam = 0; CuDuser = 0; CuDualt = 0; CuDuult = 0; CuDumax = 0; coverstraindam = 0; coverstrainser = 0; coverstrainalt = 0; coverstrainult = 0; coverstrainmax = 0; steelstraindam = 0; steelstrainser = 0; steelstrainalt = 0; steelstrainult = 0; steelstrainmax = 0; momdam = 0; momser = 0; momalt = 0; momult = 0; mommax = 0; Forcedam = 0; Forceser = 0; Forcealt = 0; Forceult = 0; Forcemax = 0; if max(coverstrain) > ecser || max(abs(steelstrain)) > abs(esser) if max(coverstrain) > ecdam || max(abs(steelstrain)) > abs(esdam) displdamc = interp1(coverstrain,displ,ecdam); displdams = interp1(steelstrain,displ,esdam); displdam = min (displdamc,displdams); Dductdam = interp1(displ,Dduct,displdam); curvdam = interp1(displ,curv,displdam); CuDudam = interp1(displ,CuDu,displdam); coverstraindam = interp1(displ,coverstrain,displdam); steelstraindam = interp1(displ,steelstrain,displdam); momdam = interp1(displ,mom,displdam); Forcedam = interp1(displ,Force,displdam); end displserc = interp1(coverstrain,displ,ecser); displsers = interp1(steelstrain,displ,esser); displser = min (displserc,displsers); Dductser = interp1(displ,Dduct,displser); curvser = interp1(displ,curv,displser); CuDuser = interp1(displ,CuDu,displser); coverstrainser = interp1(displ,coverstrain,displser); steelstrainser = interp1(displ,steelstrain,displser); momser = interp1(displ,mom,displser); Forceser = interp1(displ,Force,displser); end switch alt_message case 0 ecu_alt = ecu; case 1

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ecu_alt = ecu_alt(end); end esu_alt = interp1(coverstrain,steelstrain,ecu_alt); displaltc = interp1(coverstrain,displ,ecu_alt); displalts = interp1(steelstrain,displ,-esu); displalt = min (displaltc,displalts); Dductalt = interp1(displ,Dduct,displalt); curvalt = interp1(displ,curv,displalt); CuDualt = interp1(displ,CuDu,displalt); coverstrainalt = interp1(displ,coverstrain,displalt); steelstrainalt = -interp1(displ,steelstrain,displalt); momalt = interp1(displ,mom,displalt); Forcealt = interp1(displ,Force,displalt); displultc = interp1(coverstrain,displ,ecu); displults = interp1(steelstrain,displ,-esu); displult = min (displultc,displults); if isnan(displult)==1 steelstrainult = -min(steelstrain); displult = max(displ); end Dductult = interp1(displ,Dduct,displult); curvult = interp1(displ,curv,displult); CuDuult = interp1(displ,CuDu,displult); coverstrainult = interp1(displ,coverstrain,displult); steelstrainult = interp1(displ,steelstrain,displult); momult = interp1(displ,mom,displult); Forceult = interp1(displ,Force,displult); if momloss==0 momloss = max(mom); end displmax = interp1(mom,displ,momloss); mommax = interp1(displ,mom,displmax); Dductmax = interp1(displ,Dduct,displmax); curvmax = interp1(displ,curv,displmax); CuDumax = interp1(displ,CuDu,displmax); coverstrainmax = interp1(displ,coverstrain,displmax); steelstrainmax = interp1(displ,steelstrain,displmax); Forcemax = interp1(displ,Force,displmax); outputlimit = [ coverstrainser steelstrainser momser Forceser curvser CuDuser displser Dductser coverstraindam steelstraindam momdam Forcedam curvdam CuDudam displdam Dductdam coverstrainalt steelstrainalt momalt Forcealt curvalt CuDualt displalt Dductalt coverstrainult steelstrainult momult Forceult curvult CuDuult displult Dductult coverstrainmax steelstrainmax mommax Forcemax curvmax CuDumax displmax Dductmax];

%======pointsdam = find(displ<=displdam); pointsser = find(displ<=displser);

%======output = [coverstrain;corestrain;ejen;curv;mom;-steelstrain;displ;Force]; outputbilin = [curvbilin;mombilin;displbilin;forcebilin];

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fid = fopen([name,'.xls'],'w'); fprintf(fid, 'Wall Section\n\n'); if priestleystr >= max(-steelstrain) fprintf(fid, 'Priestley limit displacement outside M-phi analysis\n'); end if chaistr >= max(-steelstrain) fprintf(fid, 'Chai limit displacement outside M-phi analysis\n'); end fprintf(fid, ' \tChai-Elayer\tPaulay-Priestley\tAlternate\tUltimate\t@MomMax\n'); fprintf(fid, 'steel str\t%1.4f\t%1.4f\t%1.4f\t%1.4f\t%1.4f\n',priestleystr,chaistr,steelstrainalt,esu,steelstrainmax); fprintf(fid, 'displ. (m)\t%1.4f\t%1.4f\t%1.4f\t%1.4f\t%1.4f\n', priestleydisp,chaidisp,displalt,displult,displmax); fprintf(fid, 'u.disp.\t%1.4f\t%1.4f\t%1.4f\t%1.4f\t%1.4f\n',priestleyDispDu,chaiDispDu,Dductalt,Dductult,Dductmax); fprintf(fid, 'u.curv.\t%1.4f\t%1.4f\t%1.4f\t%1.4f\t%1.4f\n\n',priestleyCuDu,chaiCuDu,CuDualt,CuDuult,CuDumax); switch lower(confined) case 'mclw' fprintf(fid,'lightweight concrete\n'); otherwise fprintf(fid,'normalweight concrete\n'); end for i = 1:size(ConcMat,1) fprintf(fid, '\nSection #: %3.1f\n',i); fprintf(fid, 'Width: %5.1f mm Height: %5.1f mm\n',ConcMat(i,2),ConcMat(i,1)); fprintf(fid, 'cover to longitudinal bars: %4.1f mm\n',clb(i)); fprintf(fid, 'diameter of transverse steel (x dir|y dir): %4.1f mm %4.1f mm\n ',Dh(i,1),Dh(i,2)); fprintf(fid, 'spacing of transverse steel: %4.1f mm\n',s(i)); fprintf(fid, '# legs transv. steel x_dir (confinement): %4.1f\n',ncx(i)); fprintf(fid, '# legs transv. steel y_dir (shear): %4.1f\n',ncy(i)); end fprintf(fid, '\nDist.Top\t# Long\tDiameter\n'); fprintf(fid, '[mm]\tBars\t[mm]\n'); fprintf(fid, '%4.1f\t%3.1f\t%4.2f\n',MLR'); fprintf(fid, '\naxial load: %8.2f kN\n',P/1000); fprintf(fid, 'concrete compressive strength: %3.2f MPa\n',fpc); fprintf(fid, 'long steel yielding stress: %4.2f MPa\n',fy); fprintf(fid, 'long steel max. stress: %4.2f MPa\n',max(fs)); fprintf(fid, 'transverse steel yielding stress: %4.2f MPa\n',fyh); fprintf(fid, 'Member Length: %5.1f mm\n',L); switch lower(bending) case 'single' fprintf(fid,'Single Bending\n'); case 'double' fprintf(fid,'Double Bending\n'); case 'wall' fprintf(fid,'Wall in Single Bending\n'); end switch lower(ductilitymode) case 'uniaxial' fprintf(fid,'Uniaxial Bending\n'); case 'biaxial' fprintf(fid,'Biaxial Bending\n'); end fprintf(fid, 'Longitudinal Steel Ratio: %1.3f\n',LongSteelRatio); fprintf(fid, 'Average Transverse Steel Ratio: %1.3f\n',TransvSteelRatioAverage); fprintf(fid, 'Axial Load Ratio: %1.3f\n\n',AxialRatio); fprintf(fid, 'Cover\tCore\tN.A\tCurvature\tMoment\tSteel\tDispl.\tForce\n'); fprintf(fid, 'Strain\tStrain\t[mm\t[1/m]]\t[kN-m]\tStrain\t[m]\t[kN]\n');

202

fprintf(fid, '%1.5f\t%1.5f\t%4.2f\t%1.5f\t%8.2f\t%1.5f\t%1.5f\t%8.2f\n',output); fprintf(fid, ' \n'); fprintf(fid, 'Bilinear Approximation:\n\n'); fprintf(fid, 'Curvature\tMoment\tDispl.\tForce\n'); fprintf(fid, '[1/m]\t[kN-m]\t[m]\t[kN]\n'); fprintf(fid,'%1.5f\t%8.2f\t%1.5f\t%8.2f\n',outputbilin); fprintf(fid, ' \n'); switch message case 1 fprintf(fid,' *** concrete strain exceeds maximum ***\n'); case 2 fprintf(fid,' *** steel strain exceeds maximum ***\n'); case 3 fprintf(fid,' *** number of iteration exceeds maximum ***\n'); case 4 fprintf(fid,' *** excessive lost of strength ***\n'); end switch alt_message case 0 fprintf(fid,' *** concrete core strain is controlled by the external section ***\n'); case 1 fprintf(fid,' *** concrete core strain is controlled by a non-external section ***\n'); fprintf(fid,'Partial Limit at:%1.5f\n',ecu_alt); end fprintf(fid, '\n\nMoment for First Yielding: %8.2f kN-m\n',fyM); fprintf(fid, 'Curvature for First Yielding: %1.5f 1/m\n',fycurv); fprintf(fid, 'Potential Section Nominal Moment: %8.2f kN-m\n',Mn); fprintf(fid, 'Equivalent Curvature: %1.5f 1/m\n',eqcurv); fprintf(fid, 'Potential Section Curvature Ductility: %3.2f\n',SectionCurvatureDuctility); fprintf(fid, 'Potential Displacement Ductility: %3.2f\n',DisplDuct); fprintf(fid, ' \n'); fprintf(fid, ' \n'); if bucritMK == 1 bucklDd = interp1(CuDu,Dduct,failCuDuMK); bucklcurv = interp1(CuDu,curv,failCuDuMK); bucklmom = interp1(CuDu,mom,failCuDuMK); fprintf(fid,'Moyer - Kowalsky buckling model:\n'); fprintf(fid, '\nCurvature Ductility for Buckling: %8.2f\n',failCuDuMK); fprintf(fid, 'Curvature at Buckling: %3.5f m\n',bucklcurv); fprintf(fid, 'Displacement Ductility at Buckling: %8.2f\n',bucklDd); fprintf(fid, 'Displacement at Buckling: %3.5f m\n',buckldispl); fprintf(fid, 'Force for Buckling: %8.2f kN\n',bucklforce); fprintf(fid, 'Moment for Buckling: %8.2f kN\n',bucklmom); end fprintf(fid, '\n'); if bucritBE == 1 bucklDdBE = interp1(CuDu,Dduct,failCuDuBE); bucklcurvBE = interp1(CuDu,curv,failCuDuBE); bucklmomBE = interp1(CuDu,mom,failCuDuBE); fprintf(fid,'Berry - Eberhard buckling model:\n'); fprintf(fid, '\nCurvature Ductility for Buckling: %8.2f\n',failCuDuBE); fprintf(fid, 'Curvature at Buckling: %3.5f m\n',bucklcurvBE); fprintf(fid, 'Displacement Ductility at Buckling: %8.2f\n',bucklDdBE); fprintf(fid, 'Displacement at Buckling: %3.5f m\n',buckldisplBE); fprintf(fid, 'Force for Buckling: %8.2f kN\n',bucklforceBE); fprintf(fid, 'Moment for Buckling: %8.2f kN\n',bucklmomBE); end fprintf(fid, '\n');

203

fprintf(fid, '\n'); fprintf(fid, '== Potential Deformation Limit States (serviceability/damage control/alternate ultimate/ultimate/max) ==\n\n'); fprintf(fid, 'Cover\tSteel\tMoment\tForce\tCurvature\tCurvature\tDisplacement\tDisplacement\n'); fprintf(fid, 'Strain\tStrain\t[kN-m]\t[kN]\t[1/m]\tDuctility\t[m]\tDuctility\n'); fprintf(fid, '%1.5f\t%1.5f\t%8.2f\t%8.2f\t%1.5f\t%3.2f\t%2.5f\t%3.2f\n',outputlimit'); fprintf(fid, '\nDeformation Limit States Citeria :\n'); fprintf(fid, ' serviceability concrete strain: %1.4f\n',ecser); fprintf(fid, ' serviceability steel strain: %1.4f\n',esser); fprintf(fid, ' damage control concrete strain: %1.4f\n',ecdam); fprintf(fid, ' damage control steel strain: %1.4f\n',esdam); fprintf(fid, ' alternate web control concrete strain: %1.4f\n',ecu_alt); fprintf(fid, ' alternate web control steel strain: %1.4f\n',-esu_alt); fprintf(fid, ' ultimate control concrete strain: %1.4f\n',ecu); fprintf(fid, ' ultimate control steel strain: %1.4f\n',-esu); switch lower(confined) case 'mc' fprintf(fid,'\nOriginal Mander Model Ultimate Concrete Strain: %1.4f\n',ecumander); end fprintf(fid, '\nfor non-linear THA:\n\n'); fprintf(fid, 'E: %10.2f Pa\n',Ec*(10^6)); fprintf(fid, 'G: %10.2f Pa\n', G*(10^6)); fprintf(fid, 'A: %10.4f m2\n', Agross/(10^6)); fprintf(fid, 'I: %10.6f m4\n', Ieq); fprintf(fid, 'I-ratio: %10.6f\n', Iratio); fprintf(fid, 'Bi-Factor: %1.3f\n', Bi); fprintf(fid, 'Hinge Length: %1.3f m\n', Lp/1000); fprintf(fid, 'Tension Yield: %10.2f N\n', PTY); fprintf(fid, 'Compression Yield: %10.2f N\n', PCY); fprintf(fid, 'Moment Yield: %10.2f N-m\n\n', Mn*1000); fclose(fid);

