the effects of diaphragm flexibility on the ......the effects of diaphragm flexibility on the...

THE EFFECTS OF DIAPHRAGM FLEXIBILITY ON THE

SEISMIC PERFORMANCE OF LIGHT FRAME WOOD

STRUCTURES

Rakesh Pathak

Dissertation submitted to the faculty of the Virginia

Polytechnic Institute and State University in partial fulfillment of the requirements for the

degree of

Doctor of Philosophy

in

Civil Engineering

Dr. Finley A. Charney (Chair)

Dr. Daniel P. Hindman

Dr. Elisa D. Sotelino

Dr. Raymond H. Plaut

Dr. W. Samuel Easterling

May 1, 2008

Blacksburg, Virginia

Keywords: Diaphragm Flexibility, Object Oriented C++, Static, Nonlinear Dynamic

Analysis, Light Frame Wood Structure, Finite Element

@Copyright 2008, Rakesh Pathak

THE EFFECTS OF DIAPHRAGM FLEXIBILITY ON THE

SEISMIC PERFORMANCE OF LIGHT FRAME WOOD

STRUCTURES

Rakesh Pathak

Department of Civil Engineering

Virginia Polytechnic Institute and State University

Blacksburg, VA 24060 USA

(ABSTRACT)

This dissertation presents work targeted to study the effects of diaphragm flexibility on

the seismic performance of light frame wood structures (LFWS). The finite element

approach is considered for modeling LFWS as it is more detailed and provides a way to

explicitly incorporate individual structural elements and corresponding material

properties. It is also suitable for capturing the detailed response of LFWS components

and the structure as a whole. The finite element modeling methodology developed herein

is in general based on the work done by the other finite element researchers in this area.

However, no submodeling or substructuring of subassemblages is performed and instead

a detailed model considering almost every connection in the shear walls and diaphragms

is developed. The studs, plates, sills, blockings and joists are modeled using linear

isotropic three dimensional frame elements. A linear orthotropic shell element

incorporating both membrane and plate behavior is used for the sheathings. The

connections are modeled using oriented springs with modified Stewart hysteresis spring

stiffnesses. The oriented spring pair has been found to give a more accurate

representation of the sheathing to framing connections in shear walls and diaphragms

when compared to non-oriented or single springs typically used by most researchers in

the past. Fifty six finite element models of LFWS are created using the developed

methodology and eighty eight nonlinear response history analyses are performed using

the Imperial Valley and Northridge ground motions. These eighty eight analyses

encompass the parametric study on the house models with varying aspect ratios,

diaphragm flexibility and lateral force resisting system. Torsionally irregular house

models showed the largest range of variation in peak base shear of individual shear walls,

iii

when corresponding flexible and rigid diaphragm models are compared. It is also found

that presence of an interior shear wall helps in reducing peak base shears in the boundary

walls of torsionally irregular models. The interior walls presence was also found to

reduce the flexibility of diaphragm. A few analyses also showed that the nail connections

are the major source of in-plane flexibility compared to sheathings within a diaphragm,

irrespective of the aspect ratio of the diaphragm.

A major part of the dissertation focuses on the development of a new high performance

nonlinear dynamic finite element analysis program which is also used to analyze all the

LFWS finite element models presented in this study. The program is named

WoodFrameSolver and is written on a mixed language platform Microsoft Visual Studio

.NET using object-oriented C++, C and FORTRAN. This tool set is capable of

performing basic structural analysis chores like static and dynamic analysis of 3D

structures. It has a wide collection of linear, nonlinear and hysteretic elements commonly

used in LFWS analysis. The advanced analysis features include static, nonlinear dynamic

and incremental dynamic analysis. A unique aspect of the program lies in its capability of

capturing elastic displacement participation (sensitivity) of spring, link, frame and solid

elements in static analysis. The program’s performance and accuracy are similar to that of

SAP 2000 which is chosen as a benchmark for validating the results. The use of fast and

efficient serial and parallel solver libraries obtained from INTEL has reduced the solution

time for repetitive dynamic analysis. The utilization of the standard C++ template library

for iterations, storage and access has further optimized the analysis process, especially

when problems with a large number of degrees of freedom are encountered.

iv

I dedicate this dissertation to my loving parents

Chandra D. Pathak and Bhagwati Pathak

and my brother Naveen Pathak

Rakesh Pathak

05-01-2008

v

ACKNOWLEDGEMENTS

I would like to thank my advisor, Dr. Finley A. Charney, for his academic guidance,

patience, and financial support which guided me through this project. I am also thankful

to him for his personal advice and support on various occasions in the last five years. He

is an excellent advisor and I have really enjoyed being a part of his research group. His

hard work and intellect has been a source of motivation for me to work and learn and

shall remain all my life.

I also thank Dr. Daniel P. Hindman, Dr. Elisa D. Sotelino, Dr. W. Samuel Easterling, and

Dr. Raymond H. Plaut for their precious time to serve on my committee and reviewing

my thesis. Discussions with Dr. Hindman and the research material provided by him on

light frame wood structures have proved really fruitful in the various stages of model

development.

I would like to acknowledge the initial WoodFrameSolver development team: Dr. Finley

A. Charney, Paul W. Spears, Dr. Samuel K. Kassegne and Hariharan Iyer. Their initial

efforts were the foundation for the further development of WoodFrameSolver program.

I thank Simpson Strong Tie for their monetary grant which supported my studies and

stipend during the course of my Ph.D. I would also like to thank Mr. Steven E. Pryor and

Mr. Badri Hiriyur from Simpson Strong Tie for coming to Blacksburg for discussions and

providing useful input. I also thank Johnn P. Judd for providing me the results from his

analysis which helped me verify a few of my shear wall models.

Special thanks are due to all my friends who have made my stay in Blacksburg a

wonderful and memorable time.

Finally, I would like to thank my parents, Mr. Chandra D. Pathak and Mrs. Bhagwati

Pathak, my brother Naveen and my fiancée Priyanka for their enormous love and support.

vi

TABLE OF CONTENTS

ABSTRACT ii

DEDICATION iv

ACKNOWLEDGEMENTS v

TABLE OF CONTENTS vi

LIST OF FIGURES ix

LIST OF TABLES xv

1. CHAPTER 1: INTRODUCTION 1

MOTIVATION 1

OBJECTIVE AND SCOPE 3

ORGANIZATION 3

2. CHAPTER 2: MODELING AND ANALYSIS OF LIGHT FRAME

WOOD SHEAR WALLS, DIAPHRAGMS AND HOUSES: OVERVIEW

7

INTRODUCTION 7

NON-FINITE ELEMENT MODELS 8

FINITE ELEMENT MODELS 13

DISCUSSION 19

CONCLUSION 21

REFERENCES 23

3. CHAPTER 3: THE EFFECT OF DIAPHRAGM FLEXIBILITY ON

THE SEISMIC PERFORMANCE OF LIGHT FRAME WOOD

STRUCTURES, PART I: MODEL FORMULATION

27

INTRODUCTION 27

LFWS COMPONENT AND BEHAVIOR DESCRIPTION 29

FINITE ELEMENT MODELING METHODOLOGY 32

FINITE ELEMENTS 33

FRAME 33

SHELL 33

NLLINK 33

FE SHEAR WALL MODEL 34

FE DIAPHRAGM MODEL 35

FE HOUSE MODEL 36

vii

HOUSE MODELS BASIC DESCRIPTION 36

RECTANGULAR TYPE 1 37







MODEL GENERATION 40

SUMMARY 41

REFERENCES 43

4. CHAPTER 4: WOODFRAMESOLVER: A HIGH PERFORMANCE

NONLINEAR FINITE ELEMENT ANALYSIS PROGRAM

60

INTRODUCTION 60

WHY OOP? 62

PROGRAM ARCHITECTURE AND PERFORMANCE 63

PROGRAM FEATURES 67

ELEMENT LIBRARY 68

FRAME 68

SHELL 68

8 NODE BRICK 68

SPRING 69

NLLINK ELEMENT 69

GAP AND HOOK 70

TRILINEAR HYSTERESIS 70

MODIFIED STEWART HYSTERESIS 70

ANALYSIS CASES 72

STATIC ANALYSIS 72

EIGEN ANALYSIS 72

DYNAMIC ANALYSIS 72

INCREMENTAL DYNAMIC ANALYSIS 73

VIRTUAL WORK ANALYSIS FOR DISPAR 73

EXAMPLES 74

VERIFICATION WITH EXPERIMENTAL AND ABAQUS

ANALYTICAL MODEL

74

VERIFICATION WITH SAPWOOD 75

VERIFICATION WITH SAP2000 FRAME MODEL 76

VERIFICATION WITH SAP2000 3D HOUSE MODEL 76

SUMMARY 77

REFERENCES 79

5. CHAPTER 5: THE EFFECT OF DIAPHRAGM FLEXIBILITY ON

THE SEISMIC PERFORMANCE OF LIGHT FRAME WOOD

STRUCTURES II: PARAMETRIC STUDY

107

INTRODUCTION 107

viii

NONLINEAR RESPONSE HISTORY ANALYSIS 108

MASS MATRIX 110

DAMPING MATRIX 110

STIFFNESS MATRIX 111

LOADING 111

NONLINEAR RESPONSE HISTORY ANALYSIS OF SHEAR WALLS AND

VERIFICATION WITH DOLAN (1989) EXPERIMENTS

112

WALL DESCRIPTION 112

FINITE ELEMENT MODEL DESCRIPTION 112

RESULTS COMPARISON 113

LFWS HOUSE FE MODELS AND ANALYSIS DESCRIPTION 113

RESULTS OF THE ANALYSIS 117

FLEXIBLE AND RIGID DIAPHRAGM MODELS RESPONSE 118

INTERIOR SHEAR WALL PEAK IN-PLANE LOAD SHARING 120

TORSIONAL IRREGULARITY 121

STUDY USING THE CODE SPECIFIED MEASURE OF RIGIDITY 123

INVESTIGATION (1) 124




SUMMARY AND CONCLUSIONS 126

REFERENCES 129

6. CHAPTER 6: FUTURE WORK 166

FUTURE WORK 166

APPENDIX A: WOODFRAMESOLVER PROGRAM

ARCHITECTURE

170

APPENDIX B: WOODFRAMESOLVER USERS MANUAL AND

INPUT FILE FORMAT

181

APPENDIX C: WOODFRAMESOLVER VERIFICATION MANUAL 224

APPENDIX D: WHFEMG PROGRAM USERS MANUAL 321

APPENDIX E: ANALYSIS RESULTS 346

ix

LIST OF FIGURES

3-1 A wood house with horizontal floor and roof 45

3-2 A wood house with horizontal and sloped roof 45

3-3 Floor diaphragm with sheathing panels stacked along

the edges in a checkerboard format (no blockings

present), thick line in the figure represents panel

boundary

46

3-4 Floor diaphragm with sheathing panels put together in

non-checkerboard staggered manner (no blockings

present), thick line in the figure represents panel

boundary

46

3-5 Shear wall (no blockings) 47

3-6 Load distribution 48

3-7 Load-slip response of fastener under cyclic loading 49

3-8 Shear wall finite element model (viewed in SAP2000) 49

3-9 Different nail trajectories (shown in arrow and dotted

line) in a shear wall (α1, α2, α3). An angle α is

calculated between the horizontal and the nail

trajectory at the initial load

50

3-10 Floor or roof diaphragm finite element model (viewed

in SAP2000)

50

3-11 LFWS classification chart for the parametric study 51

3-12 Type 1, Model 1 – Floor Plan 51

3-13 Type 1, Model 1 – FE Model (viewed in SAP2000) 52

3-14 Type 2, Model I – Floor Plan 52

3-15 Type 2, Model I – FE Model (viewed in SAP2000) 53


x










3-26 WHFEMG program interface 58

4-1 Class diagram of WoodFrameSolver program 82

4-2 Benchmark problems (SAP2000 view) to compare

WoodFrameSolver performance

83

4-3 Speed comparison between SAP version 10 and

WoodFrameSolver (dense system of equations, linear

static analysis)

84

4-4 Speed comparison between SAP version 10 and

WoodFrameSolver (sparse system of equations, linear

static analysis)

84

4-5 Two node frame element 85

4-6 Three, four node shell elements 85

4-7 Eight node solid element 86

4-8 One node spring element 86

4-9 Nllink element 87

4-10 Gap spring behavior 87

xi

4-11 Hook spring behavior 88

4-12 Trilinear spring behavior 88

4-13 Modified Stewart spring behavior 89

4-14 WoodFrameSolver program interface 89

4-15 Shear wall with two sheathing panels (Dolan 1989) 90

4-16 Shear wall finite element model 90

4-17 Plywood sheathed shear wall response 91

4-18 Waferboard sheathed wall response 91

4-19 A light frame wood house floor plan 92

4-20 Analytical model – U house 92

4-21 An arbitrarily selected ground motion 93

4-22 Deformation-time histories – Wall 1 93

4-23 Force-deformation histories – Wall 1 94

4-24 Deformation-time histories – Wall 11 94

4-25 Force-deformation histories – Wall 11 95

4-26 A 3-story 1 bay moment frame 95

4-27 Nllink 6 deformation-time histories, 0% damping case 96

4-28 Nllink 6 force-deformation histories, 0% damping case 96

4-29 Nllink 1 deformation-time histories, 2% damping,

modes 1 and 3

97

4-30 Nllink 1 force-deformation histories, 2% damping,

modes 1 and 3

97

4-31 A garage type structure 98

4-32 Finite element model of garage 98

xii

4-33 Displacement response history – Joint23 99




4-37 Force-deformation response history – Nail25 101



4-40 X direction base shear response history 102

5-1 Lateral force distribution in a shear wall under rigid

and flexible diaphragm assumption

130

5-2 Newton-Raphson within a load step 131

5-3 Base shear convergence test of wall 1 Type IV model I 131

5-4 Modified Stewart spring behavior 132

5-5 Shear wall with two sheathing panels (Dolan 1989) 132

5-6 Shear wall finite element model 133

5-7 Kern County earthquake 133

5-8 Plywood wall displacement history 134

5-9 Waferboard wall displacement history 134

5-10 Type 1 floor plans 135






xiii


5-17 Type 1 houses finite element models 141







5-24 Imperial Valley earthquake 146

5-25 Northridge earthquake 147

5-26 Wall numbering 148

5-27 Ratio of interior and exterior shear wall in-plane peak

base shear per unit length vs. the X direction aspect

ratio, Flexible diaphragm model 1

149

5-28 Ratio of interior and exterior shear wall in-plane peak

base shear per unit length vs. the X direction aspect

ratio, Rigid diaphragm model 1

149

5-29 Type 1, wall 1 force-displacement response history,

Imperial Valley earthquake

150


Northridge earthquake

150



151



151



152

xiv



152



153



153

5-37 Directional rigidity criterion 154

5-38 Rigidity criterion plot for Types 2, 3 and 4 models 1

and 2

155

5-39 Rigidity criterion plot for Types 5, 6 and 7 model 1

155


with various in-plane diaphragm flexibilities

156


with various in-plane diaphragm flexibilities

156

xv

LIST OF TABLES

2-1 Non finite element models 26

2-2 Finite element models 26

3-1 Number of finite elements and degree of freedoms in

parent house models

59

4-1 Node attributes 103

4-2 Static analysis steps 103

4-3 Modified Stewart hysteresis parameters description 103

4-4 Shear wall properties 104

4-5 Shear wall results with plywood sheathings 104

4-6 Shear wall results with waferboard sheathings 104

4-7 Bilinear shear wall properties 105

4-8 Modified Stewart shear wall properties 105

4-9 Moment frame properties 105

4-10 Frame, sheathing and connection properties used in 3D

house model

106

5-1 Plywood shear wall element properties used in

verification analysis

157

5-2 Waferboard shear wall element properties used in


157

5-3 Maximum and minimum displacements used in


158

5-4 Direction aspect ratios and vibration periods of all the

models

158

5-5 Element properties used in the house models 159

5-6 Analysis cases 160

xvi

5-7 Type 1 Models, ratio of peak in-plane base shears

obtained using rigid and flexible diaphragm

assumptions for Imperial Valley earthquake loading

160



assumptions for Northridge earthquake loading

161




161




162




162




162




163




163




163




164




164

5-18 Rigidity criterion ratios for Types 2, 3 and 4 models 1

xvii

and 2 164

5-19 Rigidity criterion ratios for Types 5, 6 and 7 model 1 164

5-20 Rigidity criterion ratios for Types 2, 3 and 4 model 2

with various in-plane flexibilities

165

5-21 Rigidity criterion ratios for Types 5, 6 and 7 model 1

with various in-plane flexibilities

165

1

CHAPTER 1

INTRODUCTION

MOTIVATION: Light frame wood structures (LFWS) are increasingly being constructed

in high seismic zones of Australia, Japan, New Zealand and North America. The

symmetric plan LFWS has performed well in past earthquakes, but the last few years

have seen a rise in the construction of asymmetric plan houses, long-span floors and

roofs, and large openings. This presents a great challenge to researchers and structural

engineers, as the current analysis and design methodologies are not suitable for the new

structures being built. Moreover, the damage to wood structures that occurred in the 1994

Northridge earthquake has raised the concern for LFWS analysis that uses rigid

diaphragm or flexible diaphragm assumption.

