wind loading effects and equivalent static

WIND LOADING EFFECTS AND EQUIVALENT STATIC

WIND LOADING ON LOW-RISE BUILDINGS

by

XIAOHONG HU, M.S.

A DISSERTATION

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Douglas A. Smith, PH.D. Chairperson of the Committee

Kishor C. Mehta, PH.D.

Chris W. Letchford, PH.D.

Xinzhong Chen, PH.D.

Kathleen Gilliam, PH.D.

Accepted

John Borrelli Dean of the Graduate School

May, 2006

— — I

ACKNOWLEDGEMENT

I would like to acknowledge many people for helping me during my doctoral work. I

would especially like to thank my advisor, Dr. Smith, for his long time support and

encouragement. He always makes me feel better when I am nervous. I can still

remember what he said to me before my qualify exam, he said that he and all the

committee members are on my side to help finish my dissertation research, not to

embarrass me. He is a really nice teacher and if one day, I have the chance to be a

teacher, I would like to be a teacher like him.

Throughout my doctoral work Dr. Chen spend a lot of time to discuss with me about

my dissertation research; and I can always benefit from talk with him; from his papers,

I learned a lot of things, and his help makes my dissertation going more smoothly.

I am also very grateful for having an exceptional doctoral committee and wish to

thank Dr. Mehta, Dr. Letchford, and Dr. Gilliam. Although I didn’t talk with them

about my dissertation a lot, I once took their classes and from their classes, I learned a

lot of knowledge necessary to an engineer and I also learned the responsibility

involved to be a good engineer.

I extend many thanks to my colleagues and friends, especially Anjing Bi, Hua He,

Guoqing Huang, Dejiang Chen. They give me a lot of help in my study and life.

Finally, I'd like to thank my family. I'm grateful to my mother and my brother for their

encouragement and enthusiasm. I'm especially grateful to my husband, Rujin Ma and

my best friends, Qingfu Wang, Huaixin Zhang, Kunyu Li, Jian Dai, and Dian Wang

for their patience and for helping me keep my life in proper perspective and balance.

This research was funded by NIST and I would like to thank TTU for giving the

opportunity to study here and to see a country quite different from my own country

but as lovely as my country. And I also would like to thank Dr. Tamura for the data he

provided.

Xiaohong Hu

March 28, 2006

— — II

ABSTRACT

Wind-induced pressures acting on the Wind Engineering Research Field Laboratory

(WERFL) of Texas Tech University are integrated over each surface to obtain three

forces and moments at the base of the building along the three principal axes with its

origin at the geometric center of the building. Mean and fluctuating pressure

distributions around the WERFL building are investigated, and the pressure

distributions producing maximum fluctuating along-wind loading, across-wind

loading, and torsional moment are studied, and the correlation between these forces is

studied, a method to investigate the load combination of these forces is proposed.

WERFL building is also used for estimation of wind loading effects and

corresponding gust response factors and background factors, and a wind tunnel model

of Tokyo Polytechnic University is also utilized. The gust factors and back ground

factors of responses of these two buildings under wind loading are calculated

respectively. The responses calculated by pressure time histories are compared to

those calculated by applying ASCE7-05. Methods to investigate universal equivalent

static wind load are applied to both buildings. Several equivalent static wind loading

(ESWL) methods are compared, and the universal ESWL method is applied to

WERFL building and another modified universal ESWL method is also utilized for

WERFL building.

— — III

TABLE OF CONTENTS

ACKNOWLEDGEMENT I ABSTRACT II LIST OF TABLES V LIST OF FIGURES X CHAPTER I INTRODUCTION 1

1.1 Introduction with an Objective and Scope 1 1.1.1 Objective 2 1.1.2 Scope 2

1.2 Introduction 3 1.3 Wind Loads on Low-rise Building 6 1.4 Wind Loads on Tall Building 14

1.4.1 DGLF approach 16 1.4.2 MGLF approach 20 1.4.3 DRF approach 24 1.4.4 Linear Combination of Background and Resonant Equivalent Static Wind Loads 25 1.4.5 Across-wind and torsional wind loading 28

Reference 31 II INTEGRATED WIND LOADS ON A FULL-SCALE LOW-RISE

BUILDING 35

2.1 Introduction 35 2.2 Instantaneous Wind Pressure Distribution Causing Maximum Quasi-Steady Load Effects 36

2.2.1 Quasi-static base shear and torsional base moment 36 2.2.2 Correlation between the wind loads 39 2.2.3 Relationship between torsional moment and other forces 40

2.3 Wind induced internal stresses in structure members 41 2.4 Comparison to responses calculated by ASCE 43

2.4.1 ASCE (Figure 6-9) 43 2.4.2 ASCE (Figure 6-10) 43

2.5 Derivation of the wind load combination 44 2.6 Application of the Derived Equivalent Static Wind Loads to the Assumed Frame System 47 2.7 Conclusion Remarks 49 Reference 49

— — IV

III WIND INDUCED RESPONSES OF LOW-RISE BUILDING 72

3.1 Introduction 72 3.2 WERFL Building (Full scale building) 72

3.2.1 Gust Response Factors 73 3.2.2 Comparison to responses calculated by ASCE 75 3.2.3 Background Factors 76

3.3 Wind Tunnel Model in Tokyo Polytechnic University 78 3.3.1 Gust Factors 78 3.3.2 Background Factors 79

3.4 Background Factors Based on Four Gust Loading Envelops of WERFL Building 80 3.5 Concluding remarks 82 Reference 82

IV EQUIVALENT STATIC WIND LOAD 130

4.1 Introduction 130 4.2 LRC pressure distribution of WERFL building 131 4.3 Universal Equivalent Static Wind Load 133

4.3.1 Application of Universal ESWL to WERFL building 136 4.3.2 Modified Universal Equivalent Static Wind Load 140

4.4 Concluding remarks 144 Reference 145

V SUMMARY, CONTRIBUTIONS AND RECOMMENDATIONS 179

5.1 Summary 179 5.2 Contributions 180 5.3 Recommendations for future research 181

APPENDIX I-PART I: WERFL BUILDING 183 APPENDIX I-PART II: TAMURA WIND MODEL 199 APPENDIX II 211

— — V

LIST OF TABLES 2.1 Mean and standard deviation of the along-wind, across-wind and

torsional moment coefficients (AOA around 0º) 51 2.2 Maximum fluctuating along-wind, across-wind coefficients, torsional

moment coefficients, as well as lift force coefficient and their corresponding another three force coefficients (AOA round 0º) 52

2.2 (continued) 53 2.3 Mean and standard deviation of the along-wind, across-wind and

torsional moment coefficients (AOA around 90º) 54 2.4 Maximum fluctuating along-wind, across-wind coefficients, torsional

moment coefficients, as well as lift force coefficient and their corresponding another three force coefficients (AOA around 90º) 54

2.4 (continued) 55 2.5 The torsion coefficient of the four walls when AOA is close to zero 55 2.6 The torsion coefficient of the four walls when AOA is close to zero 56 2.7 Covariance of torsion coefficients of the four walls with along-wind

load (AOA around 0º) 57 2.8 The covariance of torsion coefficients of the four walls with across-

wind load (AOA around 0º) 57 2.9 The covariance of torsion coefficients of the four walls with along-

wind and across-wind load (AOA around 0º) 58 2.10 Influence coefficients 58 2.11 Peak normal stresses in column C1 (AOA close to zero) 59 2.12 Peak normal stresses in column C3 (AOA close to zero) 59 2.13 Peak fluctuating normal stresses in column C1 (AOA close to zero) 59 2.14 Peak fluctuating normal stresses in column C3 (AOA close to zero) 59 2.15 Comparison of actual response of normal stresses by ASCE(Fig.6-9) 60 2.16 Comparison of actual response of normal stresses by ASCE(Fig.6-10) 60 2.17 Correlation coefficients between along-wind load Fx, across-wind

load Fy, and torsional moment Mz (AOA around 0º) 60 2.18 rms value of the along-wind, across-wind and torsional moment

(AOA around 0º) 61 2.19 Mean value of the along-wind, across-wind and torsional moment

(AOA around 0º) 61 3.1 Mean, Maximum Responses and Corresponding Gust Response

Factors of Critical Section 3, 5, 7 on Frame A, (AOA=0º) 83 3.2 Mean, Maximum Responses and Corresponding Gust Response

Factors of Critical Section 3, 5, 7 on Frame B, (AOA=0º) 83 3.3 Mean, Maximum Responses and Gust Response Factors of Critical

Section 3, 5, 7 on Frame C, (AOA=0º) 84 3.4 Mean, Maximum Responses and Corresponding Gust Response

Factor of Critical Section 3, 5, 7 on Frame D, (AOA=0º) 84 3.5 Mean, Maximum Responses and Corresponding Gust Response

Factors of Critical Section 3, 5, 7 on Frame A, (AOA=90º) 85 3.6 Mean, Maximum Responses and Corresponding Gust Response

Factors of Critical Section 3, 5, 7 on Frame B, (AOA=90º) 85 3.7 Mean, Maximum Responses and Corresponding Gust Response

Factors of Critical Section 3, 5, 7 on Frame C, (AOA=90º) 86

— — VI

3.8 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 3, 5, 7 on Frame D, (AOA=90º) 86

3.9 Comparison between responses of Frame C calculated from ASCE (Figure 6-9) with actual responses 87

3.10 Comparison between responses calculated from ASCE and actual responses 87

3.11 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame A, (AOA=0º) 88

3.12 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame B, (AOA=0º) 88

3.13 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame C, (AOA=0º) 89

3.14 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame D, (AOA=0º) 89

3.15 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame A, (AOA=90º) 90

3.16 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame B, (AOA=90º) 90

3.17 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame C, (AOA=90º) 91

3.18 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame D, (AOA=90º) 91

3.19 Dynamic Responses of Critical Section 3, 5, 7 on Frames A B C D, (AOA=0º) 92

3.20 Dynamic Responses of Critical Section 3, 5, 7 on Frames A B C D, (AOA=90º) 93

3.21 Dynamic Response Factors of Critical Section 3, 5, 7 on Frames A B C D, (AOA=0º) 94

3.22 Dynamic Response Factors of Critical Section 3, 5, 7 on Frames A B C D, (AOA=90º) 95

3.23 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Across Frame A (AOA=0º) 96

3.24 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Across Frame B (AOA=0º) 97

3.25 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Across Frame C (AOA=0º) 98

3.26 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Along Frame A (AOA=0º) 99

3.27 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Along Frame B (AOA=0º) 100

3.28 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Along Frame C (AOA=0º) 101

3.29 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Across Frame A (AOA=0º) 102

3.30 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Across Frame B(AOA=0º) 103

3.31 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Across Frame C(AOA=0º) 104

3.32 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Along Frame A (AOA=0º) 105

— — VII

3.33 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Along Frame B (AOA=0º) 106

3.34 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Along Frame C (AOA=0º) 107

3.35 Maximum Responses of Critical Section 3, 5, 7 on Frames A B C D under four gust loading envelopes, (AOA=0º) 108

3.36 Maximum Responses of Critical Section 3, 5, 7 on Frames A B C D under four gust loading envelopes, (AOA=90º) 109

3.37 Background factor based on four gust loading envelopes, (AOA=0º) 110 3.38 Background factor based on four gust loading envelopes, (AOA=90º) 111 4.1 Combination coefficients and contribution factor (AOA=0º) 147 4.2 Combination coefficients and contribution factor (AOA=0º) 147 4.3 Combination coefficients and contribution factor (AOA=0º) 147 4.4 Combination coefficients and contribution factor (AOA=90º) 148 4.5 Combination coefficients and contribution factor (AOA=90º) 148 4.6 Combination coefficients and contribution factor (AOA=90º) 148 4.7 Combination coefficients and contribution coefficients (AOA=0º) 149 4.8 Combination coefficients and contribution coefficients (AOA=90º) 149 I-I-1 Mean response of Frame A (lb, lb-ft) (AOA=0º) 183 I-I-2 Absolute maximum total response of Frame A (lb, lb-ft) (AOA=0º) 183 I-I-3 Absolute maximum dynamic response of Across A (lb, lb-ft)

(AOA=0º) 183 I-I-4 Dynamic Responses under Absolute Maximum Fluctuating Pressure

Distribution Based on POD Pressure Sign, Frame A (AOA=0º) 184 I-I-5 Gust Factor, Frame A (AOA=0º) 184 I-I-6 Background Factor, Frame A (AOA=0º) 184 I-I-7 Mean response of Frame B (lb, lb-ft) (AOA=0º) 185 I-I-8 Absolute Maximum Total response of Across B (lb, lb-ft) (AOA=0º) 185 I-I-9 Absolute Maximum Dynamic response of Across B (lb, lb-ft)


Distribution Based on POD Pressure Sign, Frame B (AOA=0º) 186 I-I-11 Gust Factor, Frame B (AOA=0º) 186 I-I-12 Background Factor, Frame B (AOA=0º) 186 I-I-13 Mean response of Frame C (lb, lb-ft) (AOA=0º) 187 I-I-14 Absolute Maximum Total response of Across C (lb, lb-ft) (AOA=0º) 187 I-I-15 Absolute Maximum Dynamic response of Across C (lb, lb-ft)


Distribution Based on POD Pressure Sign, Frame C (AOA=0º) 188 I-I-17 Gust Factor, Frame C (AOA=0º) 188 I-I-18 Background Factor, Frame C (AOA=0º) 188 I-I-19 Mean response of Across D (lb, lb-ft) (AOA=0º) 189 I-I-20 Absolute Maximum Total response of Across D (lb, lb-ft) (AOA=0º) 189 I-I-21 Absolute Maximum Dynamic response of Across D (lb, lb-ft)


Distribution Based on POD Pressure Sign, Frame D (AOA=0º) 190 I-I-23 Gust Factor, Frame D (AOA=0º) 190 I-I-24 Background Factor, Frame D (AOA=0º) 190

— — VIII

I-I-25 Mean response of Across A (lb, lb-ft) (AOA=90º) 191 I-I-26 Absolute Maximum Total response of Across A (lb, lb-ft) (AOA=90º)

191 I-I-27 Absolute Maximum Dynamic response of Across A (lb, lb-ft)


Distribution Based on POD Pressure Sign, Frame A (AOA=90º) 192 I-I-29 Gust Factor, Frame A (AOA=90º) 192 I-I-30 Background Factor, Frame A (AOA=90º) 192 I-I-31 Mean response of Across B (lb, lb-ft) (AOA=90º) 193 I-I-32 Absolute Maximum Total response of Across B (lb, lb-ft) (AOA=90º)

193 I-I-33 Absolute Maximum Dynamic response of Across B (lb, lb-ft)


Distribution Based on POD Pressure Sign, Frame B (AOA=90º) 194 I-I-35 Gust Factor, Frame B (AOA=90º) 194 I-I-36 Background Factor, Frame B (AOA=90º) 194 I-I-37 Mean response of Across C (lb, lb-ft) (AOA=90º) 195 I-I-38 Absolute Maximum Total response of Across C (lb, lb-ft) (AOA=90º)

195 I-I-39 Absolute Maximum Dynamic response of Across C (lb, lb-ft)


Distribution Based on POD Pressure Sign, Frame C (AOA=90º) 196 I-I-41 Gust Factor, Frame C (AOA=90º 196 I-I-42 Background Factor, Frame C (AOA=90º) 196 I-I-43 Mean response of Across D (lb, lb-ft) (AOA=90º) 197 I-I-44 Absolute Maximum Total response of Across D (lb, lb-ft) (AOA=90º)

197 I-I-45 Absolute Maximum Dynamic response of Across D (lb, lb-ft)


Distribution Based on POD Pressure Sign, Frame D (AOA=90º) 198 I-I-47 Gust Factor, Frame D (AOA=90º) 198 I-I-48 Background Factor, Frame D (AOA=90º) 198 I-II-1 Mean response of Across A (KN, KN-m) 199 I-II-2 Absolute Maximum Total response of Across A (KN, KN-m) 199 I-II-3 Absolute Maximum Dynamic response of Across A (KN, KN-m) 199 I-II-4 Dynamic Responses under Absolute Maximum Fluctuating Pressure

Distribution Based on POD Pressure Sign, Across Frame A (KN, KN-m) 200

I-II-5 Gust Factor, Across Frame A 200 I-II-6 Background Factor, Across Frame A 200 I-II-7 Mean response of Across B (KN, KN-m) 201 I-II-8 Absolute Maximum Total response of Across B (KN, KN-m) 201 I-II-9 Absolute Maximum Dynamic response of Across B (KN, KN-m) 201 I-II-10 Dynamic Responses under Absolute Maximum Fluctuating Pressure

Distribution Based on POD Pressure Sign, Across Frame B (KN, KN-m) 202

— — IX

I-II-11 Gust Factor, Across Frame B 202 I-II-12 Background Factor, Across Frame B 202 I-II-13 Mean response of Across C (KN, KN-m) 203 I-II-14 Absolute Maximum Total response of Across C (KN, KN-m) 203 I-II-15 Absolute Maximum Dynamic response of Across C (KN, KN-m) 203 I-II-16 Dynamic Responses under Absolute Maximum Fluctuating Pressure

Distribution Based on POD Pressure Sign, Across Frame C (KN, KN-m) 204

I-II-17 Gust Factor, Across Frame C 204 I-II-18 Background Factor, Across Frame C 204 I-II-19 Mean response of Along A (KN, KN-m) 205 I-II-20 Absolute Maximum Total response of Along A (KN,KN-m) 205 I-II-21 Absolute Maximum Dynamic response of Along A (KN,KN-m) 205 I-II-22 Dynamic Responses under Absolute Maximum Fluctuating Pressure

Distribution Based on POD Pressure Sign, Along Frame A (KN, KN-m) 206

I-II-23 Gust Factor, Along Frame A 206 I-II-24 Background Factor, Along Frame A 206 I-II-25 Mean response of Along B (KN, KN-m) 207 I-II-26 Absolute Maximum Total response of Along B (KN, KN-m) 207 I-II-27 Absolute Maximum Dynamic response of Along B (KN, KN-m) 207 I-II-28 Dynamic Responses under Absolute Maximum Fluctuating Pressure

Distribution Based on POD Pressure Sign, Along Frame B (KN, KN-m) 208

I-II-29 Gust Factor, Along Frame B 208 I-II-30 Background Factor, Along Frame B 208 I-II-31 Mean response of Along C (KN, KN-M) 209 I-II-32 Absolute Maximum Total response of Along C (KN, KN-M) 209 I-II-33 Absolute Maximum Dynamic response of Along C (KN, KN-M) 209 I-II-34 Dynamic Responses under Absolute Maximum Fluctuating Pressure

Distribution Based on POD Pressure Sign, Along Frame C (KN, KN-m) 210

I-II-35 Gust Factor, Along Frame C 210 I-II-36 Background Factor, Along Frame C 210

— — X

LIST OF FIGURES

1.1 Examples of instantaneous pressure distributions causing maximum

wind force coefficients 12 1.2 Cross-correlation coefficients between wind forces 13 1.3 Cross-correlation coefficients between absolute wind forces 13 1.4. Probablistic dynamic-based approaches to gust loading 22 2.1 WERFL building of Texas Tech University 62 2.2 Pressure Tap Arrangement of WERFL Building 62 2.3 Mean pressure distribution at three pressure tap layers (AOA=0.1296º)

63 2.4 Fluctuating pressure distribution at three pressure tap layers

(AOA=0.1296º) 63 2.5 Instantaneous wall pressure distributions causing maximum

fluctuating quasi-static along-wind base shear FDmax at three pressure tap layers (AOA=0.1296º) 63

2.6 Instantaneous wall pressure distributions causing maximum fluctuating quasi-static across-wind base shear FLmax at three pressure tap layers (AOA=0.1296º) 63

2.7 Instantaneous wall pressure distributions causing maximum fluctuating quasi-steady base moment MTmax at three pressure tap layers (AOA=0.1296º) 64

2.8 Instantaneous wall pressure distributions causing maximum fluctuating quasi-steady Lift Force at three pressure tap layers (AOA=0.1296º) 64

2.9 Ensemble averaged mean wind pressure distributions at three pressure tap layers (AOA around 0º) 64

2.10 Ensemble averaged fluctuating wind pressure distributions at three pressure tap layers (AOA around 0º) 64

2.11 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static along-wind base shear at three pressure tap layers (AOA around 0º) 65

2.12 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static across-wind base shear at three pressure tap layers (AOA around 0º) 65

2.13 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static torsional base moment at three pressure tap layers (AOA around 0º) 65

2.14 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static lift force at three pressure tap layers (AOA around 0º) 65

2.15 Ensemble averaged mean wind pressure distributions at three pressure tap layers (AOA around 90º) 66

2.16 Ensemble averaged fluctuating wind pressure distributions at three pressure tap layers (AOA around 90º) 66

2.17 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static along-wind base shear at three pressure tap layers (AOA around 90º) 66

— — XI

2.18 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static across-wind base shear at three pressure tap layers (AOA around 90º) 66

2.19 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static torsional moment at three pressure tap layers (AOA around 90º) 67

2.20 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static lift force at three pressure tap layers (AOA around 90º) 67

2.21 The relationship between the maximum along-wind base shear CX and its simultaneously recorded across-wind base shear ratio CY/ CYmax and torsional base moment ratio CT/CTmax, CZ/CZmax (AOA=0º). 67

2.22 The relationship between the maximum across-wind base shear CY and its simultaneously recorded along-wind base shear ratio CX/CXmax and torsional base moment ratio CT/CTmax, CZ/CZmax (AOA=0º). 67

2.23 The relationship between the maximum torsional base moment CT and its simultaneously recorded along-wind base shear ratio CX/CXmax and across-wind base shear ratio CY/CYmax, CZ/CZmax (AOA=0º). 68

2.24 The relationship between the maximum torsional base moment CZ and its simultaneously recorded along-wind base shear ratio CX/CXmax and across-wind base shear ratio CY/CYmax, CT/CTmax (AOA=0º). 68

2.25 The relationship between the maximum along-wind base shear CY and its simultaneously recorded across-wind base shear ratio CX/CXmax and torsional base moment ratio CT/CTmax, CZ/CZmax.(AOA=90º) 68

2.26 The relationship between the maximum across-wind base shear CX and its simultaneously recorded along-wind base shear ratio CY/CYmax and torsional base moment ratio CT/CTmax, CZ/CZmax.(AOA=90º) 68

2.27 The relationship between the maximum torsional base moment CT and its simultaneously recorded along-wind base shear ratio CY/CYmax and across-wind base shear ratio CX/CXmax, CZ/CZmax.(AOA=90º) 69

2.28 The relationship between the maximum torsional base moment CZ and its simultaneously recorded along-wind base shear ratio CY/CYmax and across-wind base shear ratio CX/CXmax, CT/CTmax.(AOA=90º) 69

2.29a Cross-correlation coefficients between wind forces (AOA= 0.1297º) 69 2.29b Cross-correlation coefficients between wind forces (AOA= 0.1297º) 69 2.30a Cross-correlation coefficients between wind forces (AOA= 85.4797º) 70 2.30b Cross-correlation coefficients between wind forces (AOA= 85.4797º) 70 2.31 Ensemble averaged extreme wind pressure distributions causing

maximum quasi-static load effects at the base 70 2.32(a) Full design wind pressure of CASE1 of ASCE(Figure 6-9) 70 2.32(b) Full design wind pressure of CASE1 of ASCE(Figure 6-9) 71 2.33 Pressure distribution based on ASCE7 (Figure 6-10) for calculation 71 2.34 Frame model 71 2.35 Columns of the frame model 71 3.1 Frame System for WERFL Building 112 3.2 Critical Sections for Frames 112 3.3 Pressure Tap Arrangement 113 3.4 Frame Arrangement of Wind Tunnel Model 114 3.5 Pressure distribution base on ASCE 114

— — XII

3.6(a) Pressure distribution on Frame A based on ASCE 115 3.6(b) Pressure distribution on Frame B based on ASCE 115 3.6(c) Pressure distribution on Frame C based on ASCE 115 3.6(d) Pressure distribution on Frame D based on ASCE 115 3.7 Gust Loading Envelope Distribution Based on first POD mode on

Frame A at AOA=0º 116 3.8 Gust Loading Envelope Distribution Based on first POD mode on

Frame B at AOA=0º 116 3.9 Gust Loading Envelope Distribution Based on first POD mode on

Frame C at AOA=0º 116 3.10 Gust Loading Envelope Distribution Based on first POD mode on

Frame D at AOA=0º 117 3.11 Gust Loading Envelope Distribution Based on first POD mode on

Frame A at AOA=90º 117 3.12 Gust Loading Envelope Distribution Based on first POD mode on

Frame B at AOA=90º 117 3.13 Gust Loading Envelope Distribution Based on first POD mode on

Frame C at AOA=90º 118 3.14 Gust Loading Envelope Distribution Based on first POD mode on

Frame D at AOA=90º 118 3.15 Gust Loading Envelope 1 Distribution on Frame A at AOA=0º 118 3.16 Gust Loading Envelope 2 Distribution on Frame A at AOA=0º 119 3.17 Gust Loading Envelope 3 Distribution on Frame A at AOA=0º 119 3.18 Gust Loading Envelope 4 Distribution on Frame A at AOA=0º 119 3.19 Gust Loading Envelope 1 Distribution on Frame B at AOA=0º 120 3.20 Gust Loading Envelope 2 Distribution on Frame B at AOA=0º 120 3.21 Gust Loading Envelope 3 Distribution on Frame B at AOA=0º 120 3.22 Gust Loading Envelope 4 Distribution on Frame B at AOA=0º 121 3.23 Gust Loading Envelope 1 Distribution on Frame C at AOA=0º 121 3.24 Gust Loading Envelope 2 Distribution on Frame C at AOA=0º 121 3.25 Gust Loading Envelope 3 Distribution on Frame C at AOA=0º 122 3.26 Gust Loading Envelope 4 Distribution on Frame C at AOA=0º 122 3.27 Gust Loading Envelope 1 Distribution on Frame D at AOA=0º 122 3.28 Gust Loading Envelope 2 Distribution on Frame D at AOA=0º 123 3.29 Gust Loading Envelope 3 Distribution on Frame D at AOA=0º 123 3.30 Gust Loading Envelope 4 Distribution on Frame D at AOA=0º 123 3.31 Gust Loading Envelope 1 Distribution on Frame A at AOA=90º 124 3.32 Gust Loading Envelope 2 Distribution on Frame A at AOA=90º 124 3.33 Gust Loading Envelope 3 Distribution on Frame A at AOA=90º 124 3.34 Gust Loading Envelope 4 Distribution on Frame A at AOA=90º 125 3.35 Gust Loading Envelope 1 Distribution on Frame B at AOA=90º 125 3.36 Gust Loading Envelope 2 Distribution on Frame B at AOA=90º 125 3.37 Gust Loading Envelope 3 Distribution on Frame B at AOA=90º 126 3.38 Gust Loading Envelope 4 Distribution on Frame B at AOA=90º 126 3.39 Gust Loading Envelope 1 Distribution on Frame C at AOA=90º 126 3.40 Gust Loading Envelope 2 Distribution on Frame C at AOA=90º 127 3.41 Gust Loading Envelope 3 Distribution on Frame C at AOA=90º 127 3.42 Gust Loading Envelope 4 Distribution on Frame C at AOA=90º 127 3.43 Gust Loading Envelope 1 Distribution on Frame D at AOA=90º 128 3.44 Gust Loading Envelope 2 Distribution on Frame D at AOA=90º 128

— — XIII

3.45 Gust Loading Envelope 3 Distribution on Frame D at AOA=90º 128 3.46 Gust Loading Envelope 4 Distribution on Frame D at AOA=90º 129 4.1 Mean Pressure Distribution on Frame B (AOA=0º) 150 4.2 ESWL pressure distribution based on Conditional Sampling on

Frame B at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=0º) 150

4.3 ESWL pressure distribution based on LRC on Frame B at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=0º) 150

4.4 ESWL pressure distribution based on GLE on Frame B at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=0º) 150

4.5 Mean Pressure Distribution on Frame C (AOA=0º) 151 4.6 ESWL pressure distribution based on Conditional Sampling on

Frame C at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=0º) 151

4.7 ESWL pressure distribution based on LRC on Frame C at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=0º) 151

4.8 ESWL pressure distribution based on GLE on Frame C at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=0º) 151

4.9 Mean Pressure Distribution on Frame B (AOA=90º) 152 4.10 ESWL pressure distribution based on Conditional Sampling on

Frame B at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=90º) 152

4.11 ESWL pressure distribution based on LRC on Frame B at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=90º) 152

4.12 ESWL pressure distribution based on GLE on Frame B at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=90º) 152

4.13 Mean Pressure Distribution on Frame C (AOA=90º) 153 4.14 ESWL pressure distribution based on Conditional Sampling on

Frame C at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=90º) 153

4.15 ESWL pressure distribution based on LRC on Frame C at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=90º) 153

4.16 ESWL pressure distribution based on GLE on Frame C at critical section 3 causing maximum bending moment with support condition fix-fix (AOA=90º) 153

4.17 The relationship between the contribution coefficients and mode number at AOA=0º 154

4.18 The relationship between the contribution coefficients and mode number at AOA=90º 154

4.19 Universal ESWL distribution reproducing maximum axial force N (AOA=0º, mode number=5) 155

4.20 Universal ESWL distribution reproducing maximum shear force Q (AOA=0º, mode number=5) 155

— — XIV

4.21 Universal ESWL distribution reproducing maximum bending moment M (AOA=0º, mode number=5) 156

4.22 Universal ESWL distribution reproducing all maximum load effects simultaneously (AOA=0º, mode number=5) 156

4.23 Comparison of Actual Maximum Axial Forces and Axial Forces under Universal ESWL (AOA=0º, mode number=5) 157

4.24 Comparison of Actual Maximum Shear Forces and Shear Forces under Universal ESWL (AOA=0º, mode number=5) 157

4.25 Comparison of Actual Maximum Bending Moments and Bending Moments under Universal ESWL (AOA=0º, mode number=5) 157

4.26 Comparison of Actual Maximum Responses and Responses under Universal ESWL (AOA=0º, mode number=5) 157




















— — XV






















4.67 Modified Universal ESWL Distribution reproducing maximum axial force N (AOA=0º) 173

4.68 Modified Universal ESWL Distribution reproducing maximum shear force Q (AOA=0º) 173

4.69 Modified Universal ESWL Distribution reproducing maximum bending moment M (AOA=0º) 174

4.70 Modified Universal ESWL Distribution reproducing all the maximum internal forces N,Q,M (AOA=0º) 174

— — XVI

4.71 Comparison of Actual Maximum Axial Forces and Axial Forces under Modified Universal ESWL (AOA=0º) 175

4.72 Comparison of Actual Maximum Shear Forces and Shear Forces under Modified Universal ESWL (AOA=0º) 175

4.73 Comparison of Actual Maximum Bending Moments and Bending Moments under Modified Universal ESWL (AOA=0º) 175

4.74 Comparison of Actual Maximum Responses and Responses under Modified Universal ESWL (AOA=0º) 175

4.75 Modified Universal ESWL Distribution reproducing maximum axial force N (AOA=90º) 176

4.76 Modified Universal ESWL Distribution reproducing maximum shear force Q (AOA=90º) 176

4.77 Modified Universal ESWL Distribution reproducing maximum bending moment M (AOA=90º) 177

4.78 Modified Universal ESWL Distribution reproducing all the maximum internal forces N,Q,M (AOA=90º) 177

4.79 Comparison of Actual Maximum Axial Forces and Axial Forces under Modified Universal ESWL (AOA=90º) 178

4.80 Comparison of Actual Maximum Shear Forces and Shear Forces under Modified Universal ESWL (AOA=90º) 178

4.81 Comparison of Actual Maximum Bending Moments and Bending Moments under Modified Universal ESWL (AOA=90º) 178

4.82 Comparison of Actual Maximum Responses and Responses under Modified Universal ESWL (AOA=90º) 178

II-1 Pressure distribution on Frame A for Universal ESWL reproducing maximum axial forces N (mode=5, AOA=0º) 211

II-2 Pressure distribution on Frame B for Universal ESWL reproducing maximum axial forces N (mode=5, AOA=0º) 211

II-3 Pressure distribution on Frame C for Universal ESWL reproducing maximum axial forces N (mode=5, AOA=0º) 211

II-4 Pressure distribution on Frame D for Universal ESWL reproducing maximum axial forces N (mode=5, AOA=0º) 211

II-5 Pressure distribution on Frame A for Universal ESWL reproducing maximum shear forces Q (mode=5, AOA=0º) 212

II-6 Pressure distribution on Frame B for Universal ESWL reproducing maximum shear forces Q (mode=5, AOA=0º) 212

II-7 Pressure distribution on Frame C for Universal ESWL reproducing maximum shear forces Q (mode=5, AOA=0º) 212

II-8 Pressure distribution on Frame D for Universal ESWL reproducing maximum shear forces Q (mode=5, AOA=0º) 212

II-9 Pressure distribution on Frame A for Universal ESWL reproducing maximum bending moment M (mode=5, AOA=0º) 213

II-10 Pressure distribution on Frame B for Universal ESWL reproducing maximum bending moment M (mode=5, AOA=0º) 213

II-11 Pressure distribution on Frame C for Universal ESWL reproducing maximum bending moment M (mode=5, AOA=0º) 213

II-12 Pressure distribution on Frame D for Universal ESWL reproducing maximum bending moment M (mode=5, AOA=0º) 213

II-13 Pressure distribution on Frame A for Universal ESWL reproducing all the maximum responses NQM (mode=5, AOA=0º) 214

— — XVII

II-14 Pressure distribution on Frame B for Universal ESWL reproducing all the maximum responses NQM (mode=5, AOA=0º) 214

II-15 Pressure distribution on Frame C for Universal ESWL reproducing all the maximum responses NQM (mode=5, AOA=0º) 214

II-16 Pressure distribution on Frame D for Universal ESWL reproducing all the maximum responses NQM (mode=5, AOA=0º) 214























— — XVIII


























— — XIX


























— — XX









II-97 Pressure distribution on Frame A for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=0º) 235

II-98 Pressure distribution on Frame B for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=0º) 235

II-99 Pressure distribution on Frame C for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=0º) 236

II-100 Pressure distribution on Frame D for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=0º) 236

II-101 Pressure distribution on Frame A for Universal ESWL reproducing maximum shear forces Q based on LRC method (AOA=0º) 236

II-102 Pressure distribution on Frame B for Universal ESWL reproducing maximum shear forces Q based on LRC method (AOA=0º) 236

II-103 Pressure distribution on Frame C for Universal ESWL reproducing maximum shear forces Q based on LRC method (AOA=0º) 237

II-104 Pressure distribution on Frame D for Universal ESWL reproducing maximum shear forces Q based on LRC method (AOA=0º) 237

II-105 Pressure distribution on Frame A for Universal ESWL reproducing maximum bending moment M based on LRC method (AOA=0º) 237

II-106 Pressure distribution on Frame B for Universal ESWL reproducing maximum bending moment M based on LRC method (AOA=0º) 237

II-107 Pressure distribution on Frame C for Universal ESWL reproducing maximum bending moment M based on LRC method (AOA=0º) 238

II-108 Pressure distribution on Frame D for Universal ESWL reproducing maximum bending moment M based on LRC method (AOA=0º) 238

II-109 Pressure distribution on Frame A for Universal ESWL reproducing all the maximum responses NQM based on LRC method (AOA=0º) 238

II-110 Pressure distribution on Frame B for Universal ESWL reproducing all the maximum responses NQM based on LRC method (AOA=0º) 238

II-111 Pressure distribution on Frame C for Universal ESWL reproducing all the maximum responses NQM based on LRC method (AOA=0º) 239

II-112 Pressure distribution on Frame D for Universal ESWL reproducing all the maximum responses NQM based on LRC method (AOA=0º) 239

II-113 Pressure distribution on Frame A for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=90º) 239

— — XXI

II-114 Pressure distribution on Frame B for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=90º) 239

II-115 Pressure distribution on Frame C for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=90º) 240

II-116 Pressure distribution on Frame D for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=90º) 240

II-117 Pressure distribution on Frame A for Universal ESWL reproducing maximum shear forces Q based on LRC method (AOA=90º) 240

II-118 Pressure distribution on Frame B for Universal ESWL reproducing maximum shear forces Q based on LRC method (AOA=90º) 240

II-119 Pressure distribution on Frame C for Universal ESWL reproducing maximum shear forces Q based on LRC method (AOA=90º) 241

II-120 Pressure distribution on Frame D for Universal ESWL reproducing maximum shear forces Q based on LRC method (AOA=90º) 241

II-121 Pressure distribution on Frame A for Universal ESWL reproducing maximum bending moment M based on LRC method (AOA=90º) 241

II-122 Pressure distribution on Frame B for Universal ESWL reproducing maximum bending moment M based on LRC method (AOA=90º) 241

II-123 Pressure distribution on Frame C for Universal ESWL reproducing maximum bending moment M based on LRC method (AOA=90º) 242

II-124 Pressure distribution on Frame D for Universal ESWL reproducing maximum bending moment M based on LRC method (AOA=90º) 242

II-125 Pressure distribution on Frame A for Universal ESWL reproducing all the maximum responses NQM based on LRC method (AOA=90º) 242

II-126 Pressure distribution on Frame B for Universal ESWL reproducing all the maximum responses NQM based on LRC method (AOA=90º) 242

II-127 Pressure distribution on Frame C for Universal ESWL reproducing all the maximum responses NQM based on LRC method (AOA=90º) 243

II-128 Pressure distribution on Frame D for Universal ESWL reproducing all the maximum responses NQM based on LRC method (AOA=90º) 243

— — 1

CHAPTER I

INTRODUCTION

1.1 Introduction with an Objective and Scope

Every year, a lot of buildings are damaged due to hurricane, tornado and some other

wind loadings. In structural design, buildings are under many kinds of loading, such

as dead load, live load, snow load and so on. Besides these loads, wind loading is also

a critical loading needed to be considered. Wind loading is complicated due to several

factors, such as the surrounding category, environment, climate and other factors;

wind loading of low-rise building is even more complicated than wind loading of

high-rise building, since most low-rise buildings are immersed within the

aerodynamic roughness on earth’s surface, for them, interference and shelter effects

are also important for wind loads assessment but hard to quantify, and the wind

loading on roof is also needed to be considered, the internal pressure may also be

significant.

How to obtain wind loading of low-rise buildings has long been the question in wind

engineering. And many engineers have put a lot of effort into solving this question. In

the past years, several methods have been developed such as computational wind

engineering method, full scale experiment and wind tunnel test to obtain pressure

information around a building. Based on these pressure information, database assisted

design and equivalent static wind loading(ESWL) methods can be used to establish

wind loading for structural design.

The focus of this dissertation is equivalent static wind loading (ESWL). This

dissertation aims to give a clear understanding of wind induced loading of buildings,

and will focus on wind loading of low-rise buildings. Several methods for

investigation of equivalent static wind load are studied to give a clarified overall

review and their limitations are pointed out.

To investigate wind loading on low-rise buildings, it is necessary to have some idea

about wind pressure distribution characteristics around the building, and the

correlation between wind loadings in different directions is also worthwhile to be

studied. The wind loading correlation is a basic step to investigate wind loading

combination, for low-rise building, wind loading combination is also a topic needed to

— — 2

be considered. Equivalent static wind loading(ESWL) is a mainly method to establish

wind loading, so it is also important for wind engineers to investigate convenient and

accurate ESWL method, in this dissertation, several ESWL methods are applied to

low-rise building, their disadvantages are pointed out and some new concept of

universal ESWL is proposed, and corresponding methods are studied. Based on what

have been discussed above, the objectives and scope are stated as follows.

1.1.1 Objective

The main objectives of the thesis are,

(1) to give a view of pressure distributions characteristics around low-rise building;

(2) to investigate the mechanism of wind induced torque and its correlation with

along-wind and across-wind loads;

(3) to investigate the critical wind loads combination of wind induced torque and

other loads such as along wind and across wind loads for low rise buildings;

(4) to investigate wind induced responses in low-rise buildings;

(5) to compare several equivalent static wind load methods;

(6) to propose a new equivalent static wind load method.

1.1.2 Scope

Wind-induced pressures acting on the Wind Engineering Research Field Laboratory

(WERFL) building of Texas Tech University are integrated over each surface to

obtain three forces and moments at the base of the building along the three principal

axes with its origin at the geometric center of the base of the WERFL building. Mean

and fluctuating pressure distributions around the WERFL building are investigated,

and the pressure distributions producing maximum fluctuating along-wind, across-

wind, and torsional moment at the base of the building are studied, and the correlation

between these forces is studied, a method to investigate the load combination of these

forces is proposed.

Wind Engineering Research Field Laboratory (WERFL) of Texas Tech University is

also used for estimation of wind loading effects and corresponding gust response

factors and some other factors, and a wind tunnel model of Tokyo Polytechnic

University is also utilized. The gust factors of responses of these two buildings under

wind loading are calculated respectively. Methods to investigate universal equivalent

static wind load are applied to both buildings. Several equivalent static wind loading

— — 3

(ESWL) methods are compared, and the universal ESWL method is applied to

WERFL building and another modified universal ESWL method is also utilized for

WERFL building.

1.2 Introduction

Buildings immersed in the wind field will be subject to complicated wind loads. For a

long time, wind loads acting on buildings have been the research focus for wind

engineers. Wind induced loads are very complex and will be influenced by many

factors such as the environments where the structures locate, the geometry of

structures etc.

For wind engineering structural design, majority buildings belong to low-rise rigid

buildings. Wind loading is one of the most sophisticated requirements in building

design; and their effect on low-rise buildings is also a major concern of building

design. Accurate estimates of wind loading effect are very important and will directly

leads to reduction of wind induced damage.

Many researches have been done on wind induced effects, and several techniques

have been developed to improve the progress of wind engineering.

Utilization of manifolds and development of the well-known pneumatic averaging

technique by Surry and Stathopoulos (Surry, 1977/78) contribute a lot to physical

modeling in wind tunnel. Besides wind tunnel tests, full-scale studies of wind loads on

low-rise buildings were also been carried out. Considerable developments in

electronic instrumentation and computer based statistical analysis techniques

contributed a lot to full-scale experiments, and the full-scale studies provided a vast

body of data which challenged wind tunnel modeling techniques.

Wind tunnel and field experimentation are the traditional approaches for the

investigation of wind-induced pressure fluctuations and time histories. However, the

collection of long time histories of wind and pressure data might be time consuming

and laborious, considering the inherent variability in such time histories affected by

building geometry, measurement location, surroundings and other factors. But

computer simulation using probabilistic and statistical models can efficiently solve

those complicated factors. Generally speaking, in computer simulation, techniques

which can be used to simulate Gaussian and Non-Gaussian are divided into two

groups, that is, Faster Fourier Transformation (FFT) and Auto Regressive Moving

— — 4

Average (ARMA) (Stathopoulos, 2003). And a new technique by combining Fourier

transformation, autoregressive model with non-Gaussian input process and phase

transformation of Fourier coefficients in Fourier representation of time series has been

proposed by Seong and Peterka (Seong, 1993). Computation simulation is very

successful in generating limitless amount of data and pressure coefficient time

histories generated from direct wind tunnel experiments or via computer simulation

can be reconstituted as full-scale pressure time histories for any full-scale wind speed.

It should be noted that this computer simulation can be used in combination with the

wind tunnel generated pressure coefficient databases for purposes of interpolation

between different building geometries and exposures.

Besides these techniques discussed above, Computational Wind Engineering (CWE)

deals with the application of Computational Fluid Dynamics (CFD) methodologies,

typically numerical solutions of Navier–Stokes equations using appropriate turbulence

models into wind engineering problems. At the present time, the application of CFD

into wind engineering is limited mainly due to difficulties related to the specific

features of atmospheric boundary layer flow and structures of wind engineering

interest, such as high turbulence, high Reynolds number, 3D flow field, bluff bodies

and associated flow separation and vortex shedding. Although it has tremendous

potential for engineering use, CWE still has a long way to go to become truly useful

to the design practitioner. However, for applications to which mean wind flows and

pressures may be important, CWE can be used because of its reduced time and cost, at

least for preliminary design purposes.

All the methods mentioned above are aimed to obtain pressure data and information,

after enough pressure data are obtained, they can be used to establish wind loading for

design. And all these data are basic resources that are utilized for the formation of

most codes and standards.