204

Appendix B. Parametric Study Reference Data

Table 14 List of wall models and parameters for Phase I parametric study

Typical Wall tw Lw Hw fy fc' ρl Ρt ALR Lp Factor UBC ACI NZS PPBM CEBM

(N/A) (#) (m) (m) (m) (MPa) (MPa) (%) (%) (N/A) (N/A) Met? Met? Met? >εult? >εult? 1 0.1 4 15 450 30 2.00% 0.75% 0.1 1 No No No No No 2 0.2 4 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No * 3 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 4 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 5 0.5 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes Yes Yes 6 0.3 2 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 7 0.3 3 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 8 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 9 0.3 6 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 10 0.3 10 15 450 30 2.00% 0.75% 0.1 1 No No Yes No No 11 0.3 4 6 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 12 0.3 4 10 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 13 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 14 0.3 4 20 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 15 0.3 4 30 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 16 0.3 4 15 350 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 17 0.3 4 15 400 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 18 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 19 0.3 4 15 500 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 20 0.3 4 15 550 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 21 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 22 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 23 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 24 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 25 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 26 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes Yes Yes 27 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 28 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 29 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 30 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 31 0.3 4 15 450 30 2.00% 0.25% 0.1 1 Yes Yes Yes No No

205

Typical Wall tw Lw Hw fy fc' ρl Ρt ALR Lp Factor UBC ACI NZS PPBM CEBM

(N/A) (#) (m) (m) (m) (MPa) (MPa) (%) (%) (N/A) (N/A) Met? Met? Met? >εult? >εult? 32 0.3 4 15 450 30 2.00% 0.50% 0.1 1 Yes Yes Yes No No * 33 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 34 0.3 4 15 450 30 2.00% 1.00% 0.1 1 Yes Yes Yes No No 35 0.3 4 15 450 30 2.00% 1.25% 0.1 1 Yes Yes Yes No No 36 0.3 4 15 450 30 2.00% 0.75% -0.05 1 Yes Yes Yes No No 37 0.3 4 15 450 30 2.00% 0.75% 0 1 Yes Yes Yes No No * 38 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 39 0.3 4 15 450 30 2.00% 0.75% 0.2 1 Yes Yes Yes No No 40 0.3 4 15 450 30 2.00% 0.75% 0.3 1 Yes Yes Yes No No 41 0.3 4 15 450 30 2.00% 0.75% 0.1 0.5 Yes Yes Yes Yes Yes 42 0.3 4 15 450 30 2.00% 0.75% 0.1 0.75 Yes Yes Yes No Yes * 43 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 44 0.3 4 15 450 30 2.00% 0.75% 0.1 1.25 Yes Yes Yes No No 45 0.3 4 15 450 30 2.00% 0.75% 0.1 1.5 Yes Yes Yes No No * : Indicates the "typical" model, which was duplicated for ease of data gathering.

206

Table 15 List of wall models and parameters for Phase II parametric study

Typical Wall tw Lw Hw fy fc' ρl Ρt ALR Lp Factor UBC ACI NZS PPBM CEBM

(N/A) (#) (m) (m) (m) (MPa) (MPa) (%) (%) (N/A) (N/A) Met? Met? Met? >εult? >εult? 1 0.1 2 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 2 0.1 3 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No * 3 0.1 4 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 4 0.1 6 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 5 0.1 10 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 6 0.15 2 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 7 0.15 3 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No * 8 0.15 4 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 9 0.15 6 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 10 0.15 10 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 11 0.2 2 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 12 0.2 3 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No * 13 0.2 4 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 14 0.2 6 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 15 0.2 10 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 16 0.25 2 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 17 0.25 3 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No * 18 0.25 4 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 19 0.25 6 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 20 0.25 10 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 21 0.3 2 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No Yes 22 0.3 3 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No * 23 0.3 4 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 24 0.3 6 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 25 0.3 10 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 26 0.35 2 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes Yes Yes 27 0.35 3 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No Yes * 28 0.35 4 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No Yes 29 0.35 6 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 30 0.35 10 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 31 0.4 2 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 32 0.4 3 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No * 33 0.4 4 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No

207

Typical Wall tw Lw Hw fy fc' ρl Ρt ALR Lp Factor UBC ACI NZS PPBM CEBM

(N/A) (#) (m) (m) (m) (MPa) (MPa) (%) (%) (N/A) (N/A) Met? Met? Met? >εult? >εult? 34 0.4 6 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 35 0.4 10 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 36 0.1 4 6 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 37 0.1 4 10 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No * 38 0.1 4 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 39 0.1 4 20 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 40 0.1 4 30 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 41 0.15 4 6 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 42 0.15 4 10 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No * 43 0.15 4 15 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 44 0.15 4 20 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 45 0.15 4 30 450 30 2.00% 0.75% 0.1 1 No Yes Yes No No 46 0.2 4 6 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 47 0.2 4 10 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 48 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 49 0.2 4 20 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 50 0.2 4 30 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 51 0.25 4 6 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes Yes Yes 52 0.25 4 10 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes * 53 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 54 0.25 4 20 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 55 0.25 4 30 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 56 0.3 4 6 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes Yes Yes 57 0.3 4 10 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes Yes Yes * 58 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 59 0.3 4 20 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 60 0.3 4 30 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 61 0.35 4 6 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 62 0.35 4 10 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 63 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 64 0.35 4 20 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 65 0.35 4 30 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 66 0.4 4 6 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 67 0.4 4 10 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No

208

Typical Wall tw Lw Hw fy fc' ρl Ρt ALR Lp Factor UBC ACI NZS PPBM CEBM

(N/A) (#) (m) (m) (m) (MPa) (MPa) (%) (%) (N/A) (N/A) Met? Met? Met? >εult? >εult? * 68 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 69 0.4 4 20 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 70 0.4 4 30 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 71 0.1 4 15 250 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 72 0.1 4 15 350 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 73 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 74 0.1 4 15 550 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 75 0.1 4 15 650 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 76 0.15 4 15 250 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 77 0.15 4 15 350 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 78 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 79 0.15 4 15 550 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 80 0.15 4 15 650 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 81 0.2 4 15 250 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 82 0.2 4 15 350 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 83 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 84 0.2 4 15 550 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 85 0.2 4 15 650 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 86 0.25 4 15 250 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 87 0.25 4 15 350 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes * 88 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 89 0.25 4 15 550 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 90 0.25 4 15 650 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 91 0.3 4 15 250 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 92 0.3 4 15 350 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 93 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 94 0.3 4 15 550 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 95 0.3 4 15 650 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 96 0.35 4 15 250 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 97 0.35 4 15 350 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 98 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 99 0.35 4 15 550 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 100 0.35 4 15 650 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 101 0.4 4 15 250 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No

209

Typical Wall tw Lw Hw fy fc' ρl Ρt ALR Lp Factor UBC ACI NZS PPBM CEBM

(N/A) (#) (m) (m) (m) (MPa) (MPa) (%) (%) (N/A) (N/A) Met? Met? Met? >εult? >εult? 102 0.4 4 15 350 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 103 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 104 0.4 4 15 550 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 105 0.4 4 15 650 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 106 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 107 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 108 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 109 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 110 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 111 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 112 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 113 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 114 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 115 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 116 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 117 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 118 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 119 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 120 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 121 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 122 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 123 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 124 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 125 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 126 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 127 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 128 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 129 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 130 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 131 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 132 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 133 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 134 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 135 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No

210

Typical Wall tw Lw Hw fy fc' ρl Ρt ALR Lp Factor UBC ACI NZS PPBM CEBM

(N/A) (#) (m) (m) (m) (MPa) (MPa) (%) (%) (N/A) (N/A) Met? Met? Met? >εult? >εult? 136 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes Yes Yes 137 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 138 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 139 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 140 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 141 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes Yes Yes 142 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes * 143 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 144 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 145 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 146 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes Yes Yes 147 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes Yes Yes * 148 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 149 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 150 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 151 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 152 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 153 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 154 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 155 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 156 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 157 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 158 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 159 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 160 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 161 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 162 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 163 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 164 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 165 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 166 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 167 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 168 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 169 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No

211

Typical Wall tw Lw Hw fy fc' ρl Ρt ALR Lp Factor UBC ACI NZS PPBM CEBM

(N/A) (#) (m) (m) (m) (MPa) (MPa) (%) (%) (N/A) (N/A) Met? Met? Met? >εult? >εult? 170 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 171 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 172 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No * 173 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 174 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 175 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 176 0.1 4 15 450 30 2.00% 0.25% 0.1 1 Yes Yes Yes No Yes 177 0.1 4 15 450 30 2.00% 0.50% 0.1 1 Yes Yes Yes No Yes * 178 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 179 0.1 4 15 450 30 2.00% 1.00% 0.1 1 Yes Yes Yes No Yes 180 0.1 4 15 450 30 2.00% 1.25% 0.1 1 Yes Yes Yes No Yes 181 0.15 4 15 450 30 2.00% 0.25% 0.1 1 Yes Yes Yes No No 182 0.15 4 15 450 30 2.00% 0.50% 0.1 1 Yes Yes Yes No No * 183 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 184 0.15 4 15 450 30 2.00% 1.00% 0.1 1 Yes Yes Yes No No 185 0.15 4 15 450 30 2.00% 1.25% 0.1 1 Yes Yes Yes No No 186 0.2 4 15 450 30 2.00% 0.25% 0.1 1 Yes Yes Yes No No 187 0.2 4 15 450 30 2.00% 0.50% 0.1 1 Yes Yes Yes No No * 188 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 189 0.2 4 15 450 30 2.00% 1.00% 0.1 1 Yes Yes Yes No No 190 0.2 4 15 450 30 2.00% 1.25% 0.1 1 Yes Yes Yes No No 191 0.25 4 15 450 30 2.00% 0.25% 0.1 1 Yes Yes Yes No No 192 0.25 4 15 450 30 2.00% 0.50% 0.1 1 Yes Yes Yes No No * 193 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 194 0.25 4 15 450 30 2.00% 1.00% 0.1 1 Yes Yes Yes No No 195 0.25 4 15 450 30 2.00% 1.25% 0.1 1 Yes Yes Yes No No 196 0.3 4 15 450 30 2.00% 0.25% 0.1 1 Yes Yes Yes No No 197 0.3 4 15 450 30 2.00% 0.50% 0.1 1 Yes Yes Yes No No * 198 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 199 0.3 4 15 450 30 2.00% 1.00% 0.1 1 Yes Yes Yes No No 200 0.3 4 15 450 30 2.00% 1.25% 0.1 1 Yes Yes Yes No No 201 0.35 4 15 450 30 2.00% 0.25% 0.1 1 Yes Yes Yes No No 202 0.35 4 15 450 30 2.00% 0.50% 0.1 1 Yes Yes Yes No No * 203 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No

212

Typical Wall tw Lw Hw fy fc' ρl Ρt ALR Lp Factor UBC ACI NZS PPBM CEBM

(N/A) (#) (m) (m) (m) (MPa) (MPa) (%) (%) (N/A) (N/A) Met? Met? Met? >εult? >εult? 204 0.35 4 15 450 30 2.00% 1.00% 0.1 1 Yes Yes Yes No No 205 0.35 4 15 450 30 2.00% 1.25% 0.1 1 Yes Yes Yes No No 206 0.4 4 15 450 30 2.00% 0.25% 0.1 1 Yes Yes Yes No Yes 207 0.4 4 15 450 30 2.00% 0.50% 0.1 1 Yes Yes Yes No Yes * 208 0.4 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 209 0.4 4 15 450 30 2.00% 1.00% 0.1 1 Yes Yes Yes No Yes 210 0.4 4 15 450 30 2.00% 1.25% 0.1 1 Yes Yes Yes No Yes 211 0.1 4 15 450 30 2.00% 0.75% -0.1 1 Yes Yes Yes No No 212 0.1 4 15 450 30 2.00% 0.75% 0 1 Yes Yes Yes No No * 213 0.1 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 214 0.1 4 15 450 30 2.00% 0.75% 0.2 1 Yes Yes Yes No No 215 0.1 4 15 450 30 2.00% 0.75% 0.3 1 Yes Yes Yes No No 216 0.15 4 15 450 30 2.00% 0.75% -0.1 1 Yes Yes Yes No Yes 217 0.15 4 15 450 30 2.00% 0.75% 0 1 Yes Yes Yes No No * 218 0.15 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 219 0.15 4 15 450 30 2.00% 0.75% 0.2 1 Yes Yes Yes No No 220 0.15 4 15 450 30 2.00% 0.75% 0.3 1 Yes Yes Yes No No 221 0.2 4 15 450 30 2.00% 0.75% -0.1 1 Yes Yes Yes Yes Yes 222 0.2 4 15 450 30 2.00% 0.75% 0 1 Yes Yes Yes No No * 223 0.2 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 224 0.2 4 15 450 30 2.00% 0.75% 0.2 1 Yes Yes Yes No No 225 0.2 4 15 450 30 2.00% 0.75% 0.3 1 Yes Yes Yes No No 226 0.25 4 15 450 30 2.00% 0.75% -0.1 1 Yes Yes Yes Yes Yes 227 0.25 4 15 450 30 2.00% 0.75% 0 1 Yes Yes Yes No Yes * 228 0.25 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 229 0.25 4 15 450 30 2.00% 0.75% 0.2 1 Yes Yes Yes No No 230 0.25 4 15 450 30 2.00% 0.75% 0.3 1 Yes Yes Yes No No 231 0.3 4 15 450 30 2.00% 0.75% -0.1 1 Yes Yes Yes Yes Yes 232 0.3 4 15 450 30 2.00% 0.75% 0 1 Yes Yes Yes Yes Yes * 233 0.3 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No No 234 0.3 4 15 450 30 2.00% 0.75% 0.2 1 Yes Yes Yes No No 235 0.3 4 15 450 30 2.00% 0.75% 0.3 1 Yes Yes Yes No No 236 0.35 4 15 450 30 2.00% 0.75% -0.1 1 Yes Yes Yes Yes Yes 237 0.35 4 15 450 30 2.00% 0.75% 0 1 Yes Yes Yes Yes Yes

213

Typical Wall tw Lw Hw fy fc' ρl Ρt ALR Lp Factor UBC ACI NZS PPBM CEBM

(N/A) (#) (m) (m) (m) (MPa) (MPa) (%) (%) (N/A) (N/A) Met? Met? Met? >εult? >εult? * 238 0.35 4 15 450 30 2.00% 0.75% 0.1 1 Yes Yes Yes No Yes 239 0.35 4 15 450 30 2.00% 0.75% 0.2 1 Yes Yes Yes No No 240 0.35 4 15 450 30 2.00% 0.75% 0.3 1 Yes Yes Yes No No * : Indicates the "typical" model, which was duplicated for ease of data gathering.