To obtain an accurate seismic response of a house using a mathematical model, one

would require the model to be capable of incorporating all the structural elements

incorporated into the floors, roofs and shear walls, as well as knowledge of their true

behavior. The currently available approaches are either too simplified, with rigid

diaphragm assumptions, or do not incorporate the diaphragm details in the modeling.

Some of these approaches also neglect the appropriate inter-component interaction, and

hence the resulting models are just crude approximations lacking general applicability.

Shear wall modeling as an individual subassemblage, or as a part of a house, has been the

main attraction of research for several years, as it is the primary lateral force resisting

system in a LFWS. Currently, there is no available model that includes shear wall and

diaphragm details together in a single model. A robust approach, in which diaphragm

details can be incorporated in the mathematical model of a LFWS, is to use the finite

element method. The primary advantage of using finite elements is that the model can

incorporate various types of elements, their interaction with each other inside a

subassemblage and the subassemblage interaction. The resulting system may be used to

capture the detailed three-dimensional responses under static and dynamic loading. The

ingredients building roof or floor diaphragms are no different than what are used for

shear walls, and hence a similar modeling approach is anticipated to work quite well.

2

The finite element method is a robust and detailed solution procedure used in various

streams of engineering; however, there are not many dedicated tools to perform the

analysis of LFWS. Most of the available tools do not fulfill the current requirements of

wood structural engineers. The available research tools require a complicated approach to

incorporate hysteretic elements and do not provide detailed response data. Moreover, as

researchers continue to propose new models for wood houses, there is also a need for a

flexible and extensible platform which can accommodate new ideas instead of developing

a new program. It is found that most of the latest programs, supposedly the “state of the

art” in light frame wood structures analysis, lack the ideas of flexibility and extensibility,

as they follow a procedural approach for programming. A detailed finite element model

of a LFWS is likely to have several thousand degrees of freedom (DOF) with all the

connection DOF being nonlinear. Such a model when subjected to dynamic analysis or

repetitive dynamic analysis may easily overwhelm the memory, and the solution time

using a regular PC may be excessive.

Several experimental and mathematical studies have been performed in the last few

decades, but the current residential design guidelines still don’t consider the actual

flexibility of diaphragms in modeling LFWS when they are subjected to lateral loads. The

guidelines assume the in-plane stiffness of a diaphragm as either negligible or infinite.

The inclusion of the actual flexibility of a diaphragm is necessary to obtain an accurate

response under lateral loads. The diaphragm stiffness is a function of diaphragm

geometry, nail spacing, sheathing thickness, sheathing orientation, blocking, opening

location and size, and shear wall configurations. Practically, the combination of these

elements may result in a relatively rigid diaphragm system or a relatively flexible

diaphragm system when compared to shear walls. The answer sometimes may also lie

somewhere in between. A parametric study involving various seismic loadings with

varying diaphragm stiffness parameters presents a possible way to find the influence of

diaphragm stiffness on the seismic response of a LFWS. Unfortunately, the current

research and experiment databases are devoid of any such information.

3

OBJECTIVE AND SCOPE: The objective of this research is to understand how the

diaphragm flexibility affects the seismic response of LFWS. It has required diverse

efforts involving the development of a detailed analytical model, a new analysis program,

and performing verification with the few published experimental results. To investigate

the effect of diaphragm flexibility, the analytical models are used to perform a parametric

study with varying diaphragm geometry and shear wall configurations.

ORGANIZATION: This dissertation includes five more chapters, out of which four are

the working manuscripts of the authors. These chapters follow the Journal of Structural

Engineering manuscript format for consistency throughout; with appropriate changes

they could be submitted to the relevant journals or conferences.

CHAPTER 2 presents a detailed literature review of the mathematical models

developed for individual subassemblages and houses in the past 30

years. The chapter divides the developed models into a chronological

list of finite and non-finite element based models. It is noted that the

non-finite element based models are simple few-DOF systems when

compared to thousands of DOF in finite element based models. The

finite element modeling and analysis of LFWS is complex, but a robust

approach which can provide detailed results when studying individual

component behavior in a structure. It is also found that there are only a

few studies performed by the research and engineering community to

actually understand how diaphragm flexibility affects the seismic

response of LFWS. The main aim of this literature survey is to provide

the background material for the mathematical modeling of shear walls,

diaphragms and complete LFWS.

4

CHAPTER 3 presents nonlinear finite element models of various light frame wood

structures. The models described in this chapter are in general based on

the work done by other finite element researchers working in the field

of LFWS. However, no submodeling or substructuring of

subassemblages is performed, and instead a detailed model considering

almost every connection in the shear walls and diaphragms is

developed. The studs, plates, sills, blockings and joists are modeled

using linear isotropic three-dimensional frame elements. A linear

orthotropic shell element incorporating both membrane and plate

behavior is used for the sheathings. The connections are modeled using

oriented spring pairs with modified Stewart hysteresis spring

stiffnesses. The oriented spring pair has been found to give a more

accurate representation of the sheathing to framing connections in shear

walls and diaphragms when compared to non-oriented or single springs

typically used by previous researchers. The models presented herein are

classified into various types based on their geometry and dimensions.

They are generated using an in-house automatic customizable wood

house finite element model generator program and are manually post-

processed to modify or fill in the missing data for the analysis. These

models are further used in the parametric study of LFWS systems, the

results of which are discussed in chapter 5.

CHAPTER 4 presents WoodFrameSolver which is a high performance nonlinear

finite element analysis tool developed to analyze light frame wood

structures (LFWS). The program is written on a mixed language

platform using object-oriented C++, C and FORTRAN. This tool set is

capable of performing basic structural analysis chores like static and

dynamic analysis of 3D structures. It has a wide collection of linear,

nonlinear and hysteretic elements commonly used in LFWS analysis.

The advanced analysis features are comprised of linear and nonlinear

static, dynamic and incremental dynamic analysis. A unique aspect of

5

the program lies in its capability of capturing elastic and inelastic

displacement participation (sensitivity) of each element type in static

and dynamic analysis. The program also contains non-oriented and

oriented spring pair models to represent sheathing to framing

connections in light frame wood structures. The program’s performance

and accuracy are similar to that of SAP 2000, which is chosen as a

benchmark for validating the results. The use of fast and efficient serial

and parallel solver libraries obtained from INTEL has reduced the

solution time for repetitive dynamic analysis. The utilization of

standard C++ template library for iterations, storage and access has

further optimized the whole analysis process, especially when large-

DOF problems are encountered. This chapter discusses in detail the

program architecture, features and its applicability to light frame wood

structures. A few numerical examples discussing the advanced analysis

capabilities and their verification are also presented.

CHAPTER 5 presents the effect of diaphragm flexibility on the seismic response of

LFWS. To accomplish this, finite element models of various light

frame wood structures have been developed and analyzed using actual

flexibility and rigid diaphragm modeling. The modeling approach is

discussed in chapter 3 where these models are also discussed briefly.

The finite element models incorporate various structural elements and

their behavior in the form of beam, orthotropic shell, and nonlinear

hysteresis connector elements. The chapter presents a discussion of the

results, recommendations and conclusions of the analysis performed.

CHAPTER 6 discusses the limitations of the work undertaken in this thesis and

provides detailed recommendations for future work.

6

APPENDIX A presents the WoodFrameSolver program architecture document. The

document discusses various classes, the driver program and procedures

on how to extend the program for future development.

APPENDIX B presents the WoodFrameSolver program users’ manual and input file

format document. It contains the details on how to write input files and

how to use the WoodFrameSolver program for analysis.

APPENDIX C presents the WoodFrameSolver program verification manual. This

manual contains several example problems verifying

WoodFrameSolver program features and analysis capabilities.

APPENDIX D presents the WHFEMG program user manual. This manual discusses a

step by step procedure on how to develop a LFWS finite element model

input file to be used in analysis using the WoodFrameSolver program.

APPENDIX E presents tables of analysis results performed in chapter 5.

7

CHAPTER 2 MODELING AND ANALYSIS OF LIGHT FRAME WOOD

SHEAR WALLS, DIAPHRAGMS, AND HOUSES:

OVERVIEW

INTRODUCTION: Early research on wood houses dates back to 1927. Until today the

majority of the work in understanding the behavior of wood structures was accomplished

through experiments on individual diaphragms, shear walls and connections. Since 1980,

the main thrust of research in wood framed structures was through parametric study of

laboratory experiments. Design guidelines were based on the observations made by

varying sheathing type and orientation, fastener type and spacing, and diaphragm

geometry. A good amount of behavioral information has become available from testing

of wood houses and their performance in earthquakes. This information has provided a

background for the development of various mathematical models which have evolved in

the last 30 years. Mathematical modeling of wood structures has mainly been applied to

individual diaphragm and shear wall subassemblages. Only a very few full house models

have been developed to date. This has led to an improved understanding of the individual

components but not for structures as whole. In a loading event, the full house structure

distributes the loads among its structural components, the understanding of which cannot

be gained by analyzing just the individual components. The distribution of load is based

on the relative instantaneous stiffness of the structural elements, and in a seismic event,

progressive re-distribution of loads may also occur because of the inelastic yielding

occurring in the connections. Moreover, a good seismic design may only result when all

types of interaction occurring in the real system are considered in the modeling. For

accurate mathematical modeling and analysis, a detailed understanding of the behavior of

the structure under realistic scenarios is required.

This chapter presents a detailed literature review and discussion of the mathematical

models developed in the past 30 years for light frame wood structures (LFWS) and their

individual subassemblages. The chapter covers the articles published in journals,

8

conferences, and theses. It divides them into a list of finite and non-finite element based

models and presents them chronologically. It is noted that the non-finite element based

models are simple when compared to finite element based models. The non-finite

element models of LFWS assume floor and roof diaphragms as rigid in-plane and hence

are incapable of incorporating the actual flexibility of floor and roof elements. The finite

element models are detailed and provide a way to explicitly incorporate individual

structural elements and corresponding materials. The latter modeling approach is hence

suitable for capturing detailed response of LFWS components and the structure as a

whole. The main aim of this survey is to provide the background material for the

mathematical modeling of LFWS. The background material presented in this chapter is

used as the basis for the development of new house models. These new models are

further intended for understanding the effect of diaphragm flexibility on seismic

performance.

NON-FINITE ELEMENT MODELS: Tuomi and McCutcheon (1978) derived equations

based on an energy formulation to calculate the racking strength of sheathed-frame

panels. The model developed is based on the following assumptions: (a) the load-

deformation behavior of nails is linear; (b) the frame distorts as a parallelogram; (c) the

sheathing has all edges free and is continuous from top to bottom and never distorts; (d)

nails are spaced evenly; (e) applied loading is static; and (f) distortions and deflections

are small. The results obtained using the formulations were in close agreement with

experiments performed on two panel sizes.

Itani et al. (1982) presents a method for calculating the racking performance of sheathed-

frame panels with or without openings. In their model they replace the sheathing panels

and nails with a pair of diagonal springs. The stiffness of each spring is obtained by

calculating the internal energy due to nail deformation and setting it equal to the energy

of the equivalent diagonal brace system. The authors recognize that the load-deformation

relationship for nails is nonlinear; however they use a simple linear relationship. The

9

method presented is simple, as the complex frame-sheathing panel is reduced to an

equivalent frame-spring model, which was easily implemented and analyzed using

available computer programs at that time. The paper presents the modeling and analysis

of two shear walls with and without openings. A comparison of results for these models

with experiments showed that the model without openings overestimates the stiffness,

and the model with openings underestimates it. The racking forces required to cause a 1”

displacement at the top in both the models were within 12 percent of those from the

experiments. Also, the results indicated that the end panels in the walls are more

susceptible to damage under lateral loading.

A simple set of formulas for wood shear walls were developed by Easley et al. (1982) for

the linear stiffness of a wall, and for the nonlinear shear load-strain behavior of a wall.

These formulas are based on the deformation patterns of shear walls that were observed

in load tests. In order to validate the formulas, a comparison is made with the results

obtained from experiments and linear/nonlinear finite element analysis of walls. The

following conclusions on the derived formulas were made based on the comparison: (a)

the results obtained are within the acceptable range of accuracy, (b) are applicable to

shear walls loaded in a linear range, (c) can be used on any size of sheathing and any

types of discrete sheathing fasteners provided the deformation pattern of the walls is

similar to that discussed in paper, and (d) are valid only when there is no separation in the

framing member joints between the studs and the header or sill.

Gupta and Kuo (1985) present a simple shear wall model which is computationally less

expensive as compared to finite element models, especially in dynamic analysis. The

model incorporates the bending stiffness of studs and shear stiffness of sheathing panels.

It is recognized that these stiffnesses play a secondary role in defining the load-

deformation properties of the shear wall. The primary role is played by the nail load-

deformation characteristics.

10

McCutcheon (1985) extends the previous model developed by Tuomi and McCutcheon

(1978) to incorporate nonlinear load-slip behavior of nails. It is emphasized that racking

behavior of a sheathed wall depends primarily on the lateral load-slip characteristics of

the fasteners. The computations are simplified when nonlinear load-slip relationships are

approximated using power curves. Also, the predictions of racking strength using the

power curve method compared well with experimental results.

Gupta and Kuo (1987) later modified the previous model by Gupta and Kuo (1985) to

incorporate stud uplifting. The model presents the effect of vertical loading on the load-

deflection behavior of the wall. It is shown that the theoretical upper bound for the load-

deflection curve, for the case with very large vertical loading, matches well with the

curve with no uplift considered.