In structural design, for most buildings and structures, standards and codes provide

wind loading to use. For buildings which are not included in contemporary wind

codes and standards, database-assisted design can be utilized. The concept of

database-assisted design was proposed by Simiu and Stathopoulos(Simiu, 1997) and

Whalen et al.(Whalen, 1998) as a means of providing future code alternatives that

would make direct use of stored pressure time series for the design of low buildings.

The idea is not really new but the development of electronically scanned pressure

— — 5

measurements and increased information storage and computational capacities make it

now possible.

Besides database assisted method, equivalent static wind loading is another method to

establish wind loading. In wind engineering practice, wind induced loading is usually

represented by equivalent static wind loading. Taking advantage of the spectral

descriptions of wind loads and their effects on buildings, separation of the dynamic

response (excluding the mean component) and the associated ESWL into background

(quasi-static) and resonant components provides a more efficient response prediction

framework and a physically more meaningful description of loading.

Several methods are developed to establish equivalent static wind loading(ESWL).

Kasperski and Niemann once proposed the correlation based LRC (Load-Response

Correlation) method (Kasperski, 1992), which provides a sound theoretical basis for

estimating the expected equivalent static load distributions for the background

fluctuating wind loads. The extreme pressure distributions causing maximum

responses can be computed from Kasperski’s LRC formula. It should be noted that

LRC pressure distribution depends on influence functions of the response considered.

Different responses have different corresponding LRC pressure distributions.

Holmes also proposed some expressions for equivalent static wind load distributions,

all possible combinations of mean, background and resonant response are given

(Holmes, 2002). In his paper, structures for which the responses to wind can be

considered as quasi-static, and those for which resonant responses are significant are

considered.

Based on the characteristics of wind induced responses, wind loads on buildings can

be studied by being categorized as wind loads on low-rise buildings and high-rise

buildings respectively. A detail description of wind loading on low-rise building and

high-rise building will be given below although high-rise building is not the focus of

this dissertation, there are similarities existing between the methods to investigate

wind loading of low and high rise buildings, to make a comprehensive view of wind

loading, it is still necessary to pay some attention to wind loading on high-rise

building.

— — 6

1.3 Wind Loads on Low-rise Building

Several factors will make the estimation of wind loads for low-rise buildings as

complicated as that for tall buildings. First, usually low-rise buildings are immersed

within the aerodynamic roughness on earth’s surface, where the turbulence intensities

are rather high, and interference and shelter effects are also important for wind loads

assessment but hard to quantify; second, roof loadings of low-rise buildings are

variable due to different geometries, and are critical for the design of low-rise

buildings because many structural failures are initiated by great suctions on the roof;

thirdly, low-rise buildings usually have a single internal space, the internal pressures

can be significant, and the magnitude of internal pressure peaks, and their correlation

with external pressure peaks needs to be assessed.

A very useful and direct method which can be used to investigate wind loads on low-

rise buildings is wind tunnel test. Two earliest investigations were carried out by

Irminger in Copenhagen, Denmark, and Kernot in Melbourne, Australia. Over the

following years, many wind tunnel tests have been carried out by scholars all around

the world. It was until the 1950s that Jensen, at Technical University of Denmark,

explained the differences between full-scale and wind tunnel model measurements of

wind pressures. The work of Jensen and Franck was the precursor to a series of

extensive wind studies of wind loads on low-rise buildings in the 1970s and 1980s.

Besides wind tunnel tests, full-scale studies of wind loads on low-rise buildings were

also been carried out. Considerable developments in electronic instrumentation and

computer based statistical analysis techniques contributed a lot to full-scale

experiments, and the full-scale studies provided a vast body of data which challenged

wind tunnel modeling techniques.

For engineering practice, wind loads of low-rise buildings usually are expressed in the

form of equivalent or effective static wind loads. Equivalent static wind load

distributions are those loadings that produce the correct expected values of peak load

effects, such as the bending moments, or deflections, induced by fluctuating wind

loading.

Consequently a simplified approach to analyze the dynamic response induced by wind

was introduced, which involves the concept of gust loading factor (Davenport, 1967).

This method is first used for high-rise building, and it enables the estimation of a

— — 7

single factor, which can be applied to a calculated mean wind response, such as base

bending moment of tall buildings and allows the maximum load effects in a certain

time interval to be estimated. This approach takes both quasi-steady or background

and resonant dynamic effects into consideration. However a limitation is

automatically involved in this method, because it implies that the effective static load

distribution corresponding to the maximum structural response has the same shape as

the mean wind load distribution. For small structures, this assumption is valid, and

often gives results acceptable for the order of accuracy expected in a wind loading

standard. However in some cases, this assumption will be very misleading for

structures with complicated influence lines.

Another approach to investigate the equivalent static load distribution is to use a

conditional sampling, in which the time histories of wind pressures are searched to

identify the instantaneous pressure distribution when peak load response appears. This

direct method was once used by several wind laboratories for commercial wind-tunnel

studies of wind loads on large roofs and other structures.

Kasperski and Niemann (Kasperski, 1992) proposed the correlation based LRC

method, which provides a sound theoretical basis for estimating the expected

equivalent static load distributions for the background fluctuating wind loads. Tamura

also proved the validity of LRC method (Tamura, 2001, 2003).

For low-rise buildings, when the resonant response is negligible, the LRC method can

be used to determine the equivalent static load distributions from mean and

background components. The LRC method gives an expected load distribution

corresponding to a particular load response r, with influence coefficient )(zI r , which

means the value of r when a unit load is applied at position z. A single position

variable z is used to demonstrate this method, but this method can be easily extended

to two or three dimensions. A detailed explanation of this method is given below.

The instantaneous value of r can be expressed as

∫=L

r dzzItzptr0

)(),()( (1.1)

where ),( tzp is the fluctuating pressure at z, and L is the length of the structure,

)(zI r is the influence coefficient.

— — 8

The mean response of r(t) is

∫=L

r dzzItzpr0

)(),( (1.2)

where ),( tzp is the mean pressure at z

The standard deviation of r, Br ,σ , is given by

2/1

0 210 212'

1'

, ))()(),(),((∫ ∫=L L

rrBr dzdzzIzItzptzpσ (1.3)

where, the subscript of B represents background response, the superscript indicates a

fluctuating quantity with mean value subtracted, ),(' tzp represents the fluctuating

pressure at z.

So the expected maximum response r will be expressed as

BrBgrr ,ˆ σ+= (1.4)

Where Bg is the background peak factor, generally taken as 3.5.

Correspondingly, the LRC formula for pressure distribution is as follows

)()()()]([ zzgzpzp pprBr σρ+= (1.5)

where, )()( zzg pprB σρ is pressure distribution for background response. )(zprρ is the

correlation coefficient between ),(' tzp and )(' tr , which is given by:

Brp

L

r

pr z

dzzItzptzpz

,

0111

''

)(

)(),(),()(

σσρ

∫= =

Brp

L

r

z

dzzItzptzp

,

0111

''

)(

)(),(),(

σσ

∫ (1.6)

where Br ,σ is given by equation (1.3), )(zpσ is the standard deviation of pressure at

position z.

Generally, the fluctuating pressures are measured by wind tunnel tests or full-scale

measurements, so ),(' tzp can’t be expressed as continuous function, for a structure

surface divided into discrete panels, equation (1.6) can be put in another form as

follows:

— — 9

Brpj

N

iiji

rpj

tptp

,

1

''

,

)()(

σσ

βρ

∑== (1.7)

In this case Br ,σ can be expressed as

∑∑= =

=N

i

N

jjijiBr tptp

1 1

2/1'', ))()(( ββσ (1.8)

where i and j represent panel or tap number, and N is the total number of measure

channels. jp ' represents the fluctuating pressure at panel j or tap j; jβ is the influence

coefficient of panel j on the load effect r, and incorporates the area of the panel or tap,

which means that jβ is the value of r when a uniformly distributed patch load of unit

magnitude is applied to the panel. So the background equivalent static load

distribution is expressed as:

2/1

1 1

''

1

''

,

))()((

)()(

∑∑

∑

= =

=== N

ij

N

jiji

N

iiji

BpjpjBjB

tptp

tptpggp

ββ

βσρ (1.9)

It should be noted that LRC method provides an expected distribution for background

or quasi-static fluctuating loading. Distributions obtained directly by conditional

sampling of pressure distribution which produce the peak load effect, will converge to

that given by the LRC formula, when ensemble-averaged over a large number of

samples. LRC can’t take resonant response into consideration. And from the

procedure of LRC method, it can be easily seen that for each different response, the

LRC pressure distribution will be different, which limits the application of LRC

method.

All the above methods discussed above are mainly focused on along-wind loading.

For across-wind loading and torsional moment, until now, almost no theoretical

analysis has been done on the estimation of wind loading in these two directions of

low-rise buildings.

Due to the complexity of the across wind load and torsion, physical modeling of fluid

structure interactions seems the only feasible means of obtaining information on this

kind of wind loads, although research of computational fluid dynamics has made great

— — 10

progress in numerically generating flow field around bluff body induced by turbulent

flows. However there are difficulties involved in the wind tunnel tests since wind

tunnel tests are generally time-consuming and expensive. In practice, it is impossible

to conduct wind tunnel tests for all kinds of buildings with different shapes. Therefore,

major building codes and standards have begun to develop empirical relationships to

produce an estimation procedure to evaluate the across wind and torsional dynamic

responses in preliminary structural design, while a further wind tunnel testing for the

final design is necessary.

Among all the building codes, Japan (AIJ) (AIJ, short for Architectural Institute of

Japan, 1996) takes into consideration of along wind loading, across wind loading and

torsion, while Australia(AS1170.2-1989, 1989) and Canada (NRCC,1996) have

addressed both the along wind and across wind response in their current standards.

ASCE 7 (ASCE 7-98, 2000) specifies an unbalanced load equivalent to an eccentricity

of e/D =3.5 percent, and Eurocode 1 (ENV 1991-2-4, 1994) specifies an eccentricity

of 10 percent. Understandably, the development of generalized equations for across

wind and torsional dynamic responses, based on wind tunnel testing, is a valuable

addition to any standard, serving as a cost-effective and time-saving tool in daily

design.

As has been mentioned above, AIJ treats the torsional response of buildings; an

empirical expression for the torsional response was based on a set of wind tunnel

experiments and the experimental data of the response angle acceleration was

collected, a non-dimensional expression for the acceleration was proposed just for

buildings which have negligible eccentric effects. It can be easily seen that the

torsional effects treated in AIJ is just for relatively tall and slender buildings which

have dynamic response under the wind action. For rigid structures which have no

evident dynamic response under wind, ASCE-98 specified an eccentricity expressed

as a certain eccentricity coefficient multiplied by the characteristic length of the

structure. Although torsion effects have been mentioned in those two building codes,

a deep investigation and validation needs to be completed to make the provisions

reliable. Actually, the values produced by the equation of the torsional acceleration in

AIJ are shown to typically overestimate the measured angular tip acceleration by 30%

in comparison with full scale data (Tamura, 2001).

— — 11

To make the mechanism of wind induced torsion more clear, researches about the

wind distribution around a bluff structure have been carried out; obviously this kind of

research must base on extensive wind tunnel tests.

Boggs (Daryl, 2000) in his paper identifies some common sources of torsional loading

in terms of building shape, interfering effects of nearby buildings, and dynamic

characteristics of the structural frame. In addition, it was shown that torsional loading

is routinely larger than that provided for design in most standards. Boggs’ works are

mainly focused on tall buildings. Investigations of torsional loads on middle rise and

low rise buildings are even less.

Usually the wind induced loadings of a certain building are studied separately,

however in fact, the building is under the action of all the wind induced loading in

three dimensional directions. So besides the across-wind loads and torsional loads,

another wind loading topic that needs to be paid attention to is the wind load

combination of along-wind, across-wind loading and torsional moment.

Many researchers such as Melbourne (Melbourne, 1975), Vickery and Basu (Vickery,

1984), and Solari and Pagnini (Solari, 1999), have investigated the along-wind load

and across-wind load combinations, but they mainly focused on the behavior of tall

buildings and structures. In the design of low-rise and middle-rise buildings, for

which the along-wind response is predominant, the combination of along-wind,

across-wind loads and torque is tended to be ignored, since it is commonly known that

the along-wind force fluctuations are mainly caused by the turbulence in the

approaching wind, the across-wind load and torsional moment fluctuations are

dominantly induced by vortex shedding and thus it is supposed that the along-wind

loading has no correlation with across-wind load and torsional moment.

Wind load combination of low-rise buildings was once studied by Tamura.et.al. In his

paper (Tamura, 2003), wind pressures acting on a building are integrated over all its

surfaces and quasi-steady along-wind, across-wind loading and torque at the base of

the low rise buildings were first obtained, and then the extreme instantaneous pressure

distributions were investigated when the along-wind and across-wind loads, and the

torsional moment at the base of the low rise building are maximum.

From the selected samples, the pressure distributions when along-wind loading and

torque are maximum, are similar to each other, while the similarity of pressure

— — 12

distributions coinciding to maximum across wind and torsional moment is not

significant, which conflicts with the general knowledge that across-wind load is

correlated with torsional moment while the along-wind load is supposed to have no

correlation with across-wind load and torsional moment. The following graph shows

examples of instantaneous extreme pressure distribution for maximum along-wind,

across-wind loads and torsional moment in the paper.

(a) causing CDmax (b) causing CLmax (c) causing CMTmax

Figure 1.1 Examples of instantaneous pressure distributions causing maximum wind force

coefficients

Tamura also found out that when the along-wind load is maximum, the corresponding

torsional moment can be any values; while when the torsional moment is maximum,

the corresponding along-wind load will be very close to its maximum value, which

further proves the correlation between along-wind load and torsional moment. Tamura

explains the correlation between the along-wind loading, across-wind loading and

torsional moment by investigating the correlation of the absolute values of time

histories of these three forces, which on the surface indeed gives a good explanation,

since the correlation between the absolute value of along-wind loading and torsional

moment time histories, or along-wind loading and across-wind loading time histories

is really high with correlations close to 0.5 while the correlation coefficient between

across-wind loading and torsional moment time histories is just about 0.2. The

following graph show the cross correlation of CD, CL, CMT and cross correlation of

absolute CD, CL, CMT in Tamura’s paper.

— — 13

CD- CL CD- CMT CL- CMT

Figure 1.2 Cross-correlation coefficients between wind forces

CD- CL CD- CMT CL- CMT

Figure 1.3 cross-correlation coefficients between absolute wind forces

Tamura’s research challenged the widely accepted knowledge that along-wind loading

has no significant correlation with across-wind loading and torsional moment.

However, there are several problems involved in the above mentioned correlation

research, which need to be investigated furthermore.

First, the similarity between the instantaneous extreme distribution of along-wind load

and torsional moment is proved based on several selected samples which limited its

generality. And furthermore the absolute correlation has no physical and mathematical

meaning, because in engineering design, the sign of the along-wind and across-wind,

torsional moment do matter, for example, if the normal stress or shear stress in a

structural member is considered, the sign of the external force should be considered.

And also making absolute to data will change the properties of time series, and will

make the original zero mean fluctuating external force time history physically non-

meaningful. So a more reliable and reasonable explanation needs to be provided.

Although there are some problems involved in Tamura’s analysis, Tamura’s study did

point out the importance of considering wind load combination for structural design.

Tamura in his paper (Tamura, 2003) considered a simple frame model with four

— — 14

columns on the corner and stiff roof, it is found out that the total load effects of the six

wind load components for low-rise building models result in a 30% increase on

average of the peak normal stress in column members compared with the case only

applying the along-wind load.

Few structure codes and standards have considered wind load combination. ASCE 7-

2005 just gives simple wind load combinations for buildings higher than 60ft, in this

case, 75% of along-wind load and the same values are simultaneously applied in the

across-wind direction, while the torsional load is taken into consideration by using an

eccentricity as its previous version. Besides American Codes, AS1170.2 considers the

load combination by providing a formula for peak vector moment, and it is assumed

that the peak resultant base moment is equal to the peak along-wind moment when the

across-wind response is zero or across-wind dynamic response is less than along-wind

response. The consideration of wind load combination in Codes or Standards is really

coarse and actually few data have been reported on wind load combinations for low

and middle rise buildings.

From the discussion above, it can be seen that for low-rise buildings, wind induced

torsion is still a research field on which few study has been done. Also how to

consider its combination with along-wind and across-wind should be investigated in

detail.

1.4 Wind Loads on Tall Building

Extensive research about wind loads on tall buildings has been done in the past few

years. Tall buildings, under the action of wind, will be subject to vibrations in along-

wind, across-wind and torsional directions, and generally wind loadings in these three

different directions are studied separately. In the following, they will be discussed

respectively.

The along-wind loads have been dealt with successfully by using quasi-steady and

strip theories in terms of gust loading factors. The gust loading factor method was

originally introduced by Davenport (Davenport, 1967), and was utilized to treat wind

loads on structures under the buffeting action of wind gusts in most major codes and

standards all around the world. In this method, the equivalent static wind load is equal

— — 15

to the mean wind load multiplied by the gust loading factor. The gust loading factor

takes the dynamics of wind fluctuations and any load amplification induced by the

building dynamics into consideration. Several papers (Simiu, 1966; Holmes, 2002;

Solari, 1993; Gurley, 1993) have provided some formulations for the gust loading

factor. Because of the simplicity of this method, the gust loading factor is widely

accepted and employed in wind loading codes in almost all major countries such as

Australian, Canada, USA, Japan, and Europe. It should be noted that the traditional

gust loading is based on displacement response while the AS1170.2-89 and ACI

standard apply it to base bending moment.

Despite its simplicity, the gust loading factor method has some limitations. Although

the gust loading factor is supposed to be used for any response, it is originally defined

on the displacement response, that is, the gust factor is actually the ratio between the

maximum and the mean displacement response and usually referred to DGLF. In

practice, this DGLF is used for any response indiscriminately, which may result in

inaccurate estimation. For a given structure, the GLF is constant because it involves

only the fluctuating and mean displacement response in the first mode and when a

constant GLF independent of height is used for estimating the extreme equivalent

wind loading, the equivalent wind loading will be the same as mean wind loading,

which is conflict with the common understanding of the equivalent static wind loads

on tall, flexible buildings. For this kind of building, the resonant response is the

dominant one; therefore, the distribution of equivalent static wind loads should

depend on the mode shape and mass distribution. It is found (Zhou, 1999) that the

DGLF can give accurate estimation for displacement but for other responses such as

base shear force its assessment is less accurate. Another obvious limitation is that this

method is invalid if either the mean wind force or the mean response is zero; an

example is a cantilever bridge with an asymmetrical first mode shape, in this case, the

DGLF can’t be defined since the mean displacement response in the first mode is zero.

By recognition of the shortcoming of the traditional DGLF, Davenport (Davenport,

1999) and Drybre and Hansen (Drybre, 1997) developed refined GLF concept, which

is based on the response related to the influence function, but not limited to

displacement. Holmes (Holmes, 2001) also presented a detailed treatment of GLF for

a range of applications. All these improvements in GLF method result in a more

— — 16

accurate estimation of structural response. However, in the case of zero mean

response, these refined approaches fail like the traditional one.

Kareem provided an alternative format (Kareem, 2003), in which the GLF is based on

the base bending moment rather than displacement. The extreme base bending

moment is obtained by multiplying the mean base bending moment by the new GLF

(referred to MGLF). Then the base bending moment is distributed to each floor in

terms of floor load in a format similar to the one used in earthquake engineering to

distribute base shear.

Kareem also extended the new framework to formulate wind load effects in the

across-wind and torsion directions based on the GLF method (Kareem, 2003), which

has generally been used for along-wind response.

Chen and Kareem (Chen, 2003, 2004) recently proposed an equivalent static wind

load that linearly combines the background and resonant loading components. The

background and resonant loading components are derived by using the concept of gust

loading envelope and the distribution of inertial loads respectively. Gust loading

envelope method provides a very simple load description compared with the load-

response-correlation (LRC) method proposed by Kasperski (Kasperski, 1992), since

the background equivalent static wind loading based on LRC method depends on the

response of interest.

Based on the important pioneer role gust loading factor plays in the research of wind

loading, it is necessary to describe it in detail.

1.4.1 DGLF approach

The traditional GLF or DGLF is described as follows:

In the DGLF approach, the peak load is expressed as

)()(ˆ zPGzP •= (1.10)

where G is gust factor, which takes the dynamics of gusts and the structure into

consideration; )(zP is the mean wind force, z is the position coordinate.

In the DGLF method, G is given in terms of the displacement response

)()(ˆ

zYzYGY = (1.11)

— — 17

where YG is DGLF, )(ˆ zY , )(zY are the extreme displacement response and mean

displacement respectively. For a stationary process, YG can be expressed as:

)(/)(1 zYzgG YYY σ+= =1+ RBIg HY +2 (1.12)

where Yg is the displacement peak factor; )(zYσ is root mean square(RMS)

displacement; B and R are background and resonant factors, respectively; HI is the

turbulent intensity evaluated at the top of the building.

The mean wind load is as follows:

αρ 22 )/(2/1)( HzUWCzP HD= (1.13)

where ρ is the air density, DC drag coefficient; W the width of the structure normal

to the incoming wind; α)/()( HzUzU H= the mean wind velocity at height z above

the ground, HU is the mean wind velocity at building height H; α is the exponent of

the mean wind velocity profile.

Alternatively, the displacement gust factor can be expressed as

RgBgIG RuHY2221 ++= (1.14)

where ug is the wind velocity peak factor; Rg the resonant peak factor; for a Gaussian

process )ln(2/5772.0)ln(2 11 TfTfg R += in which T is the observation time, and

1f the natural frequency of the first mode; R= ς/SE , where S is the size reduction

factor, E the gust energy factor, and ς the critical damping ratio of the first mode.

Almost all traditional formulations of the DGLF are based on preceding expressions,

but are different in their modeling of turbulence and structure models. The

coefficients B, E and S are provided in graphs in some codes and given in close forms

in others.

Equation (1.14) can be rewritten in terms of mean, background and resonant

components

221 YRYBY GGG ++= (1.15)

— — 18

where YBG and YRG are background and resonant components of the DGLF

respectively.

The mean displacement can usually be estimated by the first mode mean displacement

response

)(/)( 111 zkPzY ϕ⋅= ∗∗ (1.16)

where ∫=∗H

dzzzPP0

11 )()( ϕ , ∗∗ = 12

11 )2( mfk π and ∫=∗H

dzzzmm0

211 )()( ϕ are the

generalized load, stiffness and mass of the first mode, respectively.

Usually the mode shape is approximated by exponential function

βϕ )/()(1 Hzcz = (1.17)

where c and β are constants, the mass is assumed to be linearly distributed as

))/(1()( 0 Hzmzm λ−= (1.18)

λ is the mass reduction factor

The fluctuating displacement can be approximated by that in the first mode

)())(()( 12/1

01

zdffSzY ϕσ ξ ⋅= ∫∞

(1.19)

where 1ξ

S is the power spectral density function of the fluctuating generalized

displacement, which can be computed following the approach given by Davenport [28]

∫∞

⋅⋅=0

2)(),()()(1

dffHffSfS du βχξ (1.20)

where )( fSu is the PSD of fluctuating wind velocity, ),( fβχ the aerodynamic

admittance function which relates the wind velocity PSD to the PSD of the resulting

fluctuating wind force; for along-wind loading, the strip and quasi-steady theories can

be regarded as valid, and by considering wind structure in terms of vertical and

horizontal correlations while ignoring the correlation between wind pressures on

windward and leeward surfaces, the following relationship holds:

)(),()(1

~ fSffS uP⋅=∗ βχ (1.21)

— — 19

where

22

2

2

),,()()1(

)(),( fJfJUWHCf ZXHD βα

βαρ

βχ++

= (1.22)

and

∫ ∫=W W

XX dxdxfxxRW

fJ0 0 21212

2 ),,(1)( (1.23)

∫ ∫ ++++=

H H

ZZ dzdzfzzRHz

Hz

HfJ

0 0 212121

2

22 ),,()()()1(),,( βαβαβαβα (1.24)

are the joint acceptance functions in the horizontal and vertical directions,

respectively.

)))(/(exp(),,( 2121 xxhUfCfxxR XX −−= (1.25)

and

)))(/(exp(),,( 2121 zzhUfCfzzR ZZ −−= (1.26)

are the horizontal and vertical coherence functions of the fluctuating wind pressure,

respectively; XC and ZC are the exponential decay coefficients, and h is the reference

height. It can be figured out that the aerodynamic admittance depends not only on the

turbulence characteristics and structure shape but also mode shape. The mechanical

admittance function for the first mode displacement is

21

21

2 /)()( ∗= kfHfH d (1.27)

where,

21

221

21 )/2(])/(1[

1)(ffff

fHς+−

= (1.28)

Then the fluctuating part of the DGLF is

∗⎟⎠⎞⎜

⎝⎛= ∫ ∗ PdffHfSzY

PY /)()()(/2

21~σ (1.29)

— — 20

The integration in the above formula is usually divided into background and resonant

response factors. The background and resonant components can be expressed,

respectively by

BIgG HuYB 2= (1.30)

RIgG HRYR 2= (1.31)

where B= ∫∞ ∗

0)(),( dffSfk uβ , R= ς/SE are background and resonant response factors;

222 ),,()()1

22(),( fJfJfk ZX βαβααβ++

+= , S= ),( 1fk β the size reduction factor;

)()4/( 11 fSfE u∗= π is the gust energy factor; )( fSu

∗ the normalized wind velocity

spectrum with respect to the mean square fluctuating wind velocity, 2uσ . In most

Codes and Standards, a linear mode shape assumption is used, that is, 1=β .

1.4.2 MGLF approach

The MGLF method proposed by Kareem is described as follows (Kareem, 2003):

The gust factor in MGLF is defined as

IIM MMG /ˆ= (1.32)

where MG is the MGLF; IM is the mean base bending moment, IM̂ is the extreme

base bending moment. The subscript ‘I’ means induced BBM, which is different from

the externally aerodynamic moment.

For a Gaussian process, the MGLF can be expressed as

IMMM MzgGI

/)(1 ~σ+= (1.33)

where Mg is the peak factor; and IM~σ is the RMS base bending moment.

The base bending moment involves dynamics of gusts and structures and can be

derived by the following mode generalized equations of structural motion:

)(~)()()( 1111111 tPtktctm ∗∗∗∗ =++ ξξξ (1.34)

— — 21

where ∗1m , ∗

1c , ∗1k , 1ξ , ∗

1~P are generalized mass, damping, stiffness, displacement, and

load in the first mode, respectively. The generalized quasi-static wind load 11ξ∗k can

be determined from the generalized displacement.

When the generalized quasi-static wind load is applied on the building, the

corresponding generalized displacement and any other response are equal to those

obtained by a detailed dynamic analysis.

The PSD of the generalized equivalent-static wind load is given by

21~

21~ )()()(

1fHfSfSkS PPe

∗∗ == ∗ξ (1.35)

where the generalized quasi-static wind load ∫=∗ dzztzPtP ee )(),(~)(~1ϕ , eP~ is the

equivalent static wind load (ESWL).

The ESWL eP~ is usually distributed along the building height in a manner different

from that of mean wind load or fluctuating externally applied aerodynamic wind loads.

If a linear mode shape is assumed, the following relationships are valid for both the

externally applied and the equivalent-static wind loads:

HMP~~ =∗ (1.36)

HM

P Ie

~~=∗ (1.37)

where, M~ , IM~ are the fluctuating components of the externally applied and the

induced base bending moment, respectively. It is important to distinguish between the

equivalent-static (induced) and the aerodynamic (externally applied wind loads) wind

loads. The former includes any amplification from building dynamics.

Substituting equation (1.36) and (1.37) into equation (1.35) leads to

21~~ )()()( fHfSfS MM I

= (1.38)

The equation provides a new probabilistic treatment of buffeting as shown in Figure

1.4 (b) (Zhou, 2001).

— — 22

Log. Frequency (a) DGLF model

Log. Frequency

(b) MGLF model

Figure 1.4.Probablistic dynamic-based approaches to gust loading

Compared to the traditional DGLF method, the MGLF has two advantages. First, it

gives a concise description of the relationship between the aerodynamic load and the

induced wind load effects. Second, in traditional method, the aerodynamic admittance

is difficult to evaluate from theoretical consideration and the resultant response

estimations are variable. The aerodynamic admittance function in the traditional

method is actually a transfer function from the input turbulence to the generalized

wind load, while the generalized wind load is dependent on the normalization used to

define the mode shape, which makes the aerodynamic admittance becomes a function

of the mode shape. This will definitely complicates the verification of its theoretical

formulation with experimental measurements. However in the MGLF method, the

aerodynamic admittance is the relationship between input turbulence and the base

bending moment, in this case, the base bending moment(BBM) can be efficiently

ascertained by some effective tools such as HFBB.

IMIM MdffHfSMI

/))()((/ 2/1210

~~ ∫∞

=σ (1.39)

The MGLF can be expressed by

RgBgIG RuHM2221 ++= =1+ 22

MRMB GG + (1.40)

— — 23

A detailed derivation of the terms in the above formula is given below.

The mean BBM on a tall building is

αρ

222/1)(

22

0 +== ∫

HWUVCzdzzPM HDH

I (1.41)

where, the subscript ‘I’ means induced base bending moment, which is different from

the externally aerodynamic moment M.

The fluctuating BBM response is divided into background and resonant components,

the background base moment can be derived by using the expression given in[34] by

employing the influence function i(z)=z;

∫ ∫ ∫ ∫ ∫∞

=0 0 0 0 0 212121

212 )()()()()()(ˆ H H W W

uXZHDuIB dfdzdzdxdxzzfSfRfRHz

Hz

UWCgM ααρ

= ∫∞ ∗

+ 0

2222

),1,()()(2

dffJfJfSWHCUIg ZXuDHH

u αα

ρ (1.42)

I

IBMB M

MGˆ

= = ∫∞ ∗

++

0

22 ),1,()()(2

222 dffJfJfSIg ZXuHu ααα (1.43)

Since the influence function is used, the contribution from higher modes and mode

coupling are automatically included.

For the resonant component, the equivalent static wind load is equal to the inertial

force. Typically only the first mode is considered. If a non-linear mode shape and a

non-uniform mass distribution are utilized, the first mode extreme resonant

displacement is given by

β

ςπ

βα

βλββαββ

πρ

⎟⎠⎞

⎜⎝⎛⋅×

+−+++++

=

∗

HzfSffJfJ

mfWCUIgzY

uZX

DHHRR

)(4

),,()(

)]21()22)[(1()22)(21(

)2()()(ˆ

112

12

1

02

1 (1.44)

It can be seen from the above formula that the displacement distribution along the

building follows the mode shape. The corresponding resonant ESWL can be

expressed as

— — 24

βλς

πβα

βλββαββρπ

))(1()(4

),,()(

)]21()22)[(1()22)(21()()(ˆ)()2()(ˆ

112

12

1

221

Hz

HzfSffJfJ

WCUIgzYzmfzP

uZX

DHHRRR

−⋅×

+−+++++

==

∗

(1.45)

From the above formulation, it can be seen that the distribution of the resonant ESWL

depends not only on the mode shape but also the mass distribution.

The BBM induced by resonant ESWL can be derived by

)(4

),,()()2)(3(

)]2()3[(

)]21()22)[(1()22)(21()(ˆˆ

112

12

1

22

0

fSffJfJ

WHCUIgzdzzPM

uZX

DHHR

H

RIR

∗

+++−+

×

+−+++++

== ∫

ςπ

βαβββλβ

βλββαββρ

(1.46)

The resonant component of the MGLF is

)(4

),,()(

)2)(3()]2()3[(

)]21()22)[(1()22)(22)(21(2

ˆ

112

12

1 fSf

fJfJ

IgMMG

uZX

HRI

IRMR

∗×

+++−+

⋅+−+++

+++==

ςπ

βα

βββλβ

βλββααββ

(1.47)

The above mentioned GLF methods, no matter based on traditional displacement

response or on the base bending moment, will result in an ESWL that has distribution

similar to the mean wind load. Although GLF can make accurate estimation of

displacement or base bending moment, it may not give good estimation to other

responses.

1.4.3 DRF approach

Besides gust response factor (GRF) method, the DRF is defined as the ratio of peak

dynamic response which includes mean, background and resonant components to the

response caused by the peak dynamic load that includes the mean and the background

load effects without any reduction due to loss of spatial correlation of wind loading.

For example, the DRF for response R is expressed as:

'

2222

xb

xrxb

Rb

RrRbxxR gR

ggRD

σ

σσ

+

++= (1.48)

— — 25

xzR

H

PxR BdzzRzuxbxxb

/)()(0

' σσ == ∫ (1.49)

∫

∫ ∫= H

Px

H H

Pxx

xz

dzzRzu

dzdzzzRzuzuB

x

xx

0111

0 0212121

)()(

),()()( (1.50)

where ),()( zzRzRxxx PP = ; zB is the background factor representing the reduction

effects of response xR due to loss of spatial correlation of wind pressure. From the

above equations, the relationship between gust response factor and dynamic response

factor can be expressed as:

zRD =zR

RR

BG

GG

xb

xrxb

/1

1 22

+

++ (1.51)

where xRbR RgGxbxb

/σ= and xRrR RgGxrxr

/σ= are background and resonant GRF’s,

respectively.

Once the DRF is decided, the equivalent static wind loading is expressed as:

))()(( ' zFzPDF ebxxReR xx+= (1.52)

Where )(' zFebx = ))(zRgxPb =gust loading envelop.

From the above formula, it can be seen that the pressure distribution of equivalent

static wind loading based on DRF is similar to the peak dynamic wind loading.

1.4.4 Linear Combination of Background and

Resonant Equivalent Static Wind Loads

All the above methods take background and resonant responses into consideration by

a single factor; and the corresponding equivalent static wind load is also considered

without separating it into background and resonant components. Alternatively, the

ESWL can be separated into background and resonant components based on their

frequency content. The background ESWL (BESWL) depends on the external wind

load characteristics. The load-response-correlation method proposed by Kasperski

will result in a background effective load distribution that varies for response under

consideration. Repetto and Solari (Repetto, 2004) proposed an ESWL distribution

— — 26

identical for all response components, expressed in terms of a polynomial expansion.

A gust loading envelop (GLE) approach was proposed by Chen and Kareem (Chen,

2004), which provides a BESWL distribution similar to the gust loading envelope.

And when the dynamic wind load distributions are unknown but the integrated base

forces are available by using the HFBB technique, the BESWL can be approximately

obtained by distributing the base bending moment over the building height.

On the other hand, the resonant ESWL can be evaluated as model inertial load. Then

the ESWL for total peak response can be expressed as a linear combination of the

background and resonant loads. The detailed description of equivalent static wind

load as the linear combination of background and resonant equivalent static wind

loads is discussed below.

Assume R is the structural response of interest at the building height z0, the mean,

background components can be calculated by static and quasi-static analysis. As far as

the resonant component is concerned, it can be analyzed by modal analysis involving

only the fundamental modes.

∫=H

xx dzzuzPR0

)()( (1.53)

∫ ∫=H H

PxxRb dzdzzzRzzxx

0 0212121

2 ),()()( µµσ (1.54)

)(4

)()(

)()()(

111

0

2

0 fSfdzzzm

dzzzzm

xr QH

x

H

xx

R ξπ

µσ

∫

∫

Θ

Θ= (1.55)

∫ ∫ ΘΘ=H H

PxxQ dzdzfzzSzzfSxxx

0 0212121 ),,()()()( (1.56)

H is the building height, ),( 21 zzRxxP is the covariance, ),,( 21 fzzS

xxP is the cross

power spectral density between ),( 1 tzPx and ),( 2 tzPx ; )( fSxQ is the power spectral

density of the generalized modal force. Peak dynamic response (excluding the mean

response), Rmax is obtained by combining the background and resonant response

components:

— — 27

2222max rb RrRb ggR σσ += (1.57)

Where, gb and gr are peak factors for the background and resonant responses,

respectively.

ESWL corresponding to maximum background response bRbg σ can be calculated by

LRC method as

∫=H

PxR

beR dzzzRz

gzF

xx

b

b

0111 ),()()( µ

σ (1.58)

The BESWL thus calculated depends on the specific response, so for different

response, the BESWL is different. This characteristic limits the application of LRC

method although the LRC method can provide accurate calculation of BESWL. Chen

proposed a method to present BESWL as the gust loading envelope (GLE) multiplied

by a background factor

)()()( ' zRgBzFBzFxb PbzebxzeR == (1.59)

'b

b

R

RzB

σσ

= ; ∫=H

bebxxR gdzzFzub

0

'' /)()(σ (1.60)

where )(' zFebx is the gust loading envelope, ),()( zzRzRxxx PP = ; '

bRbg σ is the peak

background response under the gust loading envelope which didn’t take the loss of

spatial correlation of wind load over the building height; zB represents the reduction

effect with respect to the response due to loss of wind load correlation. A extreme

situation is that the wind load is completely correlated, that is,

)()(),( 2121 zRzRzzRxxxx PPP = ; for this case, zB will reduce to unity.

The RESWL for the peak resonant response rRrr gR σ=max is given in terms of the

inertial load distribution:

)(4

)()(

)()()( 11

1

0

2

fSfdzzzm

zzmgzF

xQH

x

xrerx ξ

π

∫ Θ

Θ= (1.61)

The ESWL for total peak dynamic response, Rmax, can be provided as a linear

combination of the background and resonant loads:

— — 28

2222' /)]()([)(rbrb RrRberxRrebxzRbeR ggzFgzFBgzF σσσσ ++= (1.62)

It can also be expressed in another form

)()()(2222

'

2222zF

gg

gzFB

gg

gzF erx

RrRb

Rrebxz

RrRb

RbeR

rb

r

rb

b

σσ

σ

σσ

σ

++

+= (1.63)

)()()(2222

' zFgg

gWzFBWzF erx

RrRb

RrrebxzbeR

rb

r

σσ

σ

++= (1.64)

where

2222rb

b

RrRb

Rbb

gg

gW

σσ

σ

+= =

22

1

1

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

b

r

R

R

b

r

gg

σσ

(1.65),

2222rb

r

RrRb

Rrr

gg

gW

σσ

σ

+= =

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎠⎞

⎜⎝⎛⎟⎠⎞⎜

⎝⎛

b

r

b

r

R

R

b

r

R

R

b

r

gg

gg

σσ

σσ

2

1

(1.66)

For different responses, the weighting factors are different and so is the background

response reduction factor zB , however, by using parametric analysis, for different

responses, the weighting factors depends only on the ratio of root mean square(RMS)

values of background and resonant response.

1.4.5 Across-wind and torsional wind loading

The methods discussed above are mainly aimed to establish along-wind loading.

Under the action of strong winds, tall buildings will vibrate in along-wind, across-

wind and torsional directions. In many cases, the wind-induced responses due to

across-wind and torsional excitations are as important as those induced by along-wind

excitation in terms of both serviceability and survivability design of tall buildings.

Although the along-wind loads have been successfully treated by using quasi-static

and strip theories in terms of gust loading factors, the across-wind and torsional loads

can’t be treated in the same manner, since these loads can’t be related in a

straightforward manner to the fluctuations in the incoming flow.

— — 29

Piccardo and Solari reported a 3D closed-form GLF formulation based on a

combination of the quasi-steady theory and empirical fit to a general across-wind

spectrum utilizing the DGLF framework.

Kareem extended the new MGLF framework for the formulation of wind load effects

of along-wind response to the across-wind and torsional directions (Kareem, 2003). In

his paper, the proposed 3D GLF is an extension of a new MGLF concept based on the

base bending moment or base torque response defined as

'/ˆ MMG = (1.67)

where G is the GLF; 'M is the reference mean base bending moment or base torque,

which can be computed for the across-wind and torsional modes, respectively, by

∫=H

LD zdzzPM0

,' )( (1.68)

∫=H

T dzBzPM0

' )04.0)(( (1.69)

where )(zP is the mean along-wind load at any height z above the ground; and H and

B are the building height and width normal to the oncoming wind, respectively.

Subscript D, L, and T represent the along-wind, across-wind and torsional directions

respectively. It should be noted that the reference base bending moment in equation

(1.68) and the reference base torque in equation (1.69) are not the actual mean base

moments that act on the building. For most symmetrical buildings, the mean base

bending moment and base torque are usually close to zero. The reference base

bending moment in across-wind direction is set equal to the along-wind base bending

moment for convenience. M is the peak base bending moment or base torque, which

can be expressed as :

MgMM σ⋅+= (1.70)

where M is the mean base bending moment or base torque. g is the peak factor,

normally 3~4; and Mσ = 2/1

0

))((∫∞

dffSM is the RMS of base bending moment or base

torque. )( fSM is the power spectral density(PSD) of the fluctuating base moment or

— — 30

base torque. A usually used practice is to divide the above integration into two

portions as below:

22MRMBM σσσ += (1.71)

Where MBσ and MRσ are the background and resonant components of the base

bending moment or base torque. Consequently, the 3D GLF in equation (1.67) can be

expressed in the form:

22RB GGGG ++= (1.72)

where G , BG and RG are the mean, background and resonant components of GLF,

respectively.

'MMG = , '/ MgG MBBB σ⋅= , '/ MgG MRRR σ⋅= (1.73)

where Bg = ug is the background peak factor or peak factor for the fluctuating wind

velocity. It can be noted that when considering the along-wind response, the 3D GLF

reduces exactly to MGLF. For along-wind response, the mean component of GLF is

unity; while for across-wind and torsional responses, it is usually very small or zero.

The calculation for background and resonant components of the base bending moment

or base torque is discussed below.

Generally speaking, most GLF-based method involves the generalized wind loading,

while it is found that generalized wind loading is quite sensitive to mode shape

exponent and the aerodynamic pressure characteristics (Zhou, 2002). In engineering

practice, the mode shape exponent or the parameters used to describe the wind

pressure distribution are usually unknown and need to be estimated. And also it is

noted that the BBM based GLF approaches can greatly reduce the sensitivity. The

PSD of the fluctuating base bending moment or base torque can be expressed in the

following equation:

21 )()()( fHfSfS MMM ⋅⋅=η (1.74)

where Mη is the mode shape correction for the base moments and torque response.

For background response, both the mode shape correction parameter Mη and

— — 31

21 )( fH are equal to unity. If ideal mode shapes are assumed, that is, linear in across-

wind direction and uniform in torsional direction, Mη for the resonant response

component is also equal to unity. Unlike the procedure based on generalized wind

loading, Mη for base bending moment is relatively insensitive to mode shape exponent,

mass distribution and aerodynamic pressure field. From equation (1.74), based on

definition of background response and white noise assumption, the background and

resonant RMS components of base bending moment or base torque can be expressed

as:

MMB σσ = , )(4 1

1

1 fSfMMR ς

πσ = (1.75)

The aerodynamic base moments involve complex fluid-structure interactions, for

along-wind response, where the strip and quasi-steady theories are usually assumed,

analytical procedure can be used to determine this information based on the oncoming

velocity fluctuations and building geometry. For across-wind and torsional directions,

no acceptable analytical procedure can be used. Wind tunnel test seems the only

approach to determine the aerodynamic base moment.

Most codes and standards provide little guidance for the critical across-wind and

torsional response. This is partly due to the fact that the across-wind and torsional

responses result mainly from the aerodynamic pressure fluctuations in the separated

shear layers and wake flow fields, which have no acceptable direct analytical relation

to the oncoming wind velocity fluctuations. So wind tunnel measurements seem the

only way to determine across-wind and torsional loads. For example, the high

frequency base balance (HFBB) and aeroelastic model tests can be used as routine

tools in commercial design practice. For tall buildings, like middle-rise and low rise

buildings, wind induced torsion and across-wind loading should be paid more

attention to and detailed researches based on experimental measurements need to be

carried out for a more reasonable consideration of the across-wind loading and

torsional moment in codes and standards.