214

Appendix C. Prior Experimental Test Reference Data

Table 16 Prior Experimental Wall Tests - Loading and Geometry

Wall Geometry

Total Boundary

Pmax Hw Lw tw Lb tb Specimen # Name Loading Shape (kN) (mm) (mm) (mm) (mm) (mm) 1 Goodsir F1 Cyclic Rectangle 222 2400 1500 100 250 100 2 Goodsir F2 Cyclic Rectangle 145 2400 1500 100 250 100 3 Goodsir F3T Cyclic Flanged T 143 2400 1300 100 700 100 4 Goodsir F4 Cyclic Rectangle 152 2400 1500 100 250 100 5 He MAST1 Cyclic Flanged T 360 3200 1168 143 2640 143 6 He MAST2 Cyclic Flanged T 360 3200 1168 143 2640 143 7 He MAST3 Cyclic Flanged T 360 3200 1168 143 5080 143 8 He MAST4 Cyclic Flanged T 360 3200 1168 143 2640 143 9 He MAST5 Dynamic Flanged T 360 3200 1168 143 2640 143 10 He MAST6 Dynamic Flanged T 360 3200 1168 143 2640 143 11 He MAST7 Dynamic Flanged T 360 3200 1168 143 2640 143 12 He MAST8 Dynamic Flanged T 360 3200 1168 143 2640 143 13 He MAST9 Dynamic Flanged T 360 3200 1168 143 2640 143 14 Ji SW1 Cyclic Rectangle 220 3000 1000 60 120 60 15 Ji SW2 Cyclic Barbell 180 3000 1000 60 120 120 16 Ji SW3 Cyclic Flanged 115 3000 1000 60 420 60 17 Jiang DSW-T Cyclic Dbl. Rectangle 200 2900 495 75 150 75 18 Jiang SSW-T Cyclic Rectangle 200 2900 1000 75 150 75 19 Lefas SW-21 Monotonic Rectangle 0 1300 650 65 140 65 20 Lefas SW-22 Monotonic Rectangle 182 1300 650 65 140 65 21 Lefas SW-23 Monotonic Rectangle 343 1300 650 65 140 65 22 Lefas SW-24 Monotonic Rectangle 0 1300 650 65 140 65 23 Lefas SW-25 Monotonic Rectangle 325 1300 650 65 140 65 24 Lefas SW-26 Monotonic Rectangle 0 1300 650 65 140 65 25 Lefas SW-30 Monotonic Rectangle 0 1300 650 65 140 65 26 Lefas SW-31 Cyclic Rectangle 0 1300 650 65 140 65 27 Lefas SW-32 Cyclic Rectangle 0 1300 650 65 140 65 28 Lefas SW-33 Cyclic Rectangle 0 1300 650 65 140 65 29 Oesterle B1 Cyclic Barbell 0 4572 1905 102 305 305

215

Wall Geometry

Total Boundary

Pmax Hw Lw tw Lb tb Specimen # Name Loading Shape (kN) (mm) (mm) (mm) (mm) (mm) 30 Oesterle B2 Cyclic Barbell 0 4572 1905 102 305 305 31 Oesterle B3 Cyclic Barbell 0 4572 1905 102 305 305 32 Oesterle B4 Cyclic Barbell 0 4572 1905 102 305 305 33 Oesterle B5 Monotonic Barbell 0 4572 1905 102 305 305 34 Oesterle B6 Cyclic Barbell 1112 4572 1905 102 305 305 35 Oesterle B7 Cyclic Barbell 1426 4572 1905 102 305 305 36 Oesterle B8 Cyclic Barbell 1426 4572 1905 102 305 305 37 Oesterle B9 Cyclic Barbell 1426 4572 1905 102 305 305 38 Oesterle B10 Cyclic Barbell 1426 4572 1905 102 305 305 39 Oesterle F1 Cyclic Flanged 0 4572 1905 102 915 102 40 Oesterle F2 Cyclic Flanged 1256 4572 1905 102 915 102 41 Oesterle R1 Cyclic Rectangle 0 4572 1905 102 165 102 42 Oesterle R2 Cyclic Rectangle 0 4572 1905 102 165 102 43 Zhang SW1-1 Cyclic Rectangle 2221 2000 1000 125 190 125 44 Zhang SW1-2 Cyclic Rectangle 2221 2000 1000 125 190 125 45 Zhang SW1-3 Cyclic Rectangle 2221 2000 1000 125 190 125 46 Zhang SW1-4 Cyclic Rectangle 2221 2000 1000 125 190 125 47 Zhang SW2-2 Cyclic Rectangle 3305 1500 1000 125 190 125 48 Zhang SW2-3 Cyclic Rectangle 3305 2000 1000 125 190 125 49 Zhang SW3-1 Cyclic Rectangle 2221 2000 1000 125 190 125 50 Zhang SW3-2 Cyclic Rectangle 3305 2000 1000 125 190 125 51 Zhang SW4-1 Cyclic Rectangle 3336 2000 1000 125 190 125 52 Zhang SW4-2 Cyclic Rectangle 3305 2000 1000 125 190 125 53 Zhang SW4-3 Cyclic Rectangle 3305 2000 1000 125 190 125 54 Zhang SW5-1 Cyclic Rectangle 3305 2000 1000 125 190 125 55 Zhang SW5-2 Cyclic Rectangle 3305 2000 1000 125 190 125 56 Zhang SW5-3 Cyclic Rectangle 3305 2000 1000 125 190 125 57 Zhang SW6-1 Cyclic Rectangle 3305 2000 1000 125 190 125 58 Zhang SW6-2 Cyclic Rectangle 3305 2000 1000 125 190 125 59 Zhang SW6-3 Cyclic Rectangle 3305 2000 1000 125 190 125 60 Zhou SW1 Cyclic Rectangle 400 2250 900 75 180 75 61 Zhou SW2 Cyclic Rectangle 0 2250 900 75 180 75

216

Wall Geometry

Total Boundary

Pmax Hw Lw tw Lb tb Specimen # Name Loading Shape (kN) (mm) (mm) (mm) (mm) (mm) 62 Zhou SW3 Cyclic Rectangle 0 2250 900 75 180 75 63 Zhou SW4 Cyclic Rectangle 400 2250 900 75 180 75 *Lb is the length of the boundary element. For rectangle and barbell walls it is along the wall’s length, for flanged t-walls or double flanged walls it is the length of the flange. Similarly, tb is the thickness of the boundary element transversely for rectangle and barbell walls, but for flanged t-walls and double walls it is the flange thickness.

217

Table 17 Prior Experimental Wall Tests - Material Properties

Material Strengths

fc' fy,wl fy,wh fu,wl Specimen # Name (MPa) (MPa) (MPa) (MPa) 1 Goodsir F1 28.6 450 360 692 2 Goodsir F2 25.3 450 360 692 3 Goodsir F3T 33.8 400 380 590 4 Goodsir F4 36.5 345 335 610 5 He MAST1 19.9 490 523 687 6 He MAST2 19.9 523 523 808 7 He MAST3 15.9 523 523 808 8 He MAST4 15.9 490 523 687 9 He MAST5 14.7 475 523 740 10 He MAST6 12.0 431 523 671 11 He MAST7 12.0 475 523 740 12 He MAST8 13.9 475 523 740 13 He MAST9 13.9 475 523 740 14 Ji SW1 13.5 453 453 501 15 Ji SW2 13.5 453 453 501 16 Ji SW3 13.5 453 453 501 17 Jiang DSW-T 24.9 289 289 465 18 Jiang SSW-T 24.1 289 289 465 19 Lefas SW-21 42.8 470 520 565 20 Lefas SW-22 50.6 470 520 565 21 Lefas SW-23 47.8 470 520 565 22 Lefas SW-24 48.3 470 520 565 23 Lefas SW-25 45.0 470 520 565 24 Lefas SW-26 30.1 470 520 565 25 Lefas SW-30 30.1 470 520 565 26 Lefas SW-31 35.2 470 520 565 27 Lefas SW-32 53.6 470 520 565 28 Lefas SW-33 49.2 470 520 565 29 Oesterle B1 53.0 521 521 695 30 Oesterle B2 53.6 532 532 701

218

Material Strengths

fc' fy,wl fy,wh fu,wl Specimen # Name (MPa) (MPa) (MPa) (MPa) 31 Oesterle B3 47.3 479 479 656 32 Oesterle B4 45 450 505 707 33 Oesterle B5 45.3 502 502 672 34 Oesterle B6 21.8 512 512 676 35 Oesterle B7 49.3 490 490 696 36 Oesterle B8 42.0 454 482 616 37 Oesterle B9 44.1 461 461 613 38 Oesterle B10 45.6 475 475 633 39 Oesterle F1 38.4 525 525 705 40 Oesterle F2 45.6 464 464 607 41 Oesterle R1 44.7 522 522 700 42 Oesterle R2 46.4 535 535 691 43 Zhang SW1-1 20.7 352 392 493 44 Zhang SW1-2 20.7 352 392 493 45 Zhang SW1-3 20.7 352 392 493 46 Zhang SW1-4 20.7 352 392 493 47 Zhang SW2-2 30.8 352 392 493 48 Zhang SW2-3 30.8 352 392 493 49 Zhang SW3-1 20.7 352 392 493 50 Zhang SW3-2 30.8 352 392 493 51 Zhang SW4-1 30.8 342 392 447 52 Zhang SW4-2 30.8 352 392 493 53 Zhang SW4-3 30.8 352 392 493 54 Zhang SW5-1 30.8 352 392 493 55 Zhang SW5-2 30.8 352 392 493 56 Zhang SW5-3 30.8 352 392 493 57 Zhang SW6-1 30.8 352 348 409 58 Zhang SW6-2 30.8 352 392 493 59 Zhang SW6-3 30.8 352 392 493 60 Zhou SW1 30.0 345 345 465 61 Zhou SW2 30.0 345 345 465 62 Zhou SW3 30.0 345 345 465 63 Zhou SW4 30.0 345 345 465

219

Table 18 Prior Experimental Wall Tests - Reinforcing Details

Longitudinal Rebar Transverse Rebar Cover Boundary Web Boundary Web

cbl dbl sbl ρbl dwl swl dbh sbh dwh swh Specimen # Name (mm) (mm) (mm) (%) (mm) (mm) (mm) (mm) (mm) (mm) 1 Goodsir F1 12 12 48 4.71% 6 175 5 40 6.18 80 2 Goodsir F2 12 12 48 4.71% 6 175 5 72 6.18 80 3 Goodsir F3T 12 10 40 3.93% 6 150 5 60 6.18 100 4 Goodsir F4 12 12 48 4.71% 6 175 5 40 6.18 80 5 He MAST1 62 19 400 0.55% 19 400 13 203 - - 6 He MAST2 66 12 400 0.25% 12 400 13 203 - - 7 He MAST3 66 12 400 0.25% 12 400 13 203 - - 8 He MAST4 62 19 400 0.55% 19 400 13 203 - - 9 He MAST5 62 19 400 0.55% 19 400 13 203 - - 10 He MAST6 66 12 400 0.25% 12 400 13 203 - - 11 He MAST7 62 19 400 0.55% 19 400 13 203 - - 12 He MAST8 62 19 400 0.55% 19 400 13 203 - - 13 He MAST9 62 19 400 0.55% 19 400 13 203 - - 14 Ji SW1 8 12 40 9.40% 4 100 4 30 4 100 15 Ji SW2 8 12 40 4.70% 4 100 6 30 4 100 16 Ji SW3 8 12 40 7.10% 4 100 4 30 4 100 17 Jiang DSW-T 8 6 35 2.30% 6 90 6 90 6.24 90 18 Jiang SSW-T 8 6 35 2.30% 6 90 6 90 6.24 90 19 Lefas SW-21 8 8 62 3.30% 8 62 4 115 6.5 115 20 Lefas SW-22 8 8 62 3.30% 8 62 4 115 6.5 115 21 Lefas SW-23 8 8 62 3.30% 8 62 4 115 6.5 115 22 Lefas SW-24 8 8 62 3.30% 8 62 4 115 6.5 115 23 Lefas SW-25 8 8 62 3.30% 8 62 4 115 6.5 115 24 Lefas SW-26 8 8 62 3.30% 8 62 4 250 6.5 115 25 Lefas SW-30 8 8 62 3.30% 8 100 4 130 6.5 260 26 Lefas SW-31 8 8 62 3.30% 8 100 4 130 6.5 260 27 Lefas SW-32 8 8 62 3.30% 8 100 4 130 6.5 260 28 Lefas SW-33 8 8 62 3.30% 8 100 4 130 6.5 260 29 Oesterle B1 16 13 127 1.11% 6 229 5 203 6.35 203