Patton-Mallory and McCutcheon (1987) extended the previous model developed in

McCutcheon (1985) to incorporate sheathings on different sides of a wall with dissimilar

materials. The paper presents the results of four different mathematical approximations of

the fastener behavior; asymptotic, power, logarithmic and hyperbolic tangent curves. It is

concluded that the asymptotic form is the most accurate in predicting wall behavior over

a wide range of the data. The power and logarithmic curves fit the data only in a limited

range. The hyperbolic tangent method does not fit well. The authors recommend the use

of power curves over asymptotic and logarithmic curves, as they are relatively simple.

The racking behavior of a small shear wall is predicted well with the proposed model

using asymptotic curves for the fasteners.

Schmidt and Moody (1989) present the need to formulate a rational procedure for

analyzing three-dimensional light frame wood structures. The paper emphasizes the need

to understand subassembly interaction to predict more accurate behavior of the entire

structure. A simple analysis technique is developed to predict the nonlinear deformations

of the three dimensional light-frame structures subjected to lateral loads at and beyond

11

design levels. The shear wall models and their nail load deformations are derived from

the work of Tuomi and McCutcheon (1978), Foschi (1977) and McCutcheon (1985).

Only the in plane stiffness of shear walls is assumed, which combined with the rigid floor

and roof diaphragm assumption results in a three-degree-of-freedom system for each

story. Two example models of residential sized buildings are analyzed and compared

with the actual performance. The computed lateral translation and rotations of the floor

are reported to be in good agreement with the experimental results.

Dolan and Filiatrault (1990) present a single degree of freedom model which is derived

from the static analysis test data obtained from the experiments. The presented model is

capable of predicting the steady-state response of nailed shear walls. They divided the

hysteresis behavior into six linear regions based on certain assumptions. The paper

presents results from four test specimens, and a comparison is made. The proposed model

is found to be accurate.

Kamiya and Itani (1998) developed a simplified procedure for analyzing sheathed

diaphragms with and without openings. They proposed simple formulas which are found

to produce results that compare well for ultimate loads and unit shear around the opening.

The authors recognize that it is difficult to evaluate shear forces around the openings. The

observed deflections from experiment are smaller by 14-28 percent when compared with

the results obtained from the proposed formulas.

A three degree of freedom model for wood frame shear walls was developed by Dinehart

and Shenton (2000). The model is capable of capturing important dynamic characteristics

and the seismic response using the basic properties of the structure. The model can also

accommodate variations in wall geometry, sheathing and framing material type, fastener

type and fastener spacing. The paper also presents a method for estimating the connection

properties from the results of full scale shear wall tests. A comparison of results shows

that the model works moderately well for predicting hysteresis behavior at low to

12

moderate displacement levels; however, it fails to predict the pinched hysteretic action at

large displacements.

To facilitate understanding of the cyclic behavior of wood shear walls, Folz and

Filiatrault (2001) developed a simple numerical model. The model is capable of

predicting the load-displacement response and energy dissipation characteristics of wood

shear walls under quasi-static loading. The following elements are used in the model: (a)

rigid frame members, (b) linear elastic sheathing panels and (c) nonlinear sheathing to

framing connectors. The connector model is based on the previous research work done by

Foschi (1977) and Stewart (1987). It takes into account the pinched hysteretic behavior

and the degradation of stiffness and strength under cyclic loadings. The proposed model

is implemented in a computer program named CASHEW. The model is found to

accurately predict the load-displacement response and energy dissipation characteristics

of wood shear walls under general cyclic loading. As an example application study, the

results from the program are used to derive parameters for an equivalent SDOF model.

This model is tested for an arbitrary seismic loading and results are found to compare

well with the results from shake table tests.

Folz and Filiatrault (2004a) present a simple numerical model to predict the dynamic

characteristics, quasi-static pushover, and seismic response of wood frame buildings. The

structure is decomposed into two main components: (a) rigid horizontal diaphragms and

(b) zero height unidirectional nonlinear spring elements representing shear walls. The

spring characteristics are obtained from the associated numerical model. This simple

approach reduces the number of degrees of freedom to three per floor. The model is

implemented in a computer program called SAWS. The verification of the proposed

model is discussed in Folz and Filiatrault (2004b). The results of the model are compared

with the results obtained from shake table tests performed on a full-scale two-story wood

frame house. The program provides an accurate estimate of dynamic characteristics,

quasi-static pushover, and seismic response of the test structures.

13

A recent addition to the list of available non-finite element based programs is the

SAPWood program (Pei and Lindt 2006) which is based on the SAWS programming

structure and concept. It is a GUI based software package capable of handling bi-

directional seismic input. The program is also being designed to incorporate performance

based seismic design concepts. To model LFWS, it consists of a three-degree-of-freedom

per floor model from SAWS, and a one degree of freedom per floor shear building

model. Apart from nonlinear static and dynamic analysis, the program can also perform

incremental dynamic analysis (IDA), system identification analysis and multi-case IDA.

To model shear wall elements, the program has linear, bilinear, modified Stewart

hysteresis (same as CASHEW) and the latest Evolutionary Parameter Hysteretic Model

(EPHM) springs (Pang et al. 2007). The program also contains a utility for semi-

autofitting of shear wall test hysteresis data to any of the four spring models. This

program is an outcome of the NEESWood project, the main objective of which is “to

develop a logical and economical performance based seismic design philosophy to safely

increase the height of wood frame construction in the regions of moderate to high

seismicity”.

FINITE ELEMENT MODELS: Foschi (1977) presents one of the very first analytical

models based on finite elements taking into account the orthotropic plate action of

sheathings and the nonlinear load-deformation behavior of the connections. It is

identified that the structural analysis of diaphragms is very complex, as the system is

highly indeterminate. The model presented was implemented in a computer program

SADT which was developed at the Western Forest Products Laboratory. An example

model presented by Foschi shows that it gives reliable estimates of diaphragm

deformation and approximate ultimate loads based on connection yielding.

In an attempt to investigate static and dynamic response of diaphragms, Cheung and Itani

(1983) developed a numerical model based on finite elements. The model implements a

14

fastener element derived from nonlinear load slip properties in a computer program called

NONSAP (Bathe et al. 1974). A numerical example simulating the behavior of a light

frame wood shear wall sheathed with plywood is presented. No general conclusions are

made, as the model was developed as a part of an ongoing project. This work is later

extended and discussed in Itani and Cheung (1984). A nonlinear finite element

formulation to obtain load-deflection characteristics of diaphragms with and without

openings is presented. The model proposed is general and is applicable to any sheathing

arrangement or load application. It does not assume any particular geometry of distorted

diaphragms as was assumed in previous models proposed by other authors.

Gutokowski and Castillo (1988) presented a finite element model which is capable of

analyzing partially composite shear walls loaded in plane. The stud frames are modeled

using standard frame elements with axial and without shear deformations. The joints

between the stud frames use linear springs for translational and rotational force transfer.

Sheathings are modeled using plane stress elements, and the connection between

sheathing and studs uses nonlinear fasteners. The behavior of sheathing gaps is

recognized as discontinuous and nonlinear, however they are modeled using linear

connector elements with stiffness coming into action when the sheathing edges come into

contact with each other. The model is implemented in a computer program called

WANELS. The proposed model is found to predict the load-deformation response of

shear walls with a high degree of accuracy when compared to experimental results.

Falk and Itani (1989) present a two-dimensional finite element model for analyzing

horizontal and vertical diaphragms. This is one of the first attempts to actually model

horizontal diaphragms. The model uses beam and plane stress elements for studs and

sheathings. A new transfer element which is a spring pair is developed to represent the

fastener between the studs and the sheathing. The stiffness of each spring in the pair

corresponds to the lateral stiffness of the nail obtained from the experiments. A nonlinear

behavior is assumed for the fasteners and a good correlation for load-displacement

15

behavior is obtained between experiments and the proposed model. The paper also

performs a parametric study of floor diaphragms using varying nail stiffness, nail spacing

and the use of blocking. It is concluded that the nail spacing has more influence on the

diaphragm behavior than the nail stiffness, and the use of blocking increases the stiffness

of the diaphragm.

Dolan and Foschi (1991) developed a numerical model for nonlinear analysis of wood

shear walls. The model includes the effects of nonlinear fasteners, bearing between

adjacent sheathings and out of plane bending of sheathing elements. This paper is the first

attempt in modeling the out-of-plane bending of the sheathings. The out-of-plane bending

of sheathing may occur in walls with large stud spacing or a wall with the flexible

sheathings. It is recognized that bearing can have significant effects in very large shear

walls with staggered joints. The proposed model is implemented in a computer program

named SHWALL, which is an improved version of the SADT program developed by

Foschi (1977). The studs are modeled as two-dimensional frame elements, sheathings are

modeled using four-node plate elements, connectors are modeled using three independent

nonlinear springs (smeared connections can be used to model adhesive connections), and

the bearing is modeled as a bilinear spring with low stiffness in tension and high stiffness

in compression. Comparisons with experiments show good correlation of stiffness and

ultimate load capacity of the shear wall.

Kasal and Leichti (1992a) present a method which uses energy concepts to transfom a

three-dimensional finite element model of a wood-frame wall to a single equivalent two-

dimensional finite element model. The equivalent model behaves similarly as its three-

dimensional counterpart but has fewer degrees of freedom, reducing the number of

equations to be solved. Due to the reduced degrees of freedom, the computer analysis

time is also considerably reduced. The equivalent model results are in good agreement

with experiments and three-dimensional finite element model analyses.

16

Kasal and Leichti (1992b) discuss load sharing in shear walls for a full wood house

structure. They modeled shear walls using nonlinear quasi superelements and the roof

and floor using linear superelements. In addition to a three-dimensional finite element

model, they proposed linear and nonlinear models assuming the roof diaphragm as a rigid

beam on elastic supports. The results for the load distribution are compared with the

experimentally verified model, and the finite element analysis results are found to be the

closest compared to other proposed models. The results demonstrated that the design

procedures do not take into account the load redistribution capability of the structures.

Kasal et al. (1994) present the importance of modeling the full structure in order to

incorporate "interactive behavior of individual components and connections" and to get a

more coherent understanding of the response under static loads. It is noted that modeling

the nonlinear character of component and substructure interaction has been the biggest

deterrent for design optimization for light frame wood houses in the past. They use the

concept of superelements and quasisuperelements as discussed in Kasal and Leichti

(1992 a and b) for modeling different assemblages. In their formulation they assume the

behavior of floors and roof diaphragms (i.e. out-of-plane bending and torsion) as linear

and modeled them as superelements. However, the behavior of walls and intercomponent

connections is assumed as nonlinear and they are modeled using quasisuperelements.

White and Dolan (1995) recognized the lack of information on dynamic response of

wood frame shear walls, resulting in seismic design codes based on static analysis data.

To supplement the tool set, they developed a finite-element program named WALSEIZ,

capable of doing nonlinear analysis of wood-frame shear walls subjected to monotonic

and dynamic loads. This program is a modification of the previous program developed by

Dolan (1989). The following elements are programmed for use in modeling a shear wall:

(a) beam element for the framing, (b) plate element for the sheathing, (c) nonlinear

springs for the connectors, and (d) bilinear spring for the bearing between adjacent

17

sheathing panels. The results from the program correlate well with previous experimental

data.

Tarabia and Itani (1997a) present a methodology for the development of a general three-

dimensional computer model for light frame wood buildings. Structural components are

idealized as diaphragm elements connected by intercomponent connection elements. The

degrees of freedom are classified as master and slaves. Finite element formulations of

different components are proposed. The analysis verification of two static and dynamic

models is performed with experimental results from Dolan (1989) and Phillips (1990).

The analytical results are found to be in good agreement for ultimate loads and

deformations when compared to the corresponding experimental values. Later, Tarabia

and Itani (1997b) use this model to study seismic response of general three-dimensional

light frame wood buildings. Buildings with different geometric configurations were

considered for evaluation. They studied the load distribution of lateral forces among the

shear walls and the deflections of the diaphragms. The following conclusions are made

from the study of the analyzed models: (a) the displacements are in the range 2 to 15

percent and the reactions are within 4 percent for the applied seismic loading, (b) the

estimated load on the structure is greater than what is obtained from the code, (c) the

diaphragm rigidity is influenced by its dimensions, (4) partition walls resist significant

seismic forces depending on their stiffness and the dimensions of horizontal diaphragms,

(5) asymmetric configuration of the shear walls generate torsional forces resulting in

large rotations and displacements of the diaphragm and (6) the walls transverse to the

loading resist 17 to 22 percent of the base shear and hence cannot be neglected in the

design.

He et al. (2001) developed a nonlinear finite element analysis program called

LightFrame3D, which may be used to study light frame wood structure components or

the entire building, under static loading conditions. The main idea is to get a better

understanding of the load distribution and load paths within the structure. The sheathing

18

to frame connector properties are based on a mechanics based representation which the

authors believe to be a unique aspect of the model. The program is capable of handling

different material properties and combined loading conditions. The program results are

observed to correlate well with experiments carried out in the past.

Kasal et al. (2004) presents a comparison of different design methodologies for lateral

force distribution in a full-scale wood frame test house. The results are also compared

with a detailed three-dimensional finite element analysis of a test house. It is found that

the detailed finite element analysis most accurately predicts the experimentally measured

load distribution.

Judd and Fonseca (2005) present finite element models of wood shear walls and

diaphragms with a new sheathing to framing connector. This new connection, called the

oriented spring, is the unique feature of the model and provides an improvement over

previous single and non-oriented spring pair connection representations. The wall and

diaphragm analytical models are developed using shells, beams and various connection

elements, i.e. single spring, non-oriented spring pair and oriented spring pair. These

models are subjected to monotonic and cyclic loadings, and the results are compared with

the previous experimental responses. The oriented model is found to be the most suitable

when compared with the other connection representations.

Collins et al. (2005a) present a finite element based three-dimensional model of a light

frame wood building. The structural components are modeled using shell, beam and

nonlinear spring elements. The in-plane shear wall behavior in the structure is modeled

using the energetically equivalent nonlinear diagonal springs, making the computations

easier. To obtain a detailed response of a wall, one needs to apply the loads calculated

using the full house model back into the detailed finite element model. Also, the use of

nonlinear diagonal springs to model in-plane wall behavior uncouples the in-plane and

out-of-plane stiffness of the shear wall. The nail connections are modeled using elements

19

having spring stiffnesses in each of the orthogonal directions. This is a slightly inaccurate

representation of the in-plane (diaphragm) stiffness of the connections, as it leads to

uncoupling of the behavior in the in-plane directions. The accuracy of the model is

verified with experimental results in a companion paper (Collins et al. 2005b). The model

presented is capable of capturing response to the nail level. It is claimed to be suitable for

dynamic analysis; however, it is verified only under the static loads.

DISCUSSION: This chapter presents a detailed literature review of non-finite element

and finite element based models developed in the last 30 years. Research efforts in the

development of both types of models have progressed in parallel and there has been a

gradual increase in the level of complexity in both. The non-finite element models are

relatively simpler and are in general a tradeoff between the level of complexity and the

solution time. Tables 2-1 and 2-2 present the list of non-finite and finite element models,

respectively. These tables also show the corresponding assemblage type and analysis type

for which each model has been developed. As can be seen from these tables, most of the

mathematical models are developed for individual subassemblages (primarily the shear

walls). Very few mathematical models and experimental verifications refer to horizontal

floors, roofs and houses. All the models are verified under static loadings and only a few

are verified for dynamic loads.