Reference

Architectural Institute of Japan (1996), Recommendations for loads on buildings, 1996

— — 32

AS1170.2-1989, the Australian Wind Loading Standard. SAA, 1989

ASCE 7-98, Minimum Design Loads for Buildings and Other Structures. ASCE, 2000

Daryl W. Boggs, Noriaki Hosoya, and Leighton Cochran (2000), Sources of Torsional Wind Loading on Tall Buildings: Lessons From the Wind Tunnel, Proceedings of the 2000 Structures Congress & Exposition, Philadelphia, May 2000, ed. M. Elgaaly, SEI/ASCE, 2000

X. Chen, A. Kareem (2003), Equivalent static wind loads on structures, Proceedings of the 11th International Conference on Wind Engineering, Lubbock, Texas, June2-5, 2003

X. Chen, A. Kareem (2004), Equivalent static wind loads on buildings: New Model, Journal of Structural Engineering, Vol.130, No.10, pp.1425

A. G. Davenport (1967), A.G. Gust Loading Factors, Journal of Structural Engineering Division, ASCE, Vol.93, pp.11-34

A. G. Davenport (1995), How can we simplify and generalize wind loads, Journal of Wind Engineering and Industrial Aerodynamics, Vol.54/55, pp.657-669

A. G. Davenport (1999), Missing links in wind engineering, Proceedings of the 10th ICWE, Copenhagen, Denmark,1999,pp.1-8

C. Drybre, S. O. Hansen (1997), wind loads on structures, Wiley, New York,1997

ENV 1991-2-4, EUROCODE I: basis of design and actions on structures, Part 2.4: wind actions, 1994

K. Gurley, A. Kareem (1993), Gust loading factor for tension leg platforms, Appl. Ocean Res.Vol.15, Issue3 pp.137-154;

J. D. Holmes (2001), Wind loading on Structures, SPON Press, London, 2001

J. D. Holmes (2002), Effective Static Load Distributions in Wind Engineering, Journal of Wind Engineering and Industrial Aerodynamics, Vol.90, pp.91–109

A. Kareem, Y. Zhou (2003),Gust Loading factor-past, present and future, Journal of Wind Engineering and Industrial Aerodynamics, Vol.91, pp.1301-1328

M. Kasperski, H. J. Niemann (1992), The LRC (Load-response-correlation) method: a general method of estimating unfavorable wind load distributions for linear and nonlinear structural behavior, Journal of Wind Engineering and Industrial Aerodynamics, Vol.43, pp.1753-1763

M. Kasperski (1992), Extreme wind load distributions for linear and nonlinear design, Eng.Struct.Vol.14, pp.27-34

T. Kijewski1, A. Kareem, Dynamic Wind Effects: A Comparative Study of Provisions in Codes and Standards with Wind Tunnel Data,

— — 33

W. H. Melbourne (1975), Probability distributions of responses of BHP house to wind action and model comparisons, J.Ind.Aerodyn.1(2)(1975)167

NRCC, User’s Guide-NBC1995 Structural Commentaries (Part 4), 1996

M. P. Repetto and Solari, G.(2004) Equivalent static wind actions on structures, Journal of Wind Engineering and Industrial Aerodynamics, Vol.100, No.7, pp.1032-1040

S. H. Seong, J.A. Peterka (1993), Computer simulation of non-Gaussian wind pressure .fluctuations, Proceedings of the Seventh US National Conference on Wind Engineering, Los Angeles, CA, June 27–30, 1993.

E. Simiu, R. Scanlan (1996), Wind Effects on structures: Fundamentals and Applications to Design, 3rd Edition, Wiley, New York,1996

E. Simiu, T. Stathopoulos (1997), Codification of wind loads on buildings using bluff body aerodynamics and climatologically databases, Journal of Wind Engineering and Industrial Aerodynamics, Vol.69–71, pp.497–506.

G. Solari, L.C. Pagnini (1999), Gust buffeting and aeroelastic behavior of poles and monotubular towers , J.Fluid Struct.13 877

G. Solari (1993), Gust buffeting. I: peak wind velocity and equivalent pressure , Journal of Structural Engineering, ASCE, Vol.119, Issue2, pp.365-382

G. Solari, Gust buffeting. II: dynamic along-wind response, Journal of Structural Engineering, ASCE, Vol.119, Issue2, pp.383-397

T. Stathopoulos (2003), Wind loads on low buildings: in the wake of Alan Davenport’s contributions, Journal of Wind Engineering and Industrial Aerodynamics, Vol.91, pp.1565–1585

D. Surry, T. Stathopoulos (1977/1978), An experimental approach to the economical measurement of spatially averaged wind loads, Journal of Wind Engineering and Industrial Aerodynamics, Vol. 2 ,pp.385–397

Y. Tamura, H. Kikuchi, K. Hibi (2001), Extreme Wind Pressure Distributions on Low-rise Building Models, Journal of Wind Engineering and Industrial Aerodynamics, Vol.89, pp.1635-1646

Y. Tamura, H. Kikuchi, K. Hibi (2003), Quasi-static wind load combinations for low- and middle-rise buildings, Journal of Wind Engineering and Industrial Aerodynamics, Vol.91, pp.1613-1625

B. J. Vickery, R. I. Basu, The response of reinforced concrete chimneys to vortex shedding, Eng. Struct. 6(1984) 324-333

T. Whalen, E. Simiu, G. Harris, J. Lin, D. Surry (1998), The use of aerodynamic databases for the effective estimation of wind effects in main wind-force resisting systems: application to low buildings, Journal of Wind Engineering and Industrial Aerodynamics, Vol.77/78, pp.685–693.

— — 34

Y. Zhou, A. Kareem, M. Gu (1999), Gust loading factors for design applications, Proceedings of the 10th ICWE, Copenhagen, Denmark, 1999,pp.169-176

Y. Zhou, A. Kareem (2001), Gust loading factor: new model, Journal of Structural Engineering, ASCE, Vol.127, No.2, pp.168-175

Y. Zhou, A. Kareem, M. Gu (2002), Mode shape corrections for wind load effects on tall buildings, Journal of Engineering Mechanics, ASCE, Vo.128, No.1, pp.15-23.

— — 35

CHAPTER II

INTEGRATED WIND LOADS ON A FULL-SCALE

LOW-RISE BUILDING

2.1 Introduction

As part of the National Institute for Standards and Technology/Texas Tech University

(NIST/TTU) wind mitigation initiative (WMI), researchers at Texas Tech University

upgraded the data acquisition system at the Wind Engineering Research Field

Laboratory (WERFL) to include 204 pressure transducers relatively uniformly spaced

over all four walls and the roof. In addition to the differential pressure transducers, a

sonic anemometer is mounted at a height of 17 ft above the geometric center of the

WERFL building roof to provide high resolution wind speed and angle of attack data

for analysis. The addition of these taps and transducers allow for the investigation of

overall loads on the building including torsional loading which is the focus of this

chapter.

Wind-induced pressures acting on the WERFL building are integrated over each

surface to obtain three forces and moments at the base of the building along the three

principal axes with its origin at the geometric center of the base of the WERFL

building. Generally, wind loads for structural design are uniform and symmetrical,

and most codes disregard the torsional moments about a vertical axis, only except in

situations where there is an eccentricity between the centers of twist and the building

geometry. Actual overall wind loads are rarely uniformly distributed even for

buildings with symmetrical geometry.

Torsional moments can be divided into two parts: (1) the mean torsional moment; and

(2) the fluctuating torsional moment. Mean torsional moments can occur due to the

non-uniformities in the wind field or for wind directions not aligned with the axes of

building symmetry. The fluctuating torsional moment is caused by the unbalances in

the instantaneous pressure distributions and will be accentuated by the eccentricities

between the aerodynamic mass and elastic centers. Until now, there are no theoretical

models for the consideration of torsional wind load, especially for low-rise buildings.

In this chapter, mean and fluctuating pressure distributions around the WERFL

building are investigated, and the pressure distributions producing maximum

fluctuating along-wind, across-wind, torsional moment at the base of the building are

— — 36

studied, and the correlation between these forces is studied, finally a method to

investigate the load combination of these forces is proposed.

2.2 Instantaneous Wind Pressure Distribution Causing

Maximum Quasi-Steady Load Effects

The WERFL building is located on the campus of Texas Tech University, seen in

Figure 2.1 (Levitan, 1992). The test building has a length of 45ft, width of 30ft and

height of 13ft. The exposure category for the surrounding terrain can be regarded as

Exposure C as defined in ASCE 2002 (or 2005). The sampling duration for each run

is 15 minute and the sampling rate is 30 Hz, thus in each run there are 27000 sampling

points. Wind speed and direction for this study principally comes from a sonic

anemometer located at a height of 17 ft above the geometric center of the building.

A dataset of 26 records with angles of attack of 0º ± 5º is used for the work presented

here because the situation considered is when wind angle of attack is close to zero.

The tap pressure designations and the definition of wind angle of attack on the

building are shown in Figure 2.2. Wind direction perpendicular to wall 1 is defined as

having an angle of attack of 0º. AOA increases clockwise from wall 1 to wall2, wall3

and wall4. The pressure taps on the walls were distributed at three different levels as

shown in Figure 2.2.

2.2.1 Quasi-static base shear and torsional

base moment

To give a clear view about the quasi-static base shear and torsional moment, the mean

pressure distribution and fluctuating pressure distribution, and the actual extreme

pressure distributions causing those maximum load effects need to be investigated.

Figure 2.3 and Figure 2.4 show the mean and standard deviation pressure distributions

at three pressure tap layers when the angle of attack (AOA) is 0.1296º. Figure 2.5

shows an example of the instantaneous wall pressure distributions when the maximum

base shear FX was recorded (FX was represented by the maximum fluctuating shear

force coefficient, CX). The extreme pressure pattern of each pressure tap level is

shown in Figure 2.5 to identify any significant differences between the three pressure

tap levels. From left to right in Figure 2.5, the pictures show the pressure distribution

— — 37

patterns of the lowest, the middle and the top pressure tap layers respectively. For all

the pressure distribution graphs, lines outside wall represents positive, inside negative.

From Figure 2.5, we can see that the instantaneous pressure distributions at two side

walls and the leeward wall are similar to each other at the three levels; and although

the pressure distribution on the windward wall are slightly different from each other,

they have a common characteristic, that is, the pressure distributions generally cluster

on one side of the wall. Large local suctions tend to appear near the windward edges

of side walls. Based on the non-symmetrical features of the wind distribution pattern,

it’s not hard to image that wind induced torsion may accompany the maximum along-

wind force due to the unsymmetrical pressure distribution.

Figure 2.6 shows the example of the instantaneous wall pressure distributions when

the maximum fluctuating base shear FY was recorded (FY was represented by the

maximum shear force coefficient, CY). From left to right in Figure 2.6, the graphs

show the pressure distribution patterns of the lowest, the middle and the top wall

pressure tap levels, respectively. Figure 2.6 shows that a large negative pressure on

one side wall and nearly zero pressure on the other side occur when the maximum

across-wind force occurs. This is the typical pressure patterns for this case.

Figure 2.7 shows an example of the instantaneous wall pressure distributions when the

maximum torsional base moment MT was recorded (MT is represented by the

maximum torsional base moment coefficient, CTmax). From left to right in Figure 2.7,

the graphs show the pressure distribution patterns of the lowest, the middle and the

top pressure tap levels, respectively. For this case, a large negative suction tends to

happen near the windward edge of one of the side walls. The wind pressure

distribution on the windward wall doesn’t spread symmetrically (they generally

cluster to one side) which is also a characteristic of the wind pressure distribution

when maximum along-wind forces are recorded, but the values are much smaller than

those at maximum along-wind forces.

Figure 2.8 shows an example of the instantaneous wall pressure distributions when the

maximum lift force FZ was recorded.

To get the typical extreme pressure distributions causing the maximum load effects at

the base, all the 26 15-minute duration records with angles of attack (AOA) in the

range of 0º~5º, and 355º~360º were ensemble averaged. Figure 2.9 and Figure 2.10

— — 38

show the ensemble averaged mean pressure distribution and fluctuating pressure

distribution respectively. Figure 2.11 through Figure 2.14 show the typical

instantaneous wall pressure distributions coinciding to maximum along-wind, across-

wind base shear and torsional moment, as well as lift force. From left to right in

Figure 2.9 through Figure 2.14, the graphs show the pressure distribution patterns of

the lowest, the middle and the top pressure wall tap levels, respectively. As far as the

ensemble averaged extreme wind distribution patterns causing maximum across-wind

base shear and maximum torsional base moment are concerned, since the signs of

these force and moment are not important and we only care about the magnitude of

their value, conditional average is used, for example, if the across-wind base shear is

negative according to the coordinate system, the pressure distribution for ensemble

average will be the original pressure distribution mirrored by the centric axis.

Besides pressure distributions, the along-wind force coefficient, across-wind force

coefficient, and torsional moment coefficient, lift force coefficient are also necessary

to be investigated. Table 2.1 lists the mean and standard deviation of the along-wind,

across-wind and torsional moment coefficients, lift force coefficients of the 26

samples. Table 2.2 lists the maximum fluctuating along-wind, across-wind and

torsional moment coefficients, lift force coefficients and their corresponding another

three force coefficients respectively. Here ‘maximum’ means the maximum absolute

value.

For angle of attack (AOA) close to 90º, the mean, fluctuating pressure distributions,

and ensemble averaged extreme pressure distributions coinciding to maximum along-

wind, across-wind, torsional moment and lift force at the base of the building are also

studied. These pressure distributions are shown in Figure 2.15~Figure 2.20.

Table 2.3 lists the mean and standard deviation of the along-wind, across-wind and

torsional moment coefficients, lift force coefficients of the samples under angle of

attack close to 90º. Table 2.4 lists the maximum fluctuating along-wind, across-wind

and torsional moment coefficients, lift force coefficients and their corresponding

another three force coefficients respectively.

It should be noted that for angle of attack close to 0º and 90º, along-wind force and

across-wind force are directed to different directions, that is, for AOA=0º, along-wind

force refers to FX, while for AOA=90º, along-wind force refers to FY.

— — 39

2.2.2 Correlation between the wind loads

The relationship between the maximum fluctuating base shear FX and its

simultaneously recorded base shear FY, torsional base moment MT as well as lift force

FZ when angle of attack close to 0º is shown in Figure 2.21. CY and CT, CZ ratio is

calculated by using the accompanying CY, CT and CZ divided by their corresponding

maximum fluctuating value recorded in each run. There are 26 runs used for this

graph (AOA from 0º~5º and 355º~360º ). Figure 2.22 shows the relationship

between maximum fluctuating base shear FY and its simultaneously recorded base

shear FX and torsional base moment MT and lift force FZ. Figure 2.23 shows the

relationship between maximum torsional base moment MT and its simultaneously

recorded base shear forces FX, FY and lift force FZ. Figure 2.24 shows the relationship

between maximum lift force FZ and its simultaneously recorded base shear forces FX,

FY and torsional moment MT.

Figure 2.25 and Figure 2.26 show the corresponding relationships when angle of

attack close to 90º.

Figure 2.29(a) and (b) shows an example of cross-correlation between the three wind

forces and torsion, that is, base shear force FX, base shear force FY, lift force FZ and

torsional base moment MT for AOA=0º. Very low correlation between CX-CY and

between CX-CT is observed. A relatively high correlation is noted between CY-CT but

is still less than 0.5. These correlations agree well with the correlations of wind

induced forces acting on high-rise buildings.

Figure 2.30 (a) and (b) show an example of cross-correlation between the three wind

forces and torsion, that is, along-wind base shear, across-wind base shear and

torsional base moment, lift force when AOA is close to 90º. Correlation between CY-

CX at this angle of attack is higher than that when AOA is close to 0º, but still less

than 0.5, and correlation CY-CT is less than 0.3. A very high correlation is noted

between CY-CZ. It should be noted that in this case, the correlation between CX- CT is

low, which is a little different from that when AOA is close to 0º.

— — 40

2.2.3 Relationship between torsional moment

and other forces

To establish which wall contributes most to the torsional moment MT, the torsional

coefficients of the four walls are calculated respectively for each of the 26 runs under

AOA close to 0º. It can be seen from Table 2.5 that the mean torsion coefficients of

wall 2 are always the maximum and the mean torsion coefficients of wall 4 are always

the minimum, the standard deviations of the torsion coefficients of wall 2 and wall 4

are the largest compared to those of the other walls. The torsion coefficients of the

four walls when the total torsional moment is maximum and minimum are

investigated and are listed in Table 2.6, and it can be seen that the torsional moments

provided by wall 2 and wall 4 (both are side walls) are the main contributors to the

overall torsion. In Table 2.6, there is only one case in which the main contribution of

the torsion comes from wall 1 due to the asymmetric pressure distribution on the

windward wall. It is also noted that for this one run that the torsion contributed by

wall 2 is comparable to that of wall 1.

To make the relationship of wind induced torque with other wind loads more clear,

the correlation of the torsion parts contributed by the four walls and their

corresponding correlation with along-wind load and across-wind load need to be

studied.

From the third and fifth columns of Table 2.7, it can be seen that the torsional

moments induced by pressure distributions on wall 2 and wall 4 are highly correlated

with the along-wind load FX, which is mainly caused by pressure distribution on the

windward and leeward walls (walls 1 & 3). These high correlations are also reflected

in Figure 2.5. From Figure 2.5, it can be seen that when the along-wind load is

maximum, the pressure distributions on wall2 and wall4 are unsymmetrical and there

are large suctions clustering on the windward edges, which will cause large torsional

moments. And because opposite correlations of the along-wind load FX with torsional

moments MT contributed by wall 2 and wall 4, the correlation of the combined

torsional moments of wall 2 and wall 4(main contributors to torsional moment) with

along-wind load FX is low, so is the correlation of total torsional moment and along-

wind load, which can be reflected from the third column of Table 2.9 and the left side

graphs of Figure 2.29(a) respectively.

— — 41

From Table 2.8, we can see that the correlations between torsional moments induced

by pressure distributions on wall 1,2,3,4 and across-wind load are generally

comparable to each other; because of opposite correlations of across-wind load and

torsional moment from wall1 and wall3, the combined torsional moment of wall1 and

wall3 are almost uncorrelated with across-wind load and since torsional moments

from wall2 and wall4 are correlated with across-wind in the same way, their

combined torsional moment and thus the total torsional moment are correlated with

across-wind load.

In Tamura’s paper, a similarity between pressure distributions producing maximum

along-wind base shear and maximum torsional moment is noted (Tamura, 2001).

Figure 5 of his paper (Figure 2.31 in this Chapter) shows ensemble averaged extreme

pressure distributions causing maximum quasi-static load effects (maximum along-

wind base shear, maximum across-wind base shear and maximum torsional base

moment). From the Figure 2.31, it can be seen that the ensemble averaged wind

pressure distribution causing maximum along-wind and across-wind base shear have

the same characteristic as those of WERFL building, however for ensemble averaged

wind pressure distribution causing maximum torsional moment, there is a little

difference from that for WERFL building. The pressure distribution causing

maximum torsional moment of WERFL building has large suction near windward

edge of one of the side walls with pressure distribution on windward wall clustering to

the same side wall, but pressure distribution on another side wall is almost zero. From

Figure 2.31(c) of this graph, it can be seen that the pressure distribution causing

maximum torsional moment of Tamura’s model has large suctions on windward edge

of both side walls with pressure distribution on windward wall clustering to one side.

One possible for the difference is that mean pressure distribution is included in

ensemble averaged extreme wind pressure distribution causing maximum quasi-static

load effects in Tamura’s paper, but mean pressure distribution is extracted from

ensemble averaged pressure distributions in this chapter for WERFL building.

2.3 Wind induced internal stresses in structure members

The study of correlation of wind loads is a basic step to investigate wind load

combination. The wind induced effects in a structure are affected by pressure

distributions around the building. If the integrated wind loads at the base of the

building calculated from pressure distributions are used, the stresses in structural

— — 42

system of a building can be regarded as the simultaneous action of various wind loads

including along-wind load, across-wind load, uplift wind load, along-wind bending

moment, across-wind bending moment, and torsion. For high rise buildings, the wind

load combination of along-wind load and across-wind load has been studied, for

example by Melbourne (Melbourne, 1975), Solari and Pagnini (Solari, 1999).

However for low-rise building, wind load combination has not been investigated

thoroughly. Tamura discussed the wind load combination effects. In his paper, the

maximum normal stress in the column members of a simple frame system was studied

to check load combination effects. In this chapter, a similar structure is assumed for

the WERFL building as shown in Figure 2.34. C1, C2, C3 and C4 are four columns

set at the corner of the building model and the roof beams are assumed to be stiff. The

maximum normal stress is caused by resultant effects of surface wind pressure, which

are represented by the six force components at the base of the building, FX, FY, FZ, MX,

MY, and MZ.

Table 2.10 lists the influence coefficients of each of the six force components to the

internal forces in column1, column2, column3 and column4.

Based on the time series of these internal forces, the normal stress in the four columns

were calculated, the peak normal stresses in C1 to C4 averaged over the 26 records

with AOA in the range 0º to 5º or 355º to 360º are calculated. For simplicity, only

results of C1 and C3 are shown. From Table 2.11 and Table 2.12, we can see that the

peak compressive stress considering all wind force components is the largest one in

both columns compared to the compressive stress calculated by the other load

combination cases, while the peak tensile normal stress calculated by combination

(FX+MZ) and (FX+FY+MZ) are much larger than that considering all wind force

components in C1 and C3. This feature also points out that although along-wind load

FX is dominant in cases with AOA close to 0º, the combination of along-wind force

FX and torsional moment MZ is necessary for structure design, even for low-rise

building with relatively larger rigidity.

Another result worth noting is that for column one(C1) peak tensile stress considering

all wind force components is almost 22% larger than that only considering the along-

wind component, and the peak compressive stress is 82% larger than that considering

the along-wind force only.

— — 43

Table 2.13 and Table 2.14 list the peak fluctuating normal tensile and compressive

stresses calculated by different kinds of load combinations. All the results support the

importance of considering wind load combinations even for rigid building. For the

case when the wind angle of attack is close to 0º, other wind loads besides along-

wind force also need to be taken into account.

2.4 Comparison to responses calculated by ASCE

2.4.1 ASCE (Figure 6-9)

The normal stresses calculated by applying the pressure distribution based on Figure

6-9 of ASCE7-05 are listed in Table 2.15. The pressure distributions based on Case 1

for AOA=0º and AOA=90º are plotted Figure 2.32. The wind load distribution based

on all the other three cases can be derived from these basic pressure distributions.

From Table 2.15., it can be seen that the actual responses and responses calculated

from ASCE are very close to each other. The results calculated from ASCE can be

conservative and inconservative.

2.4.2 ASCE (Figure 6-10)

The normal stresses calculated by applying the pressure distribution based on Figure

6-10 of ASCE7-05 are listed in Table 2.16. A factor 2.34(1.532) is applied to the

pressure coefficients indicated in this figure since in ASCE the pressure coefficient is

3 second gust pressure coefficient while in WERFL full scale experiment, each run is

15 minute long, to make the calculation results from ASCE comparable to actual

responses calculated from full scale pressure time history, the dynamic pressure used

with 3 second gust pressure coefficient in ASCE should be multiplied by the square of

the ratio of 3 second wind speed to 15 minute wind speed which is equal to 1.53. The

pressure distribution from ASCE7-05 for calculation is shown in Figure 2.33.

The pressure coefficients are given by Figure6-10 of ASCE 7-02. in longitudinal

Direction(AOA=0º):

4.5a ft= , 13.0h ft= , 2.605hq psf= Zone 1: ( )0.40hp q= × + =+ 1.042 psf

Zone 2: ( )0.69hp q= × − =−1.797 psf

Zone 3: ( )0.37hp q= × − =−0.964 psf

— — 44

Zone 4: ( )0.29hp q= × − =−0.755 psf

Zone 5 and 6: ( )0.45hp q= × − =−1.172 psf Similarly,

Zone 1E: ( )0.61hp q= × + =+ 1.589 psf

Zone 2E: ( )1.07hp q= × − =−2.787 psf

Zone 3E: ( )0.53hp q= × − =−1.381 psf

Zone 4E: ( )0.43hp q= × − =−1.120 psf The normal stresses calculated by ASCE pressure distribution are compared with the

actual responses from pressure time history, and are listed in Table 2.16. It should be

noted that negative sign of percentage means that ASCE underestimates the responses.

It can be seen from Table 2.16 that for tensile stress in C1, the result given by ASCE

is conservative, but for compressive stress, the result given by ASCE underestimates

the actual response by 53.8%. For C3, both tensile stress and compressive stress are

underestimated by ASCE with percentage of 1.5% and 45.3% respectively. All the

results prove the inconsistency of ASCE, which means for some responses, it is

conservative while for certain responses, it will greatly underestimate the results. The

deficiency of ASCE load specification has also been pointed out by Simiu et al (Simiu,

2003) and Stathopoulos (Stathopoulos, 2003) and Holmes (Holmes, 2003).

2.5 Derivation of the wind load combination

In wind engineering practice, equivalent static wind loading usually is used for design.

Equivalent static wind loading (ESWL) will cause the same peak response as the

actual wind load. So for different structural response, the effective static wind loads

are different. A low-rise building immersed in a wind field is usually mainly under the

action of along-wind, across-wind and wind induced torsion. To get an equivalent

static wind load for a specific response under the action of these forces, a reasonable

combination of these three forces needs to be proposed.

To consider a reasonable combination of along-wind load, across-wind load and

torsional moment, a set of weighting factors for these three forces needs to be

established. The determination of weighting factors is based on the principle that the

combined wind loading when applied on the structure will cause the same peak

response as the actual response. The derivation of weighting factors is described as

follows.

— — 45

Suppose a structure response R is of interest; the wind induced fluctuating part of R

can be expressed in terms of fluctuating QX, QY and MT

TRtYRyXRx MQQtR µµµ ++=)( (2.1)

where Rxµ , Ryµ , Mtµ are the influence coefficients when a unit load of XQ , YQ and

TM are applied on the structure respectively, XQ , YQ and TM are fluctuating along-

wind load, across-wind load and torque.

To obtain the equivalent static wind loads based on the response R, that is to set up a

reasonable ESWL under the action of which a peak value of R will achieve, the RMS

of R needs to be incorporated:

tytytxtx

yxyxtyx

MQMQRtRyMQMQRtRx

QQQQRyRxMRtQRyQRxR

σσρµµσσρµµ

σσρµµσµσµσµσ

22

22222222

+

++++= (2.2)

The above formula can be represented in another form:

RMMRtQMQRyQMQRxRt

RQMMQRtQRyQQQRxRy

RQMMQRtQQQRyQRxRxR

ttytyxtx

yttyyxyx

xttxyyxx

σσσµσρµσρµµ

σσσρµσµσρµµ

σσσρµσρµσµµσ

/)'(

/)(

/)(

++

+++

+++=

(2.3)

It can be easily shown that:

xx QRxQR σµσ =_ ,yy QRyQR σµσ =_ ,

tt MRtMR σµσ =_ (2.4)

So

RMMRQRMQQRMQRt

RQMRMQQRQRQQRy

RQMRMQQRQQQRRxR

ttytyxtx

yttyyxyx

xttxyyxx

σσσσρσρµ

σσσρσσρµ

σσσρσρσµσ

/)('

/)(

/)(

___

___

___

++

+++

+++=

(2.5)

A simplified formula is as follows:

ttyyxx MMRtQQRyQQRxR WWW σµσµσµσ ++= (2.6)

where

RMRMQQRQQQRQ ttxyyxxxW σσρσρσ /)( ___ ++= (2.7)

RMRMQQRQRQQQ ttyyxyxyW σσρσσρ /)( ___ ++= (2.8)

— — 46

RMRQRMQQRMQM tytyxtxtW σσσρσρ /)( ___' ++= (2.9)

The weighting factor can also be expressed in terms of the response ratio and wind

load correlation coefficient

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+

=

x

t

x

y

ty

x

t

tx

x

y

yx

x

t

x

y

x

t

tx

x

y

yx

x

QR

MR

QR

QRMQ

QR

MRMQ

QR

QRQQ

QR

MR

QR

QR

QR

MRMQ

QR

QRQQ

QW

_

_

_

_

_

_

_

_2

_

_

2

_

_

_

_

_

_

22

21

1

σσ

σ

σρ

σσ

ρ

σ

σρ

σσ

σ

σ

σσ

ρσ

σρ

(2.10)

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

x

t

x

y

ty

x

t

tx

x

y

yx

x

t

x

y

x

t

ty

x

y

yx

y

QR

MR

QR

QRMQ

QR

MRMQ

QR

QRQQ

QR

MR

QR

QR

QR

MRMQ

QR

QRQQ

QW

_

_

_

_

_

_

_

_2

_

_

2

_

_

_

_

_

_

22

21

σσ

σ

σρ

σσ

ρ

σ

σρ

σσ

σ

σ

σσ

ρσ

σρ

(2.11)

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+

=

x

t

x

y

ty

x

t

tx

x

y

yx

x

t

x

y

x

t

x

y

tytx

t

QR

MR

QR

QRMQ

QR

MRMQ

QR

QRQQ

QR

MR

QR

QR

QR

MR

QR

QRMQMQ

MW

_

_

_

_

_

_

_

_2

_

_

2

_

_

_

_

_

_

22

21

σσ

σ

σρ

σσ

ρ

σ

σρ

σσ

σ

σ

σσ

σ

σρρ

(2.12)

From equation (2.10)~(2.12), it can be seen that the weighting factors for XQ , YQ and

TM depend on their correlations between each other and responses ratio between

each other but not the responses themselves. So by quantifying response ratio, the

weighting factors can be categorized into several groups, which makes this method

suitable for utilization in Codes.

After the weighting factors are established, the load combination of XQ , YQ and

TM can be established to get the equivalent static wind load for response R. The

— — 47

formula used to express the equivalent pressure distribution which will produce peak

response for R can be expressed as

maxmaxmax )()()(TTYYXX MMQQQQmeane PWPWPWPP +++= (2.13)

Where, Pe is the equivalent pressure distribution or equivalent wind loads which will

produce the same static response as the peak response, Pmean is the mean pressure

distribution or mean wind loads, XQW ,

YQW , and TMW are weighting factors obtained

from the above formula; max)(XQP , max)(

YQP and max)(TMP are the pressure distribution

when fluctuating QX, QY, and TM are maximum respectively or just the maximum

fluctuating wind loads. It should be noted that the subscript ‘max’ in the above

formulas means the maximum fluctuating value of QX, QY, and TM that happen in a

certain wind direction, not the maximum value without the consideration of wind

angle of attack. Table 2.17 lists the correlation coefficients between along-wind load

and across-wind load and torsional moment for WERFL building. Table 2.18 and

Table 2.19 list the rms value and mean of the along-wind, across-wind and torsional

moment coefficients for all the samples of WERFL building respectively.

2.6 Application of the Derived Equivalent Static Wind

Loads to the Assumed Frame System

The structure used for the application is as Figure 2.34 shows, the normal stresses in

the columns are calculated, the wind angles of attack are close to 0º, so all the

samples with AOA in the range from -5º to +5º(0º~5º and 355º~360º) are selected.

To apply the derived formula, the weighting factors need to be determined. From

equation (2.10) to (2.12), we can see that the correlation coefficients of the three

forces and their corresponding response ratios need to be computed. The correlation

coefficients of along-wind, across-wind and torsional moment for each sample

selected can be calculated, and then correlation coefficients are ensemble averaged.

Table 2.17 lists the correlation coefficients for all the samples.

From the table, we can see that the correlation between along-wind and across-wind

loads, along-wind and torsional moment are very close to zero, while across-wind

loading and torsional moment have some correlation, for this case, the correlation can

be set to be -0.37.

— — 48

Substituting YX QQρ =0, 0=

TX MQρ , 37.0−=TY MQρ to equation (2.10)~(2.12), the

weighting factors for along-wind and across-wind loading and torsional moment can

be expressed as:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+

=

X

T

X

Y

X

T

X

Y

X

QR

MR

QR

QR

QR

MR

QR

QR

QW

_

_

_

_

2

_

_

2

_

_ 74.01

1

σσ

σσ

σσ

σσ

(2.14)

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛

=

X

T

X

Y

X

T

X

Y

X

T

X

Y

Y

QR

MR

QR

QR

QR

MR

QR

QR

QR

MR

QR

QR

QW

_

_

_

_

2

_

_

2

_

_

_

_

_

_

74.01

37.0

σσ

σσ

σσ

σσ

σσ

σσ

(2.15)

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛−

=

X

T

X

Y

X

T

X

Y

X

T

X

Y

T

QR

MR

QR

QR

QR

MR

QR

QR

QR

MR

QR

QR

MW

_

_

_

_

2

_

_

2

_

_

_

_

_

_

74.01

37.0

σσ

σσ

σσ

σσ

σσ

σσ

(2.16)

For column one, C1, normal stress induced by along-wind loading FX can be

expressed as:

YFcFQR ZHADHXXX

/8//)2/(_ σσσ −= (2.17)

And the normal stress induced by across-wind loading Fy can be expressed as:

XFcFQR ZHABHYYY

/8//)2/(_ σσσ += (2.18)

The normal stress induced by torsional moment Mz can be expressed as:

XMYMMR ZRBHZRDHTTT

/)16/(/)16/( 22_ σσσ −−= (2.19)

Where P is the ensemble averaged dynamic pressure for all the samples with AOA

close to zero.

So the corresponding response ratios in expression (2.14) to expression (2.16) are

YFcF

XFcF

QR

QR

ZHADHZHABH

XX

YY

X

Y

/8//)2/(/8//)2/(

_

_

σσσσ

σσ

−+

= (2.20)

— — 49

YFcF

XMYM

QR

MR

ZHADHZRBHZRDH

XX

TT

X

T

/8//)2/(/)16/(/)16/( 22

_

_

σσσσ

σσ

−−−

= (2.21)

Substituting lbXF 36.98=σ , lb

YF 40.119=σ , 746.51TM lb ftσ = × , B=30ft, D=45ft,

H=13ft, R=27.042ft, ZX=ZY=0.2797ft3, Ac=0.4618ft2 into above expressions

(2.14)~(2.16), we can get the weighting factors:

5336.0=XQW , 8208.0−=

YQW , 4930.0=TMW (2.22)

The peak normal stress in column one, C1 can be calculated by using the derived load

combination:

σσσ gpeak ±= = psi64.24± (2.23)

Compared to Table 2.11, the peak fluctuating tensile stress in C1 corresponding to

load case FX+FY+MT is 29.97psi, while the peak fluctuating compressive stress is -

32.17 psi. The difference between the estimated peak values and the actual peak

normal stresses maybe is due to average correlation coefficients assumed.

2.7 Conclusion Remarks

The actual extreme pressure distributions causing the maximum quasi-static load

effects on a low-rise building are conditionally sampled and the extreme pressure

distributions corresponding to maximum along-wind load, across-wind load and

torsional moment at the base of the building are compared.

A frame structure is assumed for the low-rise building, and it is found that the total

load effects of the six wind load components result in a maximum 30% increase of the

peak tensile stress in column members compared with case only applying the along-

wind load.

A formula of wind load combination has been derived, and the derived wind load

combination is applied to the assumed frame structure, the peak normal stresses

caused by the wind load combination are compared to the actual one. The method

provides an effective tool to investigate wind load combination suitable for inclusion

in Codes.

Reference

ASCE 7-02 (2002), Minimum Design Loads for Buildings and Other Structures. ASCE, 2002

— — 50

E. Simiu, F. Sadek, T. M. Whalen, S. Jang, etc. (2003), Achieving safer and more economical buildings through database-assisted, reliability-based design for wind, J. Ind. Aerodyn., 91, 1587-1611

J.D. Ginger, J. D. Holmes (2003), Effect of building length on wind loads on low-rise buildings with a steep roof pitch, J. Ind. Aerodyn., 91 1377-1400

Levian M. L., K. C. Mehta (1992), Texas Tech field experiments for wind loads part I building and pressure measuring system, 43(1-3), pp.1565

T. Stathopoulos (2003), Wind Loads on Low buildings: in the wake of Alan Davenport’s Contributions, J. Ind. Aerodyn., 91, 1565-1585.

W. H. Melbourne (1975), Probability distributions of responses of BHP house to wind action and model comparisons, J. Ind. Aerodyn., Vol.1, No.2, pp.167

G. Solari, L. C. Pagnini (1999), Gust buffeting and aeroelastic behavior of poles and monotubular towers, J. Fluid Struct. Vol.13, pp.877

Y. Tamura, H. Kikuchi, K. Hibi (2001), Extreme Wind Pressure Distributions on Low-rise Building Models, Journal of Wind Engineering and Industrial Aerodynamic, Vol. 89, pp.1635-1646

— — 51

Table 2.1 Mean and standard deviation of the along-wind, across-wind and torsional moment coefficients (AOA around 0º)

Mean Coefficients Standard Deviation Coefficients AOA CX CY CT CZ CX CY CT CZ

355.2567 0.6444 0.1003 0.0038 0.4419 0.2472 0.1542 0.0619 0.1580 355.7948 0.6447 0.0940 -0.0044 0.4250 0.2339 0.1633 0.0644 0.1431 355.8156 0.6340 0.0891 -0.0062 0.4349 0.2092 0.1478 0.0579 0.1328 355.8889 0.6121 0.0940 0.0005 0.4571 0.1941 0.1152 0.0615 0.1096 356.3687 0.6677 0.0889 -0.0031 0.4657 0.2444 0.1601 0.0666 0.1619 356.4689 0.6362 0.1127 -0.0092 0.4838 0.2609 0.1841 0.0633 0.1642 356.6344 0.6291 0.0741 -0.0007 0.4971 0.1847 0.1936 0.0606 0.1140 357.0119 0.6578 0.0672 -0.0017 0.4262 0.2231 0.1461 0.0653 0.1344 357.4335 0.6666 0.0392 0.0020 0.4651 0.2494 0.2175 0.0668 0.1669 357.5329 0.6315 0.0662 0.0036 0.4436 0.2157 0.1440 0.0648 0.1252 358.0199 0.6905 0.0362 0.0005 0.4749 0.2578 0.2261 0.0638 0.1733 358.1373 0.6734 0.0343 0.0065 0.4958 0.2209 0.2712 0.0624 0.1429 358.2166 0.6317 0.0382 0.0025 0.4359 0.2240 0.1621 0.0680 0.1449 358.3164 0.6359 0.0500 -0.0058 0.4405 0.2231 0.1518 0.0645 0.1417 358.9301 0.7467 0.0279 0.0099 0.5604 0.2697 0.2972 0.0739 0.1984 359.5899 0.6490 0.0465 0.0001 0.4510 0.2175 0.2155 0.0637 0.1356

0.1296 0.6351 0.0016 0.0074 0.4393 0.2093 0.1249 0.0623 0.1235 0.3163 0.6332 -0.0059 0.0114 0.4306 0.2276 0.1528 0.0612 0.1366 0.5723 0.6459 -0.0252 0.0068 0.4510 0.2494 0.1536 0.0652 0.1577 2.1902 0.6375 -0.0393 0.0119 0.4537 0.2254 0.1509 0.0625 0.1469 3.3616 0.6816 -0.0689 0.0159 0.4787 0.2256 0.1640 0.0666 0.1385 4.0318 0.6985 -0.0881 0.0124 0.4938 0.2536 0.2812 0.0676 0.1790 4.3397 0.6440 -0.0672 0.0141 0.4613 0.2073 0.2534 0.0649 0.1367 4.4977 0.7116 -0.0686 0.0012 0.5194 0.2754 0.2595 0.0689 0.1861 4.5469 0.6690 -0.1051 0.0177 0.4626 0.2118 0.1729 0.0625 0.1328 5.0656 0.6399 -0.1059 0.0212 0.4630 0.2001 0.1614 0.0619 0.1190

— — 52

Table 2.2 Maximum fluctuating along-wind, across-wind coefficients, torsional moment coefficients, as well as lift force coefficient and their corresponding another three force coefficients (AOA round 0º)

Maximum CX Maximum CY AOA CXmax CY CT CZ CX CYmax CT CZ

355.2567 1.1059 -0.4232 -0.0334 0.4350 0.6699 0.7280 0.0312 0.2439 355.7948 1.2309 -0.1632 0.0062 0.6758 0.4510 0.6464 -0.1664 0.3875 355.8156 1.0390 -0.0340 0.0017 0.4128 0.2689 0.6284 -0.0353 0.5311 355.8889 0.9704 0.1265 0.0511 0.4845 -0.1461 -0.4359 0.1893 0.0818 356.3687 1.1164 0.1575 0.2525 0.4533 0.3627 -0.6882 -0.0302 0.4548 356.4689 1.0184 0.4344 0.0040 0.2877 0.8676 0.7783 -0.0042 0.6414 356.6344 0.8326 0.5947 -0.1944 0.3962 0.6106 0.8416 0.0123 0.3723 357.0119 1.0821 -0.0099 0.0322 0.3706 -0.2024 0.6194 -0.0481 -0.0311 357.4335 1.6124 -0.1113 -0.2330 0.7724 0.7567 -0.9779 0.1054 0.4969 357.5329 0.9463 0.3573 0.0053 0.4358 0.5141 0.5752 -0.0866 0.2790 358.0199 1.1658 0.1605 0.1199 0.5153 0.4859 -1.0126 0.1594 0.5211 358.1373 0.9962 -0.6223 0.0990 0.5528 0.2707 0.8704 -0.0622 0.2629 358.2166 0.9302 0.2792 0.0276 0.4608 0.6044 0.7017 0.0164 0.3771 358.3164 1.0389 0.4366 0.1114 0.4138 0.6803 0.6808 0.0552 0.2068 358.9301 1.0310 -1.1089 0.2074 0.9828 0.9149 -1.4333 0.2443 0.9823 359.5899 0.9184 0.0246 0.0752 0.3817 -0.2810 -0.8154 0.2145 0.1602 0.1296 1.0264 0.3643 0.0001 0.3872 0.4312 -0.8693 0.0816 0.3878 0.3163 1.6583 0.6902 0.1882 0.8005 1.5232 0.7282 0.1850 0.6840 0.5723 1.0628 -0.4392 -0.1520 0.5683 0.2031 0.6360 -0.0449 0.1552 2.1902 1.0312 0.7539 0.0836 0.4450 0.7502 1.0170 0.0722 0.4245 3.3616 1.1194 -0.3328 0.1347 0.6322 0.2223 1.3199 -0.2423 0.3990 4.0318 0.8903 -0.0617 0.1190 0.3732 0.2112 -0.7185 0.0510 0.1847 4.3397 1.0986 0.1122 0.0197 0.5030 0.3555 -0.7662 0.1450 0.3851 4.4977 0.9989 0.3022 -0.0066 0.4195 0.2688 -0.6337 -0.0123 0.2908 4.5469 1.0061 0.2230 -0.0020 0.4930 0.0643 0.6238 -0.1679 0.0708 5.0656 0.9280 -0.2109 -0.0239 0.4168 -0.1474 0.6071 -0.0375 0.0813

— — 53

Table 2.2(continued)

Maximum CT Maximum CZ AOA CX CY CTmax CZ CX CY CT CZmax

355.2567 0.1455 -0.6471 0.2827 0.4057 0.9202 0.3265 -0.0101 0.5261 355.7948 0.9804 -0.1178 -0.3162 0.6856 1.0081 -0.1551 -0.2160 0.7736 355.8156 -0.0988 0.0289 -0.2206 -0.1358 0.2256 0.6207 -0.0598 0.5420 355.8889 0.4341 0.1084 -0.2619 0.1554 0.9704 0.1265 0.0511 0.4845 356.3687 0.5908 0.3447 -0.2984 0.2646 0.6684 -0.5490 -0.2418 0.6007 356.4689 0.2645 0.1064 -0.2860 0.1542 0.8575 0.7581 0.0189 0.6819 356.6344 0.2152 0.1738 -0.2341 0.2802 0.7038 0.6774 -0.1506 0.4789 357.0119 0.4503 0.1852 0.3670 0.2371 1.0078 0.1142 -0.0093 0.5508 357.4335 1.1882 -0.2615 -0.4649 0.6588 1.3224 -0.3062 -0.1883 0.8093 357.5329 0.4345 0.0046 0.3104 0.1870 0.9046 0.3379 0.0379 0.5691 358.0199 0.5424 -0.1126 0.2827 0.2969 0.5897 -0.9110 0.1844 0.6542 358.1373 0.1818 -0.1240 0.3221 0.0422 0.7207 -0.6982 0.0154 0.5849 358.2166 0.3646 -0.3658 0.2934 0.2512 0.8034 -0.2421 -0.1339 0.4836 358.3164 0.1536 -0.0410 -0.3039 0.0278 0.7377 0.2968 0.0312 0.5205 358.9301 0.7155 -1.3170 0.3476 0.9234 0.8481 -1.4009 0.2174 1.0275 359.5899 0.2069 0.2710 -0.2787 0.1833 0.4109 0.4839 -0.0392 0.4493 0.1296 0.6216 0.1040 -0.3179 0.3132 0.8420 0.0650 -0.0476 0.4841 0.3163 1.4289 0.3234 0.3137 0.6069 0.3947 -0.6616 0.0914 0.4853 0.5723 0.3293 0.0289 -0.3272 0.1056 1.0505 0.3458 0.0879 0.9297 2.1902 0.5173 -0.0734 -0.2803 0.1364 1.0628 -0.4392 -0.1520 0.5683 3.3616 0.2646 0.8178 -0.2876 0.2327 0.9151 0.9074 0.1464 0.5258 4.0318 0.4791 0.4875 -0.2786 0.2092 1.0957 -0.2799 0.0349 0.7081 4.3397 0.3690 -0.5373 0.3245 0.2688 0.7355 -0.4744 -0.0502 0.5132 4.4977 0.3888 0.0362 -0.2537 0.2004 1.0456 0.2062 -0.0602 0.6028 4.5469 0.3480 0.1676 -0.2683 0.1539 0.9921 0.2939 -0.0713 0.4588 5.0656 0.4441 -0.2321 0.2870 0.1738 0.7236 -0.1971 -0.1035 0.5117