220

Longitudinal Rebar Transverse Rebar Cover Boundary Web Boundary Web

cbl dbl sbl ρbl dwl swl dbh sbh dwh swh Specimen # Name (mm) (mm) (mm) (%) (mm) (mm) (mm) (mm) (mm) (mm) 30 Oesterle B2 16 19 76 3.67% 6 229 5 203 6.35 102 31 Oesterle B3 16 13 127 1.11% 6 229 6 34 6 203 32 Oesterle B4 16 13 127 1.11% 6 127 6 34 6 203 33 Oesterle B5 16 19 76 3.67% 6 229 6 34 6 102 34 Oesterle B6 16 19 76 3.67% 6 229 5 34 6 102 35 Oesterle B7 16 19 76 3.67% 6 229 6 34 6 102 36 Oesterle B8 16 19 76 3.67% 6 229 6 34 10 102 37 Oesterle B9 16 19 76 3.67% 6 229 6 34 6 102 38 Oesterle B10 16 16 76 1.97% 6 229 6 34 6 102 39 Oesterle F1 16 13 64 3.89% 6 229 6 89 6 89 40 Oesterle F2 16 13 64 4.35% 6 229 6 34 6 102 41 Oesterle R1 13 10 140 1.84% 6 229 5 102 6 203 42 Oesterle R2 13 13 70 5.35% 6 229 6 34 6 203 43 Zhang SW1-1 13 10 80 1.57% 10 110 6 80 6 150 44 Zhang SW1-2 13 10 80 1.57% 10 110 6 80 6 150 45 Zhang SW1-3 13 10 80 1.57% 10 110 6 80 6 150 46 Zhang SW1-4 13 10 80 1.57% 10 110 6 80 6 150 47 Zhang SW2-2 13 10 80 1.57% 10 110 6 80 6 150 48 Zhang SW2-3 13 10 80 1.57% 10 110 6 80 6 150 49 Zhang SW3-1 13 10 80 1.57% 10 110 6 80 6 150 50 Zhang SW3-2 13 10 80 1.57% 10 110 6 80 6 150 51 Zhang SW4-1 13 8 80 1.01% 8 110 6 80 6 150 52 Zhang SW4-2 13 10 80 1.57% 10 110 6 80 6 150 53 Zhang SW4-3 13 10 80 1.57% 10 110 6 80 6 150 54 Zhang SW5-1 13 10 80 1.57% 10 110 6 80 6 150 55 Zhang SW5-2 13 10 80 1.57% 10 110 6 80 6 150 56 Zhang SW5-3 13 10 80 1.57% 10 110 6 80 6 150 57 Zhang SW6-1 13 10 80 1.57% 10 110 4 80 6 150 58 Zhang SW6-2 13 10 80 1.57% 10 110 6 80 6 150 59 Zhang SW6-3 13 10 80 1.57% 10 110 6 60 6 150 60 Zhou SW1 8 8 45 1.66% 6 90 4 30 6.18 90 61 Zhou SW2 8 8 45 1.66% 6 90 4 30 6.18 90

221

Longitudinal Rebar Transverse Rebar Cover Boundary Web Boundary Web

cbl dbl sbl ρbl dwl swl dbh sbh dwh swh Specimen # Name (mm) (mm) (mm) (%) (mm) (mm) (mm) (mm) (mm) (mm) 62 Zhou SW3 8 8 45 1.66% 6 90 4 30 6.18 90 63 Zhou SW4 8 8 45 1.66% 6 90 4 30 6.18 90

222

Table 19 Prior Experimental Wall Tests - Displacements and Normalized Results

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 1 Goodsir F1 1 17.0 No 40.1 54.6 0.42 0.31 1 Goodsir F1 2 17.0 No 40.1 54.6 0.42 0.31 1 Goodsir F1 3 37.0 No 40.1 54.6 0.92 0.68 1 Goodsir F1 4 37.0 No 40.1 54.6 0.92 0.68 1 Goodsir F1 5 65.0 Yes 40.1 54.6 1.62 1.19 2 Goodsir F2 1 8.0 No 37.7 51.9 0.21 0.15 2 Goodsir F2 2 8.0 No 37.7 51.9 0.21 0.15 2 Goodsir F2 3 25.0 No 37.7 51.9 0.66 0.48 2 Goodsir F2 4 26.0 No 37.7 51.9 0.69 0.50 2 Goodsir F2 5 46.0 No 37.7 51.9 1.22 0.89 2 Goodsir F2 6 47.0 No 37.7 51.9 1.25 0.90 2 Goodsir F2 7 72.0 No 37.7 51.9 1.91 1.39 2 Goodsir F2 8 72.0 Yes 37.7 51.9 1.91 1.39 3 Goodsir F3T 1 8.0 No 66.3 85.5 0.12 0.09 3 Goodsir F3T 2 9.0 No 66.3 85.5 0.14 0.11 3 Goodsir F3T 3 21.0 No 66.3 85.5 0.32 0.25 3 Goodsir F3T 4 22.0 No 66.3 85.5 0.33 0.26 3 Goodsir F3T 5 48.0 No 66.3 85.5 0.72 0.56 3 Goodsir F3T 6 48.0 No 66.3 85.5 0.72 0.56 3 Goodsir F3T 7 73.0 Yes 66.3 85.5 1.10 0.85 4 Goodsir F4 1 9.0 No 50.9 65.7 0.18 0.14 4 Goodsir F4 2 9.0 No 50.9 65.7 0.18 0.14 4 Goodsir F4 3 25.0 No 50.9 65.7 0.49 0.38 4 Goodsir F4 4 24.0 No 50.9 65.7 0.47 0.37 4 Goodsir F4 5 56.0 No 50.9 65.7 1.10 0.85 4 Goodsir F4 6 55.0 No 50.9 65.7 1.08 0.84 4 Goodsir F4 7 76.0 Yes 50.9 65.7 1.49 1.16 5 He F1 1 3.8 No 125.5 168.0 0.03 0.02 5 He F1 2 12.7 No 125.5 168.0 0.10 0.08 5 He F1 3 12.7 No 125.5 168.0 0.10 0.08 5 He F1 4 25.4 No 125.5 168.0 0.20 0.15 5 He F1 5 25.4 No 125.5 168.0 0.20 0.15 5 He F1 6 50.8 No 125.5 168.0 0.40 0.30

223

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 5 He F1 7 52.1 No 125.5 168.0 0.41 0.31 5 He F1 8 52.1 No 125.5 168.0 0.41 0.31 5 He F1 9 76.2 No 125.5 168.0 0.61 0.45 5 He F1 10 77.5 No 125.5 168.0 0.62 0.46 5 He F1 11 78.7 No 125.5 168.0 0.63 0.47 5 He F1 12 101.6 No 125.5 168.0 0.81 0.60 6 He F2 1 5.2 No 266.6 344.4 0.02 0.02 6 He F2 2 13.5 No 266.6 344.4 0.05 0.04 6 He F2 3 17.3 No 266.6 344.4 0.06 0.05 6 He F2 4 32.8 No 266.6 344.4 0.12 0.10 6 He F2 5 48.8 No 266.6 344.4 0.18 0.14 6 He F2 6 64.3 No 266.6 344.4 0.24 0.19 7 He F3 1 5.1 No 279.8 359.2 0.02 0.01 7 He F3 2 8.4 No 279.8 359.2 0.03 0.02 7 He F3 3 25.4 No 279.8 359.2 0.09 0.07 7 He F3 4 50.8 No 279.8 359.2 0.18 0.14 7 He F3 5 101.6 No 279.8 359.2 0.36 0.28 8 He F4 1 4.8 No 116.5 156.7 0.04 0.03 8 He F4 2 12.2 No 116.5 156.7 0.10 0.08 8 He F4 3 25.4 No 116.5 156.7 0.22 0.16 8 He F4 4 50.8 No 116.5 156.7 0.44 0.32 8 He F4 5 76.2 No 116.5 156.7 0.65 0.49 8 He F4 6 101.6 No 116.5 156.7 0.87 0.65 9 He F5 1 8.7 No 113.4 153.1 0.08 0.06 9 He F5 2 24.9 No 113.4 153.1 0.22 0.16 9 He F5 3 18.6 No 113.4 153.1 0.16 0.12 9 He F5 4 12.8 No 113.4 153.1 0.11 0.08 9 He F5 5 19.4 No 113.4 153.1 0.17 0.13 9 He F5 6 17.0 No 113.4 153.1 0.15 0.11 9 He F5 7 13.4 No 113.4 153.1 0.12 0.09 9 He F5 8 8.7 No 113.4 153.1 0.08 0.06 9 He F5 9 4.0 No 113.4 153.1 0.04 0.03 9 He F5 10 2.2 No 113.4 153.1 0.02 0.01 9 He F5 11 2.1 No 113.4 153.1 0.02 0.01

224

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 9 He F5 12 15.2 No 113.4 153.1 0.13 0.10 9 He F5 13 5.9 No 113.4 153.1 0.05 0.04 9 He F5 14 13.7 No 113.4 153.1 0.12 0.09 9 He F5 15 30.5 No 113.4 153.1 0.27 0.20 9 He F5 16 13.6 No 113.4 153.1 0.12 0.09 9 He F5 17 17.5 No 113.4 153.1 0.15 0.11 9 He F5 18 2.0 No 113.4 153.1 0.02 0.01 9 He F5 19 9.9 No 113.4 153.1 0.09 0.06 9 He F5 20 14.7 No 113.4 153.1 0.13 0.10 9 He F5 21 12.7 No 113.4 153.1 0.11 0.08 9 He F5 22 5.1 No 113.4 153.1 0.04 0.03 9 He F5 23 9.1 No 113.4 153.1 0.08 0.06 9 He F5 24 5.3 No 113.4 153.1 0.05 0.03 9 He F5 25 2.3 No 113.4 153.1 0.02 0.01 9 He F5 26 6.1 No 113.4 153.1 0.05 0.04 9 He F5 27 7.1 No 113.4 153.1 0.06 0.05 9 He F5 28 12.7 No 113.4 153.1 0.11 0.08 9 He F5 29 10.2 No 113.4 153.1 0.09 0.07 9 He F5 30 7.6 No 113.4 153.1 0.07 0.05 9 He F5 31 6.9 No 113.4 153.1 0.06 0.04 9 He F5 32 4.8 No 113.4 153.1 0.04 0.03 9 He F5 33 3.6 No 113.4 153.1 0.03 0.02 9 He F5 34 2.5 No 113.4 153.1 0.02 0.02 10 He F6 1 5.8 No 233.7 304.3 0.02 0.02 10 He F6 2 21.8 No 233.7 304.3 0.09 0.07 10 He F6 3 7.1 No 233.7 304.3 0.03 0.02 10 He F6 4 19.8 No 233.7 304.3 0.08 0.07 10 He F6 5 7.6 No 233.7 304.3 0.03 0.03 10 He F6 6 16.0 No 233.7 304.3 0.07 0.05 10 He F6 7 8.9 No 233.7 304.3 0.04 0.03 10 He F6 8 3.6 No 233.7 304.3 0.02 0.01 10 He F6 9 1.9 No 233.7 304.3 0.01 0.01 10 He F6 10 1.7 No 233.7 304.3 0.01 0.01 10 He F6 11 1.3 No 233.7 304.3 0.01 0.00

225

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 10 He F6 12 0.9 No 233.7 304.3 0.00 0.00 10 He F6 13 51.8 No 233.7 304.3 0.22 0.17 10 He F6 14 84.3 No 233.7 304.3 0.36 0.28 10 He F6 15 76.2 No 233.7 304.3 0.33 0.25 10 He F6 16 40.6 No 233.7 304.3 0.17 0.13 10 He F6 17 15.0 No 233.7 304.3 0.06 0.05 10 He F6 18 14.5 No 233.7 304.3 0.06 0.05 10 He F6 19 10.4 No 233.7 304.3 0.04 0.03 10 He F6 20 81.3 No 233.7 304.3 0.35 0.27 11 He F7 1 6.5 No 103.7 140.9 0.06 0.05 11 He F7 2 21.0 No 103.7 140.9 0.20 0.15 11 He F7 3 16.8 No 103.7 140.9 0.16 0.12 11 He F7 4 9.9 No 103.7 140.9 0.10 0.07 11 He F7 5 17.1 No 103.7 140.9 0.17 0.12 11 He F7 6 14.9 No 103.7 140.9 0.14 0.11 11 He F7 7 10.3 No 103.7 140.9 0.10 0.07 11 He F7 8 6.1 No 103.7 140.9 0.06 0.04 11 He F7 9 2.7 No 103.7 140.9 0.03 0.02 11 He F7 10 1.9 No 103.7 140.9 0.02 0.01 11 He F7 11 1.5 No 103.7 140.9 0.01 0.01 11 He F7 12 57.2 No 103.7 140.9 0.55 0.41 11 He F7 13 53.3 No 103.7 140.9 0.51 0.38 11 He F7 14 95.3 No 103.7 140.9 0.92 0.68 11 He F7 15 93.7 No 103.7 140.9 0.90 0.67 11 He F7 16 141.0 Yes 103.7 140.9 1.36 1.00 12 He F8 1 18.3 No 110.6 149.6 0.17 0.12 12 He F8 2 50.8 No 110.6 149.6 0.46 0.34 13 He F9 1 5.1 No 110.6 149.6 0.05 0.03 13 He F9 2 17.8 No 110.6 149.6 0.16 0.12 13 He F9 3 13.5 No 110.6 149.6 0.12 0.09 13 He F9 4 5.8 No 110.6 149.6 0.05 0.04 13 He F9 5 8.6 No 110.6 149.6 0.08 0.06 13 He F9 6 11.6 No 110.6 149.6 0.11 0.08 13 He F9 7 7.8 No 110.6 149.6 0.07 0.05