The load-deformation characteristic of sheathing to stud connectors in the few early shear

wall models are approximated by using a piece-wise linear or smooth polynomial curves

and are applicable only under static loadings. A nonlinear pinched hysteretic load-

deformation behavior is desired for the connections in shear wall models when they are

subjected to cyclic loadings (Stewart 1987 and Dolan 1989). The models for shear walls

are mostly governed by the fact that the primary role is played by the connection’s load-

deformation characteristics. A hysteretic load-deformation model for the connections can

be calibrated to model the load-deformation characteristics of a wood shear wall. Some of

the shear wall models presented in the literature can capture the pinched hysteretic

20

response, but only until a certain displacement level or only under a low frequency cyclic

loading. They also may not capture the stiffness and strength degradation occurring

during cyclic loadings. The models involving linear nail load-deformation behavior are

oversimplified and may lead to overestimation and underestimation of the forces in the

walls. The empirical formulas derived for shear walls are not generalized, as they are

based on a specific deformation pattern of sheathing and studs. The CASHEW program

implements the modified Stewart hysteretic model spring for shear walls and captures the

shear wall response accurately for both static and dynamic loading. The nonlinear

hysteretic behavior of walls using a modified Stewart spring is represented by using one

exponential and a few linear loading and unloading segments. These wall models require

ten input parameters and have also been successfully implemented in the SAWS

program. The linear segments in the Stewart model are assumed to be time invariant,

which is an assumption for simplification in their modeling. An improvement on this is

considered in the EPHM model for wood shear walls. This model requires seventeen

input parameters and captures the damage occurring in the walls more realistically.

The finite element models for LFWS are complex because they are highly redundant

nonlinear systems consisting of several studs, sheathings and connector elements. Early

finite element models were simplified by using model reduction techniques by creating

an energetically equivalent model or condensing out linear/non-required degrees of

freedom. The finite element models involve explicit representation of structural elements

and material properties. Beam elements are typically used to model studs or joists. The

plane stress, plate or shell elements are used to model sheathings. In a full three-

dimensional model the use of a shell element is preferred over plane stress or plate

elements, but at the same time it increases the number of degrees of freedom in the

system. The beam, plane stress, plate and shell elements are always modeled as linear but

may use orthotropic material properties. The connector between sheathing and stud may

be modeled using a single spring, non-oriented spring pair or oriented spring pair. The

oriented spring pair type connector has been found to be the most accurate among the

three and is a relatively new approach. The spring stiffnesses in the connectors may be

21

modeled using various hysteresis models. The parameters for these springs are generally

obtained from the experiments. The mechanics based connection models are

computationally complex and are difficult to incorporate at the global level.

CONCLUSION: This chapter has presented a literature review and a discussion of

mathematical models developed for light frame shear walls, roof/floor diaphragms and

houses. This survey has covered most of the journal articles discussing the development

of mathematical models for LFWS and their subassemblages. It is known from the

experiments that the diaphragm flexibility is influenced by various parameters, i.e. its

dimension, sheathing thickness, nail size, spacing, blocking, sheathing pattern and shear

wall locations. The influence of diaphragm flexibility on the seismic response of LFWS

has not yet been studied in detail. A parametric study of a few horizontal diaphragm

examples under static loading is conducted by Falk and Itani (1989). The results of

models analyzed in Tarabi and Itani (1997b) suggests that the influence of diaphragm

flexibility on the seismic response of LFWS should be subjected to a more detailed

investigation.

A detailed investigation on the influence of diaphragm flexibility can only be

successfully performed if a full three dimensional LFWS incorporating reasonably

accurate component behavior and component interaction is developed. The models

presented in this survey vary from simple single-degree-of-freedom hysteretic springs to

complex finite elements. The non finite element based house models are simple but

cannot be used to incorporate the diaphragm details and hence are unsuitable for

parametric studies. The survey has led to the conclusion that finite element modeling is

the only way to get detailed results. It is also realized that with a little difficulty one may

include all the structural details in a finite element model of a LFWS, but simultaneously

it may also overwhelm the capacity of the computer on which it is analyzed. This may

require some simplifications in the finite element models of LFWS, as has been done by

other researchers in the past.

22

The analysis of finite element models presented in the above survey has been done either

by using a commercial program or by using some in-house software developed

particularly for the presented model. As the programs are mostly in-house, they are not

available to other researchers. These programs also are written using procedural

programming concepts and may not be easy to extend for the purpose of further usage.

The use of commercial programs like ABAQUS1, ANSYS

2 or SAP

3 may be

advantageous due to several obvious reasons; however, they lack the implementation of

various hysteresis elements developed for LFWS. Moreover, any new element being

devised by the wood research community may not be easy to add to the commercial

program. Development of a general purpose finite element program for light frame wood

structures which may accommodate various types of elements and analysis, and provides

a flexible platform for extension of capabilities, may certainly prove to be useful.

In this thesis we have also developed a general purpose finite element analysis program

called WoodFrameSolver in object oriented C++ which is capable of static and dynamic

analysis of large structural systems. The program incorporates shell, frame, nonlinear

link, and spring elements which are generally used to model LFWS systems. The

nonlinear links element provides spring properties which may be bilinear, trilinear or

modified Stewart and may be used to represent connection properties in LFWS systems.

The program is capable of performing nonlinear dynamic analysis, which may be utilized

for response history analysis. The program implements fast solution algorithms and

solvers, which helps in obtaining static and dynamic analysis results in a reasonable time

frame. The program is discussed in detail in Chapter 4. Overall, a platform is developed

which is efficient and easy to use and develop, for analysis and development purposes

respectively.

1 http://www.simulia.com/ 2 http://www.ansys.com/ 3 http://www.csiberkeley.com/

23

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35. Tarabia, A. M., and Itani, R. Y. (1997b). "Seismic Response of Light-Frame Wood


36. Tuomi, R. L., and McCutcheon, W. J. (1978). "Racking Strength of Light-Frame

Nailed Walls." Journal of Structural Division, 104(ST7), 1131-1140.

37. White, M. W., and Dolan, J. D. (1995). "Nonlinear Shear-Wall Analysis." Journal of

Structural Engineering, 121(11), 1629-1635.

26

Table 2-1: Non-finite element models

YEAR SHEAR FLOOR & HOUSE STATIC DYNAMIC

WALL ROOF

Tuomi and McCutcheon 1978 √ Х Х Х Х

Itani et al. 1982 √ Х Х Х Х

Easley and Dodds 1982 √ Х Х Х Х

Gupta and Kuo 1985 √ Х Х Х Х

McCutcheon 1985 √ Х Х Х Х

Gupta and Kuo 1987 √ Х Х Х Х

Patton-Mallory and McCutcheon 1987 √ Х Х Х Х

Schmidt and Moody 1989 √ RIGID √ Х Х

Dolan and Filiatrault 1990 √ Х Х Х Х

Kamiya and Itani 1998 √ √ Х Х Х

Dinehart and Shenton 2000 √ Х Х √ √

Folz and Filiatrault 2001 √ Х Х √ √

Folz and Filiatrault 2004a √ RIGID √ √ √

Pei and Lindt 2006 √ RIGID √ √ √

ANALYSISASSEMBLAGE

AUTHORS

√ = Included, X = Not included

Table 2-2: Finite element models

YEAR SHEAR FLOOR & HOUSE STATIC DYNAMIC

WALL ROOF

Foschi 1977 √ Х Х √ Х

Cheung and Itani 1983 √ √ Х √ √

Itani and Cheung 1984 √ √ Х √ √

Gutokowski and Castillo 1988 √ Х Х √ Х

Falk and Itani 1989 √ √ Х √ Х

Dolan and Foschi 1991 √ Х Х √ Х

Kasal and Leichti 1992a √ Х Х √ Х

Kasal and Leichti 1992b √ √ √ √ Х

Kasal et al. 1994 √ √ √ √ Х

White and Dolan 1995 √ Х Х √ √

Tarabia and Itani 1997a √ √ √ √ √

He et al. 2001 √ √ √ √ Х

Judd and Fonseca 2005 √ √ Х √ √

Collins et al. 2005a √ √ √ √ √

ANALYSISASSEMBLAGE

AUTHORS

√ = Included, X = Not included

27

CHAPTER 3 THE EFFECTS OF DIAPHRAGM FLEXIBILITY ON THE SEISMIC PERFORMANCE OF LIGHT FRAME WOOD

STRUCTURES, PART I: MODEL FORMULATION

INTRODUCTION: LFWS are the most common single and multi-story residential

structures constructed in high seismic and wind zones of Australia, Japan, New Zealand

and North America. In spite of their inherent complexity, LFWS are preferred over

reinforced concrete and steel. The primary reason is the LFWS structural efficiency

owing to its low seismic mass, nonlinear inelastic response, high strength and high

stiffness. The other reasons include easy availability of wood, low production cost, low

construction cost, environmental friendliness, easy recyclability and high aesthetic

qualities. As LFWS are very common in seismic and wind zones, they are also

susceptible to damage caused by these loadings. The seismic or wind loads create lateral

forces and overturning moments in an LFWS system. These forces are resisted by the

structural members and their connections, and the actual force distributions in the

members are based on their relative stiffnesses, calculations of which are not trivial. In

fact, there exist no means by which one can calculate the exact forces in the components

of a LFWS.

Various experimental approaches have been used to understand the behavior and the

force distribution in the individual subassemblages and full scale houses. The

experimental approach has proved quite useful but cannot always be applied because of

the high cost involved. An alternative approach is to create mathematical models of the

individual subassemblages or the houses. Wood structures are highly redundant nonlinear

inelastic systems and their modeling poses several difficulties. Substantially, even the

most simplified model of a subassemblage or a single story wood house requires

numerical calibrations and a nonlinear analysis approach to obtain the responses. Several

different mathematical models have been proposed in the last 30 years and are briefly

discussed in Lindt (2004) and Chapter 2. The latter classifies the analytical models into

28

non-finite and finite element based models. The non-finite element models consist of

simple analytical and empirical formulas and procedures which are applicable only to

specific type of problem sets. These simple analytical models consist of LFWS and their

subassemblages reduced to few degree of freedom systems. On the other hand, the finite

element models are complex and form very large degree of freedom (DOF) systems

ranging from hundreds to several thousands. These models, however, are more realistic

and general, as they involve explicit modeling of the structural members and connections

using linear springs, hysteretic springs, frames and shells and their properties.

Researchers in the past have tried to simplify these large DOF systems by using

substructuring, submodeling or equivalent energy modeling approaches. Both non-finite

and finite element models, however, can represent the behavior only within the range of

the assumptions involved in their formulation. The limitations of the simplified analytical

models are obvious. The detailed finite element models are mainly affected by the

complexity of the mathematical model and the time required in obtaining their solution.

There are only a very few mathematical models which are suitable for full scale houses.

These models (Gupta and Kuo 1987, Schmidt and Moody 1989, Yoon and Gupta 1991,

Ge 1991, Kasal et al. 1994, Tarabia and Itani 1997, He et al. 2001, Folz and Filiatrault

2004 and Collins et al. 2005) belong to both the non-finite and finite element classes and

have produced reasonably accurate results. The non-finite element models are not

suitable for studying the effect of diaphragm flexibility on the seismic response of LFWS

because they lack the explicit representation of the structural elements, and hence it is

hard to incorporate variation of parameters, e.g., wall stud sizes, sheathing thickness,

diaphragm nail spacing, etc., in their models. A detailed nonlinear dynamic finite element

model is more suitable for such analyses, and Collins et al. (2005) presents one such

model in the literature. This model is based on the work of Kasal et al. (1994) and has

only been verified under static and cyclic loadings. The remaining other full house finite

element models have mainly been verified under monotonic and cyclic loadings.

29

This chapter presents nonlinear finite element models of various light frame wood

structures. The models described herein are in general based on the work done by Kasal

et al. (1994) and Collins et al. (2005). However, no substructuring or submodeling of

subassemblages is performed, and instead a detailed model considering almost every

connection in the shear walls and diaphragms is developed. The studs, plates, sills,

blockings and joists are modeled using linear isotropic 3D frame elements. A linear

orthotropic shell element incorporating both membrane and plate behavior is used for the

sheathings. The connections are modeled using oriented springs (Judd and Fonseca 2005)

with modified Stewart hysteresis spring stiffnesses. The oriented spring has been found to

give a more accurate representation of the sheathing to framing connections in shear

walls and diaphragms when compared to non-oriented or single springs typically used by

previous researchers. These elements and spring stiffness properties are discussed in

detail in Chapter 4. The modified Stewart hysteresis model is chosen because of its

computational efficiency, as it is based on mostly linear path following rules and

accurately represents the connections’ pinching behavior with strength and stiffness

degradation (Folz and Filiatrault 2001). It is also the latest state of the art model used in

representing dowel type connector stiffness in LFWS. The models presented herein are

classified into various types based on their geometry and dimensions. They are generated

using an in-house automatic customizable wood house finite element model generator

program (Appendix D) and are manually post-processed to modify or fill in the missing

data for the analysis. A high performance nonlinear finite element program named

WoodFrameSolver developed in Chapter 4 is used in the analysis of these models. These

models are further used in the parametric study of LFWS systems, the results of which

are discussed in Chapter 5 of this thesis.

LFWS COMPONENTS AND BEHAVIOR DESCRIPTION: LFWS is typically a

composite construction comprised of a skeleton of wood frames which is covered by the

sheathings. Two typical wood residential houses are shown in Figures 3-1 and 3-2. They

consist of horizontal floors, horizontal roof, sloped roof, and vertical shear walls, all of

30

which physically appear as a deep thin beam. Floors and roofs are referred to as the

diaphragms and the vertical structural elements are called shear walls. They are the

primary structural components in a LFWS and consist of frames and sheathings, which

are connected together using nails or staples or adhesives or their combination.

In a floor or roof diaphragm sheathings are placed over the joists and generally their

longer edge is placed perpendicular to the joist’s direction. If in a diaphragm all the edges

of the sheathings lie on the joists and blockings, then it is called a blocked diaphragm,

and otherwise it is called unblocked. Blocked diaphragms provide better shear transfer

over unblocked due to the additional connection elements. Sheathings in diaphragms may

be placed beside each other in stack or in staggered form as shown in Figures 3-3 and 3-

4, respectively. The different layout pattern also affects the shear strength of the

diaphragm. More than one layer of the sheathing along the thickness may also be placed

on a floor or a roof for additional shear strength. A shear wall consists of horizontal top

plates, horizontal bottom sill, vertical studs, blockings and sheathings connected via nails

along the perimeter and inside as seen in Figure 3-5. Under the lateral loading on a

LFWS, the double top plate acts as a chord or a collector depending upon the shear wall

location with respect to the load direction, and hence are the critical design elements.

Similar to the diaphragm, a shear wall may also have multiple sheathing panels on one

side, and often they are applied to both the sides. Sometimes openings in the diaphragms

or the walls are required for the staircases, skylights, windows or doors. If an opening is

large enough to reduce the subassemblage shear strength, additional framing around the

opening is used to strengthen the diaphragm or the wall. The nails connecting sheathing

to framings in diaphragms and shear walls are applied both on the perimeter and in the

field region of the sheathing panel. The sheathing panels for diaphragms and shear walls

may use a variety of panel grades; however, typically plywood and oriented strandboard

are used. The wall and diaphragm framings, e.g., studs, joists, etc., also come in different

materials, for e.g., Douglas fir-larch, Southern Pine, etc. To assemble a full house using

floors, roof and shear walls, these have to be connected to each other using nails, bolts,

31

and straps, and attached to the foundation using anchor bolts and vertical tie down

systems.