— — 54

Table 2.3 Mean and standard deviation of the along-wind, across-wind and torsional moment coefficients (AOA around 90º)

Mean Coefficients Standard Deviation Coefficients AOA CY CX CT CZ CY CX CT CZ

85.4797 0.48 0.10 0.02 0.53 0.16 0.19 0.10 0.14 85.4897 0.55 0.13 0.03 0.59 0.19 0.18 0.11 0.16 85.5346 0.46 0.14 0.01 0.51 0.15 0.14 0.09 0.13 85.5440 0.61 0.16 0.04 0.66 0.21 0.31 0.13 0.19 86.4494 0.64 0.15 0.04 0.70 0.23 0.26 0.13 0.19 86.6495 0.63 0.13 0.05 0.52 0.25 0.19 0.12 0.19 87.0289 0.61 0.07 0.01 0.65 0.21 0.23 0.13 0.17 87.1356 0.62 0.08 0.00 0.68 0.23 0.30 0.13 0.18 87.3489 0.48 0.10 0.02 0.50 0.16 0.15 0.10 0.14 87.8189 0.53 0.08 0.01 0.62 0.22 0.24 0.11 0.25 88.9611 0.53 0.08 0.01 0.56 0.19 0.15 0.11 0.20 89.0175 0.48 0.03 0.01 0.39 0.19 0.19 0.10 0.17 89.0326 0.68 -0.02 -0.01 0.73 0.23 0.25 0.13 0.20 89.5779 0.73 -0.05 0.00 0.77 0.29 0.27 0.16 0.25 89.8888 0.51 0.01 -0.02 0.57 0.16 0.21 0.10 0.13 90.3845 0.42 -0.07 0.01 0.46 0.14 0.13 0.10 0.12 90.7914 0.43 0.04 -0.01 0.45 0.17 0.19 0.09 0.15 90.7957 0.65 0.02 -0.02 0.71 0.23 0.32 0.13 0.19

Table 2.4 Maximum fluctuating along-wind, across-wind coefficients, torsional moment coefficients, as well as lift force coefficient and their corresponding another three force coefficients (AOA around

90º)

Maximum CY Maximum CX AOA CYmax CX CT CZ CY CXmax CT CZ

85.4797 0.77 -0.19 0.04 0.43 -0.25 -0.71 0.05 -0.06 85.4897 0.85 0.12 0.11 0.45 -0.62 0.75 -0.22 0.43 85.5346 0.62 -0.37 -0.20 0.23 -0.28 -0.54 0.26 0.12 85.5440 1.00 0.23 0.29 0.62 -0.21 -1.10 -0.09 0.15 86.4494 0.99 0.07 -0.30 0.75 0.02 -1.02 -0.03 0.19 86.6495 1.04 0.21 0.20 0.69 0.08 0.80 0.03 0.16 87.0289 0.91 0.06 0.10 0.49 -0.31 -0.87 -0.25 0.36 87.1356 1.31 0.42 0.18 0.81 -0.83 -1.00 -0.17 0.52 87.3489 0.66 -0.29 -0.07 0.46 -0.23 -0.51 0.04 0.11 87.8189 1.20 0.26 -0.38 1.03 -0.52 -1.11 0.08 0.79 88.9611 0.97 0.27 0.18 0.76 -0.32 0.61 -0.18 0.40 89.0175 0.68 0.19 0.16 0.58 -0.22 -0.73 -0.14 0.24 89.0326 1.13 -0.28 -0.06 0.75 -0.38 1.12 0.11 0.42 89.5779 1.59 0.00 -0.06 0.79 0.14 1.06 -0.05 -0.02 89.8888 0.89 0.19 0.07 0.47 -0.22 0.72 -0.16 0.14 90.3845 0.59 0.39 -0.22 0.34 -0.34 -0.57 -0.14 0.36 90.7914 0.75 0.20 -0.24 0.49 -0.30 0.71 -0.09 0.45 90.7957 1.00 -0.27 -0.45 0.39 -0.23 1.26 0.11 0.46

— — 55

Table 2.4 (continued)

Maximum CT Maximum CZ AOA CY CX CTmax CZ CY CX CT CZmax

85.4797 -0.23 -0.24 0.40 0.27 -0.66 -0.07 -0.08 0.60 85.4897 -0.13 0.07 -0.42 0.28 -0.59 0.53 -0.02 0.71 85.5346 -0.44 -0.41 0.42 0.35 -0.47 -0.39 0.41 0.43 85.5440 -0.28 -0.64 -0.55 0.06 -0.91 0.08 0.32 0.70 86.4494 -0.54 -0.18 0.54 0.39 -0.79 0.61 0.03 0.90 86.6495 -0.51 -0.20 0.63 0.37 -0.83 0.68 -0.13 0.72 87.0289 -0.61 -0.01 -0.62 0.25 -0.72 -0.08 0.13 0.66 87.1356 -0.62 -0.33 0.54 0.46 -1.29 0.27 0.20 0.85 87.3489 -0.08 0.26 -0.40 0.06 -0.47 0.04 0.30 0.57 87.8189 -0.68 -0.03 0.58 0.67 -1.03 0.02 -0.30 1.16 88.9611 -0.59 0.03 0.52 0.48 -0.93 0.28 0.20 0.79 89.0175 -0.28 -0.21 0.49 0.37 -0.63 0.05 0.18 0.67 89.0326 -0.45 -0.05 0.52 0.14 -1.03 -0.31 -0.04 0.79 89.5779 -1.38 -0.32 -0.60 0.92 -1.01 -0.34 -0.58 1.08 89.8888 -0.24 0.07 0.39 0.10 -0.80 0.20 0.05 0.50 90.3845 -0.32 0.12 -0.41 0.29 -0.24 -0.32 -0.14 0.71 90.7914 -0.35 -0.27 -0.39 0.18 -0.57 0.64 -0.13 0.53 90.7957 -0.83 -0.29 -0.54 0.43 -0.88 -0.43 -0.14 0.76

Table 2.5 The torsion coefficient of the four walls when AOA is close to zero

The mean torsion coefficient The rms torsion coefficient AOA Wall 1 Wall 2 Wall 3 Wall 4 Wall 1 Wall 2 Wall 3 Wall 4

355.2567 -0.0135 0.1408 0.0060 -0.1371 0.0351 0.0608 0.0119 0.0639355.7948 -0.0101 0.1436 0.0051 -0.1342 0.0350 0.0636 0.0130 0.0621355.8156 -0.0066 0.1413 0.0055 -0.1340 0.0348 0.0532 0.0116 0.0649355.8889 -0.0167 0.1389 0.0078 -0.1305 0.0319 0.0528 0.0107 0.0546356.3687 -0.0094 0.1510 0.0045 -0.1430 0.0366 0.0680 0.0127 0.0717356.4689 -0.0111 0.1367 0.0050 -0.1214 0.0378 0.0654 0.0126 0.0644356.6344 -0.0079 0.1329 0.0030 -0.1273 0.0381 0.0579 0.0125 0.0620357.0119 -0.0082 0.1510 0.0033 -0.1444 0.0351 0.0616 0.0125 0.0617357.4335 -0.0056 0.1383 0.0012 -0.1359 0.0378 0.0661 0.0133 0.0693357.5329 -0.0067 0.1410 0.0037 -0.1416 0.0349 0.0578 0.0122 0.0548358.0199 -0.0054 0.1423 0.0018 -0.1392 0.0412 0.0687 0.0144 0.0722358.1373 -0.0074 0.1305 0.0020 -0.1316 0.0386 0.0530 0.0135 0.0690358.2166 -0.0044 0.1428 0.0017 -0.1426 0.0349 0.0589 0.0126 0.0605358.3164 -0.0020 0.1490 0.0026 -0.1438 0.0378 0.0554 0.0129 0.0592358.9301 -0.0070 0.1460 0.0020 -0.1508 0.0439 0.0770 0.0157 0.0772359.5899 -0.0043 0.1317 0.0032 -0.1308 0.0357 0.0747 0.0132 0.06490.1296 -0.0002 0.1443 0.0005 -0.1520 0.0342 0.0565 0.0129 0.05830.3163 -0.0007 0.1390 0.0005 -0.1502 0.0340 0.0616 0.0131 0.05850.5723 0.0020 0.1436 -0.0029 -0.1496 0.0370 0.0620 0.0127 0.06782.1902 0.0033 0.1371 -0.0020 -0.1503 0.0340 0.0626 0.0129 0.05753.3616 0.0046 0.1381 -0.0041 -0.1546 0.0376 0.0645 0.0127 0.05834.0318 0.0077 0.1196 -0.0036 -0.1362 0.0419 0.0771 0.0146 0.06654.3397 0.0056 0.1139 -0.0004 -0.1332 0.0391 0.0761 0.0125 0.06364.4977 0.0090 0.1258 -0.0027 -0.1334 0.0404 0.0954 0.0140 0.07464.5469 0.0067 0.1278 -0.0052 -0.1470 0.0373 0.0592 0.0121 0.05245.0656 0.0061 0.1237 -0.0040 -0.1471 0.0342 0.0601 0.0113 0.0513

— — 56

Table 2.6 The torsion coefficient of the four walls when AOA is close to zero

The torsion coefficient * The torsion coefficient ** AOA Wall 1 Wall 2 Wall 3 Wall 4 Total Wall 1 Wall 2 Wall 3 Wall 4 Total

355.2567 -0.0012 0.1995 0.0133 0.0711 0.2827 0.0202 -0.2512 -0.0075 -0.0065 -0.2450355.7948 0.0968 -0.1008 -0.0118 0.3190 0.3031 -0.0695 -0.3992 0.0384 0.1141 -0.3162355.8156 0.0617 -0.0903 0.0009 0.2424 0.2147 -0.0960 -0.0899 0.0154 -0.0500 -0.2206355.8889 0.0573 0.1057 -0.0059 0.0678 0.2248 -0.0663 -0.2791 0.0148 0.0687 -0.2619356.3687 0.0568 0.0108 -0.0231 0.2432 0.2878 -0.0912 -0.2945 -0.0040 0.0914 -0.2984356.4689 0.0676 0.0501 -0.0287 0.1705 0.2595 -0.1373 -0.1962 -0.0015 0.0490 -0.2860356.6344 0.0192 0.0586 -0.0093 0.1271 0.1956 -0.0387 -0.2578 0.0110 0.0515 -0.2341357.0119 0.1470 0.0582 -0.0102 0.1720 0.3670 -0.0851 -0.2438 -0.0133 0.0656 -0.2766357.4335 0.1073 -0.0585 0.0089 0.2192 0.2769 0.0045 -0.4679 0.0463 -0.0478 -0.4649357.5329 0.0798 0.0582 -0.0321 0.2045 0.3104 -0.0726 -0.0935 -0.0264 -0.0340 -0.2264358.0199 0.0820 -0.1212 -0.0072 0.3291 0.2827 -0.0675 -0.2678 0.0198 0.0344 -0.2812358.1373 -0.0204 0.1688 -0.0204 0.1941 0.3221 -0.1143 -0.1341 0.0185 0.0035 -0.2265358.2166 0.0895 -0.0143 0.0154 0.2028 0.2934 -0.0840 -0.2684 0.0272 0.0489 -0.2762358.3164 0.1100 -0.0135 0.0017 0.2041 0.3023 -0.1589 -0.1531 0.0148 -0.0068 -0.3039358.9301 -0.0445 0.3187 0.0242 0.0493 0.3476 0.0273 -0.1987 0.0114 -0.1280 -0.2880359.5899 0.0531 0.0884 -0.0251 0.1523 0.2686 -0.0974 -0.1610 -0.0260 0.0057 -0.27870.1296 0.0577 -0.0113 -0.0151 0.2557 0.2870 0.0078 -0.3031 0.0385 0.0271 -0.22970.3163 -0.0076 0.1906 0.0524 0.0798 0.3152 -0.1052 -0.3162 0.0248 0.0788 -0.31780.5723 0.0288 0.0138 -0.0516 0.3227 0.3137 -0.0430 -0.1028 0.0173 -0.1816 -0.31012.1902 0.1059 -0.0409 0.0156 0.2285 0.3091 -0.1389 -0.1709 0.0167 -0.0342 -0.32733.3616 0.0868 0.0157 -0.0315 0.1829 0.2538 -0.0887 -0.1929 0.0150 -0.0137 -0.28034.0318 0.0944 0.0166 -0.0055 0.1653 0.2709 -0.0360 -0.0867 -0.0250 -0.1399 -0.28764.3397 0.0527 -0.1267 -0.0125 0.3358 0.2492 0.0435 -0.3543 -0.0135 0.0457 -0.27864.4977 0.0705 0.0776 0.0277 0.1487 0.3245 -0.0560 -0.2896 0.0130 0.0349 -0.29774.5469 0.0675 0.0472 0.0065 0.1173 0.2385 -0.0344 -0.1335 0.0030 -0.0888 -0.25375.0656 0.0264 -0.0189 0.0044 0.2290 0.2411 -0.0505 -0.2184 0.0211 -0.0206 -0.2684

Note: * the coefficients are corresponding to the total torsional coefficient is maximum, with mean extracted ** the coefficients are corresponding to the total torsional coefficient is minimum, with mean extracted

— 57 —

Table 2.7 Covariance of torsion coefficients of the four walls with along-wind load (AOA around 0º)

AOA W1&CX W2&CX W3&CX W4&CX 355.2567 0.0006 -0.0110 -0.0003 0.0115 355.7948 0.0014 -0.0106 -0.0001 0.0095 355.8156 0.0005 -0.0074 -0.0004 0.0093 355.8889 0.0008 -0.0065 -0.0005 0.0073 356.3687 0.0000 -0.0129 0.0002 0.0136 356.4689 0.0019 -0.0123 -0.0008 0.0101 356.6344 0.0006 -0.0068 -0.0004 0.0059 357.0119 0.0005 -0.0102 -0.0001 0.0102 357.4335 -0.0011 -0.0110 0.0010 0.0111 357.5329 0.0009 -0.0085 -0.0004 0.0080 358.0199 -0.0003 -0.0119 0.0003 0.0125 358.1373 -0.0004 -0.0066 0.0003 0.0076 358.2166 0.0003 -0.0095 0.0001 0.0100 358.3164 0.0007 -0.0085 -0.0004 0.0091 358.9301 -0.0010 -0.0102 0.0003 0.0131 359.5899 0.0012 -0.0116 -0.0006 0.0100 0.1296 -0.0005 -0.0086 0.0001 0.0094 0.3163 -0.0006 -0.0097 0.0001 0.0104 0.5723 -0.0001 -0.0111 0.0001 0.0136 2.1902 -0.0009 -0.0099 0.0005 0.0090 3.3616 0.0002 -0.0096 0.0000 0.0087 4.0318 -0.0010 -0.0100 0.0005 0.0101 4.3397 -0.0007 -0.0088 0.0001 0.0095 4.4977 -0.0008 -0.0169 0.0007 0.0153 4.5469 -0.0013 -0.0065 0.0007 0.0068 5.0656 -0.0014 -0.0080 0.0007 0.0070

Note: the numbers following ‘W’ represent the wall number, wall2 and wall 4

are side walls

Table 2.8 The covariance of torsion coefficients of the four walls with across-wind load (AOA around 0º)

AOA W1&CY W2&CY W3&CY W4&CY 355.2567 0.0023 -0.0009 -0.0009 -0.0031 355.7948 0.0026 -0.0015 -0.0009 -0.0036 355.8156 0.0025 -0.0006 -0.0007 -0.0037 355.8889 0.0012 -0.0017 -0.0003 -0.0016 356.3687 0.0021 -0.0002 -0.0011 -0.0042 356.4689 0.0035 -0.0023 -0.0011 -0.0037 356.6344 0.0044 -0.0016 -0.0012 -0.0058 357.0119 0.0018 -0.0010 -0.0007 -0.0029 357.4335 0.0044 -0.0002 -0.0017 -0.0076 357.5329 0.0021 -0.0028 -0.0005 -0.0017 358.0199 0.0053 -0.0010 -0.0020 -0.0069 358.1373 0.0063 -0.0002 -0.0025 -0.0117 358.2166 0.0018 -0.0016 -0.0010 -0.0037 358.3164 0.0023 -0.0020 -0.0009 -0.0022 358.9301 0.0078 -0.0077 -0.0031 -0.0079 359.5899 0.0040 -0.0079 -0.0017 -0.0012 0.1296 0.0011 -0.0016 -0.0006 -0.0022 0.3163 0.0018 -0.0025 -0.0009 -0.0020 0.5723 0.0023 -0.0012 -0.0008 -0.0035 2.1902 0.0020 -0.0030 -0.0008 -0.0016 3.3616 0.0027 -0.0044 -0.0009 -0.0009 4.0318 0.0074 -0.0087 -0.0025 -0.0055 4.3397 0.0060 -0.0104 -0.0020 -0.0020 4.4977 0.0056 -0.0122 -0.0019 0.0006 4.5469 0.0029 -0.0036 -0.0011 -0.0016 5.0656 0.0019 -0.0031 -0.0009 -0.0017

Note: the numbers following ‘W’ represent the wall number

— 58 —

Table 2.9 The covariance of torsion coefficients of the four walls with along-wind and across-wind load (AOA around 0º)

AOA W13&CX W24&CX W13&CY W24&CY 355.2567 0.0003 0.0005 0.0014 -0.0039 355.7948 0.0013 -0.0011 0.0017 -0.0051 355.8156 0.0001 0.0020 0.0018 -0.0043 355.8889 0.0003 0.0007 0.0010 -0.0033 356.3687 0.0003 0.0007 0.0010 -0.0044 356.4689 0.0011 -0.0022 0.0024 -0.0060 356.6344 0.0001 -0.0009 0.0032 -0.0074 357.0119 0.0004 0.0000 0.0011 -0.0039 357.4335 -0.0001 0.0001 0.0027 -0.0078 357.5329 0.0004 -0.0005 0.0015 -0.0044 358.0199 0.0000 0.0006 0.0032 -0.0079 358.1373 -0.0002 0.0010 0.0038 -0.0119 358.2166 0.0004 0.0005 0.0008 -0.0053 358.3164 0.0003 0.0006 0.0015 -0.0042 358.9301 -0.0008 0.0028 0.0047 -0.0156 359.5899 0.0005 -0.0016 0.0024 -0.0091 0.1296 -0.0003 0.0008 0.0005 -0.0038 0.3163 -0.0005 0.0006 0.0009 -0.0045 0.5723 0.0000 0.0025 0.0015 -0.0047 2.1902 -0.0003 -0.0009 0.0012 -0.0046 3.3616 0.0003 -0.0009 0.0018 -0.0053 4.0318 -0.0004 0.0001 0.0050 -0.0142 4.3397 -0.0006 0.0008 0.0040 -0.0124 4.4977 -0.0001 -0.0016 0.0037 -0.0116 4.5469 -0.0006 0.0003 0.0018 -0.0052 5.0656 -0.0008 -0.0009 0.0010 -0.0048

Note: the numbers following ‘W’ represent the wall number

Table 2.10 Influence coefficients

Fx Fy Fz Mx My Mz N FxH/(2D) FyH/(2B) -

Fz/4Mx/(2B) -

My/(2D)

Qy -Fy/4 +MzD/(8R2) Qx -Fx/4 -MzB/(8R2) Mx FyH/8 -MzDH/(16R2) My -FxH/8 -MzBH/(16R2)

C1

Mz N -

FxH/(2D)FyH/(2B) -

Fz/4Mx/(2B) My/(2D)

Qy -Fy/4 -MzD/(8R2) Qx -Fx/4 -MzB/(8R2) Mx FyH/8 +MzDH/(16R2) My -FxH/8 -MzBH/(16R2)

C2

Mz N -

FxH/(2D)-FyH/(2B)

-Fz/4

-Mx/(2B)

My/(2D)

Qy -Fy/4 -MzD/(8R2) Qx -Fx/4 +MzB/(8R2) Mx FyH/8 +MzDH/(16R2) My -FxH/8 +MzBH/(16R2)

C3

Mz N FxH/(2D) -

FyH/(2B)-Fz/4

-Mx/(2B)

-My/(2D)

Qy -Fy/4 +MzD/(8R2) Qx -Fx/4 +MzB/(8R2) Mx FyH/8 -MzDH/(16R2) My -FxH/8 +MzBH/(16R2)

C4

Mz

Note: the signs of all the internal forces are based on the global coordinate

system as shown in Figure 2.34.

— 59 —

Table 2.11 Peak normal stresses in column C1 (AOA close to zero)

Load Conditions Tensile stress(psi) Peak factor

Compressive stress(psi)Peak factor

ALL: Fx,Fy,Fz,Mx,My,Mz 38.43 (3.52) -51.46 (2.28) Fx+Fy 45.56 (2.55) -42.14 (2.62) Fx+Mz 33.88 (1.81) -30.75 (1.84) Fy+Mz 24.15 (--) -23.87 (--) Fx+Fy+Mz 46.40 (2.59) -43.52 (2.87) Along-wind Fx only 31.50 (1.66) -28.28 (1.66) Across-wind Fy only 21.32 (--) -20.83 (--) Torsional moment Mz only 7.24 (--) -7.24 (--) ALL/Along-wind only (%) 122% 182%

Table 2.12 Peak normal stresses in column C3 (AOA close to zero)

Load Conditions Tensile stress(psi) Peak factor

Compressive stress(psi) Peak factor

ALL: Fx,Fy,Fz,Mx,My,Mz 39.92 (2.95) -47.62 (2.50) Fx+Fy 42.14 (2.62) -45.56 (2.55) Fx+Mz 30.12 (1.82) -33.24 (1.79) Fy+Mz 20.85 (--) -21.59 (--) Fx+Fy+Mz 43.52 (2.87) -46.40 (2.60) Along-wind Fx only 28.28 (1.66) -31.50 (1.66) Across-wind Fy only 20.83 (--) -21.32 (--) Torsional moment Mz only 7.24 (--) -7.24 (--) ALL/Along-wind only (%) 141% 151%

Table 2.13 Peak fluctuating normal stresses in column C1 (AOA close to zero)

Load Conditions Tensile stress (psi) Compressive stress (psi) ALL: Fx,Fy,Fz,Mx,My,Mz 30.10 -35.92 Fx+Fy 32.91 -31.17 Fx+Mz 21.82 -19.99 Fy+Mz 23.41 -23.76 Fx+Fy+Mz 29.87 -32.17 Along-wind Fx only 19.54 -17.59 Across-wind Fy only 20.79 -20.44 Torsional moment Mz only 7.21 -7.21 ALL/Along-wind only (%) 154% 204%

Table 2.14 Peak fluctuating normal stresses in column C3 (AOA close to zero)

Load Conditions Tensile stress (psi) Compressive stress (psi) ALL: Fx,Fy,Fz,Mx,My,Mz 29.99 -34.32 Fx+Fy 31.17 -32.91 Fx+Mz 19.50 -21.36 Fy+Mz 20.63 -21.06 Fx+Fy+Mz 27.97 -33.58 Along-wind Fx only 17.59 -19.54 Across-wind Fy only 20.44 -20.79 Torsional moment Mz only 7.21 -7.21 ALL/Along-wind only (%) 171% 176%

— 60 —

Table 2.15 Comparison of actual response of normal stresses by ASCE(Fig. 6-9)

CASE1 CASE2 CASE3 CASE4 Actual R. Err.* C1 27.20 20.41 49.33 38.65 51.46 -4.1% C3 24.60 18.45 50.46 36.19 47.62 5.9%

Note:* Error is the actual response with the maximum value of four cases.

Table 2.16 Comparison of actual response of normal stresses by ASCE(Fig. 6-10)

Maximum normal stress Minimum normal stress ASCE R. Actual R. perc. ASCE R. Actual R. perc.

C1 42.79 38.43 11.3% -23.75 -51.46 -53.8%

C3 39.31 39.92 -1.5% -26.04 -47.62 -45.3%

Table 2.17 Correlation coefficients between along-wind load Fx, across-wind load Fy, and torsional moment Mz (AOA around 0º)

AOA Along and across Along-torsion Across-torsion 355.2567 0.1502 0.0521 -0.2652 355.7948 0.2029 0.0155 -0.3231 355.8156 0.1257 0.1680 -0.2868 355.8889 0.3004 0.0868 -0.3325 356.3687 -0.0869 0.0609 -0.3117 356.4689 0.3644 -0.0664 -0.3113 356.6344 0.1685 -0.0715 -0.3544 357.0119 0.0577 0.0313 -0.2934 357.4335 -0.2443 0.0026 -0.3524 357.5329 0.2773 -0.0084 -0.3134 358.0199 -0.0959 0.0348 -0.3218 358.1373 -0.1101 0.0575 -0.4788 358.2166 -0.0090 0.0591 -0.4111 358.3164 0.2228 0.0589 -0.2764 358.9301 -0.1001 0.1030 -0.4962 359.5899 0.2631 -0.0759 -0.4885

0.1296 -0.1045 0.0362 -0.4281 0.3163 -0.0686 0.0137 -0.3848 0.5723 -0.1158 0.1517 -0.3181 2.1902 -0.0974 -0.0887 -0.3625 3.3616 0.0080 -0.0407 -0.3206 4.0318 -0.1830 -0.0193 -0.4865 4.3397 -0.0724 0.0091 -0.5116 4.4977 -0.0400 -0.0888 -0.4429 4.5469 -0.3830 -0.0236 -0.3151 5.0656 -0.2387 -0.1385 -0.3880 average 0.0074 0.0123 -0.3683

— 61 —

Table 2.18 rms value of the along-wind, across-wind and torsional moment (AOA around 0º)

AOA Along(lb) Across(lb) Torsion(lb-ft)355.2567 106.0683 99.2517 717.8651 355.7948 100.3597 105.0691 746.6338 355.8156 89.7657 95.0908 671.9573 355.8889 83.2481 74.1594 714.0246 356.3687 104.8605 103.0384 773.1532 356.4689 111.9427 118.4686 734.6763 356.6344 79.2234 124.5949 703.1844 357.0119 95.7178 94.0020 757.2697 357.4335 106.9816 139.9360 775.3707 357.5329 92.5544 92.6769 751.3499 358.0199 110.5831 145.4685 740.6679 358.1373 94.7447 174.5004 724.0367 358.2166 96.0985 104.2833 788.6363 358.3164 95.6917 97.6555 748.6183 358.9301 115.6929 191.2257 857.5635 359.5899 93.2859 138.6720 739.1290 0.1296 89.7764 80.3914 722.4848 0.3163 97.6281 98.3323 709.4463 0.5723 106.9950 98.8336 756.0895 2.1902 96.6828 97.0749 724.7545 3.3616 96.7633 105.5163 772.6980 4.0318 108.8065 180.9540 784.2259 4.3397 88.9373 163.0745 752.9202 4.4977 118.1546 167.0185 799.4115 4.5469 90.8806 111.2453 725.0610 5.0656 85.8626 103.8466 718.1271 average 98.3579 119.3993 746.5137

Table 2.19 Mean value of the along-wind, across-wind and torsional moment (AOA around 0º)

AOA Along(lb) Across(lb) Torsion(lb-ft) 355.2567 276.4519 64.5530 44.1803 355.7948 276.5824 60.4841 -50.8892 355.8156 272.0038 57.3647 -71.6506 355.8889 262.5887 60.4772 5.6377 356.3687 286.4224 57.2381 -35.3940 356.4689 272.9136 72.5374 -106.9104 356.6344 269.8901 47.6866 -8.4930 357.0119 282.1763 43.2326 -20.0006 357.4335 285.9661 25.2180 23.0960 357.5329 270.8999 42.6136 41.2864 358.0199 296.2381 23.2846 5.7964 358.1373 288.8955 22.0491 75.1938 358.2166 271.0065 24.5706 29.4122 358.3164 272.7944 32.1951 -67.5552 358.9301 320.3438 17.9362 114.6397 359.5899 278.4352 29.8974 1.2368 0.1296 272.4679 1.0332 86.3391 0.3163 271.6582 -3.7730 132.4350 0.5723 277.1223 -16.1962 79.0863 2.1902 273.4965 -25.2931 137.4718 3.3616 292.4317 -44.3563 185.0505 4.0318 299.6674 -56.7161 143.3921 4.3397 276.2887 -43.2678 163.0639 4.4977 305.2855 -44.1427 14.0491 4.5469 287.0073 -67.6500 204.5899 5.0656 274.4969 -68.1451 246.3959 average 281.2897 12.0320 52.7485

— 62 —

Figure 2.1 WERFL building of Texas Tech University

Figure 2.2 Pressure Tap Arrangement of WERFL Building

— 63 —

Figure 2.3 Mean pressure distribution at three pressure tap layers (AOA=0.1296º)

Figure 2.4 Fluctuating pressure distribution at three pressure tap layers (AOA=0.1296º)

Figure 2.5 Instantaneous wall pressure distributions causing maximum fluctuating quasi-static along-wind base shear FDmax at three pressure tap layers (AOA=0.1296º)

Figure 2.6 Instantaneous wall pressure distributions causing maximum fluctuating quasi-static across-wind base shear FLmax at three pressure tap layers (AOA=0.1296º)

— 64 —

Figure 2.7 Instantaneous wall pressure distributions causing maximum fluctuating quasi-steady base moment MTmax at three pressure tap layers (AOA=0.1296º)

Figure 2.8 Instantaneous wall pressure distributions causing maximum fluctuating quasi-steady Lift Force at three pressure tap layers (AOA=0.1296º)

Figure 2.9 Ensemble averaged mean wind pressure distributions at three pressure tap layers (AOA around 0º)

Figure 2.10 Ensemble averaged fluctuating wind pressure distributions at three pressure tap layers (AOA around 0º)

— 65 —

Figure 2.11 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static along-wind base shear at three pressure tap layers

(AOA around 0º)

Figure 2.12 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static across-wind base shear at three pressure tap layers

(AOA around 0º)

Figure 2.13 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static torsional base moment at three pressure tap layers

(AOA around 0º)

Figure 2.14 Ensemble averaged extreme fluctuating wind pressure distributions

causing maximum quasi-static lift force at three pressure tap layers (AOA around 0º)

— 66 —

Figure 2.15 Ensemble averaged mean wind pressure distributions at three pressure tap layers (AOA around 90º)

Figure 2.16 Ensemble averaged fluctuating wind pressure distributions at three pressure tap layers (AOA around 90º)

Figure 2.17 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static along-wind base shear at three pressure tap layers

(AOA around 90º)

Figure 2.18 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static across-wind base shear at three pressure tap layers

(AOA around 90º)

— 67 —

Figure 2.19 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static torsional moment at three pressure tap layers (AOA

around 90º)

Figure 2.20 Ensemble averaged extreme fluctuating wind pressure distributions causing maximum quasi-static lift force at three pressure tap layers (AOA around 90

º)

Figure 2.21 The relationship between the maximum along-wind base shear CX and its simultaneously recorded across-wind base shear ratio CY/ CYmax and torsional

base moment ratio CT/CTmax, CZ/CZmax (AOA=0º).

Figure 2.22 The relationship between the maximum across-wind base shear CY and its simultaneously recorded along-wind base shear ratio CX/CXmax and torsional base

moment ratio CT/CTmax, CZ/CZmax (AOA=0º).

— 68 —

Figure 2.23 The relationship between the maximum torsional base moment CT and its simultaneously recorded along-wind base shear ratio CX/CXmax and across-wind

base shear ratio CY/CYmax, CZ/CZmax (AOA=0º).

Figure 2.24 The relationship between the maximum torsional base moment CZ and its simultaneously recorded along-wind base shear ratio CX/CXmax and across-wind

base shear ratio CY/CYmax, CT/CTmax (AOA=0º).

Figure 2.25 The relationship between the maximum along-wind base shear CY and its simultaneously recorded across-wind base shear ratio CX/CXmax and torsional

base moment ratio CT/CTmax, CZ/CZmax.(AOA=90º)

Figure 2.26 The relationship between the maximum across-wind base shear CX and its simultaneously recorded along-wind base shear ratio CY/CYmax and torsional base

moment ratio CT/CTmax, CZ/CZmax.(AOA=90º)

— 69 —

Figure 2.27 The relationship between the maximum torsional base moment CT and its simultaneously recorded along-wind base shear ratio CY/CYmax and across-wind

base shear ratio CX/CXmax, CZ/CZmax.(AOA=90º)

Figure 2.28 The relationship between the maximum torsional base moment CZ and its simultaneously recorded along-wind base shear ratio CY/CYmax and across-wind

base shear ratio CX/CXmax, CT/CTmax.(AOA=90º)

Figure 2.29a Cross-correlation coefficients between wind forces (AOA= 0.1297º)

Figure 2.29b Cross-correlation coefficients between wind forces (AOA= 0.1297º)

— 70 —

Figure 2.30a Cross-correlation coefficients between wind forces (AOA= 85.4797º)

Figure 2.30b Cross-correlation coefficients between wind forces (AOA= 85.4797º)

(a) maximum along-wind base shear FDmax (b) Maximum across-wind base shear

FLmax

(c) Maximum torsional base moment MTmax

Figure 2.31 Ensemble averaged extreme wind pressure distributions causing maximum quasi-static load effects at the base (7H/8, α =1/4,154 samples)

Figure 2.32(a) Full design wind pressure of CASE1 of ASCE(Figure 6-9)

WIND AOA=0º

0.8qz

0.9qh 0.5qh 0.3qh

0.4qh

— 71 —

Figure 2.32(b) Full design wind pressure of CASE1 of ASCE(Figure 6-9)

Figure 2.33 Pressure distribution based on ASCE7 (Figure 6-10) for calculation

Figure 2.34 Frame model (Tamura, 2001)

Figure 2.35 Columns of the frame model (Tamura, 2001)

WIND

Direction of MWFRS Reference

Corner

12 3

4

5

6

1E

2E 3E

4E

2a, with a=4.5ft

AOA=0º

WIND AOA=90º

0.8qz

0.9qh 0.5qh 0.3qh

0.5qh

— — 72

CHAPTER III

WIND INDUCED RESPONSES OF LOW-RISE

BUILDING

3.1 Introduction

The gust response factor method was originally introduced by Davenport (Davenport,

1967), and was utilized to treat wind loads on structures under the buffeting action of

wind gusts in most major codes and standards all around the world. In this method,

the effective static wind load is equal to the mean wind load multiplied by the gust

response factor.

In this chapter, the gust response factor method is investigated for low-rise buildings.

Gust response factor method is widely used for estimating equivalent static wind load.

In this scheme, the equivalent static wind loading used for design is equal to the mean

wind force multiplied by the GRF. Gust response factor is defined as the ratio of

maximum structural response in a certain time interval to mean structural response. It

is not hard to imagine that for different structural responses, different gust response

factors exist, which makes the corresponding equivalent static wind load variable.

Besides gust response factor method, other methods to study equivalent static wind

loads are also discussed in this chapter.

Wind Engineering Research Field Laboratory (WERFL) of Texas Tech University is

used for estimation of wind loading effects and corresponding gust response factors

and some other factors, and a wind tunnel model of Tokyo Polytechnic University is

also utilized; WERFL building is a full scale building and the other one is a wind

tunnel model. The gust factors of responses of these two buildings under wind loading

are calculated respectively. And methods to investigate universal equivalent static

wind load are applied to both buildings too.

3.2 WERFL Building (Full scale building)

To investigate the wind induced responses in low-rise buildings, a frame system is

assumed for WERFL building, several critical sections along each frame will be

checked for internal axial force N, shear force Q, and bending moment M. All those

internal forces will be investigated under two wind angle of attacks(AOA), that is,

angle of attack close(AOA) to 0º and 90º. The frame system is shown in Figure 3.1.

— — 73

For the full scale WERFL building, the pressure distribution characteristics such as

mean pressure distribution, fluctuating pressure distribution have been investigated in

Chapter II for angles of attack (AOA) around 0º and 90º. Under both angles of attack,

the pressure distributions coinciding with maximum along-wind force, across-wind

force, torsional moment, as well as lift force at the base of the building are also

studied.

To figure out a more reasonable wind load for wind resistant design, the traditional

gust response factor method is studied in detail for assumed frame structures of the

full-scale WERFL building.

3.2.1 Gust Response Factors

There are totally four across-wind frames considered for WERFL building, that is,

Frame A, B, C and D; For each of the frames, nine critical sections along frame are

investigated, and for each section, the axial force N, shear force Q and bending

moments M are calculated by using the pressure time histories of taps installed on the

frame. Since the WERFL building is a low rise building and has relatively high

rigidity, resonant responses can be ignored and the background responses are

calculated based on quasi-static theory.

The selected nine critical sections for each frame are shown in Figure 3.2. They are

the bottom point of left column(Number 1), middle point of left column(Number 2),

top point of left column(Number 3), quarter point of roof beam(Number 4), middle

point of roof beam(Number 5), three quarter point of roof beam(Number 6), top point

of right column(Number 7), middle point of right column(Number 8), bottom point of

right column(Number 9) respectively. Three kinds of support conditions, pin-pin,

pin-roller, fix-fix, are assumed for each frame. Figure 3.2 only shows support

condition pin-pin.

The mean and maximum dynamic axial forces N, shear forces Q, bending moments M

at those 9 critical sections of each frame are calculated and listed in Appendix I, and

gust response factors for those responses are also calculated and listed in Appendix I.

From those tables, it is noted that section 3, 5, 7 are the most critical sections at which

significant axial force N, shear force Q or bending moment M always happen. So for

simplicity, just internal forces at these 3 sections and corresponding gust factors of all

these frames are listed in Table 3.1~Table 3.4.

— — 74

Table 3.1 lists the mean, absolute maximum responses with mean included, and

corresponding gust factor for Frame A with AOA around 0º. From this table, it can be

seen that the largest gust factor is 5.991, and smallest one 2.305. For Frame B with

AOA around 0º, the largest gust factor is 8.184, the smallest one 2.947. When AOA

is around 0º, the largest gust factor for Frame C is 9.389, the smallest one is 2.614;

the largest gust factor for Frame D is 6.404, smallest one is 2.333. It is noted that all

the largest gust factors of all the frames happen at critical section 3 for internal

moment M with support condition pin roller. The largest gust factor among all the

responses happen at Frame C, critical section 3 for internal moment M with support

condition pin roller.

When AOA is around 90º, internal forces at these three sections and corresponding

gust factors of all these frames are listed in Table 3.5~Table 3.8. The largest gust

factor for Frame A is 5.77, the smallest one is 2.87; for Frame B, largest value is

21.17, smallest one 2.63; For Frame C, largest one is 7.62, smallest one 2.29; the

largest gust factor for Frame D is 10.76, and smallest one is 3.05. It is needed to be

pointed out that all the largest gust factors happen at critical section 5 for shear force

Q with support condition pin roller. The large gust factors of 21.17 and 10.76 among

all the responses is due to the extremely small mean response of -1.11 and -1.94

respectively for internal shear force at critical section 5, Frame B and Frame C with

support condition pin roller. So actually, the large gust factors due to small mean

response are not really meaningful, and thus can be disregarded.

Since the gust response factors vary in a large range, it is hard to quantify a uniform

gust factor for all the internal forces, which is a limitation of gust response factor

(GRF) method. If the maximum gust response factor is set for the design gust

response factor, then for many structural responses, it will be too conservative.

However, if the average gust response factor is used, then some structural responses

will be underestimated by the assumed equivalent static wind load.

— — 75

3.2.2 Comparison to responses calculated by

ASCE

3.2.2.1 ASCE (Figure 6-9) For this frame system, wind loading cases in Figure6-9 in ASCE7-05 are applied.

Table 3.9 lists the responses calculated by four cases indicated in ASCE and the actual

responses of Frame C with support condition fix-fix and AOA=90º.

From Table 3.9, it can be seen that for some responses the results from ASCE are very

conservative while for some responses, they are not necessarily larger than actual

responses, which proves the inconsistency of ASCE.

3.2.2.2 ASCE (Figure 6-10) For this frame system, wind loading cases in Figure6-10 in ASCE7-05 are also

applied. The pressure distribution based on ASCE is shown in Figure 3.5.

The pressure coefficients given by Figure6-10 of ASCE 7-05 in transverse Direction

(AOA=90º) are listed below:

3.0a ft= , 13.0h ft= , 2.019hq sf= Zone 1: ( )0.40hp q= × + =+ 0.808 psf

Zone 2: ( )0.69hp q= × − =−1.393 psf

Zone 3: ( )0.37hp q= × − =−0.747 psf

Zone 4: ( )0.29hp q= × − =−0.586 psf

Zone 5 and 6: ( )0.45hp q= × − =−0.909 psf Similarly,

Zone 1E: ( )0.61hp q= × + =+ 1.232 psf

Zone 2E: ( )1.07hp q= × − =−2.160 psf

Zone 3E: ( )0.53hp q= × − =−1.070 psf

Zone 4E: ( )0.43hp q= × − =−0.868 psf Based on the pressure coefficients listed above, the pressure distribution on Frame A,

B, C and D are shown in Figure 3.6.

The axial force, shear force and bending moment at critical section 3, 5, 7 of Frame

A,B,C and D with support condition fix-fix under the pressure distributions based on

ASCE7-05 are calculated and compared with actual responses at AOA=90 º

calculated by pressure time history, the results are listed in Table 3.10. It should be

noted that negative sign of percentage in the table means that ASCE underestimates

the responses.

— — 76

From Table 3.10, it can be seen that almost all the response calculated from ASCE

pressure distribution are underestimated compared with the actual responses. For the

shear force at critical section 5 of Frame D, it is underestimated by almost 82.5%,

which proves that ASCE can not always give results conservatively.

3.2.3 Background Factors

Besides gust response factor (GRF) method, background factor (BF) is another way to

provide equivalent static wind loading. The background factor is used to define the

influence of loss of spatial correlation on the background response. It is defined as the

ratio of peak dynamic background response to that caused by gust loading envelop.

For high rise building, the gust loading envelop (Chen, 2003 ,2004) is the background

wind loading without any reduction due to loss of spatial correlation of wind loading,

which is equal to the standard deviation of dynamic loading along the building

height. And almost all the responses calculated by the gust loading envelop are

definitely larger than the actual dynamic responses since for high rise building, the

influence functions for almost all kinds of responses are simple and without sign

change.

For low-rise buildings, different structural responses may have dramatically different

influence lines, which are particularly complicated by their changes in sign over the

structure.

A uniform dynamic loading envelop for a given frame is defined as follows. The

value of a point pressure is taken as the larger pressure magnitude among actual

maximum and minimum dynamic pressures. Its sign is determined based on the first

POD mode. The pressure distributions on Frame A, B, C and D corresponding to the

defined gust loading envelope are plotted in Figure 3.7~Figure 3.10.

Table 3.11 ~Table 3.14 list the actual maximum dynamic responses , dynamic

responses under dynamic gust loading envelop, and corresponding background factors

of critical section 3, 5, 7 on Frames A, B, C and D with angle of attack(AOA) around

0º. And the maximum dynamic responses, dynamic responses under gust loading

envelop and corresponding background factors for all the frames and all the critical

sections are listed in Appendix I.

— — 77

From these tables, it can be seen that the largest background factor for Frame A is

1.47, and smallest one 0.42. For Frame B, the largest background factor is 2.09, the

smallest one 0.55. The largest background factor for Frame C is 1.93, the smallest one

is 0.31; the largest background factor for Frame D is 1.80, smallest one is 0.51.