226

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 13 He F9 8 11.8 No 110.6 149.6 0.11 0.08 13 He F9 9 6.1 No 110.6 149.6 0.06 0.04 13 He F9 10 5.8 No 110.6 149.6 0.05 0.04 14 Ji_SW1 1 3.5 No 31.9 50.1 0.11 0.07 14 Ji_SW1 2 5.0 No 31.9 50.1 0.16 0.10 14 Ji_SW1 3 6.5 No 31.9 50.1 0.20 0.13 14 Ji_SW1 4 8.5 No 31.9 50.1 0.27 0.17 14 Ji_SW1 5 10.5 No 31.9 50.1 0.33 0.21 14 Ji_SW1 6 12.5 No 31.9 50.1 0.39 0.25 14 Ji_SW1 7 14.5 No 31.9 50.1 0.45 0.29 14 Ji_SW1 8 16.6 No 31.9 50.1 0.52 0.33 14 Ji_SW1 9 18.7 No 31.9 50.1 0.59 0.37 14 Ji_SW1 10 20.8 No 31.9 50.1 0.65 0.41 14 Ji_SW1 11 21.0 No 31.9 50.1 0.66 0.42 14 Ji_SW1 12 21.0 No 31.9 50.1 0.66 0.42 14 Ji_SW1 13 23.0 No 31.9 50.1 0.72 0.46 14 Ji_SW1 14 23.0 No 31.9 50.1 0.72 0.46 14 Ji_SW1 15 23.0 Yes 31.9 50.1 0.72 0.46 15 Ji_SW2 1 3.0 No 94.3 119.8 0.03 0.03 15 Ji_SW2 2 5.0 No 94.3 119.8 0.05 0.04 15 Ji_SW2 3 7.0 No 94.3 119.8 0.07 0.06 15 Ji_SW2 4 10.7 No 94.3 119.8 0.11 0.09 15 Ji_SW2 5 14.4 No 94.3 119.8 0.15 0.12 15 Ji_SW2 6 17.8 No 94.3 119.8 0.19 0.15 15 Ji_SW2 7 21.4 No 94.3 119.8 0.23 0.18 15 Ji_SW2 8 25.0 No 94.3 119.8 0.27 0.21 15 Ji_SW2 9 28.5 No 94.3 119.8 0.30 0.24 15 Ji_SW2 10 35.7 No 94.3 119.8 0.38 0.30 15 Ji_SW2 11 42.8 No 94.3 119.8 0.45 0.36 15 Ji_SW2 12 50.0 No 94.3 119.8 0.53 0.42 15 Ji_SW2 13 57.0 No 94.3 119.8 0.60 0.48 16 Ji_SW3 1 2.0 No 1494.8 1858.0 0.00 0.00 16 Ji_SW3 2 3.5 No 1494.8 1858.0 0.00 0.00 16 Ji_SW3 3 5.0 No 1494.8 1858.0 0.00 0.00

227

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 16 Ji_SW3 4 7.0 No 1494.8 1858.0 0.00 0.00 16 Ji_SW3 5 9.2 No 1494.8 1858.0 0.01 0.00 16 Ji_SW3 6 15.0 No 1494.8 1858.0 0.01 0.01 16 Ji_SW3 7 20.0 No 1494.8 1858.0 0.01 0.01 16 Ji_SW3 8 25.0 No 1494.8 1858.0 0.02 0.01 16 Ji_SW3 9 30.0 No 1494.8 1858.0 0.02 0.02 16 Ji_SW3 10 35.0 No 1494.8 1858.0 0.02 0.02 16 Ji_SW3 11 40.0 No 1494.8 1858.0 0.03 0.02 16 Ji_SW3 12 45.0 No 1494.8 1858.0 0.03 0.02 16 Ji_SW3 13 45.0 No 1494.8 1858.0 0.03 0.02 16 Ji_SW3 14 50.0 No 1494.8 1858.0 0.03 0.03 16 Ji_SW3 15 55.0 No 1494.8 1858.0 0.04 0.03 17 Jiang_DSW-T 1 4.0 No 373.2 476.7 0.01 0.01 17 Jiang_DSW-T 2 11.3 No 373.2 476.7 0.03 0.02 17 Jiang_DSW-T 3 24.3 No 373.2 476.7 0.07 0.05 17 Jiang_DSW-T 4 39.3 No 373.2 476.7 0.11 0.08 17 Jiang_DSW-T 5 53.5 No 373.2 476.7 0.14 0.11 17 Jiang_DSW-T 6 69.6 No 373.2 476.7 0.19 0.15 17 Jiang_DSW-T 7 87.0 No 373.2 476.7 0.23 0.18 18 Jiang_SSW-T 1 4.5 No 92.1 123.5 0.05 0.04 18 Jiang_SSW-T 2 9.0 No 92.1 123.5 0.10 0.07 18 Jiang_SSW-T 3 18.0 No 92.1 123.5 0.20 0.15 18 Jiang_SSW-T 4 28.0 No 92.1 123.5 0.30 0.23 18 Jiang_SSW-T 5 38.0 No 92.1 123.5 0.41 0.31 18 Jiang_SSW-T 6 48.0 No 92.1 123.5 0.52 0.39 18 Jiang_SSW-T 7 58.0 No 92.1 123.5 0.63 0.47 19 Lefas SW-21 1 22.0 No 108.7 140.2 0.20 0.16 20 Lefas SW-22 1 16.3 No 151.4 194.7 0.11 0.08 21 Lefas SW-23 1 14.2 No 192.5 247.7 0.07 0.06 22 Lefas SW-24 1 19.4 No 109.4 140.8 0.18 0.14 23 Lefas SW-25 1 9.8 No 185.7 239.4 0.05 0.04 24 Lefas SW-26 1 22.1 No 116.1 149.6 0.19 0.15 25 Lefas SW-30 1 21.5 No 99.0 132.8 0.22 0.16 26 Lefas SW-31 1 4.0 No 92.5 105.1 0.04 0.04

228

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 26 Lefas SW-31 2 22.0 No 92.5 105.1 0.24 0.21 27 Lefas SW-32 1 8.3 No 121.9 156.6 0.07 0.05 27 Lefas SW-32 2 24.2 No 121.9 156.6 0.20 0.15 28 Lefas SW-33 1 5.5 No 114.9 147.7 0.05 0.04 28 Lefas SW-33 2 8.4 No 114.9 147.7 0.07 0.06 28 Lefas SW-33 3 15.0 No 114.9 147.7 0.13 0.10 28 Lefas SW-33 4 24.5 No 114.9 147.7 0.21 0.17 29 Oesterle B1 1 12.7 No 905.7 1134.6 0.01 0.01 29 Oesterle B1 2 25.4 No 905.7 1134.6 0.03 0.02 29 Oesterle B1 3 50.8 No 905.7 1134.6 0.06 0.04 29 Oesterle B1 4 76.2 No 905.7 1134.6 0.08 0.07 29 Oesterle B1 5 101.6 No 905.7 1134.6 0.11 0.09 29 Oesterle B1 6 127.0 No 905.7 1134.6 0.14 0.11 29 Oesterle B1 7 152.4 No 905.7 1134.6 0.17 0.13 30 Oesterle B2 1 12.7 No 997.5 1249.1 0.01 0.01 30 Oesterle B2 2 25.4 No 997.5 1249.1 0.03 0.02 30 Oesterle B2 3 50.8 No 997.5 1249.1 0.05 0.04 30 Oesterle B2 4 76.2 No 997.5 1249.1 0.08 0.06 30 Oesterle B2 5 101.6 No 997.5 1249.1 0.10 0.08 30 Oesterle B2 6 127.0 No 997.5 1249.1 0.13 0.10 30 Oesterle B2 7 152.4 No 997.5 1249.1 0.15 0.12 31 Oesterle B3 1 6.4 No 818.9 1033.4 0.01 0.01 31 Oesterle B3 2 12.7 No 818.9 1033.4 0.02 0.01 31 Oesterle B3 3 25.4 No 818.9 1033.4 0.03 0.02 31 Oesterle B3 4 50.8 No 818.9 1033.4 0.06 0.05 31 Oesterle B3 5 76.2 No 818.9 1033.4 0.09 0.07 31 Oesterle B3 6 101.6 No 818.9 1033.4 0.12 0.10 31 Oesterle B3 7 127.0 No 818.9 1033.4 0.16 0.12 31 Oesterle B3 8 152.4 No 818.9 1033.4 0.19 0.15 31 Oesterle B3 9 177.8 No 818.9 1033.4 0.22 0.17 31 Oesterle B3 10 203.2 No 818.9 1033.4 0.25 0.20 32 Oesterle B4 1 315.0 No 853.0 1068.2 0.37 0.29 33 Oesterle B5 1 7.6 No 857.5 1072.8 0.01 0.01 33 Oesterle B5 2 15.2 No 857.5 1072.8 0.02 0.01

229

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 33 Oesterle B5 3 25.4 No 857.5 1072.8 0.03 0.02 33 Oesterle B5 4 50.8 No 857.5 1072.8 0.06 0.05 33 Oesterle B5 5 76.2 No 857.5 1072.8 0.09 0.07 33 Oesterle B5 6 101.6 No 857.5 1072.8 0.12 0.09 33 Oesterle B5 7 127.0 No 857.5 1072.8 0.15 0.12 34 Oesterle B6 1 11.4 No 690.6 871.9 0.02 0.01 34 Oesterle B6 2 19.1 No 690.6 871.9 0.03 0.02 34 Oesterle B6 3 27.9 No 690.6 871.9 0.04 0.03 34 Oesterle B6 4 50.8 No 690.6 871.9 0.07 0.06 34 Oesterle B6 5 76.2 No 690.6 871.9 0.11 0.09 35 Oesterle B7 1 5.1 No 529.3 677.4 0.01 0.01 35 Oesterle B7 2 12.7 No 529.3 677.4 0.02 0.02 35 Oesterle B7 3 27.9 No 529.3 677.4 0.05 0.04 35 Oesterle B7 4 50.8 No 529.3 677.4 0.10 0.07 35 Oesterle B7 5 76.2 No 529.3 677.4 0.14 0.11 35 Oesterle B7 6 101.6 No 529.3 677.4 0.19 0.15 35 Oesterle B7 7 127.0 No 529.3 677.4 0.24 0.19 35 Oesterle B7 8 127.0 No 529.3 677.4 0.24 0.19 35 Oesterle B7 9 127.0 No 529.3 677.4 0.24 0.19 36 Oesterle B8 1 5.1 No 758.4 957.5 0.01 0.01 36 Oesterle B8 2 8.9 No 758.4 957.5 0.01 0.01 36 Oesterle B8 3 12.7 No 758.4 957.5 0.02 0.01 36 Oesterle B8 4 17.8 No 758.4 957.5 0.02 0.02 36 Oesterle B8 5 25.4 No 758.4 957.5 0.03 0.03 36 Oesterle B8 6 50.8 No 758.4 957.5 0.07 0.05 36 Oesterle B8 7 76.2 No 758.4 957.5 0.10 0.08 36 Oesterle B8 8 101.6 No 758.4 957.5 0.13 0.11 36 Oesterle B8 9 127.0 No 758.4 957.5 0.17 0.13 36 Oesterle B8 10 152.4 No 758.4 957.5 0.20 0.16 37 Oesterle B9 1 12.7 No 723.8 912.4 0.02 0.01 37 Oesterle B9 2 127.0 No 723.8 912.4 0.18 0.14 37 Oesterle B9 3 43.2 No 723.8 912.4 0.06 0.05 37 Oesterle B9 4 116.8 No 723.8 912.4 0.16 0.13 37 Oesterle B9 5 106.7 No 723.8 912.4 0.15 0.12

230

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 38 Oesterle B10 1 10.2 No 834.8 1049.5 0.01 0.01 38 Oesterle B10 2 137.2 No 834.8 1049.5 0.16 0.13 38 Oesterle B10 3 33.0 No 834.8 1049.5 0.04 0.03 38 Oesterle B10 4 106.7 No 834.8 1049.5 0.13 0.10 38 Oesterle B10 5 129.5 No 834.8 1049.5 0.16 0.12 38 Oesterle B10 6 124.5 No 834.8 1049.5 0.15 0.12 38 Oesterle B10 7 109.2 No 834.8 1049.5 0.13 0.10 39 Oesterle F1 1 8.9 No 129.0 168.8 0.07 0.05 39 Oesterle F1 2 17.8 No 129.0 168.8 0.14 0.11 39 Oesterle F1 3 25.4 No 129.0 168.8 0.20 0.15 39 Oesterle F1 4 50.8 No 129.0 168.8 0.39 0.30 39 Oesterle F1 5 101.6 No 129.0 168.8 0.79 0.60 39 Oesterle F1 6 106.7 No 129.0 168.8 0.83 0.63 40 Oesterle F2 1 3.8 No 145.1 186.3 0.03 0.02 40 Oesterle F2 2 7.6 No 145.1 186.3 0.05 0.04 40 Oesterle F2 3 12.7 No 145.1 186.3 0.09 0.07 40 Oesterle F2 4 25.4 No 145.1 186.3 0.18 0.14 40 Oesterle F2 5 50.8 No 145.1 186.3 0.35 0.27 40 Oesterle F2 6 76.2 No 145.1 186.3 0.53 0.41 40 Oesterle F2 7 101.6 No 145.1 186.3 0.70 0.55 40 Oesterle F2 8 127.0 No 145.1 186.3 0.88 0.68 41 Oesterle R1 1 6.4 No 63.4 85.7 0.10 0.07 41 Oesterle R1 2 12.7 No 63.4 85.7 0.20 0.15 41 Oesterle R1 3 25.4 No 63.4 85.7 0.40 0.30 41 Oesterle R1 4 50.8 No 63.4 85.7 0.80 0.59 41 Oesterle R1 5 76.2 No 63.4 85.7 1.20 0.89 41 Oesterle R1 6 101.6 No 63.4 85.7 1.60 1.19 41 Oesterle R1 7 127.0 No 63.4 85.7 2.00 1.48 42 Oesterle R2 1 12.7 No 47.9 66.9 0.27 0.19 42 Oesterle R2 2 25.4 No 47.9 66.9 0.53 0.38 42 Oesterle R2 3 50.8 No 47.9 66.9 1.06 0.76 42 Oesterle R2 4 76.2 No 47.9 66.9 1.59 1.14 42 Oesterle R2 5 101.6 No 47.9 66.9 2.12 1.52 42 Oesterle R2 6 127.0 Yes 47.9 66.9 2.65 1.90