In a LFWS, diaphragms are supported by shear walls which stand on either another shear

wall (multi-storey) or the foundation. Diaphragms and shear walls are designed to carry

in-plane lateral loads, perpendicular surface pressure loads, and vertical gravity loads,

and from the functional perspective are designed to serve similar purposes. In the present

work the authors are concerned only with the effect of lateral loading occurring due to

earthquakes and hereon discuss only that. In an earthquake event, various types of forces

occur in a LFWS and they include (1) inertia forces due to self plus imposed mass, (2)

damping forces due to slipping interfaces of the connected materials, (3) elastic and

inelastic forces occurring in the structural elements and (4) an equivalent force due to the

earthquake. At the subassemblage level, the forces occurring in the roof or floor

diaphragms gets distributed to the shear walls below it via inter-component connectors,

which in turn transfer the load to either another shear wall or the foundation below it.

This is shown for a single story box shaped house in Figure 3-6. The force distribution

among the subassemblages is based on their relative stiffnessses, which is the

combination of stiffnesses of constituting elements. The force distribution keeps

changing throughout the earthquake loading due to the varying nature of the loading and

inelastic response of the connection elements. In a wood house, vertical shear walls are

the primary lateral load resisting members and the overall system is desired to act as a

unit so that proper load transfer to the foundation is ensured. The response of a wood

house under monotonic loading is generally nonlinear, and under the dynamic loading the

response is often nonlinear hysteretic.

Experiments on the diaphragms and shear walls have shown that the behavior of framings

and sheathings remains linear under static and dynamic loading, and it is the nail

connections between the sheathings and the framings that exhibit the nonlinear response.

The sheathings in a LFWS resist most of the in-plane shear acting on the diaphragms and

32

the shear walls. Connections play an important role in transferring the racking force from

sheathing to the framing. In fact, the response of walls and diaphragms is primarily

governed by their connection’s behavior. Under dynamic loading the connections exhibit

hysteretic response which is characterized by degrading stiffness, strength and pinching,

and this has been recognized experimentally by several authors (Medearis and Young

1964, Stewart 1987, Kamiya 1988, Dolan 1989, Dolan and Foschi 1991). A typical nail

connection behavior generally observed in the experiments is shown in Figure 3-7. The

lateral loading applied on a wood structure also causes sheathings to interact with each

other along their edges. This is a nonlinear interaction which is called bearing and has

been recognized by Jizba (1978). The stiffness for this type of behavior comes into action

only when the sheathings start interacting with each other.

FINITE ELEMENT MODELING METHODOLOGY: The finite element approach uses

explicit modeling of structural elements and material properties. Choosing the

appropriate finite elements to describe the behavior and to get the desired response of

individual components is an important issue in modeling light frame wood structures and

their individual subassemblages. Further, the method requires descretization of the

domain using the selected elements, which for LFWS is not a trivial task. This is

primarily due to the complex configuration of LFWS which consist of thousands of

independent structural elements (frames, sheathings, fasteners, etc.) connected together to

act as a unit. The behavior of LFWS elements has been studied experimentally and

analytically by various researchers in the past, and mainly requires frame, shell and

nonlinear link elements for modeling. These elements are implemented in the

WoodFrameSolver program and are briefly discussed below in the context of using these

elements in modeling LFWS. The three sections after that discuss how these finite

element elements are put together to create shear walls, diaphragms and the full house

models.

33

FINITE ELEMENTS:

FRAME: This is a two node, three-dimensional, isotropic, linear frame element with six

degrees of freedom at each node. It includes the effects of axial, shear, bending and

torsional deformations. This element may have the end moment releases and self mass

assigned in the form of mass per unit volume. The frame element is capable of handling

inertial, nodal, point, trapezoidal and uniformly distributed loads. It can also calculate the

element forces at the nodes for the applied load.

SHELL: This is a three or four node linear orthotropic shell element with six degrees of

freedom at each node. The element is considered to be made of a plate bending element

and a membrane element. The plate bending element is based on Discrete Kirchhoff

Theory (DKT) and has two rotations and a transverse displacement at each node. The

membrane element consists of two linear translations and a fictitious drilling degree of

freedom. It is anticipated that the membrane action is going to be predominant in the

calculation of the response of the diaphragm and shear walls under lateral loading;

however, shell elements are chosen to keep the model as general as possible. The mass of

the element is assigned as mass per unit volume. The element is capable of handling

inertial, nodal and uniform pressure loading and calculates the in-plane stresses for the

applied load.

NLLINK: This is a zero length link element which may be used to connect two overlapping

nodes or a single node to the ground (considered as the second node). The element

contains six internal springs and has six active degrees of freedom at each node. The

nllink element can be of type gap, hook, trilinear or modified Stewart. The type defines

the property of the internal springs. Also, these elements may be defined to orient

themselves about their axis, in which case they are called oriented nllink elements. This

orientation property is useful in modeling dowel type fasteners in LFWS and is discussed

in detail in the shear wall model section. The nllink element may have mass assigned to

it, which gets evenly distributed among the nodes. The element is capable of handling

inertial and nodal loads, and calculates the element nodal forces for the applied load.

34

FE SHEAR WALL MODEL: A shear wall is modeled using frame, shell and nllink

elements as shown in Figure 3-8. All these elements are put together in the center-plane

of the wall to create the finite element model. Modeling in the center-plane here implies

that the offset between sheathing and framing center-planes is not considered in the

model. The frame elements incorporating axial, bending, shear and torsional stiffnesses

are used for modeling plates, sills, blockings and studs. A shear wall typically has double

side studs and a double top plate, and if there is any opening then it is also surrounded by

the double studs. To model the double studs and double plates, a single framing element

with the combined dimensions is used and hence ignores the interaction occurring

between the studs. The shell elements represent the sheathing panels in the shear wall,

and always only one layer, with combined thickness for multiple sheathing layers

ignoring sheathing interaction, is used irrespective of the number of sheathings. The use

of the shell element ensures the modeling of in-plane and out-of-plane bending behavior

of the sheathings. For a given shear wall, shells and frames are generated such that they

have overlapping nodes at the locations of the connections. These overlapping nodes at

the connector locations are joined using the modified Stewart nllink elements, which may

have a single spring in just one direction or a pair of non-oriented uncoupled springs in

orthogonal directions or a pair of oriented uncoupled springs. The single spring

representation cannot capture the bidirectional motion under cyclic loading and may

cause numerical difficulties near the ultimate load. The disadvantage with the non-

oriented approach involves uncoupling of the connection stiffness in the orthogonal

directions. For example, if the spring in one direction fails, the spring stiffness in the

other direction may remain active, showing no loss of capacity of the connection. The

real behavior of connections is quite complicated, as each connection moves in a different

direction when the loading is applied. Judd and Fonseca (2005) present a new oriented

spring pair model which seems to correlate better with the experimental response when

compared with single spring and non-oriented spring pair models. The orientation of the

oriented spring pair in a shear wall is the angle which the nail trajectory forms with the

35

horizontal in the plane of the shear wall. This is shown in Figure 3-9 and it assumes that

the sheathing and the frame nodes connected via fasteners remain in the plane of the

shear wall. The hysteresis parameters for the modified Stewart springs are typically

obtained from the experiments. The connection between the vertical studs and the top

plate and bottom sill are modeled using moment releases. If blockings are used, then they

too are assigned the end releases. If a shear wall has two or more sheathings, then the gap

between the sheathings can also be modeled by using the gap nllink element. This

element connects the adjacent nodes of the sheathings using a single spring which is

oriented in the direction perpendicular to the overlapping edges. The stiffness value of

this spring is assigned a high number compared to other structural element stiffness

values in the system, when the gap between the sheathing reduces to zero, and otherwise

it is assigned a very small numerical value.

FE DIAPHRAGM MODEL: The floor and roof diaphragms are made of joists, blockings,

sheathings and nails. They have the same structural ingredients and similar modeling

approach as a shear wall, and they differ mainly in their boundary conditions and their

global orientation. A finite element model of a diaphragm is shown in Figure 3-10. The

joists and blockings are modeled using frame elements and have releases applied at the

ends as in the shear walls. The framing for the joist is assumed to be continuous from one

shear wall to another and no splice connections are modeled. Whenever there is an

opening in a diaphragm, a double framing is considered around it. The shell elements are

used for the sheathings which are placed over the framing in some staggered or non-

staggered pattern. The shell element nodes are connected to the framing nodes at the

location of the fasteners using modified Stewart nllink elements with oriented spring

pairs. The orientations of these springs are calculated in the plane of the diaphragm as is

done for the shear walls. A diaphragm model may also have gap nllink elements to

represent sheathing to sheathing interaction. To have this diaphragm behave as a rigid

body, it has to behave as a single unit with no internal deformations in diaphragm

36

framings, shells and nllinks. This is accomplished by assigning very high stiffness values

to the properties of these elements.

FE HOUSE MODEL: To assemble a 3D finite element house model, the diaphragm and

shear wall models as created using the above approaches are put together and connected.

These connections between diaphragms and walls in reality may be of various types;

however, in our models we consider only toe nails. We model these toe nails using

nllinks with oriented springs. These nails connect the framing of the diaphragm to the top

plates of shear walls. This connection transfers forces from the diaphragm to the walls

and completes the load path between diaphragm and shear walls. The shear wall to shear

wall connection may also be made via nllinks representing nails. To transfer the forces

between the walls and the foundation, restraints are used which approximately represent

the vertical tie downs. At the tie down locations in a shear wall, the nodes are restrained

to move in all the directions except for the out of plane rotations. These restraints may

also be replaced by the nllink elements if their force-deformation properties may be

obtained or reasonably assumed.

HOUSE MODEL’S BASIC DESCRIPTION: The above methodology is used to create

various house models for the parametric study of diaphragm flexibility under various

seismic loadings. As several house models are created, a classification of the house

models based on their shear wall configuration and aspect ratio is done in order to get a

more organized picture of the problems and the analyse’s results. Figure 3-11 shows the

classification chart of the finite element house models devised for the parametric study.

Only the rectangular box shaped house models with varying floor plan aspect ratios

ranging from 1 to 5 are chosen for the analysis. They are classified as Type 1, Type 2,

Type 3, Type 4, Type 5, Type 6 and Type 7, and within these types various models, i.e. 1,

2 etc., with varying input model parameters are created. These models are based on the

arbitrary chosen dimensions and properties; however, they do conform to the typical

LFWS constructed in various parts of North America. The model inputs are obtained

37

from the design codes, theses and journal papers published in the past. In all the models

the self-mass of the frames and sheathings is included, as the mass per unit volume of the

wood and some additional superimposed mass is also always applied on the roof

diaphragms of all the models. The details of the basic types are discussed in the following

sections. The list of models within the various types, their analysis and responses are

discussed in Chapter 5. Also, the numbers presented below for the properties, dimensions

etc. are for model 1 in each house type.

RECTANGULAR TYPE 1: This is a 20 ft x 20 ft single story boxed shaped house, the

plan configuration of which is shown in Figure 3-12. It has eight shear walls on its

perimeter and an interior shear wall, all of which support a roof diaphragm at their top.

The openings on the perimeter are covered by the vertical studs, as can be seen in the

corresponding finite element model shown in Figure 3-13. All the shear walls are 8 ft x 8

ft which are made of two oriented strand board sheathings and 1.5 in. x 3.5 in. stud

framings connected via 2.67 mm spiral nails. All the walls have a double top plate and

double side studs and have a mid height blocking. The roof diaphragm is one single

subassemblage in which the principal joists of nominal size 1.5 in. x 9.5 in. run along the

Y axis direction with 24 in. center to center spacing between them. The blockings of the

same size and spacing as the principal joist are used in the diaphragm which runs across

the principal joist. The diaphragm is composed of ten and five sheathings of sizes 4 ft x 8

ft and 4 ft x 4 ft, respectively, which are placed over the joists and blockings and are

connected together using 2.67 mm spiral nails. The perimeter and field nail spacings for

walls and diaphragms are kept as 6 in and 12 in, respectively. The diaphragm framing is

connected to the shear wall’s top plates below it using 2.67 mm spiral nails spaced at 6 in

center to center.

RECTANGULAR TYPE 2: This is a 32 ft x 16 ft single story boxed shaped house having

the plan configuration as shown in Figure 3-14. It has six shear walls on the perimeter

and one shear wall in the mid-interior, all of which support a roof diaphragm on their top.

38




framing connected via 2.67 mm spiral nails. All the walls have a double top plate and





the principal joist. The diaphragm is composed of sixteen sheathings each of size 4 ft x 8

ft which are placed over the joists and blockings and are connected together using 2.67

mm spiral nails. The perimeter and field nail spacings for walls and diaphragms are kept

as 6 in. and 12 in., respectively. The diaphragm framing is connected to the shear wall’s

top plates below it using 2.67 mm spiral nails spaced at 6 in. center to center.

RECTANGULAR TYPE 3: This is a 36 ft x 12 ft single story boxed shaped house with

the plan configuration as shown in Figure 3-16. It has six shear walls on the perimeter

and one shear wall in the mid-interior, all of which support a roof diaphragm at their top.









the principal joist. The diaphragm is composed of twelve and three sheathings of sizes 4

ft x 8 ft and 4 ft x 4 ft, respectively, which are placed over the joists and blockings and

are connected together using 2.67 mm spiral nails. The perimeter and field nail spacings

for walls, diaphragms are kept as 6 in. and 12 in., respectively. The diaphragm framing is

39

connected to the shear wall’s top plates below it using 2.67 mm spiral nails spaced at 6

in. center to center.

RECTANGULAR TYPE 4: This is a 40 ft x 8 ft single story boxed shaped house with the

plan configuration as shown in Figure 3-18. It has six shear walls on the perimeter and

one shear wall in the mid-interior all of which support a roof diaphragm at their top. The

openings on the perimeter are covered by the vertical studs, as can be seen in the





subassemblage in which the principal joists of size 1.5 in. x 9.5 in. run along the Y axis

direction with 24 in. center to center spacing between them. The blockings of the same

size and spacing as the principal joist are used in the diaphragm which runs across the

principal joist. The diaphragm is composed of ten sheathings each of size 4 ft x 8 ft which

are placed over the joists and blockings and are connected together using 2.67 mm spiral

nails. The perimeter and field nail spacings for walls, diaphragms are kept as 6 in. and 12

in., respectively. The diaphragm framing is connected to the shear wall’s top plates below

it using 2.67 mm spiral nails spaced at 6 in. center to center.

RECTANGULAR TYPE 5: This is a 20 ft x 20 ft single story boxed shaped house the

plan configuration of which is shown in Figure 3-20. It has six shear walls on its

perimeter and an interior shear wall all of which support a roof diaphragm at their top.




framings connected via 2.67 mm spiral nails. All the walls have a double top plate and



40



the principal joist. The diaphragm is composed of ten and five sheathings of sizes 4 ft x 8

ft and 4 ft x 4 ft, respectively, which are placed over the joists and blockings and are

connected together using 2.67 mm spiral nails. The perimeter and field nail spacings for

walls, diaphragms are kept as 6 in. and 12 in., respectively. The diaphragm framing is

connected to the shear wall’s top plates below it using 2.67 mm spiral nails spaced at 6

in. center to center.


the plan configuration as shown in Figure 3-22. It is the same model as Type 2 with only

its interior wall parallel to the shorter side and shorter dimension along the X axis instead

of Y. The finite element model of this house is shown in Figure 3-23.


the plan configuration as shown in Figure 3-24. It is the same model as Type 4 with only

its interior wall parallel to the shorter side and shorter dimension along the X axis instead

of Y. The finite element model of this house is shown in Figure 3-25.