When AOA is around 90º, the maximum dynamic responses, dynamic responses

under gust loading envelop and corresponding background factors for all the frames

and all the critical sections are also listed in Appendix I. The pressure distributions on

Frame A, B, C and D corresponding to the defined gust loading envelope are plotted

in Figure 3.11~Figure 3.14.

Table 3.15~Table 3.18 list the actual maximum dynamic responses, dynamic

responses under dynamic gust loading envelop, and corresponding background factors

of critical section 3, 5, 7 on Frames A B C and D with angle of attack(AOA) around

90º.

From these tables, the largest background factor for Frame A is 2.07, the smallest one

is 0.39; for Frame B, largest value is 32.17, smallest one 0.47; For Frame C, largest

one is 7.81, smallest one 0.38; the largest background factor for Frame D is 15.60, and

smallest one is 0.37. The large background factors of 32.17, 15.60 and 7.81 among all

the responses are due to the extremely small responses under the gust loading envelop.

So actually, the large background factors are not really meaningful, and thus can be

disregarded.

Because the background factors of low-rise building vary in a large range, so it also

may not be a good method to estimate equivalent static wind loading for low rise

building too.

Besides the defined gust loading envelop based on first POD mode, gust loading

envelop can also be defined according to mean pressure distribution. The value of a

point pressure on this kind of gust loading envelop is taken as the larger pressure

magnitude among actual maximum and minimum dynamic pressures. Its sign is

determined based on the mean pressure distribution.

Table 3.19 and Table 3.20 list the dynamic internal forces calculated by maximum

dynamic loading envelop based on mean sign at section 3, 5, 7. Table 3.21 and Table

3.22 list the corresponding background factors at section 3, 5, 7.

— — 78

Dynamic internal forces and background factors of all the nine critical sections of

along-wind and across-wind frames are listed in Appendix I.

For high-rise building, the responses of interest are usually the shear force or bending

moment at the base of building or the top displacement, for which the influence

functions are usually simple without sign change, so the dynamic response calculated

by gust loading envelop is usually larger than the actual maximum dynamic response

and correspondingly, background factors are less than one; and by quantifying the

background factors, a simple expression of equivalent static wind loading which can

reproduce the same maximum dynamic response can be determined by multiplying

the gust loading envelop by background factor.

However for low-rise building, the responses of interest are usually the internal forces

in structure members, and the influence functions are usually very complicated with

sign change. The responses calculated by gust loading envelop based on mean

pressure sign or first POD mode sign are not always larger than actual dynamic

responses, so for low-rise building, gust loading envelop is no longer a convenient

concept and at mean time, background factors can’t be quantified easily. Thus another

method of describing equivalent static wind load needs to be proposed.

3.3 Wind Tunnel Model in Tokyo Polytechnic University

3.3.1 Gust Factors

A wind tunnel model with L:B:H 1:1:1(20cmX20cmX20cm) is tested in Tokyo

Polytechnic University. There are 500 pressure taps uniformly distributed around the

building surface. The power law index of the wind velocity profile is 1/4; length scale

is 1/250. The sampling interval is 0.00128s, there are total 32768 data points in the

pressure time history, which means the total sampling time is about 41 second.

The tap location is shown in Figure 3.3. Since this wind tunnel model is a cubic model,

for a single angle of attack, frames in two directions (along-wind and across-wind

directions) can be assumed and studied for wind induced internal effects, which is

different from the rectangular WERFL building for which frames in one direction are

studied under two angles of attack(0º and 90º). Three across-wind frames (in x

direction) and three along-wind frames (in y direction) are assumed as members for

investigation for the wind tunnel model. Nine critical sections with the same locations

— — 79

as the WERFL building along each frame are checked for axial force, shear force and

bending moment. Totally there are three support conditions assumed for all the frames,

that is, pin-roller, pin-pin, and fix-fix.

The assumed frame arrangement of this wind tunnel model is shown in Figure 3.4.

The axial force, shear force and bending moment at these 9 critical sections are

calculated by wind-induced pressure measured in pressure taps along the frames. All

responses of these nine critical sections are listed in Appendix I. From these tables, it

can be seen that responses of critical section 1, 3, 5, 7, 9 are critical. So Table

3.23~Table 3.28 list the mean, maximum dynamic internal forces of section 1, 3, 5, 7,

9 for Across Frame and Along Frame A, B, C and corresponding gust response factors

respectively.

From these tables, it can be seen that some very large gust response factors are

obtained due to extreme small mean response. It is reasonable to disregard this kind of

gust response factors as has been discussed for WERFL building.

The largest gust factor for Across Frame A is 6.58, the smallest one is 2.16; for

Across Frame B, largest value is 9.22, smallest one 2.90; For Across Frame C, largest

one is 12.40, smallest one 3.36.It is needed to be pointed out that all the largest gust

factors happen at critical section 1 or critical section 9 for bending moment M with

support condition fix-fix.

For Along Frame A, the largest gust factor is 13.26 and smallest is 2.82. For Along

Frame B, largest value is 13.67, smallest one 2.52; For Along Frame C, largest one is

10.59, smallest one 2.52. Some extreme large gust response factors due to small mean

responses have been neglected.

As has been discussed for WERFL building, for the wind tunnel model, the gust

response factors also vary in a large range, it is hard to quantify a uniform gust factor

for all the internal forces.

3.3.2 Background Factors

Table 3.29~Table 3.34 list the maximum dynamic internal forces, and responses

under the gust loading envelop based on first POD mode of section 1, 3, 5, 7, 9 for

Across Frame A, B, C and Along Frame A, B, C and corresponding background

factors respectively.

— — 80

For Across Frame A, there are four large background factors obtained due to very

small responses calculated by gust loading envelope. Except those irregular values,

the largest background factor is 1.88, the smallest one is 0.46. For Across Frame B,

because of the small responses calculated by gust loading envelop, there are several

background factors much larger than 1, as has been discussed, this kind of factors can

be neglected. And the largest background factor is 1.57, the smallest one 0.54. For

Across Frame C, except some large background factors, the largest value is 2.05,

smallest one 0.42.

As far as Along frame A, B, C, excepts some very large background factors, the

largest background factors for Along Frame A, B, C are 1.95, 1.73, 1.88 respectively;

the smallest background factors for Frame A, B, and C are 0.50, 0.57, 0.60

respectively.

As has been discussed for WERFL building, the responses calculated by gust loading

envelop based on first POD mode sign are not always larger than actual dynamic

responses, so it is proved that for low-rise building, gust loading envelop is no longer

a convenient concept and at mean time, background factors can’t be quantified easily.

Thus another method of describing equivalent static wind load needs to be proposed.

3.4 Background Factors Based on Four Gust Loading

Envelops of WERFL Building

For WERFL building, at AOA around 0º, the building frames are under across-wind

excitation and at AOA around 90 º , the building frames are under along-wind

excitation.

From the discussion of background factors of WERFL building, it can been seen that

background factors of frames based on first POD mode under along-wind excitation

are less sensitive to specific response than those calculated under across-wind

excitation, which means that when angle of attack is close to 0o, background factors of

different responses vary in a range narrower than those when angle of attack is close

to 90º. This may be due to the fact that in the case of along-wind excitation, the

pressure distribution causing peak dynamic response usually follow the first POD

mode which has the same direction as mean pressure distribution around the building,

while in the case of across-wind excitation, the pressure distribution causing peak

— — 81

dynamic response is very different from the first POD mode or the mean pressure

distribution.

Base on the above discussion, it is necessary to give a more reasonable gust loading

envelop (GLE) and background factor (BF) definition. Four gust loading envelops can

be provided for across-wind excitations, and those gust loading envelops are defined

as those proposed by Chen. The gust loading envelope 1 and 2 are featured by the

minimum (negative peak) pressure on wall two and maximum (positive peak)

pressures on wall four, but with negative peak pressures on the roof for envelope 1

and positive peak pressures on the roof for envelope 2. The gust loading envelops 3

and 4 are characterized by the positive peak pressures on wall two and negative peak

pressures on wall four but with negative peak pressures on the roof for envelope 3 and

positive peak pressures on the roof for envelope 4. The pressure distributions on

Frame A, B, C and D corresponding to all the four gust loading envelopes at AOA=0o

and AOA=90º are plotted in Figure 3.15~Figure 3.46.

The responses under the action of all the four gust loading envelopes can be

calculated and the maximum responses can be used for the determination of

corresponding background factors, it should be noted that the background factor is

defined as the ratio of the actual peak dynamic response to the maximum response

calculated by one corresponding gust loading envelope. The maximum responses and

corresponding background factors are listed in Table 3.35 and Table 3.37 for AOA =

0º ; and for AOA=90º , the maximum responses and corresponding background

factors of section 3, 5 and 7 are listed in Table 3.36 and Table 3.38. From Table 3.37,

it can be seen that the background factors are less sensitive to specific response

compared to the background factors calculated based on first POD mode as shown in

Table 3.11 ~Table 3.14 at AOA=0º, that is, under the across-wind excitation. For

AOA=90º, or under the action of along-wind excitation, comparison between Table

3.38 and Table 3.15~Table 3.18 will give the same conclusion about the background

factors, and the background factors based on the four gust loading envelopes are

usually not larger than 1 as those calculated by gust loading envelop based on first

POD mode.

These results illustrated that using gust loading envelope combined with the

background factor as suggested by GLE approach can result in accurate and

— — 82

convenient modeling of across-wind loads on low-rise buildings if reasonable gust

loading envelopes are used.

3.5 Concluding remarks

In this chapter, the wind induced responses in a frame system of WERFL building and

a wind tunnel model are studied. The calculated wind induced responses include mean

responses, maximum dynamic responses and dynamic responses under the action of

gust loading envelop. Corresponding gust response factor and background factors are

also calculated. From the calculation results it can be seen that these factors vary in a

large range, which gives a clear review of the limitation of gust response factor (GRF)

method and background factor (BF) method. Based on the traditional background

factor (BF) method involving only one gust loading envelope, background factor

method calculated by four different gust loading envelopes is proposed, the

background factors are less sensitive to specific response than those calculated by a

unique gust loading envelope and unusually large values will not arise in these

background factors which prove that background factor method is efficient if

reasonable gust loading envelopes are used.

Reference

X. Chen, A. Kareem (2003), Equivalent static wind loads on structures, Proceedings of the 11th International Conference on Wind Engineering, Lubbock, TX, June2-5,2003

X. Chen, A. Kareem (2004), Equivalent static wind loads on buildings: New Model, Journal of Structural Engineering, Vol.130, No.10, pp.1425

Davenport, A.G. (1967), Gust Loading Factors, J. Struct. Eng. Div. ASCE, Vol.93, pp.11-34

— 83 —

Table 3.1 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 3, 5, 7 on Frame A, (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 63.546 -32.657 201.835 66.018 -36.326 249.680 62.987 -42.278 249.5355 32.657 -7.411 -251.674 36.326 -7.411 -203.830 42.278 -7.940 -196.051Mean response

(lb, lb-ft) 7 64.032 32.657 180.736 64.032 36.326 228.581 64.561 42.278 244.2623 180.760 -114.521 1209.229 184.364 -83.744 738.803 179.956 -99.222 642.2935 114.521 -41.895 -944.359 83.744 -41.895 -562.917 99.222 -37.379 -538.964

Actual Absolute Maximum Total response(lb, lb-ft) 7 180.883 114.521 796.817 180.883 83.744 658.289 185.540 99.222 611.073

3 2.845 3.507 5.991 2.793 2.305 2.959 2.857 2.347 2.574 5 3.507 5.653 3.752 2.305 5.653 2.762 2.347 4.708 2.749 Gust Factor with larger

absolute value 7 2.825 3.507 4.409 2.825 2.305 2.880 2.874 2.347 2.502

Table 3.2 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 3, 5, 7 on Frame B, (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 58.262 -40.196 202.035 62.024 -42.662 234.345 58.995 -47.221 251.698 5 40.196 -10.879 -248.422 42.662 -10.879 -216.113 47.221 -10.120 -210.176Mean response

(lb, lb-ft) 7 60.086 40.196 231.219 60.086 42.662 263.528 59.327 47.221 258.031 3 175.201 -156.477 1653.481 183.597 -128.288 976.448 177.272 -139.152 771.122 5 156.477 -68.471 -1447.947 128.288 -68.471 -683.987 139.152 -51.616 -661.151

Actual Absolute Maximum Total response (lb, lb-ft) 7 182.548 156.477 900.225 182.548 128.288 1001.960 181.417 139.152 784.499


absolute value 7 3.038 3.893 3.893 3.038 3.007 3.802 3.058 2.947 3.040

— 84 —

Table 3.3 Mean, Maximum Responses and Gust Response Factors of Critical Section 3, 5, 7 on Frame C, (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 24.675 -21.317 113.144 26.578 -20.255 99.474 24.853 -21.509 103.9695 21.317 -4.191 -70.548 20.255 -4.191 -84.218 21.509 -4.001 -82.595Mean response

(lb, lb-ft) 7 26.286 21.317 121.030 26.286 20.255 107.360 26.096 21.509 106.1023 76.213 -92.922 1062.288 78.022 -54.779 484.305 74.218 -56.217 304.6985 92.922 -32.114 -610.794 54.779 -32.114 -260.695 56.217 -21.872 -252.213



absolute value 7 2.923 4.359 4.405 2.923 2.704 4.471 2.857 2.614 2.868

Table 3.4 Mean, Maximum Responses and Corresponding Gust Response Factor of Critical Section 3, 5, 7 on Frame D, (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 15.859 -8.966 43.420 16.693 -10.551 64.077 16.125 -12.106 70.212 5 8.966 -3.616 -82.329 10.551 -3.616 -61.671 12.106 -3.343 -59.641 Mean response

(lb, lb-ft) 7 16.930 8.966 53.695 16.930 10.551 74.352 16.657 12.106 72.272 3 37.936 -28.613 278.040 39.528 -24.808 195.198 38.363 -28.247 173.262 5 28.613 -13.241 -268.249 24.808 -13.241 -153.575 28.247 -10.318 -147.856



absolute value 7 2.409 3.191 3.139 2.409 2.351 2.787 2.462 2.333 2.413

— 85 —

Table 3.5 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 3, 5, 7 on Frame A, (AOA=90º)


(lb, lb-ft) 7 19.24 7.61 44.84 19.24 10.46 82.00 19.36 13.22 87.27 3 107.82 -26.65 389.39 106.90 -30.33 360.82 105.40 -37.98 326.08 5 26.65 -25.93 -465.57 30.33 -25.93 -272.35 37.98 -23.75 -258.30



absolute value 7 2.97 3.50 4.08 2.97 2.90 3.74 3.00 2.87 3.11

Table 3.6 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 3, 5, 7 on Frame B, (AOA=90º)


(lb, lb-ft) 7 18.63 15.29 89.51 18.63 9.26 11.09 23.94 15.55 99.08 3 145.70 -41.23 937.30 137.75 -30.36 657.25 133.45 -41.78 453.35 5 41.23 -23.47 -384.52 30.36 -23.47 -358.47 41.78 -31.33 -334.55

Actual Absolute Maximum Total response(lb, lb-ft) 7 54.95 41.23 242.87 54.95 30.36 -229.19 63.87 41.78 296.53


absolute value 7 2.95 2.70 2.71 2.95 3.28 -20.66 2.67 2.69 2.99

— 86 —

Table 3.7 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 3, 5, 7 on Frame C, (AOA=90º)


(lb, lb-ft) 7 21.11 14.57 85.38 21.11 10.26 29.43 26.21 17.26 115.26 3 152.82 -34.83 931.46 144.07 -27.56 683.69 137.94 -39.55 468.13 5 34.83 -20.54 -375.40 27.56 -20.54 -357.03 39.55 -30.81 -332.33



absolute value 7 2.56 2.39 2.41 2.56 2.69 6.63 2.39 2.29 2.52

Table 3.8 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 3, 5, 7 on Frame D, (AOA=90º)


(lb, lb-ft) 7 14.83 8.56 50.05 14.83 8.89 54.33 15.28 10.56 63.20 3 81.35 -28.19 360.44 80.58 -27.10 302.93 80.19 -33.97 274.13 5 28.19 -20.86 -371.49 27.10 -20.86 -231.75 33.97 -20.33 -221.14



absolute value 7 3.26 3.29 3.74 3.26 3.05 4.39 3.20 3.22 3.61

— 87 —

Table 3.9 Comparison between responses of Frame C calculated from ASCE (Figure 6-9) with actual responses

3 5 7

N Q M Q M Q M

CASE1-AOA=90 -32.46 -66.99 -173.57 13.03 198.34 50.84 -316.11

CASE2-AOA=90 -24.34 -50.11 -128.14 9.91 148.76 38.26 -239.12

CASE3 -55.32 -116.13 -374.19 13.03 366.25 99.98 -516.74

CASE4 -41.50 -77.68 -139.46 19.19 274.69 84.40 -528.76

Actual R. 137.94 -39.55 468.13 -30.81 -332.33 39.55 290.10

Err.* -59.9% 193.6% -20.1% -37.7% 10.2% 152.8% 82.3%

Note:* Error is the actual response with the maximum value of four cases

Table 3.10 Comparison between responses calculated from ASCE and actual responses

N(lb) Q(lb) M(lb.ft) Point ASCE R. Actual R. perc. ASCE R. Actual R. perc. ASCE R. Actual R. perc. 3 81.03 105.40 -23.1% -27.95 -37.98 -26.4% 334.70 326.08 2.6% 5 27.95 37.98 -26.4% -6.46 -23.75 -72.8% -224.57 -258.30 -13.1% Frame

A 7 49.79 58.03 -14.2% 27.95 37.98 -26.4% 197.27 271.45 -27.3% 3 96.72 133.45 -27.5% -34.12 -41.78 -18.3% 404.52 453.35 -10.8% 5 34.12 41.78 -18.3% -6.50 -31.33 -79.3% -272.11 -334.55 -18.7% Frame

B 7 61.85 63.87 -3.2% 34.12 41.78 -18.3% 240.50 296.53 -18.9% 3 96.72 137.94 -29.9% -34.12 -39.55 -13.7% 404.52 468.13 -13.6% 5 34.12 39.55 -13.7% -6.50 -30.81 -78.9% -272.11 -332.33 -18.1% Frame

C 7 61.85 62.69 -1.3% 34.12 39.55 -13.7% 240.50 290.10 -17.1% 3 52.86 80.19 -34.1% -18.65 -33.97 -45.1% 221.10 274.13 -19.3% 5 18.65 33.97 -45.1% -3.55 -20.33 -82.5% -148.73 -221.14 -32.7% Frame

D 7 33.81 48.89 -30.9% 18.65 33.97 -45.1% 131.46 228.14 -42.4%

— 88 —

Table 3.11 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame A, (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 117.214 -81.864 1007.394 118.347 -47.418 489.123 116.969 -56.944 392.7585 81.864 -34.484 -692.685 47.418 -34.484 -359.087 56.944 -29.440 -342.913

Actual Absolute Maximum Dynamic response (lb, lb-ft) 7 116.850 81.864 616.081 116.850 47.418 429.708 120.979 56.944 366.811

3 -205.68 119.35 -827.91 -213.67 113.85 -756.77 -203.39 126.94 -740.805 -119.35 23.47 508.34 -113.85 23.47 579.48 -126.94 25.68 562.42

Dynamic response under gust loading envelop based on first POD pressure sign (lb, lb-ft) 7 -201.94 -119.35 -729.86 -201.94 -113.85 -658.72 -204.15 -126.94 -708.78

3 -0.57 -0.69 -1.22 -0.55 -0.42 -0.65 -0.58 -0.45 -0.53 5 -0.69 -1.47 -1.36 -0.42 -1.47 -0.62 -0.45 -1.15 -0.61 Background Factor at critical

points 7 -0.58 -0.69 -0.84 -0.58 -0.42 -0.65 -0.59 -0.45 -0.52

Table 3.12 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame B, (AOA=0º)



3 -188.33 158.25 -817.86 -199.91 154.15 -765.50 -187.84 165.73 -774.435 -158.25 27.78 573.52 -154.15 27.78 625.87 -165.73 28.19 610.84



points 7 -0.65 -0.73 -0.84 -0.65 -0.56 -0.99 -0.65 -0.55 -0.69

— 89 —

Table 3.13 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame C, (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 51.539 -71.605 949.143 51.444 -34.524 384.831 49.365 -34.709 200.7305 71.605 -27.923 590.349 34.524 -27.923 -176.477 34.709 -17.872 -169.618


3 -124.28 107.18 -493.02 -133.11 105.72 -474.69 -125.16 112.89 -498.185 -107.18 18.74 382.14 -105.72 18.74 400.47 -112.89 17.80 391.16


3 -0.41 -0.67 -1.93 -0.39 -0.33 -0.81 -0.39 -0.31 -0.40 5 -0.67 -1.49 1.54 -0.33 -1.49 -0.44 -0.31 -1.00 -0.43 Background Factor at critical

points 7 -0.40 -0.67 -0.77 -0.40 -0.33 -0.72 -0.39 -0.31 -0.39

Table 3.14 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame D, (AOA=0º)



3 -33.60 27.55 -130.49 -35.85 27.93 -135.61 -33.77 30.09 -141.165 -27.55 5.44 119.09 -27.93 5.44 113.97 -30.09 5.26 111.17



points 7 -0.65 -0.71 -0.84 -0.65 -0.51 -0.94 -0.67 -0.54 -0.72

— 90 —

Table 3.15 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame A, (AOA=90º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 73.38 -19.04 321.96 71.99 -19.86 256.24 71.08 -24.77 219.56 5 19.04 -21.12 -348.36 19.86 -21.12 -192.29 24.77 -18.82 -181.85


3 142.66 -27 155.64 148.45 -47.38 420.71 144.77 -59.18 468.68 5 27 -21.74 -625.78 47.38 -21.74 -360.71 59.18 -19.56 -345.25

Dynamic response under gust loading envelop based on first POD pressure sign (lb, lb-ft) 7 96.82 27 164.9 96.82 47.38 429.96 94.64 59.18 412.89

3 0.51 0.71 2.07 0.49 0.42 0.61 0.49 0.42 0.47 5 0.71 0.97 0.56 0.42 0.97 0.53 0.42 0.96 0.53 Background Factor at critical

points 7 0.39 0.71 0.84 0.39 0.42 0.52 0.41 0.42 0.45

Table 3.16 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame B, (AOA=90º)


Actual Absolute Maximum Dynamic response (lb, lb-ft) 7 36.32 25.93 153.35 36.32 21.10 -240.28 39.92 26.23 197.44

3 190.11 -28.82 1054.77 180.95 -17.02 900.99 169.23 -42.89 622.48 5 28.82 -1.45 -302.44 17.02 -1.45 -456.22 42.89 -22.25 -422.24

Dynamic response under gust loading envelop based on first POD pressure sign (lb, lb-ft) 7 66.91 28.82 161.25 66.91 17.02 7.47 87.71 42.89 353.91


points 7 0.54 0.90 0.95 0.54 1.24 -32.17 0.46 0.61 0.56

— 91 —

Table 3.17 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame C, (AOA=90º)


Actual Absolute Maximum Dynamic response (lb, lb-ft) 7 32.87 20.26 120.41 32.87 17.29 -199.66 36.47 22.29 174.84

3 207.21 -28.06 1019.77 198.02 -23.99 966.61 186.93 -52.98 701.76 5 28.06 -3.22 -468.38 23.99 -3.22 -521.54 52.98 -23.39 -483.48



points 7 0.38 0.72 0.83 0.38 0.76 -2.55 0.34 0.43 0.41

Table 3.18 Dynamic Responses and Dynamic Response Factors of Critical Section 3, 5, 7 on Frame D, (AOA=90º)



3 112.03 -27.35 18.25 115.97 -49.75 309.8 116.55 112.03 -27.35 5 27.35 -26.52 -627.55 49.75 -26.52 -336.01 61.08 27.35 -26.52



points 7 0.37 0.72 0.82 0.37 0.37 0.40 0.40 0.38 0.41

— 92 —

Table 3.19 Dynamic Responses of Critical Section 3, 5, 7 on Frames A B C D, (AOA=0º)


Dynamic response of Frame A under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 201.952 119.355 729.886 201.952 113.850 658.747 204.157 126.948 708.803

3 188.339 -158.259 817.886 199.914 -154.158 765.528 187.844 -165.741 774.455 5 158.259 -27.782 -573.538 154.158 -27.782 -625.896 165.741 -28.196 -610.866

Dynamic response of Frame B under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 187.454 158.259 797.134 187.454 154.158 744.775 187.868 165.741 765.853

3 124.287 -107.186 493.034 133.112 -105.724 474.705 125.163 -112.899 498.203 5 107.186 -18.743 -382.152 105.724 -18.743 -400.481 112.899 -17.804 -391.174

Dynamic response of Frame C under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 126.055 107.186 533.276 126.055 105.724 514.947 125.116 112.899 510.023

3 33.601 -27.548 130.499 35.851 -27.930 135.619 33.767 -30.088 141.164 5 27.548 -5.438 -119.096 27.930 -5.438 -113.976 30.088 -5.256 -111.172

Dynamic response of Frame D under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 36.492 27.548 136.847 36.492 27.930 141.967 36.310 30.088 142.017

— 93 —

Table 3.20 Dynamic Responses of Critical Section 3, 5, 7 on Frames A B C D, (AOA=90º)


Dynamic response of Frame A under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 92.45 26.39 159.96 92.45 45.73 411.54 90.39 56.97 395.55

3 170.62 -28.34 590.34 161.70 -33.70 659.87 159.73 -58.65 530.41 5 28.34 -16.32 -513.78 33.70 -16.32 -444.26 58.65 -27.13 -411.53

Dynamic response of Frame B under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 80.12 28.34 159.04 80.12 33.70 228.56 90.92 58.65 423.46

3 197.61 -26.73 863.09 188.57 -27.94 878.58 180.61 -56.22 661.99 5 26.73 -7.35 -525.28 27.94 -7.35 -509.78 56.22 -24.25 -472.66

Dynamic response of Frame C under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 89.97 26.73 139.94 89.97 27.94 155.44 106.87 56.22 446.23

3 103.87 -26.07 52.51 107.24 -44.82 296.50 107.25 -54.90 361.40 5 26.07 -21.66 -549.80 44.82 -21.66 -305.81 54.90 -18.21 -292.62

Dynamic response of Frame D under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 82.65 26.07 159.50 82.65 44.82 403.48 79.21 54.90 364.94

— 94 —

Table 3.21 Dynamic Response Factors of Critical Section 3, 5, 7 on Frames A B C D, (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 0.570 0.686 1.217 0.554 0.416 0.646 0.575 0.449 0.530 5 0.686 1.469 1.363 0.416 1.469 0.620 0.449 1.146 0.610

Dynamic Response Factors of Frame A under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 0.579 0.686 0.844 0.579 0.416 0.652 0.593 0.449 0.518

3 0.621 0.735 1.775 0.608 0.555 0.969 0.630 0.555 0.671 5 0.735 2.073 2.091 0.555 2.073 0.748 0.555 1.472 0.738

Dynamic Response Factors of Frame B under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 0.653 0.735 0.839 0.653 0.555 0.991 0.650 0.555 0.687

3 0.415 0.668 1.925 0.386 0.327 0.811 0.394 0.307 0.403 5 0.668 1.490 -1.545 0.327 1.490 0.441 0.307 1.004 0.434

Dynamic Response Factors of Frame C under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 0.401 0.668 0.773 0.401 0.327 0.724 0.387 0.307 0.389

3 0.657 0.713 1.798 0.637 0.510 0.967 0.659 0.536 0.730 5 0.713 1.770 1.561 0.510 1.770 0.806 0.536 1.327 0.793

Dynamic Response Factors of Frame D under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 0.653 0.713 0.839 0.653 0.510 0.936 0.671 0.536 0.719

— 95 —

Table 3.22 Dynamic Response Factors of Critical Section 3, 5, 7 on Frames A B C D, (AOA=90º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 0.54 0.72 2.10 0.51 0.43 0.63 0.51 0.43 0.49 5 0.72 1.00 0.58 0.43 1.00 0.56 0.43 0.99 0.55

Dynamic Response Factors of Frame A under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 0.41 0.72 0.86 0.41 0.43 0.55 0.43 0.43 0.47

3 0.53 0.92 1.06 0.53 0.63 0.64 0.52 0.45 0.55 5 0.92 1.37 0.68 0.63 1.37 0.55 0.45 0.92 0.56

Dynamic Response Factors of Frame B under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 0.45 0.92 0.96 0.45 0.63 -1.05 0.44 0.45 0.47

3 0.47 0.76 0.72 0.46 0.62 0.49 0.46 0.40 0.43 5 0.76 2.43 0.57 0.62 2.43 0.44 0.40 0.95 0.45

Dynamic Response Factors of Frame C under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 0.37 0.76 0.86 0.37 0.62 -1.28 0.34 0.40 0.39

3 0.54 0.75 5.42 0.52 0.41 0.75 0.52 0.43 0.55 5 0.75 0.87 0.56 0.41 0.87 0.57 0.43 0.99 0.57

Dynamic Response Factors of Frame D under fluctuating pressure on frame based on mean pressure sign (lb, lb-ft) 7 0.41 0.75 0.86 0.41 0.41 0.46 0.42 0.43 0.45

— 96 —

Table 3.23 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Across Frame A (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 1 18.27 0.06 0.00 18.27 9.96 0.00 18.28 12.38 101.203 18.27 -27.51 653.23 18.27 -17.60 157.85 18.28 -15.18 138.085 27.51 -0.08 433.92 17.60 -0.08 -61.47 15.18 -0.07 -81.63 7 18.30 27.51 654.82 18.30 17.60 159.42 18.28 15.18 138.87

Mean response (KN, KN-m)

9 18.30 0.00 0.00 18.30 -9.91 0.00 18.28 -12.33 100.431 57.70 -51.41 0.00 57.70 46.30 0.00 47.92 51.94 618.143 57.70 -76.28 3051.68 57.70 -38.67 853.04 47.92 -32.73 440.795 76.28 -27.98 2292.78 38.67 -27.99 -238.40 32.73 -11.09 -252.637 57.39 76.29 2165.05 57.39 38.67 881.23 50.97 32.73 437.56

Actual Absolute Maximum Total response (KN, KN-m)

9 57.39 0.00 0.00 57.39 -46.33 0.00 50.97 -52.05 661.161 3.16 -915.85 NaN 3.16 4.65 NaN 2.62 4.19 6.11 3 3.16 2.77 4.67 3.16 2.20 5.40 2.62 2.16 3.19 5 2.77 331.10 5.28 2.20 333.65 3.88 2.16 162.05 3.09 7 3.14 2.77 3.31 3.14 2.20 5.53 2.79 2.16 3.15

Gust Factor with larger absolute value

9 3.14 NaN NaN 3.14 4.68 NaN 2.79 4.22 6.58

— 97 —

Table 3.24 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Across Frame B (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 1 8.21 0.15 0.00 8.21 6.36 0.00 8.29 8.05 72.73 3 8.21 -16.52 392.33 8.21 -10.31 81.78 8.29 -8.62 69.77 5 16.52 -0.10 290.73 10.31 -0.10 -19.84 8.62 -0.02 -33.96 7 8.53 16.52 398.73 8.53 10.31 88.15 8.45 8.62 71.91


9 8.53 0.00 0.00 8.53 -6.21 0.00 8.45 -7.91 68.49 1 42.20 -57.68 0.00 42.20 36.84 0.00 28.91 40.80 597.553 42.20 -61.57 2867.91 42.20 -29.95 815.59 28.91 -25.39 326.385 61.58 -30.88 2094.09 29.95 -30.88 -134.97 25.39 -11.56 -144.067 40.93 61.58 1627.14 40.93 29.95 808.83 28.01 25.39 335.49


9 40.93 0.00 0.00 40.93 -41.68 0.00 28.01 -45.61 631.581 5.14 -387.17 NaN 5.14 5.79 NaN 3.49 5.07 8.22 3 5.14 3.73 7.31 5.14 2.90 9.97 3.49 2.95 4.68 5 3.73 295.62 7.20 2.90 296.77 6.80 2.95 601.28 4.24 7 4.80 3.73 4.08 4.80 2.90 9.18 3.32 2.95 4.67


9 4.80 NaN NaN 4.80 6.71 NaN 3.32 5.77 9.22

— 98 —

Table 3.25 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Across Frame C (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 1 4.45 0.08 0.00 4.45 4.33 0.00 4.48 5.48 48.36 3 4.45 -10.56 260.40 4.45 -6.31 47.91 4.48 -5.17 39.14 5 10.56 -0.02 206.75 6.31 -0.02 -5.75 5.17 0.01 -15.27 7 4.50 10.56 262.57 4.50 6.31 50.06 4.47 5.17 39.78


9 4.50 0.00 0.00 4.50 -4.25 0.00 4.47 -5.39 46.84 1 29.75 48.88 0.00 29.75 37.01 0.00 14.85 40.67 599.563 29.75 -54.08 2419.78 29.75 -21.18 689.11 14.85 -17.38 252.975 54.08 -26.23 1834.22 21.18 -26.23 -66.37 17.38 -8.24 -70.11 7 31.36 54.08 1463.31 31.36 21.18 743.70 15.37 17.38 268.18

Actual Absolute Maximum Total response (KN, KN-m) 9 31.36 0.00 0.00 31.36 -36.16 0.00 15.37 -40.16 550.10

1 6.69 590.36 NaN 6.69 8.54 NaN 3.32 7.43 12.40 3 6.69 5.12 9.29 6.69 3.36 14.38 3.32 3.36 6.46 5 5.12 1551.35 8.87 3.36 1577.13 11.54 3.36 -601.23 4.59 7 6.96 5.12 5.57 6.96 3.36 14.86 3.44 3.36 6.74


9 6.96 NaN NaN 6.96 8.51 NaN 3.44 7.45 -6.69

— 99 —

Table 3.26 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Along Frame A (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 1 23.31 -18.10 0.00 23.31 -9.29 0.00 16.42 -9.83 -194.483 23.31 -9.96 740.59 23.31 -1.16 300.31 16.42 -1.69 132.455 9.96 8.46 366.82 1.16 8.46 -73.47 1.69 1.57 -69.03 7 -1.91 9.96 246.72 -1.91 1.16 -193.59 4.98 1.69 -16.86


9 -1.91 0.00 0.00 -1.91 -8.81 0.00 4.98 -8.27 150.121 68.17 -67.46 0.00 68.18 -42.78 0.00 48.55 -46.59 -781.913 68.17 -28.63 2291.07 68.18 11.17 961.66 48.55 -9.12 379.465 28.63 34.92 1195.37 -11.17 34.92 -289.54 9.12 10.01 -260.177 -25.37 28.63 695.07 -25.37 -11.17 -865.33 24.72 9.12 -223.04


9 -25.37 0.00 0.00 -25.37 -26.63 0.00 24.72 -24.34 517.471 2.93 3.73 NaN 2.93 4.60 NaN 2.96 4.74 4.02 3 2.93 2.87 3.09 2.93 -9.67 3.20 2.96 5.40 2.87 5 2.87 4.13 3.26 -9.67 4.13 3.94 5.40 6.36 3.77 7 13.26 2.87 2.82 13.25 -9.67 4.47 4.97 -4.60 13.23


9 13.26 NaN NaN 13.25 3.02 NaN 4.97 2.94 3.45

— 100 —

Table 3.27 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Along Frame B (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 1 27.27 -26.60 0.00 27.27 -15.81 0.00 17.68 -16.97 -288.293 27.27 -9.83 921.21 27.27 0.96 381.68 17.68 -0.20 151.425 9.83 11.67 454.66 -0.96 11.67 -84.89 0.20 2.08 -75.22 7 -5.99 9.83 243.07 -5.99 -0.96 -296.49 3.61 0.20 -46.89


9 -5.99 0.00 0.00 -5.99 -10.79 0.00 3.61 -9.63 191.581 69.37 -79.15 0.00 69.38 -52.65 0.00 43.19 -57.35 -900.363 69.37 -24.78 2470.60 69.38 13.16 1040.07 43.19 8.61 383.835 24.78 36.53 1296.18 -13.16 36.53 -245.00 -8.61 9.67 -208.427 -30.54 24.78 628.22 -30.54 -13.15 -977.74 15.03 -8.61 -244.76


9 -30.54 0.00 0.00 -30.54 -29.03 0.00 15.03 -25.45 550.551 2.54 2.98 NaN 2.54 3.33 NaN 2.44 3.38 3.12 3 2.54 2.52 2.68 2.54 -7.25 2.72 2.44 -43.44 2.53 5 2.52 3.13 2.85 13.67 3.33 2.89 -43.44 4.66 2.77 7 5.10 2.52 2.58 5.10 13.67 3.30 4.16 -43.43 5.22


9 5.10 NaN NaN 5.10 2.69 NaN 4.16 2.64 2.87

— 101 —

Table 3.28 Mean, Maximum Responses and Corresponding Gust Response Factors of Critical Section 1,3,5,7,9 on Along Frame C (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 1 27.43 -27.50 0.00 27.43 -16.77 0.00 17.64 -18.08 -299.063 27.43 -9.48 922.84 27.43 1.25 386.61 17.64 -0.06 152.83 5 9.48 11.84 445.87 -1.25 11.84 -90.38 0.06 2.06 -79.50 7 -5.89 9.48 234.48 -5.89 -1.25 -301.79 3.89 0.06 -46.25


9 -5.89 0.00 0.00 -5.89 -10.73 0.00 3.89 -9.42 190.27 1 70.16 -77.11 0.00 70.16 -50.48 0.00 44.53 -55.24 -877.183 70.16 -26.10 2429.13 70.16 13.21 1038.75 44.53 8.73 391.50 5 26.10 34.86 1199.57 -13.21 34.86 -268.85 -8.73 9.04 -229.607 -29.44 26.10 694.19 -29.44 -13.21 -940.57 14.57 -8.73 -235.06


9 -29.44 0.00 0.00 -29.44 -27.81 0.00 14.57 -24.68 523.05 1 2.56 2.80 NaN 2.56 3.01 NaN 2.52 3.06 2.93 3 2.56 2.75 2.63 2.56 10.59 2.69 2.52 -152.67 2.56 5 2.75 2.94 2.69 10.59 2.94 2.97 -152.61 4.39 2.89 7 5.00 2.75 2.96 5.00 10.59 3.12 3.74 -152.52 5.08


9 5.00 NaN NaN 5.00 -5.07 NaN 3.74 2.62 2.75

— 102 —

Table 3.29 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Across Frame A (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 1 39.43 -51.46 0.00 39.43 36.34 0.00 29.63 39.56 516.94 3 39.43 -48.77 2398.45 39.43 -21.07 695.19 29.63 -17.55 302.71 5 48.77 -27.90 1858.86 21.07 -27.90 -176.93 17.55 -11.02 -170.997 39.09 48.77 1510.22 39.09 21.07 721.81 32.69 17.55 298.69

Actual Absolute Maximum Dynamic response (KN, KN-m)

9 39.09 0.00 0.00 39.09 -36.43 0.00 32.69 -39.72 560.73 1 43.55 0.28 0.00 43.55 38.73 0.00 44.29 47.73 393.91 3 43.55 -85.86 2303.06 43.55 -47.41 380.68 44.29 -38.41 324.27 5 85.86 -1.26 1828.55 47.41 -1.26 -93.90 38.41 -0.51 -168.957 48.52 85.86 2383.99 48.52 47.41 461.48 47.78 38.41 367.78

Dynamic response under gust loading envelop based on first POD mode (KN, KN-m) 9 48.52 0.00 0.00 48.52 -38.45 0.00 47.78 -47.46 356.62

1 0.91 -185.00 NaN 0.91 0.94 NaN 0.67 0.83 1.31 3 0.91 0.57 1.04 0.91 0.44 1.83 0.67 0.46 0.93 5 0.57 22.09 1.02 0.44 22.13 1.88 0.46 21.39 1.01 7 0.81 0.57 0.63 0.81 0.44 1.56 0.68 0.46 0.81

Background factor

9 0.81 NaN NaN 0.81 0.95 NaN 0.68 0.84 1.57

— 103 —

Table 3.30 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Across Frame B(AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 1 33.99 -57.83 0.00 33.99 30.48 0.00 20.62 32.74 524.82 3 33.99 -45.05 2475.58 33.99 -19.64 733.81 20.62 -16.77 326.38 5 45.05 -30.77 1803.36 19.64 -30.77 123.98 16.77 -11.54 -144.067 32.40 45.06 1228.41 32.40 19.64 720.68 19.56 16.77 335.49


9 32.40 0.00 0.00 32.40 -35.47 0.00 19.56 -37.70 631.58 1 29.38 -7.39 0.00 29.38 19.43 0.00 27.36 25.33 194.99 3 29.38 -63.20 1679.86 29.38 -36.38 338.92 27.36 -30.48 239.16 5 63.20 2.46 1296.58 36.38 2.46 -44.39 30.48 0.43 -93.52 7 24.45 63.20 1552.72 24.44 36.38 211.70 26.47 30.48 213.21


1 1.16 7.83 NaN 1.16 1.57 NaN 0.75 1.29 2.69 3 1.16 0.71 1.47 1.16 0.54 2.17 0.75 0.55 1.07 5 0.71 -12.53 1.39 0.54 -12.52 -2.79 0.55 -26.70 1.18 7 1.33 0.71 0.79 1.33 0.54 3.40 0.74 0.55 1.24

Background factor

9 1.33 NaN NaN 1.33 1.32 NaN 0.74 1.15 1.90

— 104 —

Table 3.31 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Across Frame C(AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 1 25.30 48.80 0.00 25.30 32.68 0.00 10.38 35.20 551.20 3 25.30 -43.51 2159.39 25.30 -14.86 641.20 10.38 -12.21 213.82 5 43.52 -26.21 1627.47 14.86 -26.21 66.17 12.21 -8.25 -54.84 7 26.86 43.52 1200.74 26.86 14.86 693.64 10.90 12.21 228.40


9 26.86 0.00 0.00 26.86 -31.91 0.00 10.90 -34.77 503.26 1 16.66 -0.39 0.00 16.66 23.75 0.00 17.04 30.36 284.72 3 16.66 -59.69 1415.62 16.66 -35.55 208.35 17.04 -28.94 162.72 5 59.69 -0.46 1235.64 35.55 -0.45 28.33 28.94 -0.08 -26.73 7 17.39 59.70 1453.70 17.39 35.55 246.35 17.01 28.94 181.86


1 1.52 -124.47 NaN 1.52 1.38 NaN 0.61 1.16 1.94 3 1.52 0.73 1.53 1.52 0.42 3.08 0.61 0.42 1.31 5 0.73 57.52 1.32 0.42 57.72 2.34 0.42 107.20 2.05 7 1.54 0.73 0.83 1.54 0.42 2.82 0.64 0.42 1.26

Background factor

9 1.54 NaN NaN 1.54 1.32 NaN 0.64 1.13 1.89

— 105 —

Table 3.32 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Along Frame A (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 1 44.87 -49.36 0.00 44.87 -33.49 0.00 32.14 -36.77 -587.433 44.87 -18.67 1550.48 44.87 12.33 661.35 32.14 9.46 247.01 5 18.67 26.45 828.54 -12.33 26.45 -216.07 -9.46 8.43 -191.137 -23.46 18.67 448.35 -23.46 -12.33 -671.73 19.74 -9.46 -206.19


9 -23.46 0.00 0.00 -23.46 -17.82 0.00 19.74 -16.06 367.35 1 83.42 -83.26 0.00 83.42 -51.18 0.00 52.60 -55.42 -947.433 83.42 -26.74 2829.73 83.42 5.34 1225.62 52.60 1.10 490.33 5 26.75 41.05 1306.15 -5.34 41.05 -298.02 -1.10 10.23 -262.677 -12.04 26.75 656.58 -12.04 -5.34 -947.64 18.79 -1.10 -141.64

Dynamic response under gust loading envelop based on first POD mode (KN, KN-m) 9 -12.04 0.00 0.00 -12.04 -32.08 0.00 18.79 -27.84 593.89

1 0.54 0.59 NaN 0.54 0.65 NaN 0.61 0.66 0.62 3 0.54 0.70 0.55 0.54 2.31 0.54 0.61 8.63 0.50 5 0.70 0.64 0.63 2.31 0.64 0.73 8.64 0.82 0.73 7 1.95 0.70 0.68 1.95 2.31 0.71 1.05 8.64 1.46

Background factor

9 1.95 NaN NaN 1.95 0.56 NaN 1.05 0.58 0.62

— 106 —

Table 3.33 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Along Frame B (AOA=0º)