231

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 43 Zhang SW1-1 1 2.0 No 139.7 240.7 0.01 0.01 43 Zhang SW1-1 2 3.5 No 139.7 240.7 0.03 0.01 43 Zhang SW1-1 3 5.0 No 139.7 240.7 0.04 0.02 43 Zhang SW1-1 4 6.5 No 139.7 240.7 0.05 0.03 43 Zhang SW1-1 5 8.0 No 139.7 240.7 0.06 0.03 43 Zhang SW1-1 6 10.0 No 139.7 240.7 0.07 0.04 43 Zhang SW1-1 7 11.5 No 139.7 240.7 0.08 0.05 43 Zhang SW1-1 8 13.0 No 139.7 240.7 0.09 0.05 43 Zhang SW1-1 9 15.0 No 139.7 240.7 0.11 0.06 43 Zhang SW1-1 10 16.5 No 139.7 240.7 0.12 0.07 43 Zhang SW1-1 11 18.5 No 139.7 240.7 0.13 0.08 43 Zhang SW1-1 12 20.5 No 139.7 240.7 0.15 0.09 44 Zhang SW1-2 1 2.5 No 236.6 298.6 0.01 0.01 44 Zhang SW1-2 2 5.0 No 236.6 298.6 0.02 0.02 44 Zhang SW1-2 3 6.3 No 236.6 298.6 0.03 0.02 44 Zhang SW1-2 4 7.6 No 236.6 298.6 0.03 0.03 44 Zhang SW1-2 5 9.4 No 236.6 298.6 0.04 0.03 44 Zhang SW1-2 6 11.2 No 236.6 298.6 0.05 0.04 44 Zhang SW1-2 7 12.7 No 236.6 298.6 0.05 0.04 44 Zhang SW1-2 8 14.6 No 236.6 298.6 0.06 0.05 44 Zhang SW1-2 9 16.5 No 236.6 298.6 0.07 0.06 44 Zhang SW1-2 10 18.3 No 236.6 298.6 0.08 0.06 44 Zhang SW1-2 11 20.4 No 236.6 298.6 0.09 0.07 44 Zhang SW1-2 12 22.0 No 236.6 298.6 0.09 0.07 45 Zhang SW1-3 1 2.0 No 328.3 414.4 0.01 0.00 45 Zhang SW1-3 2 3.0 No 328.3 414.4 0.01 0.01 45 Zhang SW1-3 3 3.5 No 328.3 414.4 0.01 0.01 45 Zhang SW1-3 4 4.0 No 328.3 414.4 0.01 0.01 45 Zhang SW1-3 5 5.0 No 328.3 414.4 0.02 0.01 45 Zhang SW1-3 6 6.5 No 328.3 414.4 0.02 0.02 45 Zhang SW1-3 7 8.0 No 328.3 414.4 0.02 0.02 45 Zhang SW1-3 8 9.5 No 328.3 414.4 0.03 0.02 45 Zhang SW1-3 9 10.5 No 328.3 414.4 0.03 0.03 45 Zhang SW1-3 10 12.0 No 328.3 414.4 0.04 0.03

232

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 45 Zhang SW1-3 11 14.0 No 328.3 414.4 0.04 0.03 45 Zhang SW1-3 12 15.5 No 328.3 414.4 0.05 0.04 45 Zhang SW1-3 13 17.0 No 328.3 414.4 0.05 0.04 45 Zhang SW1-3 14 19.0 No 328.3 414.4 0.06 0.05 45 Zhang SW1-3 15 20.5 No 328.3 414.4 0.06 0.05 45 Zhang SW1-3 16 21.5 No 328.3 414.4 0.07 0.05 46 Zhang SW1-4 1 1.0 No 458.5 578.7 0.00 0.00 46 Zhang SW1-4 2 1.7 No 458.5 578.7 0.00 0.00 46 Zhang SW1-4 3 2.6 No 458.5 578.7 0.01 0.00 46 Zhang SW1-4 4 3.3 No 458.5 578.7 0.01 0.01 46 Zhang SW1-4 5 4.1 No 458.5 578.7 0.01 0.01 46 Zhang SW1-4 6 5.2 No 458.5 578.7 0.01 0.01 46 Zhang SW1-4 7 7.0 No 458.5 578.7 0.02 0.01 46 Zhang SW1-4 8 8.5 No 458.5 578.7 0.02 0.01 46 Zhang SW1-4 9 11.0 No 458.5 578.7 0.02 0.02 46 Zhang SW1-4 10 13.0 No 458.5 578.7 0.03 0.02 47 Zhang SW2-2 1 2.0 No 374.3 469.1 0.01 0.00 47 Zhang SW2-2 2 3.0 No 374.3 469.1 0.01 0.01 47 Zhang SW2-2 3 4.0 No 374.3 469.1 0.01 0.01 47 Zhang SW2-2 4 4.6 No 374.3 469.1 0.01 0.01 47 Zhang SW2-2 5 5.4 No 374.3 469.1 0.01 0.01 47 Zhang SW2-2 6 6.0 No 374.3 469.1 0.02 0.01 47 Zhang SW2-2 7 6.9 No 374.3 469.1 0.02 0.01 47 Zhang SW2-2 8 7.8 No 374.3 469.1 0.02 0.02 47 Zhang SW2-2 9 8.5 No 374.3 469.1 0.02 0.02 47 Zhang SW2-2 10 9.4 No 374.3 469.1 0.03 0.02 47 Zhang SW2-2 11 10.3 No 374.3 469.1 0.03 0.02 47 Zhang SW2-2 12 11.2 No 374.3 469.1 0.03 0.02 48 Zhang SW2-3 1 1.2 No 432.2 543.7 0.00 0.00 48 Zhang SW2-3 2 1.8 No 432.2 543.7 0.00 0.00 48 Zhang SW2-3 3 2.5 No 432.2 543.7 0.01 0.00 48 Zhang SW2-3 4 3.1 No 432.2 543.7 0.01 0.01 48 Zhang SW2-3 5 3.7 No 432.2 543.7 0.01 0.01 48 Zhang SW2-3 6 4.8 No 432.2 543.7 0.01 0.01

233

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 48 Zhang SW2-3 7 6.4 No 432.2 543.7 0.01 0.01 48 Zhang SW2-3 8 7.8 No 432.2 543.7 0.02 0.01 48 Zhang SW2-3 9 8.8 No 432.2 543.7 0.02 0.02 48 Zhang SW2-3 10 10.7 No 432.2 543.7 0.02 0.02 49 Zhang SW3-1 1 20.4 No 236.6 298.6 0.09 0.07 50 Zhang SW3-2 1 10.9 No 432.2 543.7 0.03 0.02 51 Zhang SW4-1 1 1.0 No 612.3 764.7 0.00 0.00 51 Zhang SW4-1 2 2.4 No 612.3 764.7 0.00 0.00 51 Zhang SW4-1 3 3.6 No 612.3 764.7 0.01 0.00 51 Zhang SW4-1 4 5.0 No 612.3 764.7 0.01 0.01 51 Zhang SW4-1 5 6.5 No 612.3 764.7 0.01 0.01 51 Zhang SW4-1 6 8.2 No 612.3 764.7 0.01 0.01 51 Zhang SW4-1 7 10.2 No 612.3 764.7 0.02 0.01 51 Zhang SW4-1 8 12.2 No 612.3 764.7 0.02 0.02 52 Zhang SW4-2 1 1.0 No 432.2 543.7 0.00 0.00 52 Zhang SW4-2 2 1.7 No 432.2 543.7 0.00 0.00 52 Zhang SW4-2 3 2.5 No 432.2 543.7 0.01 0.00 52 Zhang SW4-2 4 3.5 No 432.2 543.7 0.01 0.01 52 Zhang SW4-2 5 4.5 No 432.2 543.7 0.01 0.01 52 Zhang SW4-2 6 5.7 No 432.2 543.7 0.01 0.01 52 Zhang SW4-2 7 7.6 No 432.2 543.7 0.02 0.01 52 Zhang SW4-2 8 9.0 No 432.2 543.7 0.02 0.02 52 Zhang SW4-2 9 10.7 No 432.2 543.7 0.02 0.02 53 Zhang SW4-3 1 1.2 No 432.2 543.7 0.00 0.00 53 Zhang SW4-3 2 1.5 No 432.2 543.7 0.00 0.00 53 Zhang SW4-3 3 2.5 No 432.2 543.7 0.01 0.00 53 Zhang SW4-3 4 3.0 No 432.2 543.7 0.01 0.01 53 Zhang SW4-3 5 3.5 No 432.2 543.7 0.01 0.01 53 Zhang SW4-3 6 4.8 No 432.2 543.7 0.01 0.01 53 Zhang SW4-3 7 6.2 No 432.2 543.7 0.01 0.01 53 Zhang SW4-3 8 7.8 No 432.2 543.7 0.02 0.01 53 Zhang SW4-3 9 8.7 No 432.2 543.7 0.02 0.02 53 Zhang SW4-3 10 10.5 No 432.2 543.7 0.02 0.02 54 Zhang SW5-1 1 2.2 No 432.2 543.7 0.01 0.00

234

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 54 Zhang SW5-1 2 2.6 No 432.2 543.7 0.01 0.00 54 Zhang SW5-1 3 3.7 No 432.2 543.7 0.01 0.01 54 Zhang SW5-1 4 5.4 No 432.2 543.7 0.01 0.01 54 Zhang SW5-1 5 7.3 No 432.2 543.7 0.02 0.01 54 Zhang SW5-1 6 9.3 No 432.2 543.7 0.02 0.02 54 Zhang SW5-1 7 11.0 No 432.2 543.7 0.03 0.02 55 Zhang SW5-2 1 1.2 No 432.2 543.7 0.00 0.00 55 Zhang SW5-2 2 1.5 No 432.2 543.7 0.00 0.00 55 Zhang SW5-2 3 2.5 No 432.2 543.7 0.01 0.00 55 Zhang SW5-2 4 3.0 No 432.2 543.7 0.01 0.01 55 Zhang SW5-2 5 3.5 No 432.2 543.7 0.01 0.01 55 Zhang SW5-2 6 4.8 No 432.2 543.7 0.01 0.01 55 Zhang SW5-2 7 6.2 No 432.2 543.7 0.01 0.01 55 Zhang SW5-2 8 7.8 No 432.2 543.7 0.02 0.01 55 Zhang SW5-2 9 8.7 No 432.2 543.7 0.02 0.02 55 Zhang SW5-2 10 10.5 No 432.2 543.7 0.02 0.02 56 Zhang SW5-3 1 5.0 No 432.2 543.7 0.01 0.01 56 Zhang SW5-3 2 6.0 No 432.2 543.7 0.01 0.01 56 Zhang SW5-3 3 7.5 No 432.2 543.7 0.02 0.01 56 Zhang SW5-3 4 9.0 No 432.2 543.7 0.02 0.02 56 Zhang SW5-3 5 10.5 No 432.2 543.7 0.02 0.02 56 Zhang SW5-3 6 12.1 No 432.2 543.7 0.03 0.02 56 Zhang SW5-3 7 13.7 No 432.2 543.7 0.03 0.03 56 Zhang SW5-3 8 16.0 No 432.2 543.7 0.04 0.03 57 Zhang SW6-1 1 4.5 No 1080.5 1348.4 0.00 0.00 57 Zhang SW6-1 2 5.8 No 1080.5 1348.4 0.01 0.00 57 Zhang SW6-1 3 7.2 No 1080.5 1348.4 0.01 0.01 57 Zhang SW6-1 4 9.0 No 1080.5 1348.4 0.01 0.01 57 Zhang SW6-1 5 10.5 No 1080.5 1348.4 0.01 0.01 57 Zhang SW6-1 6 12.0 No 1080.5 1348.4 0.01 0.01 57 Zhang SW6-1 7 14.0 No 1080.5 1348.4 0.01 0.01 57 Zhang SW6-1 8 16.0 No 1080.5 1348.4 0.01 0.01 57 Zhang SW6-1 9 18.2 No 1080.5 1348.4 0.02 0.01 58 Zhang SW6-2 1 1.2 No 432.2 543.7 0.00 0.00

235

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 58 Zhang SW6-2 2 1.5 No 432.2 543.7 0.00 0.00 58 Zhang SW6-2 3 2.5 No 432.2 543.7 0.01 0.00 58 Zhang SW6-2 4 3.0 No 432.2 543.7 0.01 0.01 58 Zhang SW6-2 5 3.5 No 432.2 543.7 0.01 0.01 58 Zhang SW6-2 6 4.8 No 432.2 543.7 0.01 0.01 58 Zhang SW6-2 7 6.2 No 432.2 543.7 0.01 0.01 58 Zhang SW6-2 8 7.8 No 432.2 543.7 0.02 0.01 58 Zhang SW6-2 9 8.7 No 432.2 543.7 0.02 0.02 58 Zhang SW6-2 10 10.5 No 432.2 543.7 0.02 0.02 59 Zhang SW6-3 1 1.0 No 432.2 543.7 0.00 0.00 59 Zhang SW6-3 2 2.0 No 432.2 543.7 0.00 0.00 59 Zhang SW6-3 3 3.0 No 432.2 543.7 0.01 0.01 59 Zhang SW6-3 4 4.0 No 432.2 543.7 0.01 0.01 59 Zhang SW6-3 5 5.0 No 432.2 543.7 0.01 0.01 59 Zhang SW6-3 6 6.5 No 432.2 543.7 0.02 0.01 59 Zhang SW6-3 7 7.5 No 432.2 543.7 0.02 0.01 59 Zhang SW6-3 8 9.0 No 432.2 543.7 0.02 0.02 59 Zhang SW6-3 9 10.8 No 432.2 543.7 0.02 0.02 59 Zhang SW6-3 10 12.7 No 432.2 543.7 0.03 0.02 59 Zhang SW6-3 11 14.2 No 432.2 543.7 0.03 0.03 59 Zhang SW6-3 12 16.0 No 432.2 543.7 0.04 0.03 59 Zhang SW6-3 13 18.0 No 432.2 543.7 0.04 0.03 59 Zhang SW6-3 14 19.8 No 432.2 543.7 0.05 0.04 60 Zhou SW1 1 14.1 No 117.1 140.0 0.12 0.10 60 Zhou SW1 2 18.4 No 117.1 140.0 0.16 0.13 60 Zhou SW1 3 22.0 No 117.1 140.0 0.19 0.16 60 Zhou SW1 4 26.0 No 117.1 140.0 0.22 0.19 60 Zhou SW1 5 28.5 No 117.1 140.0 0.24 0.20 60 Zhou SW1 6 32.5 No 117.1 140.0 0.28 0.23 60 Zhou SW1 7 36.4 No 117.1 140.0 0.31 0.26 60 Zhou SW1 8 40.5 No 117.1 140.0 0.35 0.29 60 Zhou SW1 9 43.2 No 117.1 140.0 0.37 0.31 60 Zhou SW1 10 47.5 No 117.1 140.0 0.41 0.34 61 Zhou SW2 1 11.7 No 134.5 171.7 0.09 0.07