MODEL GENERATION: A finite element automatic model generator program named

Wood House Finite Element Model Generator (WHFEMG) is employed to create all the

above house models. This program is developed in Visual Basic 6.0 and implements the

modeling methodology described above to create finite element models of shear walls,

diaphragms and wood houses. The interface of the WHFEMG program is shown in

Figure 3-26 and it requires a variety of information before it creates a FE model of an

individual subassemblage or a house. The inputs required are the properties of the

elements and the subassemblage’s geometric information. The element properties include

material, frame section, sheathing, nail and connector details. The geometric information

contains the boundary corner co-ordinates, sheathing center location, sheathing size, stud

41

properties, nail spacing, etc. of the constituting subassemblages in global space. The

house models are generated by assembling the individual diaphragm and shear walls

finite element models using intercomponent connectors. Currently, the program can only

create models for box shaped houses with various wall locations both at the boundaries

and the interior. More details on the program can be obtained from the program’s users

manual described in Appendix D. The output generated is in the form of nodes, frames,

shells, nllinks, materials, etc. and are stored in an .S2K extension file which serves as the

input for the WoodFrameSolver program. The mesh sizes for the models are

automatically calculated by the program and are based on the perimeter, field nail, and

stud/joist spacing of the walls and diaphragms. Generally, the model dimensions, stud

spacing, nail spacing and, sheathing sizes are chosen such that a conformal mesh with a

reasonable number of elements is generated. Table 3-1 presents the list of number of

elements and degrees of freedom in all the finite element house models presented above.

SUMMARY: This chapter discusses the various structural ingredients and behavior of

LFWS under lateral loading. The analytical models of LFWS and their subassemblages

are obtained by explicit modeling of framing, sheathing and connections using finite

elements. The modeling approach is adapted from the work done by other finite element

researchers in the past. However, the presented models are more detailed, as they

incorporate almost every connection and capture full 3D behavior of all the structural

elements. The connections are modeled using more accurate oriented springs, which is

also an improvement over previous models. The models devised for the parametric study

are rectangular box shaped houses with different floor plan aspect ratios. The house

dimensions are randomly chosen and the properties of the structural elements are

obtained from various published texts. As creating a detailed finite element model by

hand is a very time consuming task, a special automatic wood house modeler is

developed for this purpose. This has resulted in much time saving from the model

development perspective. Seven types of rectangular house models are discussed in this

42

chapter. These rectangular house models are subjected to detailed parametric study under

seismic loadings in Chapter 5.

43

REFERENCES:

1. Collins, M., Kasal, B., Paevere, P., and Foliente, G. C. (2005). "Three Dimensional


Engineering, 131(4), 676-683.







5. Folz, B., and Filiatrault, A. (2004). "Seismic Analysis of Woodframe Structures I:

Model Formulation." Journal of Structural Engineering, 130(9), 1353-1360.

6. Ge, Y. Z. (1991). "Response of Wood-Frame Houses to Lateral Loads," MS Thesis,

University of Missouri, Columbia.

7. Gupta, A. K., and Kuo, G. P. (1987). "Modeling of a Wood-Framed House." Journal


8. He, M., Lam, F., and Foschi, R. O. (2001). "Modeling Three-Dimensional Timber

Light-Frame Buildings." Journal of Structural Engineering, 127(8), 901-913.

9. Jizba, T. D. (1978). "Sheathing Joint Stiffness for Wood Joist Floors," MS Thesis,

Colorado State University, Fort Collins.



Engineering, 131(2), 345-352.

11. Kamiya, F. (1988). "Nonlinear Earthquake Response Analysis of Sheathed Wood

Walls by a Computer-Actuator On-Line System." International Conference on

Timber Engineering.

12. Kasal, B., Leichti, R. J., and Itani, R. Y. (1994). "Nonlinear Finite-Element Model of

Complete Light-Frame Wood Structures." Journal of Structural Engineering, 120(1),

100-119.

44

13. Lindt, J. W. v. d. (2004). "Evolution of Wood Shear Wall Testing, Modeling and

Reliability Analysis: Bibliography." Practice Periodical on Structural Design and

Construction, 9(1), 44-53.

14. Medearis, K., and Young, D. H. (1964). "Energy Absorption of Structures under

Cyclic Loading." Journal of Structural Division, 90(1), 61-91.

15. Schmidt, R. J., and Moody, R. C. (1989). "Modeling Laterally Loaded Light-Frame


16. Stewart, W. G. (1987). "The Seismic Design of Plywood-Sheathed Shear Walls,"

PhD Dissertation, University of Canterbury, Christchurch, New Zealand.

17. Tarabia, A. M., and Itani, R. Y. (1997). "Static and Dynamic Modeling of Light-

Frame Wood Buildings." Computers and Structures, 63(2), 319-334.

18. Yoon, T. Y., and Gupta, A. (1991). "Behavior and Failure Modes of Low-Rise Wood-

Framed Buildings Subjected to Seismic and Wind Forces," Final Report Submitted to

National Science Foundation, Department of Civil Environmental Engineering, North

Carolina State University, Raleigh.

45

Figure 3-1: A wood house with horizontal floor and roof

Figure 3-2: A wood house with horizontal and sloped roof

Garage Door

Door

Window

Roof

Floor

Wall

Foundation

Sloped Roof

Wall

Door

Garage Door

Foundation

Roof

Window

Sloped Roof

Wall

Door

Garage Door

Foundation

Roof

Window

46

Figure 3-3: Floor diaphragm with sheathing panels stacked along the edges in a

checkerboard format (no blockings present), thick line in the figure represents panel

boundary

Figure 3-4: Floor diaphragm with sheathing panels put together in non-checkerboard

staggered manner (no blockings present), thick line in the figure represents panel

boundary

Joists

Sheathing Panel

Panel Edge Nails

Panel Interior Nails

Joists

Sheathing Panel

Panel Edge Nails

Panel Interior Nails

47

Figure 3-5: Shear wall (no blockings)

SillTie Down Anchor

Double Studs

Sheathing Panel

Sheathing to Framing Connector

Double Plate

Anchor Bolt

48

Figure 3-6: Load distribution

49

-5

-4

-3

-2

-1

0

1

2

3

4

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

SLIP (in)

LO

AD

(K

ip)

Figure 3-7: Load-Slip response of fastener under cyclic loading

Figure 3-8: Shear wall finite element model (viewed in SAP2000)

NLLINK

FRAME

SHELL

50

Figure 3-9: Different nail trajectories (shown in arrow and dotted line) in a shear

wall (α1, α2, α3). An angle α is calculated between the horizontal and the nail trajectory

at the initial load

Figure 3-10: Floor or roof diaphragm finite element model (viewed in SAP2000)

NLLINK

FRAME

SHELL

α1

α3

α2

SHEATHING

FRAMING

NAIL

X

Z

α1

α3

α2

SHEATHING

FRAMING

NAIL

X

Z

51

RECTANGULAR

LIGHT FRAME WOOD HOUSE

TYPE 1 TYPE 2 TYPE 3 TYPE 4

1

2

1

2

1

2

1

2

n n n n

Type 5

1

2

n

Type 6

1

2

n

Type 7

1

2

n

Figure 3-11: LFWS classification chart for the parametric study

Figure 3-12: Type 1, Model 1 – Floor Plan

20’

20’

8’

8’

8’

8’

ROOF

WALL

X

Y

10’

8’

52

Figure 3-13: Type 1, Model 1 – FE Model (viewed in SAP2000)

Figure 3-14: Type 2, Model 1– Floor Plan

32’

16’

8’

8’

8’

ROOF

WALL

X

Y

8’

8’

12’

53



36’

12’

8’

8’

8’

ROOF

WALL

X

Y

8’

6’

14’

54



40’

8’

8’

8’

ROOF

WALL

8’

4’

16’

X

Y

55



20’

20’

8’

8’

8’

ROOF

WALL

X

Y

10’

8’

56



X

Y

32’

16’

8’

8’

8’

WALL

4’

ROOF

57



40’

8’

8’

ROOF

WALL

X

Y

58


Figure 3-26: WHFEMG program interface

59

Table 3-1: Number of finite elements and degrees of freedoms in parent house models

MODEL NAME FRAME SHELL NLLINKS DOF

TYP1M1 2192 3904 2002 40104

TYP2M1 2656 3840 1987 42030

TYP3M1 2496 3520 1854 38976

TYP4M1 2272 3072 1621 34526

TYP5M1 2192 3392 1904 36972

TYP6M1 2720 3840 2011 42366

TYP7M1 2272 3072 1621 34526

NUMBER OF

60

CHAPTER 4 WOODFRAMESOLVER: A HIGH PERFORMANCE

NONLINEAR FINITE ELEMENT ANALYSIS PROGRAM

INTRODUCTION: An experiment performed on a full scale wood house or its

subassemblages is an expensive procedure compared to the cost involved in the

analytical modeling. On one hand, the experiments provide information which is

useful in understanding the behavior and design of wood structural systems. On the

other hand, the analytical modeling is used to implement the observed behavior so

that a wider variety of structures and their response may be studied. Analytical

modelings of wood structures have evolved only in the last 35 years and have

undergone various stages of refinements. It started from development of simple

formulas and has reached a stage where researchers are proposing complex finite

element models. Finite element analysis of LFWS may play an important role in

developing more rational design procedures. Kasal et al. (2004) present various

methods of LFWS lateral force analysis and shows that the finite element method is

the most inclusive and accurate of all the existing analysis procedures. The finite

element approach involves explicit modeling of structural members with their

geometric and material properties, and hence captures the behavior more realistically.

The structural elements required in the construction of a LFWS are (1) various types

of stud and sill framings, (2) sheathing panels and (3) various types of component

connections. These elements are put together to behave as a unit, and simultaneously

form a very complex and highly redundant system. LFWS have irregular geometries,

anisotropic material properties of the wood, and the connectors’ behavior is generally

nonlinear inelastic. The explicit modeling of these elements and their material

properties makes the problem challenging for finite element modeling and analysis.

The early finite element models of LFWS were simplified due to the high cost of

computation involved in solving complex models. Today, even the desktop CPU

architectures support high performance computing, and this has led to incorporation

of more details in the latest models. The latest models are capable of simulating full

three dimensional behaviors of structural elements with complex material properties

61

and hysteresis behavior of the connections. However, there are not many dedicated

tools to perform such analysis for LFWS. In recent publications by Collins et al.

(2005) and Lam et al. (2004) and in Chapter 2, it is noted that most of the available

tools do not fulfill the current requirements of wood structural engineers. The

available research tools require a complicated approach to incorporate hysteretic

elements and do not provide detailed response data. Moreover, as researchers

continue to propose new models for wood houses, there is also a need for a flexible

and extensible development platform which may accommodate the new formulations.

It is found that most of the latest programs, supposedly the “state of the art” in light

frame wood structures analysis, lack the characteristics of flexibility and extensibility

as they follow a procedural programming approach.

The objective of this chapter is to present the development of a high performance

nonlinear static and dynamic finite element analysis program called

WoodFrameSolver. This program is written on the Visual Studio .Net platform using

mixed language programming involving object-oriented C++, C and FORTRAN. It

has a wide collection of linear, nonlinear and hysteresis elements commonly used in

LFWS analysis. A unique aspect of the program lies in its capability of capturing

elastic displacement participation (sensitivity) of spring, link, frame and solid

elements in static analysis. The program can easily accommodate the numerical

solution procedures, materials and elements being devised by the research

community. The chapter presents various aspects of the program, which are divided

into four sections. The first section describes the importance of using object-oriented

concepts in a finite element program development. The section does not cover the full

details on object-oriented programming (OOP), but appropriate references are

provided for the more interested readers. This is followed by a discussion of the

WoodFrameSolver architecture and its performance. The program’s performance and

accuracy are similar to that of SAP 2000 (CSI 2000), which is chosen as the

benchmark for validating the results. The third section presents the program features,

where all the elements and analysis capabilities are discussed. The fourth section

presents references and example problems of the program verification, and is

followed by the summary and conclusions.

62

WHY OOP? In object oriented design, all entities are denoted as objects. Each object

constitutes a number of attributes and behavior that define the purpose and state of the

object. An attribute of an object may be represented as a simple integer, a floating

point, a string or another object. These are mostly private to the object and are

invisible to the outside world. This aspect of information hiding is known as

encapsulation. The behavior of an object is defined by methods, which are the

procedures which manipulate or return the state of the object. Objects interact by

sending messages to each other. Messages along with any arguments that accompany

them constitute the public interface of an object. An object reacts to the message by

executing one or more statements, commands, or methods and may return a response

depending on the interface. Objects that have similar attributes and methods are

grouped together into a class. A popular feature in which one object inherits the non-

private attributes and methods of another object is called inheritance. This requires

creating a new class as an extension of an existing class. The existing class and the

new class are also referred to as the base and child classes, respectively. The feature

of extending the base class saves additional programming effort and avoids

duplication. The child class can also be treated as an object of the base class, but not

vice-versa. In object oriented programming it is possible to have methods with the

same name, inside different classes, or inside the same class with different type of

data passed to it. This feature is called polymorphism. Polymorphism can be applied

to operators, too, and in such a case it is known as operator overloading. Writing an

object oriented program requires detailed planning and a thorough understanding of

the problem to ensure the right selection of classes, messages, attributes, responses

and so on. Only the interface is visible to the other parts of the program, and any

changes performed underneath don’t affect the other parts of the code. An object

oriented design provides a sound platform for maintenance, reusability and

extensibility, which are very important norms to judge the performance of a program.

Most of the finite element programs developed in the past consisted of several

thousand lines of procedural code which are difficult to modify and extend. The

object oriented programming paradigm is relatively new and it provides a list of

63

advantages over procedural programming (Archer 1996, Lu 1994, Mackie 1992,

McKenna 1997 and Pidaparti and Hudli 1993). Finite element analysis requires

domain discretization of a problem, which naturally lends itself to the creation of

objects, i.e. nodes, elements, restraints, constraints, sections, materials, etc. The

different loads and analysis types can also be broken down into objects. This idea has

been recognized by the researchers in the past (Dubois-Pelerin and Pegon 1998,

Menetrey and Zimmermann 1993, Miller 1988, Peskin and Russo 1988). Moreover,

light frame wood finite element analysis tools are still evolving and OOP seems to be

a remarkably good design approach for them.

PROGRAM ARCHITECTURE AND PERFORMANCE: The fundamental

architecture for the WoodFrameSolver program is mainly derived from the works of

Archer (1996), Fenves (1990), Forde et al. (1990) and McKenna (1997). The essential

components of the program are abstracted out into a class structure and are presented

in Figure 4-1. This is also called an entity relationship diagram (Rumbaugh et al.

1991) and is used to represent the relationship between the various classes. A box

with a diamond and a link below it is an aggregate class and an object of this class

contains objects of all the classes connected to the link. For example, the static

analysis class is an aggregate class of numberer and SOE solver classes. A box with a

link and a small circle is a class having many pointers of other classes connected to

the circle. For example, an element class object may have pointers to many node

objects. A box with a link and a small triangle is a class having just one pointer of

other classes connected to the triangle. For example, an element class object will

always have a pointer to just one material object. Finally, a box with a dotted link and

arrow connecting the link to other boxes represents a base class. The classes at the

other end of the link are called the child classes. For example, an element class is a

base class for shell, frame, spring, solid and nllink element classes. The public

interfaces of the classes depicted in Figure 4-1 are presented in Appendix A, which is

the architecture document of this program.