9 -24.55 0.00 0.00 -24.55 -18.24 0.00 11.42 -15.82 358.97 1 69.90 -74.70 0.00 69.90 -47.39 0.00 43.01 -51.50 -843.603 69.90 -20.28 2408.88 69.90 7.04 1043.20 43.01 2.92 405.20 5 20.28 35.89 1117.28 -7.04 35.89 -248.45 -2.92 9.00 -214.197 -14.31 20.28 513.81 -14.32 -7.04 -851.97 12.57 -2.92 -145.44


1 0.60 0.70 NaN 0.60 0.78 NaN 0.59 0.78 0.73 3 0.60 0.74 0.64 0.60 1.73 0.63 0.59 3.01 0.57 5 0.74 0.69 0.75 1.73 0.69 0.64 3.01 0.84 0.62 7 1.72 0.74 0.75 1.72 1.73 0.80 0.91 3.01 1.36

Background factor

9 1.72 NaN NaN 1.72 0.67 NaN 0.91 0.68 0.72

— 107 —

Table 3.34 Dynamic Responses and background factors of Critical Section 1,3,5,7,9 of Critical Section 1,3,5,7,9 on Along Frame C (AOA=0º)



9 -23.55 0.00 0.00 -23.55 -17.09 0.00 10.68 -15.26 332.79 1 69.81 -72.84 0.00 69.81 -45.85 0.00 43.85 -49.84 -814.973 69.81 -20.41 2366.21 69.81 6.57 1017.04 43.85 2.59 401.10 5 20.41 33.79 1098.31 -6.57 33.79 -250.89 -2.59 7.83 -217.737 -12.55 20.41 537.59 -12.55 -6.57 -811.67 13.42 -2.59 -129.39


1 0.61 0.68 NaN 0.61 0.74 NaN 0.61 0.75 0.71 3 0.61 0.81 0.64 0.61 1.82 0.64 0.61 3.39 0.60 5 0.81 0.68 0.69 1.82 0.68 0.71 3.39 0.89 0.69 7 1.88 0.81 0.86 1.88 1.82 0.79 0.80 3.39 1.46

Background factor

9 1.88 NaN NaN 1.88 0.63 NaN 0.80 0.66 0.69

— 108 —

Table 3.35 Maximum Responses of Critical Section 3, 5, 7 on Frames A B C D under four gust loading envelopes, (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 238.05 -119.35 1800.12 234.89 -74.33 1214.61 210.90 -98.31 839.56 5 119.35 -60.54 -2041.71 74.33 -60.54 -610.50 98.31 -34.50 -577.84Frame

A 7 239.01 119.35 729.86 239.01 74.33 1184.31 212.97 98.31 825.70 3 246.54 -158.25 2565.64 240.19 -83.83 1598.14 202.90 -110.57 978.96 5 158.25 -87.05 -2676.97 83.83 -87.05 -667.97 110.57 -43.56 -632.85Frame

B 7 246.72 158.25 797.10 246.72 83.83 1592.49 203.23 110.57 974.16 3 164.45 -107.18 1699.20 160.51 -59.07 1050.30 135.73 -75.98 642.28 5 107.18 -57.38 -1737.32 59.07 -57.38 -427.94 75.98 -27.89 -405.59Frame

C 7 164.69 107.18 533.26 164.69 59.07 1067.94 135.19 75.98 647.00 3 43.50 -27.55 427.65 42.72 -16.24 277.09 36.31 -20.81 175.66 5 27.55 -15.12 -455.95 16.24 -15.12 -121.08 20.81 -7.79 -114.80Frame

D 7 46.17 27.55 136.84 46.17 16.24 280.62 38.85 20.81 176.53

— 109 —

Table 3.36 Maximum Responses of Critical Section 3, 5, 7 on Frames A B C D under four gust loading envelopes, (AOA=90º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 126.14 -80.01 1304.60 122.87 -43.47 829.52 103.78 -57.61 512.90 5 80.01 -46.03 -1468.39 43.47 -46.03 -369.95 57.61 -24.82 -348.44 Frame

A 7 130.59 80.01 417.89 130.59 43.47 800.03 109.38 57.61 503.07 3 -145.02 90.73 -1649.23 -138.72 38.68 -972.42 -115.68 55.64 -555.05 5 -90.73 59.53 1622.26 -38.68 59.53 364.85 -55.64 31.15 343.16 Frame

B 7 -134.76 -90.73 -480.04 -134.76 -38.68 -950.63 -106.39 -55.64 -546.24 3 -128.39 98.24 -1733.62 -121.86 37.05 -938.09 -98.53 50.03 -507.42 5 -98.24 -55.13 -1785.97 -37.05 -55.13 342.37 -50.03 26.86 321.75 Frame

C 7 120.02 -98.24 -549.03 120.02 -37.05 913.78 -90.81 -50.03 481.79 3 80.69 -48.05 -869.83 77.48 -24.07 545.46 65.19 -34.01 326.26 5 48.05 30.20 -1005.48 24.07 30.20 -221.42 34.01 14.78 -207.32 Frame

D 7 87.20 48.05 246.84 87.20 24.07 559.13 71.24 34.01 333.50

— 110 —

Table 3.37 Background factor based on four gust loading envelopes, (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 0.49 0.69 0.56 0.50 0.64 0.40 0.55 0.58 0.47 5 0.69 0.57 0.34 0.64 0.57 0.59 0.58 0.85 0.59 Frame

A 7 0.49 0.69 0.84 0.49 0.64 0.36 0.57 0.58 0.44 3 0.47 0.73 0.57 0.51 1.02 0.46 0.58 0.83 0.53 5 0.73 0.66 0.45 1.02 0.66 0.70 0.83 0.95 0.71 Frame

B 7 0.50 0.73 0.84 0.50 1.02 0.46 0.60 0.83 0.54 3 0.31 0.67 0.56 0.32 0.58 0.37 0.36 0.46 0.31 5 0.67 0.49 -0.34 0.58 0.49 0.41 0.46 0.64 0.42 Frame

C 7 0.31 0.67 0.77 0.31 0.58 0.35 0.36 0.46 0.31 3 0.51 0.71 0.55 0.53 0.88 0.47 0.61 0.78 0.59 5 0.71 0.64 0.41 0.88 0.64 0.76 0.78 0.90 0.77 Frame

D 7 0.52 0.71 0.84 0.52 0.88 0.47 0.63 0.78 0.58

— 111 —

Table 3.38 Background factor based on four gust loading envelopes, (AOA=90º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 3 0.58 0.24 0.25 0.59 0.46 0.31 0.69 0.43 0.43 5 0.24 0.46 0.24 0.46 0.46 0.52 0.43 0.76 0.52 Frame

A 7 0.29 0.24 0.33 0.29 0.46 0.28 0.35 0.43 0.37 3 -0.62 -0.29 -0.38 -0.61 -0.55 -0.43 -0.72 -0.47 -0.52 5 -0.29 -0.38 -0.22 -0.55 -0.38 -0.67 -0.47 -0.80 -0.67 Frame

B 7 -0.27 -0.29 -0.32 -0.27 -0.55 0.25 -0.38 -0.47 -0.36 3 -0.72 -0.21 -0.36 -0.71 -0.47 -0.46 -0.84 -0.45 -0.56 5 -0.21 0.32 0.17 -0.47 0.32 -0.66 -0.45 -0.86 -0.66 Frame

C 7 0.27 -0.21 -0.22 0.27 -0.47 -0.22 -0.40 -0.45 0.36 3 0.70 0.41 -0.33 0.71 0.76 0.41 0.85 0.69 0.61 5 0.41 -0.63 0.31 0.76 -0.63 0.79 0.69 -1.21 0.80 Frame

D 7 0.38 0.41 0.55 0.38 0.76 0.33 0.47 0.69 0.49

— — 112

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— — 113

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Figure 3.3 Pressure Tap Arrangement

— — 114

wind

Across Frame C

Across Frame B

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Along Frame C

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Wall One

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Figure 3.4 Frame Arrangement of Wind Tunnel Model

Figure 3.5 Pressure distribution base on ASCE figure 6-10

WIND

Direction of MWFRS Reference

Corner

1 2 3

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2a, with a=3ft

AOA=90º

— — 115

Figure 3.6(a) pressure distribution on Frame A based on ASCE figure 6-10

Figure 3.6(b) pressure distribution on Frame B based on ASCE figure 6-10

Figure 3.6(c) pressure distribution on Frame C based on ASCE figure 6-10

Figure 3.6(d) pressure distribution on Frame D based on ASCE figure 6-10

WIND Frame D

2.18

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f

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-1.582lb/f

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f

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-2.895lb/f

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f

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-2.895lb/f

WIND Frame A

3.32

6lb/

f

-5.832lb/f -2.889lb/f

-2.344lb/f

— — 116

Figure 3.7 Gust Loading Envelope Distribution Based on first POD mode on Frame A at AOA=0º

Figure 3.8 Gust Loading Envelope Distribution Based on first POD mode on Frame B at AOA=0º

Figure 3.9 Gust Loading Envelope Distribution Based on first POD mode on Frame C at AOA=0º

— — 117

Figure 3.10 Gust Loading Envelope Distribution Based on first POD mode on Frame D at AOA=0º

Figure 3.11 Gust Loading Envelope Distribution Based on first POD mode on Frame A at AOA=90º

Figure 3.12 Gust Loading Envelope Distribution Based on first POD mode on Frame B at AOA=90º

— — 118

Figure 3.13 Gust Loading Envelope Distribution Based on first POD mode on Frame C at AOA=90º

Figure 3.14 Gust Loading Envelope Distribution Based on first POD mode on Frame D at AOA=90º

Figure 3.15 Gust Loading Envelope 1 Distribution on Frame A at AOA=0º

— — 119




— — 120

Figure 3.19 Gust Loading Envelope 1 Distribution on Frame B at AOA=0º



— — 121


Figure 3.23 Gust Loading Envelope 1 Distribution on Frame C at AOA=0º


— — 122



Figure 3.27 Gust Loading Envelope 1 Distribution on Frame D at AOA=0º

— — 123




— — 124




— — 125




— — 126




— — 127




— — 128




— — 129


— — 130

CHAPTER IV

EQUIVALENT STATIC WIND LOAD

4.1 Introduction

Wind load has long been the investigation topic for wind engineers. The complication

of geometry of structures and field category makes the wind induced responses on

structures hard to estimate. And for simplicity, in current design practice,

spatiotemporally varying wind loads on buildings are usually modeled as equivalent

static wind loads. This loading description can be used for estimating response under

the combined action of wind and other loads.

Equivalent static wind load (ESWL) has been investigated by several scholars. The

traditional gust response factor (GRF) method has been proposed by Davenport

(Davenport, 1967) and is widely used in most current building design codes and

standards which results in a pressure distribution similar to mean pressure distribution.

The GRF method has been improved (Piccardo, 2000; Zhou, 2001; Kareem, 2003).

The GRF is simple to use but from chapter III, it can be seen that GRF’s vary in a

large range for different responses of a structure and for structures with similar

geometric profile and similar associated wind load characteristics but different

structure systems, the GRF’s may be quite different. Similar to GRF method, a

dynamic response factor (DRF) (Holmes, 2002a) method is proposed. However the

DRF’s may have the same kind of problem as GRF’s and will vary in a large range for

different structure responses. In Repetto and Solari (Repetto, 2004), an identical

ESWL distribution for all response components was suggested by utilizing a

polynomial expansion determined on the premise that the ESWL results in accurate

estimates of a limited number of pre-selected peak responses. ESWL associated with a

certain response is also investigated by dividing into background (BESWL) and

resonant components (RESWL) which makes description of wind loading physically

more meaningful. For low-rise building with relatively large rigidity, resonant

response can be neglected and corresponding resonant equivalent static wind load

(RESWL) is not necessary to be considered. And background equivalent static wind

load (BESWL) is the main focus of wind loading on low-rise building. BESWL

depends on the external wind load characteristics.

Kasperski proposed a load-response correlation (LRC) approach for determining

— — 131

BESWL that results in different spatial load distributions for different response

components (Kasperski, 1992). The LRC approach provides a most probable load

distribution for a desired peak response, which has been experimentally proved by

Tamura et al (Tamura, 2002).

Universal equivalent static wind load is once proposed by Katsumura and Tamura el

(Katsumura, 2005). The assumed universal equivalent static wind load is actually a

combination of POD modes of random fluctuating pressure field. The combination

coefficients are decided by least square method and the main idea of this method is

that the estimated universal ESWL will produce responses fitting the actual response

well.

In this chapter, several equivalent static wind loading (ESWL) methods are compared,

and the universal ESWL method is applied to WERFL building and another modified

universal ESWL method is also utilized for WERFL building.

4.2 LRC pressure distribution of WERFL building

For low-rise buildings with large rigidity, resonant responses can be disregarded.

Wind loading which should be considered for low-rise building is the background

wind loading. Background wind loading is the quasi-static loading produced by

fluctuations due to turbulence, but with frequencies too low to excite any resonant

response.

During a windstorm, because of the incomplete correlations of pressures at various

points on a structure, loadings varying both in space and time will be experienced. For

wind engineering design, it is necessary to identify the instantaneous loadings which

produce the critical load effects in a structure.

Kasperski and Niemann once proposed Load-Response Correlation method to obtain

expected instantaneous pressure distribution associated with the maximum or

minimum load effect. For maximum value r̂ of a load effect r , the corresponding

pressure distribution is:

piprBiri igpp σρ ,ˆ)( += (4.1)

where ip and piσ are the mean and root-mean-square(r.m.s) pressures at point or

panel, i. pir ,ρ is the correlation coefficient between the fluctuating loading effect and

— — 132

the fluctuating pressure at point i, and Bg is the peak factor for background response.

In equation (4.1) the correlation coefficient pir ,ρ can be determined from the

correlation coefficients for the fluctuating pressures at all points on the tributary area,

and from the influence coefficients. It is given by the following equation.

∑=k

rpikkipir Itptp )/(])()([, σσρ (4.2)

Where kI is the influence coefficient for a pressure applied at position k.

The standard deviation of the structural load effect, rσ , is given by

kii k

kir IItptp∑∑= )()(2σ (4.3)

For the assumed frame system of WERFL building, LRC method is used to find

corresponding pressure distributions causing maximum axial force N, shear force Q

and bending moment M at nine critical sections along each frame for both AOA

around 0º and around 90º. Since Frame B and Frame C are usually the frames on

which large wind induced responses happen, the ESWL’s are focused on these two

frames.

Figure 4.1~Figure 4.4 show mean pressure distribution, ESWL pressure distribution

based on conditional sampling, ESWL pressure distribution based on LRC method,

ESWL pressure distribution based on gust loading envelop (GLE) for bending

moment M at critical section 3 on Frame B at AOA=0º with support condition fix-fix.

For clarity, it is necessary to explain the ESWL’s pressure distributions mentioned

above. ESWL pressure distribution based on conditional sampling is obtained by

averaging the pressure distributions causing a certain maximum response of all the

sampling runs. And the ESWL pressure distribution based on gust loading envelop

(GLE) is determined by multiplying the gust loading envelop (referring to the gust

loading envelop based on the first POD mode as defined in chapter III) by the

corresponding background factor for a certain response which has been calculated in

chapter III.



ESWL pressure distribution based on GLE for bending moment M at critical section 3

— — 133

on Frame C at AOA=0º with support condition fix-fix.



ESWL pressure distribution based on GLE for bending moment M at critical section 3

on Frame B at AOA=90º with support condition fix-fix.

Figure 4.13~Figure 4.16 show mean pressure distribution, ESWL pressure

distribution based on conditional sampling, ESWL pressure distribution based on

LRC method, ESWL pressure distribution based on GLE for bending moment M at

critical section 3 on Frame C at AOA=90º with support condition fix-fix.

From these figures, it can be seen that when AOA around 0º, the mean pressure

distribution and ESWL pressure distribution based on GLE are symmetrical for both

Frame B and Frame C.

For AOA around 0º, ESWL pressure distribution based on conditional sampling and

LRC method are similar to each other which proves that the ensemble averaged

extreme pressure distribution corresponding to a certain maximum response will

converge to ESWL pressure distribution based on LRC method.

For AOA around 90º, mean pressure distribution, ESWL pressure distribution based

on conditional sampling, ESWL pressure distribution based on LRC method and

ESWL pressure distribution based on GLE are similar to each other for both Frame B

and Frame C and it should be noted that in this case, the frames are parallel to the

wind direction.

4.3 Universal Equivalent Static Wind Load

Universal equivalent static wind load is once proposed by Katsumura and Tamura el

(Katsumura, 2005). The assumed universal equivalent static wind load is actually a

combination of POD modes of random fluctuating pressure field. The POD modes can

be determined once the covariance matrix of the fluctuating pressure field is formed.

The key of the method is to determine the combination coefficients of the POD modes.

The derivation of universal equivalent static wind load is simply described as follows.

In engineering practice, discrete structure model is usually used. For a static wind load

fi acting on structure at a point i, a load effect R induced by this load can be expressed

— — 134

as:

iti rfR *= (4.4)

where ir is the influence function value of R at point i.

The actual maximum load effect R can be expressed in terms of equivalent static wind

load and influence function.

R= fei * ri (4.5)

where fei is the equivalent static wind load which will produce the same maximum

response as the actual wind loading.

If a group of load effects need to be taken into consideration in design, it is desirable

that a universal load distribution can reproduce all the maximum load effects. This

relationship can be expressed in a matrix form.

Since the equivalent static wind load is for a group of load responses, several

representative load distributions can first be obtained and then a combination of these

load distributions is determined to satisfy all the maximum load effects. The universal

ESWL is expressed as

nne fcfcfcF +++= 2211 = FC (4.6)

where eF is the equivalent static wind load, F is the basic load distribution matrix, C

is the combination vector.

The eigen modes of norm 1 obtained from POD analysis of fluctuating pressure field

are employed as load distribution matrix F, each vector in matrix F is a basic load

distribution for combination. For this case, F is actually the eigen matrix Φ . The

reason of utilization of POD eigen modes is that eigen modes are physically

significant distributions which may indicate the ESWL effectively. The responses

caused by wind load combination of eigen modes can be expressed as:

CIR Φ= = CR0 (4.7)

Where I is the influence function matrix, Φ is the eigen mode matrix, 0R is the

product of influence function matrix and eigen mode matrix; C ={ }'21 ,, nccc is the

combination factor vector or contribution factor vector.

— — 135

If 0R is regular, and an inverse matrix exists, a unique solution is obtained. If the

number of unknown coefficients C is larger than that of the group of load effects

considered, several combinations of solution will be obtained. And if the number of

load effects considered is larger than that of the contribution coefficients, no precise

solution exists; this condition is the most frequently encountered case in practice,

since usually the load effects considered are much greater than the basic load

distributions. In this case, a least square approximation solution can be obtained for

contribution factor matrix C; to get the least square approximation solution, singular

value decomposition is applied to 0R , it is decomposed to the normalized orthogonal

matrix and singular values:

),(),(),(),(0 NNVNNSNMUNMR T= (4.8)

where M is the number of subjective load effects, N is the number of combination

coefficients.

So the combination factors { }C can be derived as

{ } { }RUVSC T ˆ1−= (4.9)

After the combination factors { }C are obtained, the universal ESWL can be decided

and it will reproduce approximately the subjective maximum load effects.

The universal ESWL method was once applied to a cantilevered large roof in

Katsumura’s paper (Katsumura, 2005). Two models with different connection joint

conditions are assumed for this roof model; Model I is pin-jointed, Model II is rigid-

jointed. There are totally 96 pressure taps uniformly distributed on the inside and

outside roof. The pressures at equivalent topside and underside points were measured

simultaneously in order to obtain the net pressure force. The spatial correlation matrix

of the fluctuating field is calculated by excluding the mean pressure. The eigen modes

determined by POD analysis with larger eigen values are utilized as the basic

combination load. The corresponding combination coefficients are determined for

maximum shear forces Q and maximum bending moments M in structural members

separately. For the pin-jointed model, the universal EWSL’s causing maximum shear

forces Q and maximum bending moment M in all the members are almost the same;

while for rigid-jointed model, these two universal ESWL’s are different, which

demonstrates that different ESWL’s need to be considered not only for different

— — 136

structural types but also for different load effects.

Also a contribution factor is defined in his paper (Katsumura, 2005):

i-th mode contribution=∑ i

i

CC

(4.10)

If the maximum shear forces and bending moments are considered simultaneously, for

model I, the contribution factor of the first eigen mode is about 37.7%, while for

model II, the first eigen mode contribution factor is 14.8%; for both models, the first

eigen mode is predominant. The cumulative contribution factor for model I in the

range from first mode to 15th mode is almost 80% while that for model II is only 53%.

It is clarified that in order to reproduce accurately the maximum load effects of all

members, the superposition of higher normalized modes is necessary. And it is found

that for model I, the universal EWSL causing all the maximum responses is the same

as those calculated for shear force and bending moment respectively, while for model

II, the universal ESWL causing all the maximum responses agrees well with that for

maximum shear force effects but different from the universal ESWL causing

maximum bending moment.

4.3.1 Application of Universal ESWL to

WERFL building

The universal Equivalent Static Wind Load (ESWL) of WERFL building is studied by

using this method. The assumed frame system for WERFL building which has been

pointed out in Chapter III is also utilized here. The axial force N, shear force Q and

bending moment M at these nine critical sections are the load effects subjective to

consideration. The universal EWSL causing maximum axial forces N, maximum

shear forces Q and maximum bending moments M in all frames with all the three

support conditions is calculated separately for both AOA=0º and AOA=90º. The

universal ESWL for all the load effects including maximum axial force N, maximum

shear force Q and maximum bending moment M in all frames is determined. The

eigen modes selected as the basic wind load distribution are the first 5, 10, and 15

modes with norm 1 and large corresponding eigen values. The reason of use of

different eigen modes is to investigate the influence of number of eigen modes on

universal equivalent static wind load.

— — 137

Suppose N={ }nNNN ,, 21’ which is the axial force vector composed of maximum

axial forces at nine critical sections of each frame, the total maximum axial force

considered is n. ],[ ',2

'1

'NnNNN IIII = , are the influence matrix, each row of the

matrix is influence function of axial force at those 9 critical sections of each frame,

],,[ 021 nφφφ=Φ , is the eigen mode matrix which is composed of 0n selected eigen

modes for calculation, in this case, 0n is 5, 10 and 15 respectively.

So the expression of determination of contribution vector CN = { }'21 ,, nccc for

estimating maximum axial force is as follow:

NN CIN Φ= = NCN0 (4.11)

Where 0N is the product of influence matrix and eigen mode matrix, ith row of which

is the corresponding ith axial force produced by the 0n eigen modes respectively.

For shear force and bending moment, the same kind of procedure can be used and

different contribution vector SC and MC are determined. The universal equivalent

static wind load for maximum axial force, shear force and bending moment are NCΦ ,

SCΦ and MCΦ respectively. Besides these three sets of universal equivalent static

wind load, another universal equivalent static wind load causing all the maximum

internal forces of interest, that is, the maximum axial forces N, maximum shear forces

Q and maximum bending moments M can also be determined, in this case, the

expression of determination of contribution vector C is as follows.

CRCIR 0=Φ= (4.12)

where { }'212121 ,,,,,,,, nnn MMMQQQNNNR = ,

⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

=

'

'

'

'

'

'

1

1

1 ,

n

n

n

M

M

Q

Q

N

N

I

I

I

I

I

I

I ,

],,[021 nφφφ=Φ

— — 138

Once combination factor C is decided, the universal equivalent static wind load for all

the maximum internal forces is CΦ .

Table 4.1~Table 4.3 list combination coefficients and the corresponding contribution

factors of the 5, 10 and 15 eigen modes for maximum axial force, shear force, bending

moment, and also those for all the maximum load effects when AOA=0 º .

Combination coefficients are listed in the left column and contribution factors are

listed in the right column for maximum axial force N, maximum shear force Q,

maximum bending moment M, and all the responses NQM respectively.

Table 4.4~Table 4.6 list combination coefficients and the corresponding contribution

factors of the 5, 10 and 15 eigen modes for maximum axial force, shear force, bending

moment, and also those for all the maximum load effects when AOA=90º . The

relationship between the contribution coefficients and mode number at AOA=0º and

AOA=90º is shown in Figure 4.17 and Figure 4.18 respectively.

It should be noted that in the first row of Table 4.1~Table 4.6, N means ‘only axial

forces in the frames are considered’, Q means ‘only shear forces in the frames are

considered’, M means ‘only bending moments in the frames are considered’, NQM

means ‘all the axial forces, shear forces and bending moments in the frames are

considered’.

From Table 4.1~Table 4.6, it can be seen that the contribution factor of the first mode

is not always the largest one, and the contribution factor is not necessarily decreasing

with mode number, for example, the contribution factor of mode 9 of universal ESWL

for maximum bending moment M when AOA=0º is the largest one.

The contour plots of universal equivalent static wind loads for maximum axial force,

maximum shear force, maximum bending moment and all the responses when five,

ten and fifteen eigen modes are used are shown in Figure 4.19~Figure 4.22, Figure

4.27~Figure 4.30, and Figure 4.35~Figure 4.38 respectively for AOA=0 º . The

corresponding pressure distributions on the four frames corresponding to these

universal ESWLs are shown in Appendix II.

The comparisons between actual responses and responses calculated by universal

ESWL when five, ten and fifteen eigen modes are used for AOA=0º are shown in

Figure 4.23~Figure 4.26, Figure 4.31~Figure 4.34 and Figure 4.39~Figure 4.42

— — 139

respectively. From these figures, it can be seen that the axial responses of all the

critical sections calculated by universal ESWL fit actual axial forces well since the

scatter data are always around the straight line which represents that the x axis and y

axis are equal. But for shear force and bending moment, data are not scattered close to

the straight line and it is found that the actual responses are usually larger than the

corresponding responses calculated by universal ESWL. It can also be seen that for

different responses, corresponding universal ESWL’s are also different and for the

same kind of response, universal ESWL’s based on different eigen modes are also

different, but the difference can be neglected.

The contour plots of universal equivalent static wind loads for maximum axial force,

maximum shear force, maximum bending moment and all the responses when only

five, ten and fifteen eigen modes are used are shown in Figure 4.43~Figure 4.46,

Figure 4.51~Figure 4.54, and Figure 4.59~Figure 4.62 respectively for AOA=90º.

The corresponding pressure distributions on the four frames corresponding to these

universal ESWL’s are also shown in Appendix II. The comparisons between actual

responses and responses calculated by universal ESWL when five, ten and fifteen

eigen modes are used for AOA=90º are shown in Figure 4.47~Figure 4.50, Figure

4.55~Figure 4.58 and Figure 4.63~Figure 4.66 respectively. For the case with

AOA=90 º , similar results about the comparison between actual responses and

responses calculated by universal ESWL’s when AOA=0º can be obtained.

This method for determination of universal equivalent static wind load (ESWL) gives

some insight into the exploration of wind loads on buildings, and the utilization of

eigen modes is also physically meaningful, since eigen modes can represent some

significant fluctuating pressure phenomena in many cases; however eigen modes

seems a little bit complicated for application.

From the results for both cases with AOA=0º and AOA=90º, it seems a little bit

difficult to quantify a universal ESWL for all the responses which can produce

responses fitting the actual responses quite well. The difficulty to quantify a universal

ESWL for all the responses is due to the variety of influence functions of responses

considered. For responses with similar influence functions such as axial forces at

these critical sections, universal equivalent static wind load will produce responses

which fit well with the actual ones. However for responses with quite different

— — 140

influence functions such as shear force and bending moment, it is hard to find a

universal EWSL to satisfy the requirement that it will produce responses fitting well

with the actual responses.

It should be pointed out that the universal ESWL’s for axial force, shear force, and

bending moment thus calculated in the above take all the nine critical sections and

three support conditions into consideration. For example, if universal ESWL for axial

force N is calculated, the axial forces include axial force at nine critical sections with

three support conditions under wind load action at a specific angle of attack (AOA).

The reason not to divide responses according to different support condition is that a

uniform universal ESWL suitable for all the three kind of structure system is the

object wind load. The responses for each kind of support condition can also be fitted

by the same kind of universal ESWL method, but it is found that thus calculated

universal ESWL’s based on five, ten, and fifteen eigen modes are not practical since

for axial force, shear force and bending moment, all the universal ESWL’s pressure

distributions have some very large values at some points which makes the universal

ESWL method unfeasible; so the results are not listed here.

4.3.2 Modified Universal Equivalent Static

Wind Load

The universal equivalent static wind load proposed by Katsumura and Tamura at el

(Katsumura, 2005) is based on determination of a set of combination coefficients for

eigen modes of the covariance matrix of the fluctuating pressure field. However for

fluctuating pressure field, the eigen modes are a little bit complicated and it is hard to

predict the distribution of eigen modes, even for the first several eigen modes with

large eigen value. And it can be seen that the pressure distribution of universal

ESWL’s based on eigen modes for each kind of structure response is not regular and

is a little bit hard to be used for practice use.

So maybe pressure distributions or wind loading distributions used for combination

can be substituted by another set of wind pressure distributions. For the WERFL

building, maximum base shear force FX, maximum base shear force FY, maximum lift

force FZ and maximum torsional moment MT at the base of the building are responses

which have no relationship with structure systems. And in Chapter II, these wind

forces are also used to derive the wind load combination formula. The pressure

— — 141

distributions which will produce maximum base shear force FX, maximum base shear

force FY, maximum lift force FZ and maximum torsional moment MT can be

determined by Load-Response Correlation(LRC) method (Kasperski, 1992) or by

conditional sampling.

Once the pressure distributions based on LRC method which can produce maximum

base shear force FX, maximum base shear force FY, maximum lift force FZ and

maximum torsional moment MT at the base of the building are decided, these four

wind loads or wind pressure distributions can be utilized as basic loads for

combination in the universal ESWL method. In this case, four coefficients need to be

determined for the four basic pressure distributions.

For example, the expression of determination of contribution vector CN

={ }'21 ,, nccc for estimating maximum axial force is as follow:

NN PCIN = = NCN0 (4.12)

where N is the maximum axial forces at 9 critical sections of each frame with three

support conditions, IN is the influence matrix for axial forces,

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

=

'

'

'

2

1 ,

nN

N

N

N

I

II

I , and P is

the combination wind load matrix or combination pressure distribution matrix with

four columns, ],,,[tzyx CCCC PPPPP = ,

xCP ,yCP ,

zCP , tCP are the pressure distribution

based on LRC method which can produce maximum base shear force FX, maximum

base shear force FY, maximum lift force FZ and maximum torsional moment MT at the

base of the building respectively. Now 0N is the product of influence matrix and wind

pressure distribution matrix, ith row of which is the corresponding ith axial force

produced by pressure distributions calculated based on LRC formula for maximum

base shear force FX, maximum base shear force FY, maximum lift force FZ and

maximum torsional moment MT at the base of the building respectively. And also by

using singular value decomposition, a coefficient vector with four elements can be

decided, a corresponding universal equivalent static wind load for maximum axial

force can be determined as NPC .

For shear forces and bending moments, the same kind of procedure can be used and

— — 142

different contribution vector SC and MC are determined. The universal ESWL for

maximum axial force, maximum shear force and maximum bending moment are

NPC , SPC and MPC respectively. Besides the three sets of universal equivalent static

wind load, another universal equivalent static wind load can also be determined based

on all the maximum internal forces of interest including the maximum axial forces N,

maximum shear forces Q and maximum bending moments M of the nine critical

sections with three different support conditions, in this case, the expression of

determination of contribution vector C is as follows.

CRIPCR 0== (4.13)

where { }'212121 ,,,,,,,, nnn MMMQQQNNNR = ,

⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

=

'

'

'

'

'

'

1

1

1 ,

n

n

n

M

M

Q

Q

N

N

I

I

I

I

I

I

I ,

],,,[tzyx CCCC PPPPP =

Once combination factor C is decided, the universal equivalent static wind load for all

the maximum internal forces is PC .

Table 4.7 lists the combination coefficients and corresponding contribution factors of

the 4 basic combination pressure distributions for maximum axial force N, shear force

Q, bending moment M, and also those for all the maximum internal forces NQM for

AOA=0o respectively.

In Table 4.7, combination coefficients are listed in the left column and contribution

factors are listed in the right column for universal ESWL of maximum axial force N,

maximum shear force Q, maximum bending moment M, and all the responses NQM

respectively.

From Table 4.7, it can be seen that the contribution coefficients of extreme pressure

distribution causing maximum base shear force FX at the building base is almost the

— — 143

largest one except for universal ESWL of maximum axial force N. Besides

contribution coefficient of extreme pressure distribution causing maximum base shear

FX, the coefficient of extreme pressure distribution causing maximum lift force FZ is

also large except for universal ESWL of maximum shear force Q. For extreme

pressure distribution causing maximum base shear force FY, the corresponding

contribution coefficient is always small. As far as contribution coefficient of pressure

distribution causing maximum torsional moment, it is relatively large for universal

ESWL of shear forces at critical sections.

Table 4.8 lists the combination coefficients and corresponding contribution factors of

the 4 basic combination pressure distributions for maximum axial force N, shear force

Q, bending moment M, and also those for all the maximum internal forces NQM for

AOA=90º respectively.

As Table 4.7, in Table 4.8, combination coefficients are listed in the left column and

contribution factors are listed in the right column.

From Table 4.8, it can be seen that the contribution factors of extreme pressure

distribution causing maximum base shear force FY and maximum lift force FZ at the

building base are always the largest ones for universal ESWL of maximum axial force

N, maximum shear force and bending moments. The contribution factor of extreme

pressure distribution causing maximum base shear force FX and maximum torsional

moment MZ are relatively small for all the universal EWSL’s of maximum axial force,

maximum shear force and maximum bending moment.

Figure 4.67~Figure 4.70 show the universal equivalent static pressure distributions

around the building surface for maximum axial forces N, maximum shear forces Q,

and maximum bending moments M, and all the responses at 9 critical sections of

frames based on combination of pressure distributions calculated by LRC formula

which will produce maximum base shear FX, maximum base shear FY, maximum lift

force MZ and maximum torsional moment MT respectively for AOA=0 º . The

universal ESWL pressure distributions of maximum internal forces for AOA=0º

along the four frames are shown in the Appendix II.

Figure 4.71~Figure 4.74 compare the responses under the action of universal ESWL’s

and corresponding actual maximum responses for AOA=0º.

— — 144

From Figure 4.71~Figure 4.74, it can be seen that compared to universal ESWL based

on combination of POD eigen modes, universal ESWL based on combination of

maximum pressure distribution of base shear forces FX and FY, lift force FZ and

torsional moment MT can produce responses fitting better with actual responses.

Figure 4.75~Figure 4.78 show the universal equivalent static pressure distributions

around the building surface for maximum axial forces N, maximum shear forces Q,

and maximum bending moments M, and all the responses at nine critical sections of

frames based on combination of pressure distributions calculated by LRC formula

which will produce maximum base shear FX, maximum base shear FY, maximum lift

force MZ and maximum torsional moment MT respectively for AOA=90º . The

universal ESWL pressure distributions of maximum internal forces for AOA=90º

along the four frames are also shown in the Appendix II.

Figure 4.79~Figure 4.82 compare the responses under the action of universal ESWL’s

and corresponding actual maximum responses for AOA=90º.

From Figure 4.79~Figure 4.82, it can be seen that compared to universal ESWL based

on combination of POD eigen modes, the modified universal ESWL based on

combination of maximum pressure distribution of base shear forces FX and FY, lift

force FZ and torsional moment MT can produce responses fitting better with actual

responses. And it seems that the fitting condition for AOA=90º is much better than

that for case AOA=0º . For both case AOA=0º , and AOA=90º , the modified

universal ESWL method looks much better than the usually used universal ESWL

method. So for low-rise building, the modified universal ESWL method may be used

instead of the usually used universal ESWL method.

4.4 Concluding remarks

In this chapter, equivalent static wind load for low-rise building is investigated.

Pressure distributions based on load and response correlation(LRC) method are

discussed and compared to extreme pressure distribution based on conditional

sampling, equivalent pressure distribution based on gust loading envelop(GLE)

multiplied by background factor and equivalent pressure distribution based on mean

pressure distribution multiplied by gust response factor. The ESWL pressure

distributions based on conditional sampling are similar to the ESWL pressure

— — 145

distribution based on LRC method. ESWL pressure distributions based on GLE and

ESWL pressure distributions based on mean pressure distribution have some common

characteristics, both of them are obtained by some basic pressure distributions

multiplied by a certain factor; for ESWL pressure distribution based on mean pressure

distribution, a gust response factor is used, for ESWL pressure distribution based on

GLE, a background factor is used. ESWL pressure distributions based on LRC

method and conditional sampling are dependent on individual response which limits

the application of these methods. Usually it is supposed that ESWL pressure

distribution based on a common basic pressure distribution is more usable. However

from the study, it is noted that although a common basic pressure distribution can be

decided, a certain factor such as gust response factor or background factor is also

dependent on individual responses, and it is hard to quantify a uniform factor for all

the responses. Besides those common ESWL methods, a universal ESWL method is

studied. The universal ESWL method is based on the idea that a common universal

ESWL for a certain structure system can produce responses fitting all the actual

responses well. A set of basic pressure distribution must be decided first and then a set

of combination coefficients is decided based on least square method. The universal

ESWL proposed by Tamura is based on eigen modes of the covariance matrix of the

fluctuating pressure around the structure, however, the eigen modes of the covariance

matrix are a little bit difficult to decide; besides this universal ESWL, a modified

universal ESWL method is proposed in this chapter. Compared to the usually defined

universal ESWL, this modified universal ESWL utilized the pressure distributions

which can cause maximum base shear forces, maximum lift force and torsional

moment as the combination pressure distribution set. Compared to the combination

pressure distributions consisted of eigen modes of a certain structure system, the

extreme pressure distributions corresponding to maximum shear forces, lift force and

torsional moment are more easily to be determined. In this chapter, both methods are

applied to WEFL building, and the responses under the modified universal ESWL are

much closer to the actual responses than those under the universal ESWL based on

eigen modes.

Reference

Davenport, A. G. (1967). Gust loading factors. J. Struct. Div. ASCE, Vol.93, No.1, pp.11–34.

— — 146

Holmes, J. D. (2002a). “Gust loading factor to dynamic response factor (1967-2002).” Symp. Preprints, Engineering, Symp. to Honor Alan G. Davenport for his 40 years of Contributions, The Univ. of Western Ontario, London, Ont., Canada, June 20-22, A1–1–A1–8.

Kareem, A., and Zhou, Y. (2003). “Gust loading factor—Past, present, and future.” J. Wind. Eng. Ind. Aerodyn., Vol.91, No.12–15, pp.1301–1328.

Kasperski, M. (1992). “Extreme wind load distributions for linear and nonlinear design.” Eng. Struct., Vol.14, pp.27–34.

A. Katsumura, Y. Tamura, O. Nakamura (2005), Universal wind load distribution simultaneously reproducing maximum load effects in all subject members on large-span cantilevered roof, EACWE4 — The Fourth European & African Conference on Wind Engineering J. Naprstek & C. Fischer (eds); ITAM AS CR, Prague, 11-15 July, 2005

Piccardo, G., and Solari, G. (2000). Three-dimensional wind-excited response of slender structures: Closed-form solution. J. Struct. Eng., Vo.126, No.8, pp.936–943.

Repetto, M. P., and Solari, G. (2004). “Equivalent static wind actions on vertical structures.” J. Wind. Eng. Ind. Aerodyn., Vol.100, No.7, pp.1032–1040.

Zhou, Y., and Kareem, A. (2001). “Gust loading factor: New model.” J.Struct. Eng., Vol.127, No.2, pp.168–175.