236

Wall Name Cycle Δexp Buckle Δppbm Δcebm Δexp/Δppbm Δexp/Δcebm (#) (N/A) (#) (mm) (N/A) (mm) (mm) (N/A) (N/A) 61 Zhou SW2 2 15.0 No 134.5 171.7 0.11 0.09 61 Zhou SW2 3 19.0 No 134.5 171.7 0.14 0.11 61 Zhou SW2 4 23.0 No 134.5 171.7 0.17 0.13 61 Zhou SW2 5 30.0 No 134.5 171.7 0.22 0.17 61 Zhou SW2 6 33.0 No 134.5 171.7 0.25 0.19 61 Zhou SW2 7 37.5 No 134.5 171.7 0.28 0.22 61 Zhou SW2 8 42.0 No 134.5 171.7 0.31 0.24 61 Zhou SW2 9 44.5 No 134.5 171.7 0.33 0.26 61 Zhou SW2 10 48.6 No 134.5 171.7 0.36 0.28 62 Zhou SW3 1 10.0 No 134.5 171.7 0.07 0.06 62 Zhou SW3 2 14.0 No 134.5 171.7 0.10 0.08 62 Zhou SW3 3 16.0 No 134.5 171.7 0.12 0.09 62 Zhou SW3 4 18.0 No 134.5 171.7 0.13 0.10 62 Zhou SW3 5 20.0 No 134.5 171.7 0.15 0.12 62 Zhou SW3 6 22.0 No 134.5 171.7 0.16 0.13 62 Zhou SW3 7 25.0 No 134.5 171.7 0.19 0.15 62 Zhou SW3 8 29.0 No 134.5 171.7 0.22 0.17 62 Zhou SW3 9 30.0 No 134.5 171.7 0.22 0.17 62 Zhou SW3 10 32.0 No 134.5 171.7 0.24 0.19 62 Zhou SW3 11 34.0 No 134.5 171.7 0.25 0.20 62 Zhou SW3 12 36.0 No 134.5 171.7 0.27 0.21 62 Zhou SW3 13 38.0 No 134.5 171.7 0.28 0.22 62 Zhou SW3 14 40.0 No 134.5 171.7 0.30 0.23 63 Zhou SW4 1 9.0 No 117.1 140.0 0.08 0.06 63 Zhou SW4 2 12.0 No 117.1 140.0 0.10 0.09 63 Zhou SW4 3 14.0 No 117.1 140.0 0.12 0.10 63 Zhou SW4 4 18.0 No 117.1 140.0 0.15 0.13 63 Zhou SW4 5 22.0 No 117.1 140.0 0.19 0.16 63 Zhou SW4 6 26.0 No 117.1 140.0 0.22 0.19 63 Zhou SW4 7 28.0 No 117.1 140.0 0.24 0.20 63 Zhou SW4 8 32.0 No 117.1 140.0 0.27 0.23 63 Zhou SW4 9 36.0 No 117.1 140.0 0.31 0.26 63 Zhou SW4 10 40.0 No 117.1 140.0 0.34 0.29 63 Zhou SW4 11 42.5 No 117.1 140.0 0.36 0.30

237

Table 20 Prior Experimental Prism Tests - Loading and Geometry

Geometry

Hw Lw tw Specimen # Loading Notes Shape Specimen (mm) (mm) (mm) 1 Comp. - Prism Azimikor MASP1 3800 590 140 2 Cyclic - Prism Azimikor MASP2 3800 590 140 3 Cyclic GFRP 338 mm Prism Azimikor MASP3 3800 590 140 4 Cyclic GFRP 1415 mm Prism Azimikor MASP4 3800 590 140 5 Cyclic GFRP 1600 mm Prism Azimikor MASP5 3800 590 140 6 Cyclic - Prism ChaiElayer P1 1199 203 102 7 Cyclic - Prism ChaiElayer P2 1199 203 102 8 Cyclic - Prism ChaiElayer P3 1505 203 102 9 Cyclic - Prism ChaiElayer P4 1505 203 102 10 Cyclic - Prism ChaiElayer P5 1505 203 102 11 Cyclic - Prism ChaiElayer P6 1505 203 102 12 Cyclic - Prism ChaiElayer P7 1505 203 102 13 Cyclic - Prism ChaiElayer P8 1505 203 102 14 Cyclic - Prism ChaiElayer P9 1811 203 102 15 Cyclic - Prism ChaiElayer P10 1811 203 102 16 Cyclic - Prism ChaiElayer P11 1811 203 102 17 Cyclic - Prism ChaiElayer P12 1811 203 102 18 Cyclic - Prism ChaiElayer P13 1811 203 102 19 Cyclic - Prism ChaiElayer P14 1811 203 102 20 1 Cycle - Prism Creagh P1 915 305 152 21 Comp. - Prism Creagh P2 915 305 152 22 Comp. - Prism Crysandis PER1 600 150 75 23 1 Cycle - Prism Crysandis PER2 600 150 75 24 1 Cycle - Prism Crysandis PER3 600 150 75 25 1 Cycle - Prism Crysandis PER4 600 150 75 26 1 Cycle - Prism Crysandis PER5 600 150 75 27 1 Cycle - Prism Crysandis PSR1 600 150 75 28 1 Cycle - Prism Crysandis PSR2 600 150 75 29 1 Cycle - Prism Crysandis PSR3 600 150 75 30 1 Cycle - Prism Crysandis PSR4 600 150 75 31 1 Cycle - Prism Crysandis PSR5 600 150 75 32 1 Cycle - Prism Crysandis PSR6 600 150 75

238

Geometry

Hw Lw tw Specimen # Loading Notes Shape Specimen (mm) (mm) (mm) 33 1 Cycle - Prism Crysandis PSR7 600 150 75 34 1 Cycle - Prism Crysandis PSR8 600 150 75 35 1 Cycle - Prism Crysandis PSR9 600 150 75 36 1 Cycle - Prism Crysandis PSR10 600 150 75 37 1 Cycle - Prism Crysandis PSR11 600 150 75 38 Cyclic - Prism Goodsir P1 1120 480 160 39 Cyclic - Prism Goodsir P2 1120 480 160 40 Comp. - Prism Goodsir P3 1120 480 160 41 Cyclic - Prism Goodsir P4 1120 480 160 42 Cyclic - Prism Goodsir P5 1120 480 160 43 Cyclic - Prism Goodsir P6 880 480 160 44 Cyclic - Prism Goodsir P7 880 480 160 45 Cyclic - Prism Goodsir P8 640 480 160 46 Cyclic - Prism Goodsir P9 640 480 160

239

Table 21 Prior Experimental Prism Tests - Material Properties

Material Strengths

fc' fy,wl fy,wh fu,wl Specimen # Specimen (MPa) (MPa) (MPa) (MPa) 1 Azimikor MASP1 23.2 527 527 750 2 Azimikor MASP2 23.2 464 464 646 3 Azimikor MASP3 23.2 527 527 750 4 Azimikor MASP4 23.2 527 527 750 5 Azimikor MASP5 23.2 584 584 835 6 ChaiElayer P1 34.1 375 683 563 7 ChaiElayer P2 34.1 455 683 683 8 ChaiElayer P3 34.1 375 683 563 9 ChaiElayer P4 34.1 375 683 563 10 ChaiElayer P5 34.1 375 683 563 11 ChaiElayer P6 34.1 455 683 683 12 ChaiElayer P7 34.1 455 683 683 13 ChaiElayer P8 34.1 455 683 683 14 ChaiElayer P9 34.1 375 683 563 15 ChaiElayer P10 34.1 375 683 563 16 ChaiElayer P11 34.1 375 683 563 17 ChaiElayer P12 34.1 455 683 683 18 ChaiElayer P13 34.1 455 683 683 19 ChaiElayer P14 34.1 455 683 683 20 Creagh P1 30.0 460 460 690 21 Creagh P2 30.0 460 460 690 22 Crysandis PER1 24.9 604 604 906 23 Crysandis PER2 24.9 604 604 906 24 Crysandis PER3 24.9 604 604 906 25 Crysandis PER4 24.9 604 604 906 26 Crysandis PER5 24.9 604 604 906 27 Crysandis PSR1 23.3 604 604 906 28 Crysandis PSR2 22.2 604 604 906 29 Crysandis PSR3 22.8 604 604 906 30 Crysandis PSR4 22.8 604 604 906

240

Material Strengths

fc' fy,wl fy,wh fu,wl Specimen # Specimen (MPa) (MPa) (MPa) (MPa) 31 Crysandis PSR5 23.3 604 604 906 32 Crysandis PSR6 23.3 604 604 906 33 Crysandis PSR7 23.3 604 604 906 34 Crysandis PSR8 23.3 604 604 906 35 Crysandis PSR9 23.3 604 604 906 36 Crysandis PSR10 23.3 604 604 906 37 Crysandis PSR11 23.3 604 604 906 38 Goodsir P1 24.1 442 290 660 39 Goodsir P2 24.1 442 350 660 40 Goodsir P3 24.1 442 350 660 41 Goodsir P4 24.1 442 290 660 42 Goodsir P5 29.0 442 290 660 43 Goodsir P6 29.0 442 290 660 44 Goodsir P7 29.0 442 290 660 45 Goodsir P8 29.0 442 290 660 46 Goodsir P9 29.0 442 290 660

Table 22 Prior Experimental Prism Tests - Reinforcing Details

Cover Longitudinal Transverse

c dl sl ρl dh sh Specimen # Specimen (mm) (mm) (mm) (%) (mm) (mm) 1 Azimikor MASP1 70 15 55 0.71% 0 43 2 Azimikor MASP2 70 20 55 1.07% 0 43 3 Azimikor MASP3 70 15 55 0.71% 0 43 4 Azimikor MASP4 70 15 55 0.48% 0 43 5 Azimikor MASP5 70 10 55 0.24% 0 43 6 ChaiElayer P1 12.5 9.525 75 2.10% 6 57 7 ChaiElayer P2 12.5 12.7 75 3.80% 6 76 8 ChaiElayer P3 12.5 9.525 75 2.10% 6 57 9 ChaiElayer P4 12.5 9.525 75 2.10% 6 57 10 ChaiElayer P5 12.5 9.525 75 2.10% 6 57

241

Cover Longitudinal Transverse

c dl sl ρl dh sh Specimen # Specimen (mm) (mm) (mm) (%) (mm) (mm) 11 ChaiElayer P6 12.5 12.7 75 3.80% 6 76 12 ChaiElayer P7 12.5 12.7 75 3.80% 6 76 13 ChaiElayer P8 12.5 12.7 75 3.80% 6 76 14 ChaiElayer P9 12.5 9.525 75 2.10% 6 57 15 ChaiElayer P10 12.5 9.525 75 2.10% 6 57 16 ChaiElayer P11 12.5 9.525 75 2.10% 6 57 17 ChaiElayer P12 12.5 12.7 75 3.80% 6 76 18 ChaiElayer P13 12.5 12.7 75 3.80% 6 76 19 ChaiElayer P14 12.5 12.7 75 3.80% 6 76 20 Creagh P1 19 19 105 3.68% 9.53 50 21 Creagh P2 19 19 105 3.68% 9.53 50 22 Crysandis PER1 8 8 55 2.68% 4.2 33 23 Crysandis PER2 8 8 55 2.68% 4.2 33 24 Crysandis PER3 8 8 55 2.68% 4.2 33 25 Crysandis PER4 8 8 55 2.68% 4.2 33 26 Crysandis PER5 8 8 55 2.68% 4.2 33 27 Crysandis PSR1 8 8 55 1.79% 4.2 33 28 Crysandis PSR2 8 8 55 2.68% 4.2 33 29 Crysandis PSR3 8 8 55 3.18% 4.2 33 30 Crysandis PSR4 8 10 55 3.68% 4.2 33 31 Crysandis PSR5 8 12 55 4.02% 4.2 33 32 Crysandis PSR6 8 10 55 4.19% 4.2 33 33 Crysandis PSR7 8 14 55 5.47% 4.2 33 34 Crysandis PSR8 8 12 55 6.03% 4.2 33 35 Crysandis PSR9 8 16 55 7.15% 4.2 33 36 Crysandis PSR10 8 14 55 8.21% 4.2 33 37 Crysandis PSR11 8 16 55 10.72% 4.2 33 38 Goodsir P1 17.5 16 85 3.14% 5 64 39 Goodsir P2 18.5 16 85 3.14% 6 64 40 Goodsir P3 18.5 16 85 3.14% 6 96 41 Goodsir P4 17.5 16 85 3.14% 5 64 42 Goodsir P5 17.5 16 85 3.14% 5 64 43 Goodsir P6 17.5 16 85 3.14% 5 64

242

Cover Longitudinal Transverse

c dl sl ρl dh sh Specimen # Specimen (mm) (mm) (mm) (%) (mm) (mm) 44 Goodsir P7 17.5 16 85 3.14% 5 64 45 Goodsir P8 17.5 16 85 3.14% 5 64 46 Goodsir P9 17.5 16 85 3.14% 5 64

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Table 23 Prior Experimental Prism Tests - Strains and Normalized Results