The implementation of the program is facilitated by the use of object oriented C++.

The program requires containers to store the input/output data, iterators to iterate over

64

the input to perform the analysis and various other operations, and algorithms to

process the output. The standard template library (Stepanov and Lee 1995),

abbreviated as STL, is used for these purposes in all parts of the program. STL is a

generic library and provides container classes, algorithms, and iterators. The library is

not just functionally useful, but extremely powerful when used for computational

purposes.

Three main steps are involved in the solution of a problem using WoodFrameSolver

and they are as follows: (a) read the input and populate the model object, (b) perform

the analysis and (c) print the response. A successful execution of these three steps

involves the interaction of filereader, echo, modelbuilder, model, analysis cases, and

response classes at the top level. The model class is a key component as it is the most

accessed object during the execution of the program. A model object interacts with a

modelbuilder, analysis and response objects. It is the container for all the attributes of

nodes, elements, restraints, constraints, sections, materials and load cases, etc. Table

4-1 shows a list of few selected node attributes defined inside the WoodFrameSolver

node class. An associative container called map from STL is used for storing the

objects inside the model class. Map requires the user to define a key (accessor) object

as the first entry and the input object (value) as the second. The input objects inside

the map are stored through their memory address, i.e. pointers. This provides fast

access, as the related objects (e.g. element and nodes) are dynamically linked to each

other. The modelbuilder class is an interface between the filereader class and the

model class. The modelbuilder object populates the model object with the input data

for the analysis. These data are read inside the filereader class. The other function a

modelbuilder performs is that it interacts with an echo object on the fly to echo the

read data in a text file and to check if the input information is correct or not. If the

modelbuilder object finds any discrepancy with the input data, it passes an error

message to the echo object. The echo object stores the list of error messages to be

printed at the end of all the echoed data inside an echo file. If there are any input

discrepancies, the program flags an error and discontinues.

65

The analysis objects are used to perform the requested analysis, which could be of

type static or dynamic or both. A static analysis object interacts with the model,

degree of freedom numberer, system of equations solver and the response objects. The

model object provides access to the input data, which are worked upon by the

numberer and the system of equation solver objects to obtain the responses. The

numberer object uses the requested numbering scheme to number the nodal degrees of

freedom. This is useful when one desires to reduce the bandwidth or the profile of the

global stiffness matrix of the input model. The solver object performs three tasks, i.e.

(1) create the nodal load vectors using nodal and element loads, (2) assemble the

global stiffness matrix using the element global stiffness matrix and (3) solve the

resulting system of equations to obtain the nodal displacements. The numberer and

solver steps are orchestrated inside the static analysis class object, the few steps of

which are shown in Table 4-2. The obtained displacements are mapped to the

corresponding nodes using the degree of freedom numbering. As all the elements

contain the pointers to their nodes, these displacements are directly accessed inside

the elements to calculate the element forces and stresses. The static analysis responses

are printed inside an output file using various methods from the response class. These

methods are also called inside the static analysis class.

A dynamic analysis object interacts with model, numerical integrator, modal analysis

and the response objects. The modal analysis object is of type Eigen and is used to

obtain the natural frequencies and mode shapes of the input model. This object

assembles the mass and stiffness matrix which are solved using an Eigen solver. The

numerical integrator object is used to solve the dynamic equilibrium equations either

by using mode superposition or direct integration as desired. The integrator object

may be of type linear, central difference, Newmark α, β or Runge-Kutta. In

WoodFrameSolver all these integrators can solve a linear dynamic problem using

mode superposition. However, for a nonlinear dynamic problem only the Newmark

approach with direct integration is applicable. The mathematical steps involved in

mode superposition and direct integration can be found in any structural dynamics

book. The outputs from an integrator object are the displacement, velocity and

acceleration at the time step it is solved. These responses correspond to various

66

degrees of freedom and are mapped back to the corresponding nodes and stored with

them. The element uses these displacements to obtain the element forces and stresses

at a time step, and stores it within the element. At the end of a dynamic analysis load

case, these results are printed by the response object into output files. All the calls to

the modal, numerical integrator and response objects methods are made inside the

dynamic analysis class object.

All finite element programs require solving of a system of equations (SOEs) while

performing static and dynamic analysis. The time taken to solve SOEs is a critical

measure of the program’s performance, especially in repetitive analysis like dynamic

analysis. The WoodFrameSolver program has four different types of solvers. Out of

these four, the Banded, Direct Sparse Solver (DSS) and Parallel Direct Solver

(PARDISO) are direct solvers and are purchased from INTEL (INTEL 2005). They

are embedded in the form of a dynamic link library inside the program. These direct

solvers are designed to be super-efficient in solving large sparse symmetric and

unsymmetric linear system of equations on Intel Pentium processors. The PARDISO

solver is written using OpenMP and reduces the equation solution time on a

symmetric multiprocessor (SMP) or multi-core systems. The numerical experiments

using this solver have shown that the scalability of the parallel algorithm is almost

independent of the multiprocessing architecture and gives a speedup of up to seven

using eight processors (INTEL 2005). The steps involved in solving SOEs using DSS

and PARDISO are primarily broken down into six phases, i.e. solver initialization,

non-zero structure definition, reordering to reduce fill-in, numerical factorization and

solve equations. As the user has the flexibility of accessing these phases

independently, it provides a passage to omit the phases required just once in the

repetitive analysis. For example, modified Newton Raphson inside a time step may be

accomplished by just using the solve equations phase. In some dynamic analysis

problems the non-zero structure of the global tangent matrix may never change

throughout, and hence a non-zero structure definition method needs to be executed

only once. This decomposition has been efficiently used inside the WoodFrameSolver

program and has resulted in reduced solution time when compared with SAP 2000

without compromising any accuracy. It is also noted that DSS and PARDISO solvers

67

do not require renumbering of equations, which results in additional time saving in

solving big problems. The fourth SOE solver inside WoodFrameSolver is based on

the conjugate gradient (CG) scheme with a basic Jacobi preconditioning. For big

problem sizes where computer memory limit becomes an issue, this solver has been

found experimentally useful as it is very memory efficient when compared to its Intel

counterparts, but sometimes may take more time depending upon the problem type.

The solve time for a conjugate gradient method depends upon its solution

convergence rate, which is dependent upon the type of preconditioning scheme used.

The Jacobi preconditioner is the simplest and the least efficient of other available

preconditioners (Golub and Loan 1996). If unlimited memory is assumed, the Intel

direct solvers would be the most solve-time efficient for bigger problems because the

CG solver is often limited by their convergence rate. Hence, this solver is not

implemented for dynamic analysis in the program. In fact, currently only the DSS or

the PARDISO solvers may be used for dynamic analysis in WoodFrameSolver.

The WoodFrameSolver’s performance and accuracy are compared for various

benchmark models with SAP 2000. These benchmark models include problems

resulting in large-degree-of-freedom and bandwidth systems. Some of these

benchmark models are depicted in Figure 4-2. Figures 4-3 present a comparison of

problems of large bandwidth systems with bandwidth ranging from 3624 to 8448 and

the corresponding number of equations varying from 217,08 to 118,188. Figure 4-4

presents a comparison of large equation systems ranging from 30,651 to 275,451 and

the corresponding bandwidth varying from 607 to 5407. The plots show that a good

performance ratio is obtained for WoodFrameSolver. The comparisons of

displacements and forces for all the models are also at par when compared with

SAP2000 and are presented in Appendix C, which is the verification manual of

WoodFrameSolver program.

PROGRAM FEATURES: The WoodFrameSolver program is written specifically

for analyzing light wood frame structures; however, it is equally applicable to other

types of structures that can be modeled within the element and analysis capabilities of

the program. This section describes the element library and analysis cases that can be

68

handled by the program. The material property is always linear elastic and of type

isotropic for the frame elements. The shell and solid elements may have linear elastic

and orthotropic properties. If the loading applied on the structure being analyzed is of

type nodal, then the frame, spring and link elements in the element library below can

compute their displacement participation contribution to the displacement at any node

in the structure. This requires setting up a virtual work load case in the input model.

ELEMENT LIBRARY:

FRAME: This is a two node line element as shown in Figure 4-5 and may be used to

model two and three-dimensional trusses, beams and columns. It includes the effects

of axial, shear, bending and torsional deformations. The user can set end offsets in the

element to account for the actual dimensions when used in modeling real structures.

These offsets can be made fully or partially rigid. One can apply any valid releases at

the ends of this element. The element can also handle self weight, nodal loads, point

loads and trapezoidal loads. The element calculates the frame end forces and prints it

out in the output file for various load cases applied.

SHELL: This is a three or four node triangular or quadrilateral element as shown in

Figure 4-6. It may be used to model plate, membrane or shell behavior in two-or-three

dimensional structures. The element has six degrees of freedom at each node and is

considered to be made of a plate bending element with two rotations and a transverse

displacement, and a membrane element with two in-plane translations and a fictitious

drilling degree of freedom (θz). The plate bending element is based on a Discrete

Kirchoff Theory (DKT) element (Cook et al. 1989). The element is capable of

handling nodal loads and uniform pressure loading on any of its faces. Currently, it

only generates the membrane stresses for the load cases applied and prints them in the

output file.

8 NODE BRICK: This is an eight-node three-dimensional isoparametric linear brick

element as shown in Figure 4-7 and is used to model solid structures. The element has

only three translational degrees of freedom at each node and has nine incompatible

bending modes. The element contributes stiffness to all the translational degrees of

69

freedom. The element can handle only the nodal loads in the translational directions.

The element generates three normal and three shear stresses for the applied loads and

prints them in the output file.

SPRING: This is a one-node three-dimensional linear spring element. The spring

element connects the joint to the ground and contributes to the three translational and

three rotational stiffnesses at the joint. The element has six uncoupled deformational

degrees of freedom. Figure 4-8 shows three of the six independent springs. The

element generates the spring forces for the loads applied and prints them in the output

file.

NLLINK ELEMENT: This is a one or two-node, zero or finite length nonlinear link

element. In general, nllink elements are used to model local structural nonlinearities

when a nonlinear analysis is performed. A link element constitutes six spring

elements, each representing a deformational degree of freedom, which may or may

not be coupled depending upon the type of behavior modeled. Figure 4-9 shows three

of six springs in an nllink element. Joint 1 in the figure is grounded in case the

element has just one joint. The zero-length link elements may also be defined to rotate

themselves about their local axis to a certain angle during a nonlinear static or

dynamic analysis. This is referred to as an oriented link element and may be required

when modeling connections in LFWS. When the link element is defined as oriented,

the program assumes that the element lies in the plane perpendicular to its axis

direction and hence calculates its initial trajectory in the perpendicular plane for the

initial load in the nonlinear static and dynamic analysis. The angle which the initial

trajectory makes with the local y-axis of the element is the orientation angle for

further analysis steps. This type of behavior represents nail connections more

realistically when used in modeling diaphragms (Judd and Fonseca 2005). The

nonlinear spring property for degrees of freedom inside the nllink element could be of

type gap, hook, simple trilinear hysteretic or modified Stewart hysteretic (Folz and

Filiatrault 2001a). These degree of freedom properties of an element behave

independently of each other. The spring stiffnesses inside an nllink element may have

different numerical values and also may be locked to behave linearly in a nonlinear

70

analysis, but the type remains the same for all six springs. The next few paragraphs

discuss these properties in detail.

GAP and HOOK: The stiffness property for Gap and Hook gets activated only when

the element goes into compression and tension, respectively. This is depicted in

Figures 4-10 and 4-11. These types of spring behavior are only exhibited when the

spring is defined as nonlinear and a nonlinear analysis is performed, in which case the

user defined nonlinear properties are used. If the spring is defined as linear, it uses the

linear spring stiffness. The nonlinear force-deformation relationship for Gap and

Hook is given by the Equations 1 and 2, respectively. Note that δ0 must be either zero

or any positive value.

( ) 00, 000 =<++= PelseifKP δδδδ (1)

( ) 00, 000 =>−−= PelseifKP δδδδ (2)

TRILINEAR HYSTERESIS: The trilinear property for any deformational degree of

freedom inside an nllink element is defined using one primary (K0) and two secondary

stiffnesses (K1 and K2). The secondary stiffnesses come into action only after the yield

points (Fyp and Fyn), as shown in Figure 4-12. Overall, five parameters are required to

define this property. The unloading of the spring always follows the primary stiffness

curve from the unloading point. If the secondary stiffnesses are equal, then a bilinear

hysteresis is obtained.

MODIFIED STEWART HYSTERESIS: This hysteresis model is developed by Folz and

Filiatrault (2001a) as a part of the CASHEW wood frame project and is useful in

modeling dowel type connections in wood structures. This property may be assigned

to any deformational degree of freedom of an nllink element. This is a ten parameter

model and its force-deformation behavior under monotonic and cyclic loading is as

shown in Figure 4-13. The monotonic loading portion of this model was initially

proposed by Foschi (1977) for modeling connections and requires six parameters

which may be obtained from the experimental data to characterize the curve. This is

71

depicted in Equation 3 and phenomenologically captures the crushing of wood,

connector yielding and connector removal. The cyclic response of this model is path

dependent and is capable of capturing pinched hysteretic behavior with stiffness and

strength degradation. The monotonic loading curve (J0J1 and J3J4) remains the

envelope curve for the cyclic loading, too. Mathematically, the paths J0J1 and J3J4 are

assumed as exponential; the remaining paths are all assumed as linear. If unloading

occurs from J0J1, the path J1J2 is followed, which has stiffness r3K0 and becomes J2J3

with stiffness r4K0 under continued unloading. On the path J1J2 the dowel connector

and the surrounding wood is assumed to unload elastically. If loading in the opposite

direction (negative displacement) is occurring for the first time, then the envelope

path J3J4 is traversed; else path J2J3 continues to J9. The unloading path J2J3

represents pinching response as the dowel connector loses contact with the

surrounding wood due to permanent deformation during previous loading. When

unloading from J3J4 occurs, the path J4J5 with stiffness r3K0, assumed as elastic, is

followed, which on further unloading becomes J5J6J7. The path J5J6J7, which has

stiffness r4K0, depicts pinched response and passes through FI (at zero displacement).

The reloading along J5J6J7 becomes J7J8, which has the degrading stiffness Kd as

shown in Equation 4. This stiffness degradation also produces strength degradation

when the connector is displaced to point J8, as can be noticed by comparing points J1

and J8 in Figure 4-13. Table 4-3 shows the complete list of parameters required by

this model, and these may be obtained by fitting this model to the actual connection

data.

ult

P

K

ifeKrPP δδδ

δ

≤

−+=

−

,1)( 0

0

010 (3a)

failultultult ifKrPP δδδδδ ≤<−+= ,)(02 (3b)

failifP δδ >= ,0 (3c)

72

0

00

max

00 ,

K

FwhereKKd =

= δ

δ

δα

(4)

ntdisplacemeunloadingpreviouswhere unun == δβδδ ,max (5)

ANALYSIS CASES: Five types of analysis cases may be performed using

WoodFrameSolver program. Some of these analysis cases require either a static load

case or a dynamic load case to be defined by the user in the input model. A static load

case may consist of nodal, frame point, frame uniformly distributed, frame trapezoidal

or shell pressure loading or any combination of these. A dynamic load case currently

may only have a combination of ground acceleration loads. The user may also define

various static and dynamic load cases for a particular input model. For a given

structure, the following are the details of the types of analysis which can be performed

using WoodFrameSolver program.