— — 147

Table 4.1 Combination coefficients and contribution factor (AOA=0º)

Mode Number N Q M NQM

1 -13.03 78.74% -1.39 7.28% -2.92 9.69% -10.56 85.33%2 -0.1 0.59% 7.1 37.18% -8.99 29.79% 0.65 5.21%3 0.65 3.92% 2.32 12.14% 3.8 12.59% 0.07 0.56%4 -1.26 7.59% -4.82 25.23% 5.53 18.32% -0.3 2.41%5 -1.52 9.17% 3.47 18.17% -8.93 29.61% -0.8 6.49%



1 -12.09 43.35% 7.53 4.79% -4.71 6.57% -10.08 59.77%2 -0.27 0.96% 30.79 19.59% -2.01 2.8% 0.66 3.91%3 -2.4 8.62% 20.16 12.83% 13.47 18.79% -1.43 8.48%4 -0.52 1.85% -12.03 7.65% -2.93 4.09% -0.1 0.59%5 -1.97 7.06% -4.04 2.57% -9.27 12.93% -0.48 2.82%6 -0.27 0.96% 50.11 31.88% -15.18 21.17% 1.07 6.32%7 4.53 16.23% -8.68 5.52% 7.9 11.02% 0.43 2.53%8 -1.14 4.08% -0.85 0.54% 7.65 10.68% -1.3 7.73%9 -2.68 9.61% 7.91 5.03% 2.92 4.08% -0.35 2.05%10 -2.03 7.3% -15.09 9.6% 5.65 7.88% -0.98 5.81%



1 -13.25 20.32% 2.11 1.17% -23.16 9.45% -10.2 10.49%2 -3 4.61% 17.91 9.91% -5.5 2.25% -3.69 3.79%3 -5.33 8.18% -2.08 1.15% -37.36 15.25% -14.19 14.6%4 -0.25 0.39% 5.95 3.29% 10.82 4.42% 6.83 7.03%5 -1.13 1.73% 11.16 6.17% 23.69 9.67% 0.77 0.79%6 -1.64 2.51% 62.68 34.67% 1.57 0.64% 2.26 2.32%7 2.55 3.92% -14.48 8.01% 8.8 3.59% 4.02 4.14%8 -0.33 0.51% -5.01 2.77% -0.66 0.27% -3.06 3.15%9 -16.58 25.42% 0.59 0.33% -46.36 18.93% -21.87 22.5%10 6.12 9.39% -21.96 12.15% 42.46 17.33% 6.49 6.68%11 -7.13 10.93% 6.4 3.54% -16.02 6.54% -5.83 6.00%12 -0.93 1.42% -9.44 5.22% -16.59 6.77% -7.96 8.19%13 -1.19 1.83% -0.19 0.11% -3.08 1.26% 2.06 2.12%14 -1.17 1.8% -6.46 3.58% 1.64 0.67% -6.02 6.19%15 -4.6 7.05% 14.35 7.94% -7.26 2.96% -1.95 2.01%

— — 148



1 -13.89 57.37% 10.04 44.18% -15.64 31.26% -10.83 47.74%2 -1.53 6.30% 5.39 23.72% -13.32 26.62% -2.51 11.09%3 -3.86 15.95% -2.57 11.32% -3.64 7.28% -6.35 28% 4 -2.18 9% -0.81 3.57% 4.29 8.58% -0.61 2.68%5 -2.75 11.37% -3.91 17.21% 13.14 26.26% 2.38 10.49%



1 -12.78 45.58% 9.12 8.39% -34.58 28.65% -15.01 27.83%2 -0.51 1.81% 25.37 23.35% -4.69 3.89% -5.59 10.37%3 -0.37 1.31% -15.93 14.66% -6.28 5.2% -1.95 3.62%4 -0.62 2.23% -4.69 4.32% 12.89 10.68% 3.71 6.87%5 -1.28 4.56% -3.12 2.87% 23.58 19.54% 4.17 7.74%6 -4.53 16.14% 1.27 1.17% -4.25 3.52% -0.76 1.4% 7 1.78 6.34% 10.48 9.64% -3.79 3.14% -4.21 7.81%8 1.51 5.38% -11.96 11.01% 12.87 10.66% 8.42 15.61%9 -1.42 5.06% 10.15 9.34% -12.48 10.34% -10.1 18.72%10 3.25 11.6% -16.57 15.25% -5.28 4.38% 0.01 0.02%



1 -18.41 29.45% 28.21 15.07% -77.16 27.83% -35.12 26.12%2 4.75 7.59% 4.15 2.22% 4.28 1.54% -2.79 2.08%3 -1.91 3.05% -14.83 7.92% -15.23 5.5% -8.85 6.58%4 2.35 3.75% -18.07 9.65% -4.43 1.6% 2.22 1.65%5 2.51 4.01% -12.73 6.8% 13.43 4.84% 5.48 4.08%6 -3.17 5.07% 3.25 1.74% 14.32 5.16% 5.49 4.08%7 1.84 2.94% 8.42 4.5% -5.55 2% -2.7 2.01%8 2.67 4.27% -23.22 12.41% 12.3 4.44% -6.27 4.66%9 -3.53 5.64% 16.26 8.69% -34.52 12.45% -9.78 7.27%10 5.6 8.95% -16.84 8.99% -37.73 13.61% -9.67 7.19%11 -0.99 1.58% -13.07 6.98% 28.1 10.13% 3.91 2.91%12 0.83 1.33% -10.28 5.49% -5.61 2.02% -19.42 14.45%13 4.73 7.57% -7.96 4.25% -1.03 0.37% 0.81 0.60%14 -2.39 3.82% -8.12 4.34% -22.02 7.94% -8.8 6.54%15 6.86 10.96% -1.8 0.96% -1.51 0.55% 13.15 9.78%

— — 149

Table 4.7 Combination coefficients and contribution coefficients (AOA=0º)

N Q M NQM LRCx 0.54 30.83% 1.35 63.74% 5.42 57.76% 1.04 52.30%LRCy -0.22 12.23% -0.18 8.43% 0.00 0.05% -0.07 3.69% LRCz 1.00 56.76% 0.21 9.96% -3.55 37.82% 0.52 26.09%LRCt 0.00 0.18% 0.38 17.88% 0.41 4.37% 0.35 17.92%

Table 4.8 Combination coefficients and contribution coefficients (AOA=90º)

N Q M NQM LRCx 0.03 0.37% -0.44 6.57% -0.19 5.76% -0.33 7.17%LRCy 3.97 53.99% 2.90 42.81% 0.81 24.54% 1.78 39.19%LRCz 3.34 45.49% 3.32 49.06% 2.18 66.32% 2.42 53.35%LRCt -0.01 0.15% -0.11 1.57% -0.11 3.38% 0.01 0.29%

— 150 —

Figure 4.1 Mean Pressure Distribution on Frame B (AOA=0º)

Figure 4.2 ESWL pressure distribution based on Conditional Sampling on Frame B at critical section 3 causing maximum bending moment with support

condition fix-fix (AOA=0º)

Figure 4.3 ESWL pressure distribution based on LRC on Frame B at critical section 3 causing maximum bending moment with support condition fix-fix

(AOA=0º)

Figure 4.4 ESWL pressure distribution based on GLE on Frame B at critical section 3 causing maximum bending moment with support condition fix-fix

(AOA=0º)

— 151 —

Figure 4.5 Mean Pressure Distribution on Frame C (AOA=0º)

Figure 4.6 ESWL pressure distribution based on Conditional Sampling on Frame C at critical section 3 causing maximum bending moment with support


Figure 4.7 ESWL pressure distribution based on LRC on Frame C at critical section 3 causing maximum bending moment with support condition fix-fix

(AOA=0º)

Figure 4.8 ESWL pressure distribution based on GLE on Frame C at critical section 3 causing maximum bending moment with support condition fix-fix

(AOA=0º)

— 152 —

Figure 4.9 Mean Pressure Distribution on Frame B (AOA=90º)

Figure 4.10 ESWL pressure distribution based on Conditional Sampling on Frame B at critical section 3 causing maximum bending moment with support


Figure 4.11 ESWL pressure distribution based on LRC on Frame B at critical section 3 causing maximum bending moment with support condition fix-fix

(AOA=90º)

Figure 4.12 ESWL pressure distribution based on GLE on Frame B at critical section 3 causing maximum bending moment with support condition fix-fix

(AOA=90º)

— 153 —

Figure 4.13 Mean Pressure Distribution on Frame C (AOA=90º)

Figure 4.14 ESWL pressure distribution based on Conditional Sampling on Frame C at critical section 3 causing maximum bending moment with support


Figure 4.15 ESWL pressure distribution based on LRC on Frame C at critical section 3 causing maximum bending moment with support condition fix-fix

(AOA=90º)

Figure 4.16 ESWL pressure distribution based on GLE on Frame C at critical section 3 causing maximum bending moment with support condition fix-fix

(AOA=90º)

— 154 —

1 2 3 4 50

20

40

60

80

C

ontri

butio

n Fa

ctor

(%)

Mode Number

MCforN MCforQ MCforM MCforNQM

1 2 3 4 5 6 7 8 9 10

0

10

20

30

40

50

Con

tribu

tion

Fact

or (%

)

Mode Number


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

10

20

30

Con

tribu

tion

Fact

or (%

)

Mode Number


(a) mode number=5 (b) mode number=10 (c) mode number=15

Figure 4.17 The relationship between the contribution coefficients and mode number at AOA=0º

1 2 3 4 50

20

40

60

Con

tribu

tion

Fact

or (%

)

Mode Number


1 2 3 4 5 6 7 8 9 10

0

10

20

30

40

50

Con

tribu

tion

Fact

or (%

)

Mode Number


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

10

20

30

Con

tribu

tion

Fact

or (%

)

Mode Number


(a) mode number=5 (b) mode number=10 (c) mode number=15

Figure 4.18 The relationship between the contribution coefficients and mode number at AOA=90º

— — 155

Figure 4.19 Universal ESWL distribution reproducing maximum axial force N (AOA=0º, mode

number=5)

Figure 4.20 Universal ESWL distribution reproducing maximum shear force Q (AOA=0º, mode

number=5)

— — 156

Figure 4.21 Universal ESWL distribution reproducing maximum bending moment M (AOA=0º,

mode number=5)

Figure 4.22 Universal ESWL distribution reproducing all maximum load effects simultaneously (AOA=0º, mode number=5)

— 157 —

Figure 4.23 Comparison of Actual Maximum Axial Forces and Axial Forces

under Universal ESWL (AOA=0º, mode number=5)

Figure 4.24 Comparison of Actual Maximum Shear Forces and Shear Forces


Figure 4.25 Comparison of Actual Maximum Bending Moments and Bending

Moments under Universal ESWL (AOA=0º, mode number=5)

Figure 4.26 Comparison of Actual Maximum Responses and Responses under

Universal ESWL (AOA=0º, mode number=5)

— — 158


number=10)


number=10)

— — 159


mode number=10)


— 160 —









— — 161

Figure 4.35 Universal ESWL distribution reproducing maximum axial force N (AOA=0º, mode number=15)

Figure 4.36 Universal ESWL distribution reproducing maximum shear force Q (AOA=0º, mode number=15)

— — 162


mode number=15)


— 163 —









— — 164


number=5)


number=5)

— — 165


mode number=5)


— 166 —









— — 167


number=10)


number=10)

— — 168


mode number=10)


— 169 —









— — 170


number=15)


number=15)

— — 171


mode number=15)


— 172 —









— — 173

Figure 4.67 Modified Universal ESWL Distribution reproducing maximum axial force N (AOA=0º)

Figure 4.68 Modified Universal ESWL Distribution reproducing maximum shear force Q (AOA=0º)

— — 174

Figure 4.69 Modified Universal ESWL Distribution reproducing maximum bending moment M (AOA=0º)

Figure 4.70 Modified Universal ESWL Distribution reproducing all the maximum internal forces N,Q,M (AOA=0º)

— 175 —


under Modified Universal ESWL (AOA=0º)




Moments under Modified Universal ESWL (AOA=0º)


Modified Universal ESWL (AOA=0º)

— — 176

Figure 4.75 Modified Universal ESWL Distribution reproducing maximum axial force N (AOA=90º)

Figure 4.76 Modified Universal ESWL Distribution reproducing maximum shear force Q (AOA=90º)

— — 177

Figure 4.77 Modified Universal ESWL Distribution reproducing maximum bending moment M (AOA=90º)

Figure 4.78 Modified Universal ESWL Distribution reproducing all the maximum internal forces N,Q,M (AOA=90º)

— 178 —






Moments under Modified Universal ESWL (AOA=90º)


Modified Universal ESWL (AOA=90º)

— — 179

CHAPTER V

SUMMARY, CONTRIBUTIONS AND

RECOMMENDATIONS

5.1 Summary

The primary goal of this dissertation is to investigate wind induced responses of low-

rise buildings and associated wind loading. Actually, this topic has been studied for a

long time and it has long been the main focus of wind engineering. The study on wind

induced response and wind loading is very complicated due to various structure

systems and environments around the structures and some other factors. In wind

engineering, a simple and efficient wind loading description is usually the target for

engineering practice.

In chapter I, the evolution of investigation of wind induced responses and wind

loading is described as a literature review. Some methods proposed for wind loading

and wind induced responses have been discussed in detail in chapter I, and their

advantages and disadvantages are compared to each other. In chapter II, some basic

pressure distributions such as mean pressure distribution, fluctuating pressure

distribution, and extreme pressure distributions corresponding to maximum base shear

forces, lift force and base torsional moment are investigated to give some insight into

the relationship between these forces and their correlation to each other. Besides these

basic pressure distribution characteristics, a frame structure system is assumed for

WERFL building of Texas Tech University, the integrated six forces at the base of the

building including two base shear forces, lift force, two bending moments, and

torsional moment are applied to WERFL building, the induced normal and shear

stresses in four columns are calculated, and the combination of these forces is

discussed and a formula of wind loading combination is proposed and shows some

potential to be used in codes and standards. In chapter III, another frame system is

assumed for WERFL building, four frames are considered in this system. Axial force,

shear force and bending moment at nine critical sections along each frame under wind

action with AOA=0o and AOA=90o are considered, and mean responses, maximum

dynamic responses, and responses under gust loading envelop are calculated.

Corresponding gust response factors and background factors are obtained, some

characteristics of these factors are discussed. In chapter IV, equivalent static wind

— — 180

load is discussed for WERFL building, equivalent static wind load based on LRC

method, equivalent static wind load equal to mean pressure distribution multiplied by

gust response factor, and equivalent static wind load equal to gust loading envelop

multiplied by background factor are compared. Besides all these equivalent static

wind methods, universal equivalent static wind load method proposed by Tamura and

a proposed modified universal equivalent static wind load method are applied to

WERFL building, the responses calculated by both universal equivalent static wind

loads are compared and discussed.

5.2 Contributions

For wind induced responses and wind loading, this dissertation has made the

following contributions:

1. For building with certain geometry, some basic pressure distributions around

the building are investigated in detail.

2. From these pressure distributions, the relationship between along-wind,

across-wind, lift-force and torsional moment at the base of the building is

studied and the correlation between these forces is investigated. And it is

found that torsional moment is more correlated with across-wind loading than

with along-wind loading.

3. Wind load combination between integrated shear forces, lift force and

torsional moment at the base of a building is studied and a wind load

combination formula is derived with a potential to be used in codes and

standards

4. Wind induced responses for a building with a frame system are studied for two

wind angle of attack, AOA=0o and AOA=90o. For each frame, internal forces

of nine critical sections are investigated, and mean responses, maximum

dynamic responses and responses under gust loading envelop are calculated in

detail. Corresponding gust response factors and background factors are also

calculated which provides a clear view of the variance of these factors and

proves that gust response factor and background factor are sensitive to the

specific response considered. The internal responses calculated from pressure

time history are compared to the responses calculated by ASCE7-05, and It is

found that ASCE may underestimate some responses, and it is not necessarily

— — 181

conservative for all the responses.

5. Equivalent static wind load methods are studied in detail based on a full scale

building, which include equivalent static wind loads assumed as mean pressure

distribution multiplied by gust response factor, as gust loading envelop

multiplied by background factor, and equivalent static wind load based on

LRC method. The study of equivalent static wind load provides a clear view

about these widely used methods and the disadvantage involved in each

method is pointed out.

6. Universal equivalent static wind load method proposed by Tamura is applied

to a real building. Different universal equivalent static wind loads based on

different number of eigen modes are compared and it is found that the involve

of more eigen modes can’t make the results fitting better than actual responses.

7. A modified universal equivalent static wind load method is proposed which

makes use of extreme pressure distributions causing maximum base shear

forces, lift force and torsional moment as the basic combination pressure

distribution set. The responses under the action of this universal equivalent

static wind load are compared with actual responses, which shows that the

modified universal equivalent static wind load method is much better than the

usually used one for a given building.

5.3 Recommendations for future research

The following topics are recommended for future research

1. For different low-rise buildings with various geometries, some basic pressure

distributions such as mean pressure distribution, fluctuating pressure

distribution, extreme pressure distributions causing maximum base shear

forces, lift force and base torsional moment are needed to be studied in more

detail.

2. The relationship between along-wind, across-wind, torsional moment of low-

rise buildings needs to be investigated more deeply, since it relates to wind

load combination.

3. Wind load combination is a topic of important, in which the role of torsional

moment is also needed to be studied.

— — 182

4. Wind induced responses of a building with different structural systems need to

be investigated, corresponding gust response factors and background factors

are important for providing a deep insight in the GRF method and BF method

of equivalent static wind loading.

5. A uniform equivalent static wind load is necessary to be developed which can

produce responses fitting the actual responses well for a building with a certain

structure system. The proposal of this kind of method will greatly simplify

wind loading and will have great potential to be used in wind engineering

practice.

— — 183

APPENDIX I-PART I: WERFL BUILDING

Table I-I-1 Mean response of Frame A (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 63.546 -2.199 0.000 66.018 -5.868 0.000 62.987 -11.820 -77.032 2 63.546 -15.873 50.117 66.018 -19.542 74.059 62.987 -25.494 35.412 3 63.546 -32.657 201.835 66.018 -36.326 249.680 62.987 -42.278 249.5354 32.657 30.910 -126.373 36.326 30.910 -78.432 42.278 30.381 -74.675 5 32.657 -7.411 -251.674 36.326 -7.411 -203.830 42.278 -7.940 -196.0516 32.657 -26.607 -140.952 36.326 -26.607 -92.986 42.278 -27.136 -81.295 7 64.032 32.657 180.736 64.032 36.326 228.581 64.561 42.278 244.2628 64.032 14.169 37.124 64.032 36.326 61.066 64.561 23.791 38.299 9 64.032 0.000 0.000 64.032 36.326 0.000 64.561 9.622 -61.192

Table I-I-2 Absolute maximum total response of Frame A (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M

1 180.760 76.199 0.000 184.364 49.105 0.000 179.956 -62.668 -346.0312 180.760 -93.338 544.374 184.364 -57.041 323.983 179.956 -73.968 129.1523 180.760 -114.521 1209.229 184.364 -83.744 738.803 179.956 -99.222 642.2934 114.521 94.426 -757.773 83.744 94.426 -307.469 99.222 89.447 -248.1755 114.521 -41.895 -944.359 83.744 -41.895 -562.917 99.222 -37.379 -538.9646 114.521 -79.482 -645.908 83.744 -79.482 -341.998 99.222 -77.029 -257.4327 180.883 114.521 796.817 180.883 83.744 658.289 185.540 99.222 611.0738 180.883 74.850 196.107 180.883 83.744 272.602 185.540 66.318 132.4369 180.883 0.000 0.000 180.883 83.744 0.000 185.540 53.346 -287.729

Table I-I-3 Absolute maximum dynamic response of Across A (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 117.214 78.397 0.000 118.347 54.973 0.000 116.969 59.701 -268.9992 117.214 -77.466 494.257 118.347 -37.499 249.924 116.969 -48.473 93.740 3 117.214 -81.864 1007.394 118.347 -47.418 489.123 116.969 -56.944 392.758 4 81.864 63.516 785.094 47.418 63.516 -229.037 56.944 59.066 -173.5005 81.864 -34.484 -692.685 47.418 -34.484 -359.087 56.944 -29.440 -342.9136 81.864 -52.875 587.152 47.418 -52.875 -249.012 56.944 -49.893 -176.1377 116.850 81.864 616.081 116.850 47.418 429.708 120.979 56.944 366.811 8 116.850 60.680 158.983 116.850 47.418 211.536 120.979 42.527 94.137 9 116.850 0.000 0.000 116.850 47.418 0.000 120.979 -57.415 -226.537

— — 184

Table I-I-4 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Frame A (AOA=0º)

Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M

1 -205.68 10.82 0.00 -213.67 5.31 0.00 -203.39 18.41 184.50 2 -205.68 69.18 -223.22 -213.67 63.67 -187.72 -203.39 76.77 -87.30 3 -205.68 119.35 -827.91 -213.67 113.85 -756.77 -203.39 126.94 -740.80 4 -119.35 -91.46 151.53 -113.85 -91.46 222.36 -126.94 -89.26 221.97 5 -119.35 23.47 508.34 -113.85 23.47 579.48 -126.94 25.68 562.42 6 -119.35 74.86 184.75 -113.85 74.86 255.50 -126.94 77.07 222.03 7 -201.94 -119.35 -729.86 -201.94 -113.85 -658.72 -204.15 -126.94 -708.78 8 -201.94 -60.68 -158.98 -201.94 -113.85 -123.48 -204.15 -68.27 -89.25 9 -201.94 0.00 0.00 -201.94 -113.85 0.00 -204.15 -7.59 118.43

Table I-I-5 Gust Factor, Frame A (AOA=0º) Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M 1 2.845 35.380 NaN 2.793 -8.368 NaN 2.857 5.302 4.492 2 2.845 5.880 10.862 2.793 2.919 4.375 2.857 2.901 3.647 3 2.845 3.507 5.991 2.793 2.305 2.959 2.857 2.347 2.574 4 3.507 3.055 5.996 2.305 3.055 3.920 2.347 2.944 3.323 5 3.507 5.653 3.752 2.305 5.653 2.762 2.347 4.708 2.749 6 3.507 2.987 4.582 2.305 2.987 3.678 2.347 2.839 3.167 7 2.825 3.507 4.409 2.825 2.305 2.880 2.874 2.347 2.502 8 2.825 5.282 5.282 2.825 2.305 4.464 2.874 2.788 3.458 9 2.825 NaN NaN 2.825 2.305 NaN 2.874 5.544 4.702

Table I-I-6 Background Factor, Frame A (AOA=0º) Pin-roller Pin-pin Fix-fix points N Q M N Q M N Q M

1 -0.57 7.25 NaN -0.55 10.35 NaN -0.58 3.24 -1.46 2 -0.57 -1.12 -2.21 -0.55 -0.59 -1.33 -0.58 -0.63 -1.07 3 -0.57 -0.69 -1.22 -0.55 -0.42 -0.65 -0.58 -0.45 -0.53 4 -0.69 -0.69 5.18 -0.42 -0.69 -1.03 -0.45 -0.66 -0.78 5 -0.69 -1.47 -1.36 -0.42 -1.47 -0.62 -0.45 -1.15 -0.61 6 -0.69 -0.71 3.18 -0.42 -0.71 -0.97 -0.45 -0.65 -0.79 7 -0.58 -0.69 -0.84 -0.58 -0.42 -0.65 -0.59 -0.45 -0.52 8 -0.58 -1.00 -1.00 -0.58 -0.42 -1.71 -0.59 -0.62 -1.05 9 -0.58 NaN NaN -0.58 -0.42 NaN -0.59 7.56 -1.91

— — 185

Table I-I-7 Mean response of Frame B (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 58.262 3.996 0.000 62.024 1.530 0.000 58.995 -3.029 -41.284 2 58.262 -15.843 26.004 62.024 -18.308 42.177 58.995 -22.867 30.149 3 58.262 -40.196 202.035 62.024 -42.662 234.345 58.995 -47.221 251.698 4 40.196 32.777 -123.122 42.662 32.777 -90.723 47.221 33.536 -79.127 5 40.196 -10.879 -248.422 42.662 -10.879 -216.113 47.221 -10.120 -210.176 6 40.196 -32.202 -105.525 42.662 -32.202 -73.111 47.221 -31.443 -72.923 7 60.086 40.196 231.219 60.086 42.662 263.528 59.327 47.221 258.031 8 60.086 17.982 47.113 60.086 42.662 63.286 59.327 25.007 28.465 9 60.086 0.000 0.000 60.086 42.662 0.000 59.327 7.025 -64.123

Table I-I-8 Absolute Maximum Total response of Across B (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 175.201 123.930 0.000 183.597 -68.858 0.000 177.272 -83.756 -488.643 2 175.201 -129.759 790.864 183.597 -77.218 458.287 177.272 -91.594 138.633 3 175.201 -156.477 1653.481 183.597 -128.288 976.448 177.272 -139.152 771.122 4 156.477 116.539 -1379.950 128.288 116.539 -463.262 139.152 108.813 -306.775 5 156.477 -68.471 -1447.947 128.288 -68.471 -683.987 139.152 -51.616 -661.151 6 156.477 -115.588 -914.549 128.288 -115.588 -455.764 139.152 -103.621 -296.544 7 182.548 156.477 900.225 182.548 128.288 1001.960 181.417 139.152 784.499 8 182.548 76.811 201.246 182.548 128.288 472.239 181.417 94.969 132.184 9 182.548 0.000 0.000 182.548 128.288 0.000 181.417 88.182 -502.633

Table I-I-9 Absolute Maximum Dynamic response of Across B (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 116.939 -122.823 0.000 121.574 -70.388 0.000 118.277 -80.728 -447.3592 116.939 -113.917 764.860 121.574 -58.910 416.111 118.277 -68.727 108.484 3 116.939 -116.281 1451.446 121.574 -85.627 742.104 118.277 -91.931 519.424 4 116.281 83.762 -1256.828 85.627 83.762 -372.539 91.931 75.278 -227.6485 116.281 -57.593 -1199.525 85.627 -57.593 -467.874 91.931 -41.496 -450.9756 116.281 -83.386 -809.023 85.627 -83.386 -382.653 91.931 -72.178 -223.6207 122.462 116.281 669.006 122.462 85.627 738.432 122.090 91.931 526.468 8 122.462 58.829 154.133 122.462 85.627 408.953 122.090 69.962 103.719 9 122.462 0.000 0.000 122.462 85.627 0.000 122.090 81.157 -438.510

— — 186

Table I-I-10 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Frame B (AOA=0º)


1 -188.33 -0.05 0.00 -199.91 -4.15 0.00 -187.84 7.43 139.47 2 -188.33 64.17 -167.93 -199.91 60.07 -141.78 -187.84 71.65 -76.35 3 -188.33 158.25 -817.86 -199.91 154.15 -765.50 -187.84 165.73 -774.43 4 -158.25 -99.48 188.50 -154.15 -99.48 240.58 -165.73 -99.07 228.75 5 -158.25 27.78 573.52 -154.15 27.78 625.87 -165.73 28.19 610.84 6 -158.25 88.46 190.78 -154.15 88.46 242.80 -165.73 88.87 224.84 7 -187.45 -158.25 -797.10 -187.45 -154.15 -744.75 -187.86 -165.73 -765.82 8 -187.45 -58.83 -154.13 -187.45 -154.15 -127.98 -187.86 -66.31 -74.87 9 -187.45 0.00 0.00 -187.45 -154.15 0.00 -187.86 -7.48 127.27

Table I-I-11 Gust Factor, Frame B (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 3.007 31.015 NaN 2.960 -44.999 NaN 3.005 27.654 11.8362 3.007 8.191 30.413 2.960 4.218 10.866 3.005 4.006 4.598 3 3.007 3.893 8.184 2.960 3.007 4.167 3.005 2.947 3.064 4 3.893 3.556 11.208 3.007 3.556 5.106 2.947 3.245 3.877 5 3.893 6.294 5.829 3.007 6.294 3.165 2.947 5.100 3.146 6 3.893 3.589 8.667 3.007 6.294 6.234 2.947 3.295 4.067 7 3.038 3.893 3.893 3.038 3.007 3.802 3.058 2.947 3.040 8 3.038 4.272 4.272 3.038 3.007 7.462 3.058 3.798 4.644 9 3.038 NaN NaN 3.038 3.007 NaN 3.058 12.553 7.839

Table I-I-12 Background Factor, Frame B (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 -0.62 2556.48 NaN -0.61 16.97 NaN -0.63 -10.86 -3.21 2 -0.62 -1.78 -4.55 -0.61 -0.98 -2.93 -0.63 -0.96 -1.42 3 -0.62 -0.73 -1.77 -0.61 -0.56 -0.97 -0.63 -0.55 -0.67 4 -0.73 -0.84 -6.67 -0.56 -0.84 -1.55 -0.55 -0.76 -1.00 5 -0.73 -2.07 -2.09 -0.56 -2.07 -0.75 -0.55 -1.47 -0.74 6 -0.73 -0.94 -4.24 -0.56 -0.94 -1.58 -0.55 -0.81 -0.99 7 -0.65 -0.73 -0.84 -0.65 -0.56 -0.99 -0.65 -0.55 -0.69 8 -0.65 -1.00 -1.00 -0.65 -0.56 -3.20 -0.65 -1.06 -1.39 9 -0.65 NaN NaN -0.65 -0.56 NaN -0.65 -10.85 -3.45

— — 187

Table I-I-13 Mean response of Frame C (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 24.675 0.960 0.000 26.578 2.022 0.000 24.853 0.768 -11.500 2 24.675 -8.752 19.206 26.578 -7.690 12.377 24.853 -8.943 8.851 3 24.675 -21.317 113.144 26.578 -20.255 99.474 24.853 -21.509 103.969 4 21.317 12.855 -20.553 20.255 12.855 -34.185 21.509 13.045 -31.145 5 21.317 -4.191 -70.548 20.255 -4.191 -84.218 21.509 -4.001 -82.595 6 21.317 -12.153 -16.234 20.255 -12.153 -29.857 21.509 -11.963 -29.686 7 26.286 21.317 121.030 26.286 20.255 107.360 26.096 21.509 106.102 8 26.286 9.334 24.454 26.286 20.255 17.625 26.096 9.525 8.371 9 26.286 0.000 0.000 26.286 20.255 0.000 26.096 0.192 -17.247

Table I-I-14 Absolute Maximum Total response of Across C (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 76.213 80.326 0.000 78.022 51.528 0.000 74.218 54.017 249.675 2 76.213 -81.843 507.997 78.022 -37.583 -232.874 74.218 -39.932 53.820 3 76.213 -92.922 1062.288 78.022 -54.779 484.305 74.218 -56.217 304.698 4 92.922 46.713 725.850 54.779 46.713 -219.698 56.217 40.090 -124.802 5 92.922 -32.114 -610.794 54.779 -32.114 -260.695 56.217 -21.872 -252.213 6 92.922 -44.595 445.673 54.779 -44.595 -224.363 56.217 -38.137 -127.978 7 76.844 92.922 533.185 76.844 54.779 480.025 74.569 56.217 304.249 8 76.844 42.853 112.276 76.844 54.779 226.069 74.569 39.919 51.680 9 76.844 0.000 0.000 76.844 54.779 0.000 74.569 37.319 -238.326

Table I-I-15 Absolute Maximum Dynamic response of Across C (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 51.539 79.366 0.000 51.444 49.507 0.000 49.365 53.249 261.175 2 51.539 -73.091 488.791 51.444 -29.893 -245.251 49.365 -30.989 44.968 3 51.539 -71.605 949.143 51.444 -34.524 384.831 49.365 -34.709 200.730 4 71.605 33.858 746.402 34.524 33.858 -185.513 34.709 27.044 -93.657 5 71.605 -27.923 590.349 34.524 -27.923 -176.477 34.709 -17.872 -169.618 6 71.605 -32.441 461.907 34.524 -32.441 -194.507 34.709 -26.174 -98.292 7 50.558 71.605 412.155 50.558 34.524 372.665 48.473 34.709 198.147 8 50.558 33.520 87.822 50.558 34.524 -230.626 48.473 30.393 43.309 9 50.558 0.000 0.000 50.558 34.524 0.000 48.473 -49.131 243.458

— — 188

Table I-I-16 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Frame C (AOA=0º)


1 -124.28 -3.58 0.00 -133.11 -5.04 0.00 -125.16 2.13 68.34 2 -124.28 36.15 -80.80 -133.11 34.69 -71.66 -125.16 41.86 -49.14 3 -124.28 107.18 -493.02 -133.11 105.72 -474.69 -125.16 112.89 -498.18 4 -107.18 -59.45 150.39 -105.72 -59.45 168.53 -112.89 -60.39 152.23 5 -107.18 18.74 382.14 -105.72 18.74 400.47 -112.89 17.80 391.16 6 -107.18 58.70 126.76 -105.72 58.70 144.85 -112.89 57.76 142.73 7 -126.05 -107.18 -533.26 -126.05 -105.72 -514.93 -125.11 -112.89 -510.00 8 -126.05 -39.22 -102.77 -126.05 -105.72 -93.62 -125.11 -44.94 -42.80 9 -126.05 0.00 0.00 -126.05 -105.72 0.00 -125.11 -5.71 96.73

Table I-I-17 Gust Factor, Frame C (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 3.089 83.700 NaN 2.936 25.488 NaN 2.986 70.322 -21.7112 3.089 9.352 26.450 2.936 4.887 18.053 2.986 4.465 6.081 3 3.089 4.359 9.389 2.936 2.704 4.869 2.986 2.614 2.931 4 4.359 3.634 -35.317 2.704 3.634 6.427 2.614 3.073 4.007 5 4.359 7.663 8.658 2.704 7.663 3.095 2.614 5.467 3.054 6 4.359 3.669 -27.453 2.704 3.669 7.515 2.614 3.188 4.311 7 2.923 4.359 4.405 2.923 2.704 4.471 2.857 2.614 2.868 8 2.923 4.591 4.591 2.923 2.704 12.827 2.857 4.191 6.174 9 2.923 NaN NaN 2.923 2.704 NaN 2.857 -255.491 13.819

Table I-I-18 Background Factor, Frame C (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 -0.41 -22.15 NaN -0.39 -9.81 NaN -0.39 24.99 3.82 2 -0.41 -2.02 -6.05 -0.39 -0.86 3.42 -0.39 -0.74 -0.92 3 -0.41 -0.67 -1.93 -0.39 -0.33 -0.81 -0.39 -0.31 -0.40 4 -0.67 -0.57 4.96 -0.33 -0.57 -1.10 -0.31 -0.45 -0.62 5 -0.67 -1.49 1.54 -0.33 -1.49 -0.44 -0.31 -1.00 -0.43 6 -0.67 -0.55 3.64 -0.33 -0.55 -1.34 -0.31 -0.45 -0.69 7 -0.40 -0.67 -0.77 -0.40 -0.33 -0.72 -0.39 -0.31 -0.39 8 -0.40 -0.85 -0.85 -0.40 -0.33 2.46 -0.39 -0.68 -1.01 9 -0.40 NaN NaN -0.40 -0.33 NaN -0.39 8.60 2.52

— — 189

Table I-I-19 Mean response of Across D (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix


Table I-I-20 Absolute Maximum Total response of Across D (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 37.936 21.143 0.000 39.528 11.339 0.000 38.363 -12.775 -81.440 2 37.936 -21.956 118.001 39.528 -15.664 76.385 38.363 -19.231 29.875 3 37.936 -28.613 278.040 39.528 -24.808 195.198 38.363 -28.247 173.262 4 28.613 25.438 -238.737 24.808 25.438 -93.075 28.247 24.626 -63.771 5 28.613 -13.241 -268.249 24.808 -13.241 -153.575 28.247 -10.318 -147.856 6 28.613 -22.446 -151.872 24.808 -22.446 -82.250 28.247 -20.996 -59.490 7 40.777 28.613 168.564 40.777 24.808 207.244 41.003 28.247 174.360 8 40.777 14.185 37.166 40.777 24.808 86.041 41.003 19.651 29.540 9 40.777 0.000 0.000 40.777 24.808 0.000 41.003 14.405 -92.324

Table I-I-21 Absolute Maximum Dynamic response of Across D (lb, lb-ft) (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 22.076 19.710 0.000 22.834 11.492 0.000 22.238 11.760 -67.505 2 22.076 -18.364 114.146 22.834 -10.487 62.197 22.238 -12.499 19.604 3 22.076 -19.647 234.620 22.834 -14.257 131.121 22.238 -16.140 103.0504 19.647 16.134 -190.317 14.257 16.134 -65.335 16.140 15.049 -40.101 5 19.647 -9.626 -185.920 14.257 -9.626 -91.904 16.140 -6.976 -88.214 6 19.647 -13.796 -111.129 14.257 -13.796 -62.192 16.140 -12.619 -39.400 7 23.847 19.647 114.869 23.847 14.257 132.892 24.346 16.140 102.0898 23.847 9.905 25.951 23.847 14.257 64.493 24.346 12.230 20.108 9 23.847 0.000 0.000 23.847 14.257 0.000 24.346 11.264 -70.177

— — 190

Table I-I-22 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Frame D (AOA=0º)


1 -33.60 -2.04 0.00 -35.85 -1.66 0.00 -33.77 0.50 22.13 2 -33.60 10.26 -18.97 -35.85 10.64 -21.53 -33.77 12.80 -13.21 3 -33.60 27.55 -130.49 -35.85 27.93 -135.61 -33.77 30.09 -141.16 4 -27.55 -18.10 51.44 -27.93 -18.10 46.27 -30.09 -18.28 42.12 5 -27.55 5.44 119.09 -27.93 5.44 113.97 -30.09 5.26 111.17 6 -27.55 15.72 48.77 -27.93 15.72 43.59 -30.09 15.54 42.18 7 -36.49 -27.55 -136.84 -36.49 -27.93 -141.96 -36.31 -30.09 -142.01 8 -36.49 -9.90 -25.95 -36.49 -27.93 -28.52 -36.31 -12.44 -14.73 9 -36.49 0.00 0.00 -36.49 -27.93 0.00 -36.31 -2.54 27.62

Table I-I-23 Gust Factor, Frame D (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 2.392 14.763 NaN 2.368 -74.352 NaN 2.379 7.479 5.844 2 2.392 6.112 30.607 2.368 3.026 5.384 2.379 2.856 2.909 3 2.392 3.191 6.404 2.368 2.351 3.046 2.379 2.333 2.468 4 3.191 2.734 4.930 2.351 2.734 3.355 2.333 2.572 2.694 5 3.191 3.662 3.258 2.351 3.662 2.490 2.333 3.087 2.479 6 3.191 2.595 3.728 2.351 2.595 4.101 2.333 2.507 2.961 7 2.409 3.191 3.139 2.409 2.351 2.787 2.462 2.333 2.413 8 2.409 3.314 3.314 2.409 2.351 3.993 2.462 2.648 3.132 9 2.409 NaN NaN 2.409 2.351 NaN 2.462 4.587 4.169

Table I-I-24 Background Factor, Frame D (AOA=0º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 -0.66 -9.66 NaN -0.64 -6.93 NaN -0.66 -22.16 -3.05 2 -0.66 -1.79 -6.02 -0.64 -0.99 -2.89 -0.66 -0.98 -1.48 3 -0.66 -0.71 -1.80 -0.64 -0.51 -0.97 -0.66 -0.54 -0.73 4 -0.71 -0.89 -3.70 -0.51 -0.89 -1.41 -0.54 -0.82 -0.95 5 -0.71 -1.77 -1.56 -0.51 -1.77 -0.81 -0.54 -1.33 -0.79 6 -0.71 -0.88 -2.28 -0.51 -0.88 -1.43 -0.54 -0.81 -0.93 7 -0.65 -0.71 -0.84 -0.65 -0.51 -0.94 -0.67 -0.54 -0.72 8 -0.65 -1.00 -1.00 -0.65 -0.51 -2.26 -0.67 -0.98 -1.37 9 -0.65 NaN NaN -0.65 -0.51 NaN -0.67 -4.43 -2.54

— — 191

Table I-I-25 Mean response of Across A (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix


Table I-I-26 Absolute Maximum Total response of Across A (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix


Table I-I-27 Absolute Maximum Dynamic response of Across A (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 73.38 45.98 0.00 71.99 -33.72 0.00 71.08 -41.72 -176.08 2 73.38 -27.19 -234.41 71.99 -22.27 178.48 71.08 -29.71 66.83 3 73.38 -19.04 321.96 71.99 -19.86 256.24 71.08 -24.77 219.56 4 19.04 34.97 -336.54 19.86 34.97 -130.99 24.77 33.70 -109.54 5 19.04 -21.12 -348.36 19.86 -21.12 -192.29 24.77 -18.82 -181.85 6 19.04 -27.59 -219.38 19.86 -27.59 -113.13 24.77 -26.83 -74.11 7 38.00 19.04 137.94 38.00 19.86 224.37 38.67 24.77 184.18 8 38.00 13.25 34.72 38.00 19.86 109.45 38.67 21.69 46.35 9 38.00 0.00 0.00 38.00 19.86 0.00 38.67 20.21 -116.11

— — 192

Table I-I-28 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Frame A (AOA=90º)


1 -142.66 -17.73 0.00 -148.45 2.64 0.00 -144.77 14.45 104.91 2 -142.66 13.33 33.89 -148.45 33.70 -98.69 -144.77 45.51 -70.12 3 -142.66 27.00 -155.64 -148.45 47.38 -420.71 -144.77 59.18 -468.68 4 -27.00 -50.72 468.14 -47.38 -50.72 202.90 -59.18 -52.91 171.29 5 -27.00 21.74 625.78 -47.38 21.74 360.71 -59.18 19.56 345.25 6 -27.00 51.07 378.60 -47.38 51.07 113.31 -59.18 48.89 114.18 7 -96.82 -27.00 -164.90 -96.82 -47.38 -429.96 -94.64 -59.18 -412.89 8 -96.82 -13.80 -36.15 -96.82 -47.38 -168.73 -94.64 -45.98 -75.20 9 -96.82 0.00 0.00 -96.82 -47.38 0.00 -94.64 -32.18 169.92

Table I-I-29 Gust Factor, Frame A (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 3.13 355.78 NaN 3.06 13.38 NaN 3.07 8.61 6.22 2 3.13 5.16 13.73 3.06 3.37 6.07 3.07 3.45 4.47 3 3.13 3.50 5.77 3.06 2.90 3.45 3.07 2.87 3.06 4 3.50 -0.25 5.17 2.90 4.10 4.01 2.87 4.02 3.69 5 3.50 5.39 3.97 2.90 5.39 3.40 2.87 4.82 3.38 6 3.50 3.59 4.41 2.90 3.59 5.17 2.87 3.49 4.26 7 2.97 3.50 4.08 2.97 2.90 3.74 3.00 2.87 3.11 8 2.97 4.73 4.73 2.97 2.90 4.92 3.00 3.37 4.02 9 2.97 NaN NaN 2.97 2.90 NaN 3.00 4.60 4.82

Table I-I-30 Background Factor, Frame A (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 0.51 2.59 NaN 0.49 12.75 NaN 0.49 2.89 1.68 2 0.51 2.04 6.92 0.49 0.66 1.81 0.49 0.65 0.95 3 0.51 0.71 2.07 0.49 0.42 0.61 0.49 0.42 0.47 4 0.71 0.69 0.72 0.42 0.69 0.65 0.42 0.64 0.64 5 0.71 0.97 0.56 0.42 0.97 0.53 0.42 0.96 0.53 6 0.71 0.54 0.58 0.42 0.54 1.00 0.42 0.55 0.65 7 0.39 0.71 0.84 0.39 0.42 0.52 0.41 0.42 0.45 8 0.39 0.96 0.96 0.39 0.42 0.65 0.41 0.47 0.62 9 0.39 NaN NaN 0.39 0.42 NaN 0.41 0.63 0.68

— — 193

Table I-I-31 Mean response of Across B (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 55.41 -34.52 0.00 52.75 -28.49 0.00 50.07 -34.78 -153.24 2 55.41 -24.96 199.31 52.75 -18.93 160.11 50.07 -25.21 47.72 3 55.41 -15.29 313.79 52.75 -9.26 235.37 50.07 -15.55 163.87 4 15.29 22.18 50.90 9.26 22.18 -27.46 15.55 16.87 -59.13 5 15.29 -1.11 -35.46 9.26 -1.11 -113.88 15.55 -6.42 -105.63 6 15.29 -8.19 -6.89 9.26 -8.19 -85.25 15.55 -13.51 -37.15 7 18.63 15.29 89.51 18.63 9.26 11.09 23.94 15.55 99.08 8 18.63 7.00 18.35 18.63 9.26 -20.85 23.94 7.26 26.27 9 18.63 0.00 0.00 18.63 9.26 0.00 23.94 0.26 6.27

Table I-I-32 Absolute Maximum Total response of Across B (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 145.70 -116.17 0.00 137.75 -93.99 0.00 133.45 -112.78 -482.48 2 145.70 -77.32 640.61 137.75 -55.50 497.33 133.45 -73.46 149.12 3 145.70 -41.23 937.30 137.75 -30.36 657.25 133.45 -41.78 453.35 4 41.23 73.10 442.19 30.36 73.10 -160.56 41.78 60.10 -208.03 5 41.23 -23.47 -384.52 30.36 -23.47 -358.47 41.78 -31.33 -334.55 6 41.23 -35.49 -234.42 30.36 -35.49 -268.67 41.78 -45.61 -126.04 7 54.95 41.23 242.87 54.95 30.36 -229.19 63.87 41.78 296.53 8 54.95 19.50 51.08 54.95 30.36 -156.51 63.87 28.39 92.05 9 54.95 0.00 0.00 54.95 30.36 0.00 63.87 21.25 137.04

Table I-I-33 Absolute Maximum Dynamic response of Across B (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 90.30 -81.65 0.00 85.01 -65.50 0.00 83.38 -78.00 -329.25 2 90.30 -52.36 441.30 85.01 -36.58 337.21 83.38 -48.24 101.40 3 90.30 -25.93 623.51 85.01 -21.10 421.88 83.38 -26.23 289.48 4 25.93 50.92 391.29 21.10 50.92 -133.09 26.23 43.23 -148.91 5 25.93 -22.36 -349.06 21.10 -22.36 -244.59 26.23 -24.91 -228.92 6 25.93 -27.30 -227.53 21.10 -27.30 -183.42 26.23 -32.10 -88.89 7 36.32 25.93 153.35 36.32 21.10 -240.28 39.92 26.23 197.44 8 36.32 12.49 32.73 36.32 21.10 -135.66 39.92 21.13 65.78 9 36.32 0.00 0.00 36.32 21.10 0.00 39.92 -21.02 130.77

— — 194

Table I-I-34 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Frame B (AOA=90º)


1 -190.11 121.80 0.00 -180.95 110.00 0.00 -169.23 135.87 615.10 2 -190.11 82.84 -689.63 -180.95 71.03 -612.79 -169.23 96.91 -165.87 3 -190.11 28.82 -1054.77 -180.95 17.02 -900.99 -169.23 42.89 -622.48 4 -28.82 -92.00 -84.92 -17.02 -92.00 68.67 -42.89 -71.20 191.08 5 -28.82 1.45 302.44 -17.02 1.45 456.22 -42.89 22.25 422.24 6 -28.82 31.71 204.95 -17.02 31.71 358.51 -42.89 52.51 168.38 7 -66.91 -28.82 -161.25 -66.91 -17.02 -7.47 -87.71 -42.89 -353.91 8 -66.91 -12.68 -33.23 -66.91 -17.02 43.61 -87.71 -26.75 -134.53 9 -66.91 0.00 0.00 -66.91 -17.02 0.00 -87.71 -14.07 -9.91

Table I-I-35 Gust Factor, Frame B (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 2.63 3.37 NaN 2.61 3.30 NaN 2.67 3.24 3.15 2 2.63 3.10 3.21 2.61 2.93 3.11 2.67 2.91 3.12 3 2.63 2.70 2.99 2.61 3.28 2.79 2.67 2.69 2.77 4 2.70 3.30 8.69 3.28 3.30 5.85 2.69 3.56 3.52 5 2.70 21.17 10.84 3.28 21.17 3.15 2.69 4.88 3.17 6 2.70 4.33 34.01 3.28 4.33 3.15 2.69 3.38 3.39 7 2.95 2.70 2.71 2.95 3.28 -20.66 2.67 2.69 2.99 8 2.95 2.78 2.78 2.95 3.28 7.51 2.67 3.91 3.50 9 2.95 NaN NaN 2.95 3.28 NaN 2.67 82.34 21.87

Table I-I-36 Background Factor, Frame B (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 0.48 0.67 NaN 0.47 0.60 NaN 0.49 0.57 0.54 2 0.48 0.63 0.64 0.47 0.51 0.55 0.49 0.50 0.61 3 0.48 0.90 0.59 0.47 1.24 0.47 0.49 0.61 0.47 4 0.90 0.55 4.61 1.24 0.55 1.94 0.61 0.61 0.78 5 0.90 15.42 1.15 1.24 15.42 0.54 0.61 1.12 0.54 6 0.90 0.86 1.11 1.24 0.86 0.51 0.61 0.61 0.53 7 0.54 0.90 0.95 0.54 1.24 -32.17 0.46 0.61 0.56 8 0.54 0.98 0.98 0.54 1.24 3.11 0.46 0.79 0.49 9 0.54 NaN NaN 0.54 1.24 NaN 0.46 1.49 13.20