Prism Name Cycle εexp Buckle εppbm εcebm εexp/εppbm εexp/εcebm (#) (N/A) (#) (mm/mm) (N/A) (mm/mm) (mm/mm) (N/A) (N/A) 1 Azimikor MASP1 1 0 No 0.001166 0.009343 0.00 0.00 2 Azimikor MASP2 1 0.001429 No 0.001034 0.008235 1.38 0.17 2 Azimikor MASP2 2 0.003214 No 0.001034 0.008235 3.11 0.39 2 Azimikor MASP2 3 0.005 No 0.001034 0.008235 4.84 0.61 2 Azimikor MASP2 4 0.006429 No 0.001034 0.008235 6.22 0.78 2 Azimikor MASP2 5 0.009464 No 0.001034 0.008235 9.15 1.15 2 Azimikor MASP2 6 0.012857 Yes 0.001034 0.008235 12.44 1.56 3 Azimikor MASP3 1 0.0025 No 0.001404 0.009637 1.78 0.26 3 Azimikor MASP3 2 0.003393 No 0.001404 0.009637 2.42 0.35 3 Azimikor MASP3 3 0.005179 No 0.001404 0.009637 3.69 0.54 3 Azimikor MASP3 4 0.006786 No 0.001404 0.009637 4.83 0.70 3 Azimikor MASP3 5 0.009643 No 0.001404 0.009637 6.87 1.00 3 Azimikor MASP3 6 0.010714 Yes 0.001404 0.009637 7.63 1.11 4 Azimikor MASP4 1 0.00125 No 0.003417 0.012121 0.37 0.10 4 Azimikor MASP4 2 0.001964 No 0.003417 0.012121 0.57 0.16 4 Azimikor MASP4 3 0.003036 No 0.003417 0.012121 0.89 0.25 4 Azimikor MASP4 4 0.004 No 0.003417 0.012121 1.17 0.33 4 Azimikor MASP4 5 0.006143 No 0.003417 0.012121 1.80 0.51 4 Azimikor MASP4 6 0.007143 No 0.003417 0.012121 2.09 0.59 4 Azimikor MASP4 7 0.008036 No 0.003417 0.012121 2.35 0.66 4 Azimikor MASP4 8 0.009107 No 0.003417 0.012121 2.67 0.75 4 Azimikor MASP4 9 0.010179 No 0.003417 0.012121 2.98 0.84 4 Azimikor MASP4 10 0.01125 Yes 0.003417 0.012121 3.29 0.93 5 Azimikor MASP5 1 0.001607 No 0.004817 0.014703 0.33 0.11 5 Azimikor MASP5 2 0.003643 No 0.004817 0.014703 0.76 0.25 5 Azimikor MASP5 3 0.004286 No 0.004817 0.014703 0.89 0.29 5 Azimikor MASP5 4 0.005714 No 0.004817 0.014703 1.19 0.39 5 Azimikor MASP5 5 0.007143 No 0.004817 0.014703 1.48 0.49 5 Azimikor MASP5 6 0.008571 No 0.004817 0.014703 1.78 0.58 5 Azimikor MASP5 7 0.01 No 0.004817 0.014703 2.08 0.68 5 Azimikor MASP5 8 0.011429 No 0.004817 0.014703 2.37 0.78 5 Azimikor MASP5 9 0.012214 No 0.004817 0.014703 2.54 0.83 5 Azimikor MASP5 10 0.012857 No 0.004817 0.014703 2.67 0.87

244

Prism Name Cycle εexp Buckle εppbm εcebm εexp/εppbm εexp/εcebm (#) (N/A) (#) (mm/mm) (N/A) (mm/mm) (mm/mm) (N/A) (N/A) 6 ChaiElayer P1 1 0.0218 No 0.008853 0.016547 2.46 1.32 6 ChaiElayer P1 2 0.0256 Yes 0.008853 0.016547 2.89 1.55 7 ChaiElayer P2 1 0.0167 No 0.005691 0.013846 2.93 1.21 7 ChaiElayer P2 2 0.0187 Yes 0.005691 0.013846 3.29 1.35 8 ChaiElayer P3 1 0.0078 No 0.005618 0.012556 1.39 0.62 8 ChaiElayer P3 2 0.0108 No 0.005618 0.012556 1.92 0.86 8 ChaiElayer P3 3 0.0133 No 0.005618 0.012556 2.37 1.06 8 ChaiElayer P3 4 0.0161 Yes 0.005618 0.012556 2.87 1.28 9 ChaiElayer P4 1 0.00956 No 0.005618 0.012556 1.70 0.76 9 ChaiElayer P4 2 0.0122 Yes 0.005618 0.012556 2.17 0.97 10 ChaiElayer P5 1 0.00848 No 0.005618 0.012556 1.51 0.68 10 ChaiElayer P5 2 0.0121 Yes 0.005618 0.012556 2.15 0.96 11 ChaiElayer P6 1 0.0107 No 0.003611 0.01128 2.96 0.95 11 ChaiElayer P6 2 0.0135 Yes 0.003611 0.01128 3.74 1.20 12 ChaiElayer P7 1 0.0062 No 0.003611 0.01128 1.72 0.55 12 ChaiElayer P7 2 0.0088 No 0.003611 0.01128 2.44 0.78 12 ChaiElayer P7 3 0.0115 No 0.003611 0.01128 3.18 1.02 12 ChaiElayer P7 4 0.0143 Yes 0.003611 0.01128 3.96 1.27 13 ChaiElayer P8 1 0.00876 No 0.003611 0.01128 2.43 0.78 13 ChaiElayer P8 2 0.0118 Yes 0.003611 0.01128 3.27 1.05 14 ChaiElayer P9 1 0.0104 No 0.00388 0.010411 2.68 1.00 14 ChaiElayer P9 2 0.0139 Yes 0.00388 0.010411 3.58 1.34 15 ChaiElayer P10 1 0.0055 No 0.00388 0.010411 1.42 0.53 15 ChaiElayer P10 2 0.0081 No 0.00388 0.010411 2.09 0.78 15 ChaiElayer P10 3 0.0121 No 0.00388 0.010411 3.12 1.16 15 ChaiElayer P10 4 0.0131 Yes 0.00388 0.010411 3.38 1.26 16 ChaiElayer P11 1 0.00914 No 0.00388 0.010411 2.36 0.88 16 ChaiElayer P11 2 0.0125 Yes 0.00388 0.010411 3.22 1.20 17 ChaiElayer P12 1 0.00792 No 0.002494 0.009902 3.18 0.80 17 ChaiElayer P12 2 0.0111 Yes 0.002494 0.009902 4.45 1.12 18 ChaiElayer P13 1 0.00792 No 0.002494 0.009902 3.18 0.80 18 ChaiElayer P13 2 0.0111 Yes 0.002494 0.009902 4.45 1.12 19 ChaiElayer P14 1 0.0061 No 0.002494 0.009902 2.45 0.62 19 ChaiElayer P14 2 0.0087 Yes 0.002494 0.009902 3.49 0.88

245

Prism Name Cycle εexp Buckle εppbm εcebm εexp/εppbm εexp/εcebm (#) (N/A) (#) (mm/mm) (N/A) (mm/mm) (mm/mm) (N/A) (N/A) 19 ChaiElayer P14 3 0.0117 No 0.002494 0.009902 4.69 1.18 20 Creagh P1 1 0.036111 Yes 0.020331 0.031982 1.78 1.13 21 Creagh P2 1 0 No 0.020331 0.031982 0.00 0.00 22 Crysandis PER1 1 0 No 0.010788 0.022369 0.00 0.00 23 Crysandis PER2 1 0.010658 No 0.010788 0.022369 0.99 0.48 24 Crysandis PER3 1 0.02 No 0.010788 0.022369 1.85 0.89 25 Crysandis PER4 1 0.030658 Yes 0.010788 0.022369 2.84 1.37 26 Crysandis PER5 1 0.046316 Yes 0.010788 0.022369 4.29 2.07 27 Crysandis PSR1 1 0.030263 Yes 0.013359 0.025541 2.27 1.18 28 Crysandis PSR2 1 0.031053 Yes 0.009991 0.021386 3.11 1.45 29 Crysandis PSR3 1 0.030263 Yes 0.009029 0.020199 3.35 1.50 30 Crysandis PSR4 1 0.030395 Yes 0.007986 0.018913 3.81 1.61 31 Crysandis PSR5 1 0.031053 Yes 0.007459 0.018262 4.16 1.70 32 Crysandis PSR6 1 0.031053 Yes 0.007345 0.018122 4.23 1.71 33 Crysandis PSR7 1 0.031053 Yes 0.005752 0.016156 5.40 1.92 34 Crysandis PSR8 1 0.030921 Yes 0.005393 0.015713 5.73 1.97 35 Crysandis PSR9 1 0.030987 Yes 0.004504 0.014616 6.88 2.12 36 Crysandis PSR10 1 0.031053 Yes 0.004045 0.014051 7.68 2.21 37 Crysandis PSR11 1 0.031053 Yes 0.003083 0.012864 10.07 2.41 38 Goodsir P1 1 0.0029 No 0.015265 0.025463 0.19 0.11 38 Goodsir P1 2 0.0032 No 0.015265 0.025463 0.21 0.13 38 Goodsir P1 3 0.0066 No 0.015265 0.025463 0.43 0.26 38 Goodsir P1 4 0.0072 No 0.015265 0.025463 0.47 0.28 38 Goodsir P1 5 0.0135 No 0.015265 0.025463 0.88 0.53 38 Goodsir P1 6 0.0145 No 0.015265 0.025463 0.95 0.57 38 Goodsir P1 7 0.0235 Yes 0.015265 0.025463 1.54 0.92 39 Goodsir P2 1 0.0029 No 0.015152 0.025323 0.19 0.11 39 Goodsir P2 2 0.0032 No 0.015152 0.025323 0.21 0.13 39 Goodsir P2 3 0.0066 No 0.015152 0.025323 0.44 0.26 39 Goodsir P2 4 0.0072 No 0.015152 0.025323 0.48 0.28 39 Goodsir P2 5 0.0135 No 0.015152 0.025323 0.89 0.53 39 Goodsir P2 6 0.0145 No 0.015152 0.025323 0.96 0.57 39 Goodsir P2 7 0.0235 Yes 0.015152 0.025323 1.55 0.93 40 Goodsir P3 1 0 No 0.015152 0.025323 0.00 0.00

246

Prism Name Cycle εexp Buckle εppbm εcebm εexp/εppbm εexp/εcebm (#) (N/A) (#) (mm/mm) (N/A) (mm/mm) (mm/mm) (N/A) (N/A) 41 Goodsir P4 1 0.0029 No 0.015265 0.025463 0.19 0.11 41 Goodsir P4 2 0.0032 No 0.015265 0.025463 0.21 0.13 41 Goodsir P4 3 0.0066 No 0.015265 0.025463 0.43 0.26 41 Goodsir P4 4 0.0072 No 0.015265 0.025463 0.47 0.28 41 Goodsir P4 5 0.0135 No 0.015265 0.025463 0.88 0.53 41 Goodsir P4 6 0.0145 No 0.015265 0.025463 0.95 0.57 41 Goodsir P4 7 0.0235 Yes 0.015265 0.025463 1.54 0.92 42 Goodsir P5 1 0.0029 No 0.017127 0.02776 0.17 0.10 42 Goodsir P5 2 0.0032 No 0.017127 0.02776 0.19 0.12 42 Goodsir P5 3 0.0066 No 0.017127 0.02776 0.39 0.24 42 Goodsir P5 4 0.0072 No 0.017127 0.02776 0.42 0.26 42 Goodsir P5 5 0.0135 No 0.017127 0.02776 0.79 0.49 42 Goodsir P5 6 0.0145 No 0.017127 0.02776 0.85 0.52 42 Goodsir P5 7 0.0235 Yes 0.017127 0.02776 1.37 0.85 43 Goodsir P6 1 0.00195 No 0.027743 0.040857 0.07 0.05 43 Goodsir P6 2 0.00285 No 0.027743 0.040857 0.10 0.07 43 Goodsir P6 3 0.0058 No 0.027743 0.040857 0.21 0.14 43 Goodsir P6 4 0.0086 No 0.027743 0.040857 0.31 0.21 43 Goodsir P6 5 0.01425 No 0.027743 0.040857 0.51 0.35 43 Goodsir P6 6 0.01725 No 0.027743 0.040857 0.62 0.42 43 Goodsir P6 7 0.02425 No 0.027743 0.040857 0.87 0.59 44 Goodsir P7 1 0.00195 No 0.027743 0.040857 0.07 0.05 44 Goodsir P7 2 0.00285 No 0.027743 0.040857 0.10 0.07 44 Goodsir P7 3 0.0058 No 0.027743 0.040857 0.21 0.14 44 Goodsir P7 4 0.0086 No 0.027743 0.040857 0.31 0.21 44 Goodsir P7 5 0.01425 No 0.027743 0.040857 0.51 0.35 44 Goodsir P7 6 0.01725 No 0.027743 0.040857 0.62 0.42 44 Goodsir P7 7 0.02425 No 0.027743 0.040857 0.87 0.59 45 Goodsir P8 1 0.001 No 0.052452 0.07134 0.02 0.01 45 Goodsir P8 2 0.0025 No 0.052452 0.07134 0.05 0.04 45 Goodsir P8 3 0.005 No 0.052452 0.07134 0.10 0.07 45 Goodsir P8 4 0.01 No 0.052452 0.07134 0.19 0.14 45 Goodsir P8 5 0.015 No 0.052452 0.07134 0.29 0.21 45 Goodsir P8 6 0.02 No 0.052452 0.07134 0.38 0.28

247

Prism Name Cycle εexp Buckle εppbm εcebm εexp/εppbm εexp/εcebm (#) (N/A) (#) (mm/mm) (N/A) (mm/mm) (mm/mm) (N/A) (N/A) 45 Goodsir P8 7 0.025 No 0.052452 0.07134 0.48 0.35 46 Goodsir P9 1 0.001 No 0.052452 0.07134 0.02 0.01 46 Goodsir P9 2 0.0025 No 0.052452 0.07134 0.05 0.04 46 Goodsir P9 3 0.005 No 0.052452 0.07134 0.10 0.07 46 Goodsir P9 4 0.01 No 0.052452 0.07134 0.19 0.14 46 Goodsir P9 5 0.015 No 0.052452 0.07134 0.29 0.21 46 Goodsir P9 6 0.02 No 0.052452 0.07134 0.38 0.28 46 Goodsir P9 7 0.025 No 0.052452 0.07134 0.48 0.35

248