— — 195

Table I-I-37 Mean response of Across C (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 60.48 -33.26 0.00 57.96 -28.96 0.00 55.34 -35.95 -158.40 2 60.48 -24.58 193.47 57.96 -20.28 165.50 55.34 -27.27 52.53 3 60.48 -14.57 307.88 57.96 -10.26 251.93 55.34 -17.26 184.44 4 14.57 23.50 15.68 10.26 23.50 -40.21 17.26 18.39 -69.41 5 14.57 -2.70 -74.48 10.26 -2.70 -130.44 17.26 -7.80 -121.26 6 14.57 -10.75 -31.22 10.26 -10.75 -87.10 17.26 -15.86 -39.62 7 21.11 14.57 85.38 21.11 10.26 29.43 26.21 17.26 115.26 8 21.11 6.62 17.34 21.11 10.26 -10.62 26.21 9.31 29.76 9 21.11 0.00 0.00 21.11 10.26 0.00 26.21 2.69 -5.06

Table I-I-38 Absolute Maximum Total response of Across C (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 152.82 -109.82 0.00 144.07 -91.11 0.00 137.94 -110.00 -481.62 2 152.82 -76.62 622.43 144.07 -57.36 497.55 137.94 -76.11 150.41 3 152.82 -34.83 931.46 144.07 -27.56 683.69 137.94 -39.55 468.13 4 34.83 70.88 317.60 27.56 70.88 -152.83 39.55 56.82 -201.75 5 34.83 -20.54 -375.40 27.56 -20.54 -357.03 39.55 -30.81 -332.33 6 34.83 -35.25 -226.90 27.56 -35.25 -260.24 39.55 -45.85 -119.31 7 53.97 34.83 205.79 53.97 27.56 195.15 62.69 39.55 290.10 8 53.97 16.27 42.63 53.97 27.56 -121.89 62.69 28.09 92.12 9 53.97 0.00 0.00 53.97 27.56 0.00 62.69 21.14 99.64

Table I-I-39 Absolute Maximum Dynamic response of Across C (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 92.34 -76.56 0.00 86.11 -62.15 0.00 82.61 -74.05 -323.22 2 92.34 -52.04 428.96 86.11 -37.08 332.05 82.61 -48.83 97.88 3 92.34 -20.26 623.58 86.11 -17.29 431.76 82.61 -22.29 283.68 4 20.26 47.38 301.92 17.29 47.38 -112.62 22.29 38.42 -132.33 5 20.26 -17.85 -300.92 17.29 -17.85 -226.59 22.29 -23.01 -211.07 6 20.26 -24.50 -195.69 17.29 -24.50 -173.14 22.29 -29.99 -79.69 7 32.87 20.26 120.41 32.87 17.29 -199.66 36.47 22.29 174.84 8 32.87 9.65 25.29 32.87 17.29 -111.27 36.47 18.78 62.36 9 32.87 0.00 0.00 32.87 17.29 0.00 36.47 18.44 104.69

— — 196

Table I-I-40 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Frame C (AOA=90º)


1 -208.15 117.87 0.00 -198.96 112.71 0.00 -187.26 141.72 651.16 2 -208.15 83.84 -677.00 -198.96 78.68 -643.39 -187.26 107.69 -180.68 3 -208.15 28.06 -1047.93 -198.96 22.90 -980.59 -187.26 51.91 -706.60 4 -28.06 -98.84 31.45 -22.90 -98.84 98.56 -51.91 -78.07 216.69 5 -28.06 2.28 454.31 -22.90 2.28 521.64 -51.91 23.06 483.56 6 -28.06 40.25 328.38 -22.90 40.25 395.47 -51.91 61.03 201.48 7 -87.16 -28.06 -145.70 -87.16 -22.90 -78.36 -107.94 -51.91 -428.48 8 -87.16 -11.44 -29.97 -87.16 -22.90 3.64 -107.94 -35.29 -157.89 9 -87.16 0.00 0.00 -87.16 -22.90 0.00 -107.94 -23.85 26.99

Table I-I-41 Gust Factor, Frame C (AOA=90º Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 2.53 3.30 NaN 2.49 3.15 NaN 2.49 3.06 3.04 2 2.53 3.12 3.22 2.49 2.83 3.01 2.49 2.79 2.86 3 2.53 2.39 3.03 2.49 2.69 2.71 2.49 2.29 2.54 4 2.39 3.02 20.26 2.69 3.02 3.80 2.29 3.09 2.91 5 2.39 7.62 5.04 2.69 7.62 2.74 2.29 3.95 2.74 6 2.39 3.28 7.27 2.69 3.28 2.99 2.29 2.89 3.01 7 2.56 2.39 2.41 2.56 2.69 6.63 2.39 2.29 2.52 8 2.56 2.46 2.46 2.56 2.69 11.47 2.39 3.02 3.10 9 2.56 NaN NaN 2.56 2.69 NaN 2.39 7.85 -19.70

Table I-I-42 Background Factor, Frame C (AOA=90º)


1 0.44 0.65 NaN 0.43 0.55 NaN 0.44 0.52 0.50 2 0.44 0.62 0.63 0.43 0.47 0.52 0.44 0.45 0.54 3 0.44 0.72 0.60 0.43 0.76 0.44 0.44 0.43 0.40 4 0.72 0.48 -9.60 0.76 0.48 1.14 0.43 0.49 0.61 5 0.72 7.81 0.66 0.76 7.81 0.43 0.43 1.00 0.44 6 0.72 0.61 0.60 0.76 0.61 0.44 0.43 0.49 0.40 7 0.38 0.72 0.83 0.38 0.76 -2.55 0.34 0.43 0.41 8 0.38 0.84 0.84 0.38 0.76 30.60 0.34 0.53 0.40 9 0.38 NaN NaN 0.38 0.76 NaN 0.34 0.77 -3.88

— — 197

Table I-I-43 Mean response of Across D (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 25.05 -1.45 0.00 25.28 -1.78 0.00 24.58 -3.45 -26.04 2 25.05 -7.18 24.44 25.28 -7.50 26.59 24.58 -9.17 11.30 3 25.05 -8.56 75.77 25.28 -8.89 80.05 24.58 -10.56 75.55 4 8.56 6.16 -25.15 8.89 6.16 -20.83 10.56 5.71 -22.01 5 8.56 -1.94 -61.10 8.89 -1.94 -56.82 10.56 -2.39 -54.64 6 8.56 -7.24 -31.33 8.89 -7.24 -27.01 10.56 -7.69 -21.50 7 14.83 8.56 50.05 14.83 8.89 54.33 15.28 10.56 63.20 8 14.83 3.94 10.33 14.83 8.89 12.48 15.28 5.94 10.57 9 14.83 0.00 0.00 14.83 8.89 0.00 15.28 2.00 -12.67

Table I-I-44 Absolute Maximum Total response of Across D (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix


Table I-I-45 Absolute Maximum Dynamic response of Across D (lb, lb-ft) (AOA=90º) Pin-roller Pin-pin Fix-fix


— — 198

Table I-I-46 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Frame D (AOA=90º)


1 -112.03 -20.02 0.00 -115.97 2.39 0.00 -116.55 13.72 63.22 2 -112.03 2.09 72.20 -115.97 24.50 -73.60 -116.55 35.83 -83.65 3 -112.03 27.35 -18.25 -115.97 49.75 -309.80 -116.55 61.08 -393.29 4 -27.35 -45.37 516.46 -49.75 -45.37 224.77 -61.08 -49.95 175.70 5 -27.35 26.52 627.55 -49.75 26.52 336.01 -61.08 21.94 321.18 6 -27.35 51.45 357.07 -49.75 51.45 65.35 -61.08 46.87 84.92 7 -89.64 -27.35 -166.83 -89.64 -49.75 -458.38 -85.06 -61.08 -404.51 8 -89.64 -13.79 -36.14 -89.64 -49.75 -181.94 -85.06 -47.53 -54.72 9 -89.64 0.00 0.00 -89.64 -49.75 0.00 -85.06 -33.74 200.54

Table I-I-47 Gust Factor, Frame D (AOA=90º) Pin-roller Pin-pin Fix-fix


Table I-I-48 Background Factor, Frame D (AOA=90º) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 0.50 -1.63 NaN 0.48 11.67 NaN 0.48 2.51 2.36 2 0.50 11.64 -2.50 0.48 0.83 2.02 0.48 0.74 0.70 3 0.50 0.72 15.60 0.48 0.37 0.72 0.48 0.38 0.50 4 0.72 0.76 0.55 0.37 0.76 0.56 0.38 0.67 0.63 5 0.72 0.71 0.49 0.37 0.71 0.52 0.38 0.82 0.52 6 0.72 0.47 0.54 0.37 0.47 1.55 0.38 0.51 0.79 7 0.37 0.72 0.82 0.37 0.37 0.40 0.40 0.38 0.41 8 0.37 0.96 0.96 0.37 0.37 0.48 0.40 0.42 0.74 9 0.37 NaN NaN 0.37 0.37 NaN 0.40 0.56 0.51

— — 199

APPENDIX I-PART II: TAMURA WIND MODEL

Table I-II-1 Mean response of Across A (KN, KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 18.27 0.06 0.00 18.27 9.96 0.00 18.28 12.38 101.20 2 18.27 -12.76 156.52 18.27 -2.85 -91.17 18.28 -0.43 -50.46 3 18.27 -27.51 653.23 18.27 -17.60 157.85 18.28 -15.18 138.08 4 27.51 6.74 483.80 17.60 6.74 -11.58 15.18 6.75 -31.55 5 27.51 -0.08 433.92 17.60 -0.08 -61.47 15.18 -0.07 -81.63 6 27.51 -10.43 485.49 17.60 -10.43 -9.91 15.18 -10.42 -30.26 7 18.30 27.51 654.82 18.30 17.60 159.42 18.28 15.18 138.87 8 18.30 12.82 156.75 18.30 2.91 -90.95 18.28 0.49 -51.01 9 18.30 0.00 0.00 18.30 -9.91 0.00 18.28 -12.33 100.43

Table I-II-2 Absolute Maximum Total response of Across A (KN, KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 57.70 -51.41 0.00 57.70 46.30 0.00 47.92 51.94 618.14 2 57.70 -61.01 1378.45 57.70 -17.59 -638.07 47.92 15.31 -150.36 3 57.70 -76.28 3051.68 57.70 -38.67 853.04 47.92 -32.73 440.79 4 76.28 36.41 2593.51 38.67 36.41 -378.65 32.73 22.17 -207.50 5 76.28 -27.98 2292.78 38.67 -27.99 -238.40 32.73 -11.09 -252.63 6 76.28 -43.48 2116.09 38.67 -43.48 -335.89 32.73 -30.32 -192.10 7 57.39 76.29 2165.05 57.39 38.67 881.23 50.97 32.73 437.56 8 57.39 46.93 646.56 57.39 17.51 -679.45 50.97 14.77 -162.40 9 57.39 0.00 0.00 57.39 -46.33 0.00 50.97 -52.05 661.16

Table I-II-3 Absolute Maximum Dynamic response of Across A (KN, KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 39.43 -51.46 0.00 39.43 36.34 0.00 29.63 39.56 516.94 2 39.43 -48.25 1221.92 39.43 14.97 -546.90 29.63 15.74 -99.90 3 39.43 -48.77 2398.45 39.43 -21.07 695.19 29.63 -17.55 302.71 4 48.77 29.68 2109.70 21.07 29.68 -367.07 17.55 15.41 -175.95 5 48.77 -27.90 1858.86 21.07 -27.90 -176.93 17.55 -11.02 -170.99 6 48.77 -33.05 1630.60 21.07 -33.05 336.76 17.55 -19.90 -161.84 7 39.09 48.77 1510.22 39.09 21.07 721.81 32.69 17.55 298.69 8 39.09 34.11 489.81 39.09 14.60 -588.50 32.69 -14.42 -111.39 9 39.09 0.00 0.00 39.09 -36.43 0.00 32.69 -39.72 560.73

— — 200

Table I-II-4 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Across Frame A (KN, KN-m)


1 43.55 0.28 0.00 43.55 38.73 0.00 44.29 47.73 393.91 2 43.55 -49.07 628.07 43.55 -10.62 -333.13 44.29 -1.61 -164.38 3 43.55 -85.86 2303.06 43.55 -47.41 380.68 44.29 -38.41 324.27 4 85.86 13.43 1923.01 47.41 13.43 0.59 38.41 14.17 -65.14 5 85.86 -1.26 1828.55 47.41 -1.26 -93.90 38.41 -0.51 -168.95 6 85.86 -24.98 1958.11 47.41 -24.98 35.63 38.41 -24.23 -48.74 7 48.52 85.86 2383.99 48.52 47.41 461.48 47.78 38.41 367.78 8 48.52 51.22 650.76 48.52 12.77 -310.49 47.78 3.77 -179.03 9 48.52 0.00 0.00 48.52 -38.45 0.00 47.78 -47.46 356.62

Table I-II-5 Gust Factor, Across Frame A Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 3.16 -915.85 NaN 3.16 4.65 NaN 2.62 4.19 6.11 2 3.16 4.78 8.81 3.16 6.17 7.00 2.62 -35.48 2.98 3 3.16 2.77 4.67 3.16 2.20 5.40 2.62 2.16 3.19 4 2.77 5.41 5.36 2.20 5.40 32.71 2.16 3.28 6.58 5 2.77 331.10 5.28 2.20 333.65 3.88 2.16 162.05 3.09 6 2.77 4.17 4.36 2.20 4.17 33.90 2.16 2.91 6.35 7 3.14 2.77 3.31 3.14 2.20 5.53 2.79 2.16 3.15 8 3.14 3.66 4.12 3.14 6.02 7.47 2.79 30.18 3.18 9 3.14 NaN NaN 3.14 4.68 NaN 2.79 4.22 6.58

Table I-II-6 Background Factor, Across Frame A Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 0.91 -185.00 NaN 0.91 0.94 NaN 0.67 0.83 1.31 2 0.91 0.98 1.95 0.91 -1.41 1.64 0.67 -9.75 0.61 3 0.91 0.57 1.04 0.91 0.44 1.83 0.67 0.46 0.93 4 0.57 2.21 1.10 0.44 2.21 -618.03 0.46 1.09 2.70 5 0.57 22.09 1.02 0.44 22.13 1.88 0.46 21.39 1.01 6 0.57 1.32 0.83 0.44 1.32 9.45 0.46 0.82 3.32 7 0.81 0.57 0.63 0.81 0.44 1.56 0.68 0.46 0.81 8 0.81 0.67 0.75 0.81 1.14 1.90 0.68 -3.83 0.62 9 0.81 NaN NaN 0.81 0.95 NaN 0.68 0.84 1.57

— — 201

Table I-II-7 Mean response of Across B (KN, KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 8.21 0.15 0.00 8.21 6.36 0.00 8.29 8.05 72.73 2 8.21 -7.71 88.64 8.21 -1.49 -66.64 8.29 0.20 -36.28 3 8.21 -16.52 392.33 8.21 -10.31 81.78 8.29 -8.62 69.77 4 16.52 3.29 315.01 10.31 3.29 4.45 8.62 3.38 -8.61 5 16.52 -0.10 290.73 10.31 -0.10 -19.84 8.62 -0.02 -33.96 6 16.52 -5.17 317.70 10.31 -5.17 7.13 8.62 -5.08 -8.06 7 8.53 16.52 398.73 8.53 10.31 88.15 8.45 8.62 71.91 8 8.53 7.85 93.15 8.53 1.64 -62.14 8.45 -0.06 -36.02 9 8.53 0.00 0.00 8.53 -6.21 0.00 8.45 -7.91 68.49

Table I-II-8 Absolute Maximum Total response of Across B (KN, KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 42.20 -57.68 0.00 42.20 36.84 0.00 28.91 40.80 597.55 2 42.20 -57.52 1426.10 42.20 -17.24 -599.12 28.91 15.97 -127.95 3 42.20 -61.57 2867.91 42.20 -29.95 815.59 28.91 -25.39 326.38 4 61.58 33.55 2450.39 29.95 33.56 388.16 25.39 16.37 -147.79 5 61.58 -30.88 2094.09 29.95 -30.88 -134.97 25.39 -11.56 -144.06 6 61.58 -35.94 1797.93 29.95 -35.94 387.11 25.39 -20.45 -144.23 7 40.93 61.58 1627.14 40.93 29.95 808.83 28.01 25.39 335.49 8 40.93 34.48 447.27 40.93 16.99 -635.10 28.01 14.48 -133.97 9 40.93 0.00 0.00 40.93 -41.68 0.00 28.01 -45.61 631.58

Table I-II-9 Absolute Maximum Dynamic response of Across B (KN, KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 33.99 -57.83 0.00 33.99 30.48 0.00 20.62 32.74 524.82 2 33.99 -49.81 1337.46 33.99 -15.74 -532.48 20.62 15.78 -91.67 3 33.99 -45.05 2475.58 33.99 -19.64 733.81 20.62 -16.77 326.38 4 45.05 30.26 2135.37 19.64 30.26 383.71 16.77 12.99 -147.79 5 45.05 -30.77 1803.36 19.64 -30.77 123.98 16.77 -11.54 -144.06 6 45.05 -30.77 1480.23 19.64 -30.77 379.98 16.77 -15.37 -144.23 7 32.40 45.06 1228.41 32.40 19.64 720.68 19.56 16.77 335.49 8 32.40 26.63 354.12 32.40 15.35 -572.96 19.56 -15.08 -133.97 9 32.40 0.00 0.00 32.40 -35.47 0.00 19.56 -37.70 631.58

— — 202

Table I-II-10 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Across Frame B (KN, KN-m)


1 29.38 -7.39 0.00 29.38 19.43 0.00 27.36 25.33 194.99 2 29.38 -32.58 509.49 29.38 -5.76 -160.98 27.36 0.13 -113.37 3 29.38 -63.20 1679.86 29.38 -36.38 338.92 27.36 -30.48 239.16 4 63.20 12.49 1402.48 36.38 12.49 61.53 30.48 10.46 -12.91 5 63.20 2.46 1296.58 36.38 2.46 -44.39 30.48 0.43 -93.52 6 63.20 -12.61 1336.16 36.38 -12.61 -4.84 30.48 -14.64 -28.65 7 24.45 63.20 1552.72 24.44 36.38 211.70 26.47 30.48 213.21 8 24.45 31.53 397.37 24.44 4.71 -273.14 26.47 -1.19 -124.25 9 24.45 0.00 0.00 24.44 -26.82 0.00 26.47 -32.72 296.27

Table I-II-11 Gust Factor, Across Frame B Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 5.14 -387.17 NaN 5.14 5.79 NaN 3.49 5.07 8.22 2 5.14 7.46 16.09 5.14 11.53 8.99 3.49 79.99 3.53 3 5.14 3.73 7.31 5.14 2.90 9.97 3.49 2.95 4.68 4 3.73 10.19 7.78 2.90 10.18 87.19 2.95 4.84 17.17 5 3.73 295.62 7.20 2.90 296.77 6.80 2.95 601.28 4.24 6 3.73 6.95 5.66 2.90 6.95 54.32 2.95 4.02 17.90 7 4.80 3.73 4.08 4.80 2.90 9.18 3.32 2.95 4.67 8 4.80 4.39 4.80 4.80 10.36 10.22 3.32 272.28 3.72 9 4.80 NaN NaN 4.80 6.71 NaN 3.32 5.77 9.22

Table I-II-12 Background Factor, Across Frame B Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 1.16 7.83 NaN 1.16 1.57 NaN 0.75 1.29 2.69 2 1.16 1.53 2.63 1.16 2.73 3.31 0.75 120.59 0.81 3 1.16 0.71 1.47 1.16 0.54 2.17 0.75 0.55 1.07 4 0.71 2.42 1.52 0.54 2.42 6.24 0.55 1.24 10.78 5 0.71 -12.53 1.39 0.54 -12.52 -2.79 0.55 -26.70 1.18 6 0.71 2.44 1.11 0.54 2.44 -78.52 0.55 1.05 4.75 7 1.33 0.71 0.79 1.33 0.54 3.40 0.74 0.55 1.24 8 1.33 0.84 0.89 1.33 3.26 2.10 0.74 12.70 0.79 9 1.33 NaN NaN 1.33 1.32 NaN 0.74 1.15 1.90

— — 203

Table I-II-13 Mean response of Across C (KN, KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 4.45 0.08 0.00 4.45 4.33 0.00 4.48 5.48 48.36 2 4.45 -5.27 62.78 4.45 -1.02 -43.47 4.48 0.12 -23.67 3 4.45 -10.56 260.40 4.45 -6.31 47.91 4.48 -5.17 39.14 4 10.56 1.68 219.28 6.31 1.68 6.78 5.17 1.71 -2.36 5 10.56 -0.02 206.75 6.31 -0.02 -5.75 5.17 0.01 -15.27 6 10.56 -2.66 220.22 6.31 -2.66 7.72 5.17 -2.63 -2.18 7 4.50 10.56 262.57 4.50 6.31 50.06 4.47 5.17 39.78 8 4.50 5.26 64.37 4.50 1.01 -41.89 4.47 -0.14 -23.60 9 4.50 0.00 0.00 4.50 -4.25 0.00 4.47 -5.39 46.84

Table I-II-14 Absolute Maximum Total response of Across C (KN, KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 29.75 48.88 0.00 29.75 37.01 0.00 14.85 40.67 599.56 2 29.75 -48.58 1144.40 29.75 -14.08 -606.96 14.85 15.53 -103.40 3 29.75 -54.08 2419.78 29.75 -21.18 689.11 14.85 -17.38 252.97 4 54.08 26.12 2094.46 21.18 26.12 339.28 17.38 10.01 107.97 5 54.08 -26.23 1834.22 21.18 -26.23 -66.37 17.38 -8.24 -70.11 6 54.08 -29.49 1618.07 21.18 -29.50 369.10 17.38 -12.35 117.44 7 31.36 54.08 1463.31 31.36 21.18 743.70 15.37 17.38 268.18 8 31.36 31.75 402.13 31.36 15.46 -551.29 15.37 -14.16 -109.53 9 31.36 0.00 0.00 31.36 -36.16 0.00 15.37 -40.16 550.10

Table I-II-15 Absolute Maximum Dynamic response of Across C (KN, KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 25.30 48.80 0.00 25.30 32.68 0.00 10.38 35.20 551.20 2 25.30 -43.31 1081.62 25.30 13.22 -563.49 10.38 15.41 -79.73 3 25.30 -43.51 2159.39 25.30 -14.86 641.20 10.38 -12.21 213.82 4 43.52 -25.32 1875.18 14.86 -25.32 332.50 12.21 8.30 110.32 5 43.52 -26.21 1627.47 14.86 -26.21 66.17 12.21 -8.25 -54.84 6 43.52 -26.83 1397.84 14.86 -26.83 361.38 12.21 -9.72 119.62 7 26.86 43.52 1200.74 26.86 14.86 693.64 10.90 12.21 228.40 8 26.86 26.49 337.77 26.86 14.45 -509.41 10.90 -14.03 -85.92 9 26.86 0.00 0.00 26.86 -31.91 0.00 10.90 -34.77 503.26

— — 204

Table I-II-16 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Across Frame C (KN, KN-m)


1 16.66 -0.39 0.00 16.66 23.75 0.00 17.04 30.36 284.72 2 16.66 -27.53 351.09 16.66 -3.38 -252.55 17.04 3.22 -133.00 3 16.66 -59.69 1415.62 16.66 -35.55 208.35 17.04 -28.94 162.72 4 59.69 5.26 1272.46 35.55 5.26 65.17 28.94 5.64 14.82 5 59.69 -0.46 1235.64 35.55 -0.45 28.33 28.94 -0.08 -26.73 6 59.70 -10.48 1290.10 35.55 -10.48 82.77 28.94 -10.10 23.00 7 17.39 59.70 1453.70 17.39 35.55 246.35 17.01 28.94 181.86 8 17.39 29.23 363.74 17.39 5.08 -239.94 17.01 -1.53 -139.25 9 17.39 0.00 0.00 17.39 -24.15 0.00 17.01 -30.75 265.86

Table I-II-17 Gust Factor, Across Frame C Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 6.69 590.36 NaN 6.69 8.54 NaN 3.32 7.43 12.40 2 6.69 9.22 18.23 6.69 13.80 13.96 3.32 127.61 4.37 3 6.69 5.12 9.29 6.69 3.36 14.38 3.32 3.36 6.46 4 5.12 15.57 9.55 3.36 15.57 50.03 3.36 5.86 -45.77 5 5.12 1551.35 8.87 3.36 1577.13 11.54 3.36 -601.23 4.59 6 5.12 11.07 7.35 3.36 11.07 47.80 3.36 4.69 -53.93 7 6.96 5.12 5.57 6.96 3.36 14.86 3.44 3.36 6.74 8 6.96 6.04 6.25 6.96 15.35 13.16 3.44 104.86 4.64 9 6.96 NaN NaN 6.96 8.51 NaN 3.44 7.45 -6.69

Table I-II-18 Background Factor, Across Frame C Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 1.52 -124.47 NaN 1.52 1.38 NaN 0.61 1.16 1.94 2 1.52 1.57 3.08 1.52 -3.91 2.23 0.61 4.78 0.60 3 1.52 0.73 1.53 1.52 0.42 3.08 0.61 0.42 1.31 4 0.73 -4.81 1.47 0.42 -4.81 5.10 0.42 1.47 7.44 5 0.73 57.52 1.32 0.42 57.72 2.34 0.42 107.20 2.05 6 0.73 2.56 1.08 0.42 2.56 4.37 0.42 0.96 5.20 7 1.54 0.73 0.83 1.54 0.42 2.82 0.64 0.42 1.26 8 1.54 0.91 0.93 1.54 2.84 2.12 0.64 9.20 0.62 9 1.54 NaN NaN 1.54 1.32 NaN 0.64 1.13 1.89

— — 205

Table I-II-19 Mean response of Along A (KN, KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 23.31 -18.10 0.00 23.31 -9.29 0.00 16.42 -9.83 -194.48 2 23.31 -15.37 423.13 23.31 -6.56 202.99 16.42 -7.10 21.82 3 23.31 -9.96 740.59 23.31 -1.16 300.31 16.42 -1.69 132.45 4 9.96 13.01 505.39 1.16 13.01 65.10 1.69 6.12 -16.61 5 9.96 8.46 366.82 1.16 8.46 -73.47 1.69 1.57 -69.03 6 9.96 3.98 285.83 1.16 3.98 -154.47 1.69 -2.91 -63.88 7 -1.91 9.96 246.72 -1.91 1.16 -193.59 4.98 1.69 -16.86 8 -1.91 4.95 59.90 -1.91 -3.86 -160.26 4.98 -3.33 3.17 9 -1.91 0.00 0.00 -1.91 -8.81 0.00 4.98 -8.27 150.12

Table I-II-20 Absolute Maximum Total response of Along A (KN,KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 68.17 -67.46 0.00 68.18 -42.78 0.00 48.55 -46.59 -781.91 2 68.17 -48.16 1458.53 68.18 -21.73 809.31 48.55 -25.04 133.02 3 68.17 -28.63 2291.07 68.18 11.17 961.66 48.55 -9.12 379.46 4 28.63 42.54 1651.18 -11.17 42.54 327.72 9.12 20.82 -121.80 5 28.63 34.92 1195.37 -11.17 34.92 -289.54 9.12 10.01 -260.17 6 28.63 27.65 836.85 -11.17 27.65 -557.60 9.12 -16.01 -225.30 7 -25.37 28.63 695.07 -25.37 -11.17 -865.33 24.72 9.12 -223.04 8 -25.37 14.25 164.81 -25.37 -17.43 -554.28 24.72 -14.72 61.17 9 -25.37 0.00 0.00 -25.37 -26.63 0.00 24.72 -24.34 517.47

Table I-II-21 Absolute Maximum Dynamic response of Along A (KN,KN-m) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 44.87 -49.36 0.00 44.87 -33.49 0.00 32.14 -36.77 -587.43 2 44.87 -32.79 1035.39 44.87 -15.16 606.32 32.14 -17.94 111.20 3 44.87 -18.67 1550.48 44.87 12.33 661.35 32.14 9.46 247.01 4 18.67 29.53 1145.79 -12.33 29.53 262.61 -9.46 14.70 -105.19 5 18.67 26.45 828.54 -12.33 26.45 -216.07 -9.46 8.43 -191.13 6 18.67 23.67 551.01 -12.33 23.67 -403.14 -9.46 -13.10 -161.42 7 -23.46 18.67 448.35 -23.46 -12.33 -671.73 19.74 -9.46 -206.19 8 -23.46 9.30 104.92 -23.46 -13.57 -394.02 19.74 -11.40 58.00 9 -23.46 0.00 0.00 -23.46 -17.82 0.00 19.74 -16.06 367.35

— — 206

Table I-II-22 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Along Frame A (KN, KN-m)


1 83.42 -83.26 0.00 83.42 -51.18 0.00 52.60 -55.42 -947.43 2 83.42 -57.73 1773.50 83.42 -25.65 971.44 52.60 -29.89 130.08 3 83.42 -26.74 2829.73 83.42 5.34 1225.62 52.60 1.10 490.33 4 26.74 56.25 1932.08 -5.34 56.25 327.94 -1.10 25.43 -22.03 5 26.75 41.05 1306.15 -5.34 41.05 -298.02 -1.10 10.23 -262.67 6 26.75 22.84 884.02 -5.34 22.84 -720.18 -1.10 -7.99 -299.50 7 -12.04 26.75 656.58 -12.04 -5.34 -947.64 18.79 -1.10 -141.64 8 -12.04 13.03 150.69 -12.04 -19.05 -651.42 18.79 -14.81 48.53 9 -12.04 0.00 0.00 -12.04 -32.08 0.00 18.79 -27.84 593.89

Table I-II-23 Gust Factor, Along Frame A Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 2.93 3.73 NaN 2.93 4.60 NaN 2.96 4.74 4.02 2 2.93 3.13 3.45 2.93 3.31 3.99 2.96 3.53 6.10 3 2.93 2.87 3.09 2.93 -9.67 3.20 2.96 5.40 2.87 4 2.87 3.27 3.27 -9.67 3.27 5.03 5.40 3.40 7.33 5 2.87 4.13 3.26 -9.67 4.13 3.94 5.40 6.36 3.77 6 2.87 6.94 2.93 -9.67 6.94 3.61 5.40 5.50 3.53 7 13.26 2.87 2.82 13.25 -9.67 4.47 4.97 -4.60 13.23 8 13.26 2.88 2.75 13.25 4.52 3.46 4.97 4.43 19.28 9 13.26 NaN NaN 13.25 3.02 NaN 4.97 2.94 3.45

Table I-II-24 Background Factor, Along Frame A Pin-roller Pin-pin Fix-fix


— — 207

Table I-II-25 Mean response of Along B (KN, KN-m) Pin-roller Pin-pin Fix-fix


Table I-II-26 Absolute Maximum Total response of Along B (KN, KN-m) Pin-roller Pin-pin Fix-fix


Table I-II-27 Absolute Maximum Dynamic response of Along B (KN, KN-m) Pin-roller Pin-pin Fix-fix


— — 208

Table I-II-28 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Along Frame B (KN, KN-m)


1 69.90 -74.70 0.00 69.90 -47.39 0.00 43.01 -51.50 -843.60 2 69.90 -48.76 1544.69 69.90 -21.45 861.85 43.01 -25.56 121.05 3 69.90 -20.28 2408.88 69.90 7.04 1043.20 43.01 2.92 405.20 4 20.28 47.42 1650.39 -7.04 47.42 284.68 -2.92 20.53 -17.19 5 20.28 35.89 1117.28 -7.04 35.89 -248.45 -2.92 9.00 -214.19 6 20.28 21.74 744.71 -7.04 21.74 -621.04 -2.92 -5.15 -250.64 7 -14.31 20.28 513.81 -14.32 -7.04 -851.97 12.57 -2.92 -145.44 8 -14.31 10.25 130.55 -14.32 -17.06 -552.34 12.57 -12.95 51.40 9 -14.31 0.00 0.00 -14.32 -27.32 0.00 12.57 -23.20 500.96

Table I-II-29 Gust Factor, Along Frame B Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 2.54 2.98 NaN 2.54 3.33 NaN 2.44 3.38 3.12 2 2.54 2.71 2.83 2.54 2.78 3.10 2.44 2.89 3.82 3 2.54 2.52 2.68 2.54 -7.25 2.72 2.44 -43.44 2.53 4 2.52 2.77 2.82 13.67 2.78 3.56 -43.44 2.85 5.46 5 2.52 3.13 2.85 13.67 3.33 2.89 -43.44 4.66 2.77 6 2.52 4.20 2.63 13.67 2.78 2.94 -43.43 5.93 2.86 7 5.10 2.52 2.58 5.10 13.67 3.30 4.16 -43.43 5.22 8 5.10 2.61 2.67 5.10 3.27 2.86 4.16 3.26 6.48 9 5.10 NaN NaN 5.10 2.69 NaN 4.16 2.64 2.87

Table I-II-30 Background Factor, Along Frame B Pin-roller Pin-pin Fix-fix


— — 209

Table I-II-31 Mean response of Along C (KN, KN-M) Pin-roller Pin-pin Fix-fix


Table I-II-32 Absolute Maximum Total response of Along C (KN, KN-M) Pin-roller Pin-pin Fix-fix


Table I-II-33 Absolute Maximum Dynamic response of Along C (KN, KN-M) Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 42.73 -49.61 0.00 42.73 -33.71 0.00 26.89 -37.16 -578.12 2 42.73 -31.35 993.57 42.73 -14.75 590.84 26.89 -17.69 105.57 3 42.73 -16.63 1506.29 42.73 11.97 652.15 26.89 8.78 238.67 4 16.63 29.94 1050.90 -11.97 29.94 227.73 -8.78 13.11 -72.74 5 16.63 23.01 753.70 -11.97 23.01 -178.47 -8.78 6.98 -150.10 6 16.63 23.41 574.13 -11.97 23.41 -387.40 -8.78 8.43 -153.74 7 -23.55 16.63 459.71 -23.55 -11.97 -638.78 10.68 -8.78 -188.80 8 -23.55 9.69 126.68 -23.55 -12.73 -362.64 10.68 -10.10 57.46 9 -23.55 0.00 0.00 -23.55 -17.09 0.00 10.68 -15.26 332.79

— — 210

Table I-II-34 Dynamic Responses under Absolute Maximum Fluctuating Pressure Distribution Based on POD Pressure Sign, Along Frame C (KN, KN-m)


1 69.81 -72.84 0.00 69.81 -45.85 0.00 43.85 -49.84 -814.97 2 69.81 -47.91 1508.92 69.81 -20.93 834.34 43.85 -24.91 118.88 3 69.81 -20.41 2366.21 69.81 6.57 1017.04 43.85 2.59 401.10 4 20.41 46.20 1610.33 -6.57 46.20 261.15 -2.59 20.24 -30.24 5 20.41 33.79 1098.31 -6.57 33.79 -250.89 -2.59 7.83 -217.73 6 20.41 19.76 745.88 -6.57 19.76 -603.35 -2.59 -6.20 -245.62 7 -12.55 20.41 537.59 -12.55 -6.57 -811.67 13.42 -2.59 -129.39 8 -12.55 11.05 142.06 -12.55 -15.94 -532.56 13.42 -11.96 50.22 9 -12.55 0.00 0.00 -12.55 -26.99 0.00 13.42 -23.01 483.28

Table I-II-35 Gust Factor, Along Frame C Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 2.56 2.80 NaN 2.56 3.01 NaN 2.52 3.06 2.93 2 2.56 2.67 2.72 2.56 2.84 2.91 2.52 2.90 3.44 3 2.56 2.75 2.63 2.56 10.59 2.69 2.52 -152.67 2.56 4 2.75 2.76 2.66 10.59 2.76 3.40 -152.64 2.81 5.44 5 2.75 2.94 2.69 10.59 2.94 2.97 -152.61 4.39 2.89 6 2.75 4.03 2.78 10.59 4.03 2.81 -152.57 4.94 2.89 7 5.00 2.75 2.96 5.00 10.59 3.12 3.74 -152.52 5.08 8 5.00 3.07 3.22 5.00 4.03 2.72 3.74 3.13 5.81 9 5.00 NaN NaN 5.00 -5.07 NaN 3.74 2.62 2.75

Table I-II-36 Background Factor, Along Frame C Pin-roller Pin-pin Fix-fix

points N Q M N Q M N Q M 1 0.61 0.68 NaN 0.61 0.74 NaN 0.61 0.75 0.71 2 0.61 0.65 0.66 0.61 0.70 0.71 0.61 0.71 0.89 3 0.61 0.81 0.64 0.61 1.82 0.64 0.61 3.39 0.60 4 0.81 0.65 0.65 1.82 0.65 0.87 3.39 0.65 2.41 5 0.81 0.68 0.69 1.82 0.68 0.71 3.39 0.89 0.69 6 0.81 1.18 0.77 1.82 1.18 0.64 3.39 -1.36 0.63 7 1.88 0.81 0.86 1.88 1.82 0.79 0.80 3.39 1.46 8 1.88 0.88 0.89 1.88 0.80 0.68 0.80 0.84 1.14 9 1.88 NaN NaN 1.88 0.63 NaN 0.80 0.66 0.69

— 211 —

APPENDIX II

Figure II-1 Pressure distribution on Frame A for Universal ESWL reproducing maximum axial forces N (mode=5, AOA=0º)

Figure II-2 Pressure distribution on Frame B for Universal ESWL reproducing maximum axial forces N (mode=5, AOA=0º)

Figure II-3 Pressure distribution on Frame C for Universal ESWL reproducing maximum axial forces N (mode=5, AOA=0º)

Figure II-4 Pressure distribution on Frame D for Universal ESWL reproducing maximum axial forces N (mode=5, AOA=0º)

— 212 —

Figure II-5 Pressure distribution on Frame A for Universal ESWL reproducing maximum shear forces Q (mode=5, AOA=0º)

Figure II-6 Pressure distribution on Frame B for Universal ESWL reproducing maximum shear forces Q (mode=5, AOA=0º)

Figure II-7 Pressure distribution on Frame C for Universal ESWL reproducing maximum shear forces Q (mode=5, AOA=0º)

Figure II-8 Pressure distribution on Frame D for Universal ESWL reproducing maximum shear forces Q (mode=5, AOA=0º)

— 213 —

Figure II-9 Pressure distribution on Frame A for Universal ESWL reproducing maximum bending moment M (mode=5, AOA=0º)

Figure II-10 Pressure distribution on Frame B for Universal ESWL reproducing maximum bending moment M (mode=5, AOA=0º)

Figure II-11 Pressure distribution on Frame C for Universal ESWL reproducing maximum bending moment M (mode=5, AOA=0º)

Figure II-12 Pressure distribution on Frame D for Universal ESWL reproducing maximum bending moment M (mode=5, AOA=0º)

— 214 —

Figure II-13 Pressure distribution on Frame A for Universal ESWL reproducing all the maximum responses NQM (mode=5, AOA=0º)

Figure II-14 Pressure distribution on Frame B for Universal ESWL reproducing all the maximum responses NQM (mode=5, AOA=0º)

Figure II-15 Pressure distribution on Frame C for Universal ESWL reproducing all the maximum responses NQM (mode=5, AOA=0º)

Figure II-16 Pressure distribution on Frame D for Universal ESWL reproducing all the maximum responses NQM (mode=5, AOA=0º)

— 215 —





— 216 —




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— 219 —





— 220 —


Figure II-37 Pressure distribution on Frame A for Universal ESWL

reproducing maximum shear forces Q (mode=15, AOA=0º)



— 221 —





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— 232 —


Figure II-85 Pressure distribution on Frame A for Universal ESWL



— 233 —


Figure II-88 Pressure distribution on Frame D for Universal ESWL




— 234 —





— 235 —



Figure II-97 Pressure distribution on Frame A for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=0

º)

Figure II-98 Pressure distribution on Frame B for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=0

º)

— 236 —

Figure II-99 Pressure distribution on Frame C for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=0

º)

Figure II-100 Pressure distribution on Frame D for Universal ESWL reproducing maximum axial forces N based on LRC method (AOA=0

º)

Figure II-101 Pressure distribution on Frame A for Universal ESWL reproducing maximum shear forces Q based on LRC method

(AOA=0º)

Figure II-102 Pressure distribution on Frame B for Universal ESWL reproducing maximum shear forces Q based on LRC method

(AOA=0º)

— 237 —

Figure II-103 Pressure distribution on Frame C for Universal ESWL reproducing maximum shear forces Q based on LRC method

(AOA=0º)

Figure II-104 Pressure distribution on Frame D for Universal ESWL reproducing maximum shear forces Q based on LRC method

(AOA=0º)

Figure II-105 Pressure distribution on Frame A for Universal ESWL reproducing maximum bending moment M based on LRC method

(AOA=0º)

Figure II-106 Pressure distribution on Frame B for Universal ESWL reproducing maximum bending moment M based on LRC method

(AOA=0º)

— 238 —

Figure II-107 Pressure distribution on Frame C for Universal ESWL reproducing maximum bending moment M based on LRC method

(AOA=0º)

Figure II-108 Pressure distribution on Frame D for Universal ESWL reproducing maximum bending moment M based on LRC method

(AOA=0º)

Figure II-109 Pressure distribution on Frame A for Universal ESWL reproducing all the maximum responses NQM based on LRC method

(AOA=0º)

Figure II-110 Pressure distribution on Frame B for Universal ESWL reproducing all the maximum responses NQM based on LRC method

(AOA=0º)

— 239 —

Figure II-111 Pressure distribution on Frame C for Universal ESWL reproducing all the maximum responses NQM based on LRC method

(AOA=0º)

Figure II-112 Pressure distribution on Frame D for Universal ESWL reproducing all the maximum responses NQM based on LRC method

(AOA=0º)

Figure II-113 Pressure distribution on Frame A for Universal ESWL reproducing maximum axial forces N based on LRC method

(AOA=90º)

Figure II-114 Pressure distribution on Frame B for Universal ESWL reproducing maximum axial forces N based on LRC method

(AOA=90º)

— 240 —

Figure II-115 Pressure distribution on Frame C for Universal ESWL reproducing maximum axial forces N based on LRC method

(AOA=90º)

Figure II-116 Pressure distribution on Frame D for Universal ESWL reproducing maximum axial forces N based on LRC method

(AOA=90º)

Figure II-117 Pressure distribution on Frame A for Universal ESWL reproducing maximum shear forces Q based on LRC method

(AOA=90º)

Figure II-118 Pressure distribution on Frame B for Universal ESWL reproducing maximum shear forces Q based on LRC method

(AOA=90º)

— 241 —

Figure II-119 Pressure distribution on Frame C for Universal ESWL reproducing maximum shear forces Q based on LRC method

(AOA=90º)

Figure II-120 Pressure distribution on Frame D for Universal ESWL reproducing maximum shear forces Q based on LRC method

(AOA=90º)

Figure II-121 Pressure distribution on Frame A for Universal ESWL reproducing maximum bending moment M based on LRC method

(AOA=90º)

Figure II-122 Pressure distribution on Frame B for Universal ESWL reproducing maximum bending moment M based on LRC method

(AOA=90º)

— 242 —

Figure II-123 Pressure distribution on Frame C for Universal ESWL reproducing maximum bending moment M based on LRC method

(AOA=90º)

Figure II-124 Pressure distribution on Frame D for Universal ESWL reproducing maximum bending moment M based on LRC method

(AOA=90º)

Figure II-125 Pressure distribution on Frame A for Universal ESWL reproducing all the maximum responses NQM based on LRC method

(AOA=90º)

Figure II-126 Pressure distribution on Frame B for Universal ESWL reproducing all the maximum responses NQM based on LRC method

(AOA=90º)

— 243 —

Figure II-127 Pressure distribution on Frame C for Universal ESWL reproducing all the maximum responses NQM based on LRC method

(AOA=90º)

Figure II-128 Pressure distribution on Frame D for Universal ESWL reproducing all the maximum responses NQM based on LRC method

(AOA=90º)

wind loading effects and equivalent static

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