STATIC ANALYSIS: A static analysis is performed when a static load case or a

combination of static load cases are applied to the structure. The program

automatically creates the load vector/s. It also creates the global stiffness matrix from

the joint connectivity, element section and material information. The system of linear

equations is then solved to obtain the displacement vector/s. This is also the default

analysis type, and if no loads are specified it solves the problem for zero load vectors.

EIGEN ANALYSIS: An Eigen analysis is performed to obtain the undamped free

vibration mode shapes and frequencies of the system. These mode shapes are used to

calculate the modal participation factors and participating mass ratios of all the

requested modes in the global X, Y and Z direction. These mode shapes are also used

to uncouple of the dynamic system of equations when performing linear dynamic

analysis using mode superposition.

DYNAMIC ANALYSIS: A dynamic analysis is performed when a dynamic load case or a

combination of dynamic load cases and some inertial masses is applied to the

structure. To obtain the inertial force, the mass contribution by each of the elements is

73

lumped at the element joints which may also have the masses of their own, i.e. joint

mass. The program performs response history analysis using mode superposition or

direct integration. For nonlinear response history analysis, only direct integration can

be used. The user may also define a damping model to be used in a dynamic load

case. The available damping models include user defined damping coefficients, mass

proportional, stiffness proportional and Rayleigh proportional damping. If any of the

last three damping models is set to be used in a load case and a nonlinear analysis is

performed, the user may also select to update the damping matrix based on updated

stiffness matrix or damping coefficients or both. However, the program currently does

not update the effective load term when damping matrix is updated, and hence some

errors may be introduced when the damping matrix is opted to be updated. The

nonlinear dynamic analysis using Newmark direct integration always uses the Newton

Raphson (modified or full NR) method for solution convergence within a load step.

The program currently does not perform any substepping procedures combined with

NR, on the needed basis within a time step, and instead performs NR iteration in each

time step for convergence. This may result in more time consumption for certain

nonlinear problems.

INCREMENTAL DYNAMIC ANALYSIS: Incremental dynamic analysis (IDA) is one of

the popular methods in the current state of the art which is used for predicting seismic

demand and capacity of a building structure (Vamvatsikos and Cornell 2002a).

WoodFrameSolver can perform IDA analysis by running nonlinear dynamic analyses

under a series of scaled ground motions. Applying IDA to finite element models of

light wood frame structures is not yet cited in the literature and is one of the powerful

aspects of this program. However, one should note that it is a computationally

demanding procedure and may take several days to produce the desired results for full

finite element models.

VIRTUAL WORK ANALYSIS FOR DISPAR: Applying virtual work to obtain

displacement participation (DISPAR) of components is a popular concept

(Velivasakis and DeScenza 1983, Charney 1990, 1991, 1993). This requires

integration of real stresses over virtual strains to compute displacement participation

74

of the individual elements. The integration scheme is different for each element and

has been programmed in WoodFrameSolver to compute various DISPAR components

(axial, flexural, shear, torsional) of the elements in the structure to a desired nodal

displacement in a particular direction. The use of virtual work analysis to calculate

DISPAR for problems involving finite elements is relatively new (Charney et al.

2005, Charney and Pathak 2008). Also, in the author’s knowledge WoodFrameSolver

is the only program which has extended this capability to solid finite elements.

The WoodFrameSolver program is currently DOS based and requires input in a text

format which is similar to the .S2K format for SAP version 7. Most of the

WoodFrameSolver input files can be executed in SAP 2000. The format was chosen

because of the absence of any GUI available for the creation and viewing of the input

models and also for the convenience of verification with the already existing popular

program. The interface for the WoodFrameSolver program is shown in Figure 4-14.

The details on the input format and how to use the program are discussed in Appendix

B, which is the program’s user’s manual.

EXAMPLES: The program has been successfully used to analyze finite element

models of steel beam column joints (Charney and Pathak 2008) and to study detailed

wind drift of steel framed structures (Berding 2006). This section presents four

example problems where the program’s nonlinear static and dynamic analysis

capabilities are exercised. The examples also contain the verification of the

WoodFrameSolver results with some other popular programs.

VERIFICATION WITH EXPERIMENTAL AND ABAQUS ANALYTICAL MODEL:

The program is used to analyze two 8 ft x 8 ft full scale timber shear walls. These two

walls are identical except for their sheathings which are plywood and waferboard.

These walls have been experimentally tested by Dolan (1989) and have been used by

Dolan and Foschi (1991) to validate their numerical model. Judd (2005) has also used

these results to validate his oriented spring model for the sheathing to framing

connections. The structural model of the shear wall is shown in Figure 4-15, which is

modeled using frame, shell and non-oriented/oriented nllink elements. The resulting

75

finite element model is shown Figure 4-16. The actual dimensions and properties of

the framing, sheathing and connections used in the wall models are presented in Table

4-4. A nonlinear static analysis of the shear walls is performed by using

WoodFrameSolver program’s nonlinear dynamic analysis capability. To do this, a

single degree-of-freedom (sdof) spring mass system is attached to the wall using a

massless axially flexible (rigid in bending, shear and torsion) rod at the top left node

of the shear wall. It is also assumed that no wall mass is being considered in the

analysis. The sdof spring mass system is then excited with a constant dynamic load

over a certain time interval. The mass spring system properties and loading are chosen

such that the wall top displacement is much smaller than the spring mass. This

relative displacement between the end nodes of the rod generates axial force in it

which is recorded at each time step along with the displacement of the wall top node.

The plot of this displacement versus the axial force gives the pushover curve. This

approach is displacement based and is adopted from Charney (1986), where it is

discussed in more detail and applied to framed structures. The results of nonlinear

static analysis of the analytical shear wall model are compared with the experimental

response measured by Dolan and analytical response calculated by Judd using his

oriented model in ABAQUS. These responses for plywood and waferboard are plotted

in Figures 4-17 and 4-18, and a quantitative comparison of the ultimate loads and

displacements is shown in Tables 4-5 and 4-6. The experimental results from Dolan

(1989) show a large variability in the measured ultimate loads and corresponding

displacements for the similar shear walls. The oriented model using

WoodFrameSolver overestimates the ultimate load when compared to the average

experimental responses of both plywood (9.28%) and waferboard (21.82%). The non-

oriented model also overestimates the ultimate load for both wall types. The ultimate

load comparison of the oriented model obtained using WoodFrameSolver and

ABAQUS lies within 1% of each other. Overall, the oriented spring models predict

the responses closer to experiments when compared to non-oriented spring models.

VERIFICATION WITH SAPWOOD: The analysis of a light frame wood house having

a U shaped floor plan and twelve perimeter shear walls as shown in Figure 4-19 is

considered for verification with SAPWOOD (Pei and Lindt 2006). Figure 4-20

76

depicts the analytical model of the house with the floor modeled using rigid shells and

walls represented using one joint single spring nllinks in the direction of actual

orientation of shear walls. Six bilinear and six Stewart hysteretic spring properties are

used for these walls in the model. The properties for these springs are selected

arbitrarily and are presented in Tables 4-7 and 4-8, respectively. The mass density of

the rigid floor shell elements is chosen as 0.001736 kips-sec2/in. This house model is

subjected to an arbitrarily selected bidirectional earthquake loading with PGA scaled

to 0.12g and 0.012g along X and Y directions, respectively. The applied earthquake

loading is shown in Figure 4-21. The force deformation and displacement responses

of all the walls are recorded from both the programs and compared. The displacement

and force response comparison for Wall 1 and Wall 11 are presented in Figures 4-22

to 4-25. The results obtained from WoodFrameSolver compare quite well with the

SAPWOOD response, as can be noticed from these figures.

VERIFICATION WITH SAP2000 FRAME MODEL: A three-story one-bay moment

frame with bilinear rotational hinges as shown in Figure 4-26 is considered for

verification with SAP2000. The material, section and spring properties used in the

model for frames and nllinks are presented in Table 4-9. The model is subjected to the

Loma Prieta earthquake loading and the responses are recorded for one case with 0%

damping and the other with 2% damping in modes 1 and 3. The results obtained from

both the programs for these cases are compared in Figures 4-27 to 4-30. All the

responses obtained from WoodFrameSolver are within 1% of the responses obtained

from SAP.

VERIFICATION WITH SAP2000 3D HOUSE MODEL: A light frame wood garage

type structure of size 8 ft x 8 ft x 8 ft is considered for analysis and verification with

SAP. The structure has three similar walls and one roof diaphragm as shown in Figure

4-31. The wall framing consists of a double top plate, two double side studs, three

single intermediate studs, four blockings at mid height and a bottom sill. The studs are

spaced at 2 ft. Each wall has two sheathings each of size 4 ft x 8 ft which are attached

to the framing using the nail connections spaced at 6” on the perimeter and 12” in the

field. The roof joists and blockings have similar cross-sections and their lengths are

77

spaced at 2 ft intervals. The roof has four sheathings each of size 4 ft x 4 ft which are

attached to its framing similar to wall. The roof is attached to the walls on its

perimeter with the toe nails spaced at 6” intervals. The finite element model of this

house is created using the method described in Chapter 3 and is depicted in Figure 4-

32. The studs, plates, sills, joists and blockings are modeled using frame elements.

The sheathings are modeled using the shell elements and they are attached to the

frames using nllinks. The nllinks have non-oriented bilinear spring pair in the plane of

the subassemblage and a linear spring in the perpendicular direction. The material,

section and spring properties for frames, shells and nllinks used in the FE model are

presented in Table 4-10. The mass density of the frames and sheathings is taken as

5.144 x 10-8

kips-sec2/in

4 and an additional mass of 0.06 kips-sec

2/in is distributed on

the roof. The model is subjected to Imperial Valley ground motions with PGA scaled

to 0.5g. A user defined Rayleigh proportional damping with coefficients α=0.001 and

β=0.000001 are also used in the model. The models are analyzed in SAP2000 and

WoodFrameSolver for the first 8.3 seconds of the earthquake. The results for the X

direction displacement time history at joints 23, 450, 1322 and 1499 from both the

programs are shown in Figures 4-33 to 4-36, respectively. The force-deformation

responses of the nails 25, 383 and 505 are bilinear hysteretic and are presented in

Figures 4-37 to 4-39, respectively. Figure 4-40 presents a comparison of X direction

base shear. As can be seen from all the plots, a good match is obtained between the

results from the two programs.

SUMMARY: This chapter presented a newly developed finite element analysis

program for analyzing LFWS. The program is designed and developed in object

oriented C++, C and FORTRAN. An object oriented design philosophy is adopted to

keep the program architecture reusable, flexible, extendible and easily maintainable

by further developers. The program has all the ingredients one may require to perform

the nonlinear finite element analysis of LFWS. The verification of the static and

nonlinear dynamic analysis capabilities of the program has been done and the results

are found at par with program like SAP2000, SAPWOOD and ABAQUS. The

program has specifically been exercised to verify results of FE models of LFWS and

the prediction has been found to be in the ballpark. The program is fine tuned to solve

78

large-degree-of-freedom problems subjected to nonlinear static and dynamic loadings.

The SOEs are always solved using highly optimized equation solvers from the Intel

math kernel library, resulting in high performance. The program may also be used to

run the models that can be analyzed using the SAWS and SAPWOOD programs.

Apart from its application to LFWS, the program has been successfully used in

solving framed structures with various types of joint models. The program has also

been used in finding displacement participation of web and flanges in various finite

element models of beam column joint subassemblages. Currently, the program is

being used to investigate the diaphragm flexibility of LFWS subjected to seismic

loads.

79

REFERENCES:

1. Archer, G. (1996). "Object-Oriented Nonlinear Dynamic Finite Element Analysis,"

University of California at Berkeley.

2. Berding, D. C. (2006). "Wind Drift Design of Steel Framed Buildings: An Analytical

Study and a Survey of the Practice," MS Thesis, Virginia Polytechnic Institute and

State University, Blacksburg.

3. Charney, F. A. (1986). “Correlation of the Analytical and Experimental Seismic

Response of a 1/5th Scale Seven Story Reinforced Concrete Frame-Wall Structure,”

PhD Dissertation, University of California, Berkeley, USA.

4. Charney, F. A. (1990). "Sources of Elastic Deformations in Laterally Loaded Steel

Frame and Tube Structures." Proceedings of the Fourth World Congress on Tall

Buildings, Hong Kong, 893-915.

5. Charney, F. A. (1991). "The use of Displacement Participation Factors in the

Optimization of Wind Drift Controlled Buildings." Proceedings of the Second

Conference on Tall Buildings in Seismic Regions, 91-98.

6. Charney, F. A. (1993). "Economy of Steel Frames through Identification of Structural

Behavior." Proceedings of the National Steel Construction Conference, Orlando,

Florida, 12-1 to 13-33.

7. Charney, F. A., Iyer, H., and Spears, P. W. (2005). "Computation of Major Axis

Shear Deformations in Wide Flange Steel Girders and Columns." Journal of

Constructional Steel Research, 61, 1525-1558.

8. Charney, F. A., and Pathak, R. (2008). "Sources of Elastic Deformations in Steel

Frame and Tube Structures. Part 2: Detailed Subassemblage Models." Journal of

Constructional Steel Research, 64, 101-117.

9. Collins, M., Kasal, B., Paevere, P., and Foliente, G. C. (2005). "Three Dimensional


Engineering, 131(4), 676-683.

10. Cook, R. D., Malkus, D. S., and Plesha, M. E. (1989). "Concepts and Applications of

Finite Element Analysis." John Wiley & Sons, New York.

11. CSI. (2000). "SAP2000: Integrated Software for Structural Analysis and Design."

Computers and Structures, Inc., Berkeley.

80





14. Dubois-Pelerin, Y., and Pegon, P. (1998). "Object-Oriented Programming in

Nonlinear Finite Element Analysis." Computers and Structures, 67, 225-241.

15. Fenves, G. L. (1990). "Object-Oriented Programming for Engineering Software

Development." Computers and Structures, 6, 1-15.

16. Folz, B., and Filiatrault, A. (2001a). "Cyclic Analysis of Wood Shear Walls." Journal


17. Forde, B. W. R., Foschi, R. O., and Stiemer, S. F. (1990). "Object-Oriented Finite

Element Analysis." Computers and Structures, 34(3), 355-374.

18. Foschi, R. O. (1977). "Analysis of Wood Diaphragms and Trusses. Part I:

Diaphragms.” Canadian Journal of Civil Engineering, 4(3), 345-362.

19. Golub, G. H., and Loan, C. F. V. (1996). Matrix Computations, Johns Hopkins

University Press, Baltimore, Maryland.

20. INTEL. (2005). "Intel Math Kernel Library: Reference Manual"

21. Judd, J. P. (2005). "Analytical Modeling of Wood-Frame Shear Walls and

Diaphragms," MS Thesis, Brigham Young University, Provo, Utah.



Engineering, 131(2), 345-352.

23. Jun, L., (1994). “An Object-Oriented Application Framework for Finite Element

Analysis in Structural Engineering,” PhD Dissertation, Purdue University, West

Lafayette, Indiana.

24. Kasal, B., Collins, M. S., Paevere, P. J., and Foliente, G. C. (2004). "Design Models

of Light Frame Wood Buildings under Lateral Loads." Journal of Structural

Engineering, 130(8), 1263-1271.

25. Lam, F., Filiatrault, A., Kawai, N., Nakajima, S., and Yamaguchi, N. (2004).

"Performance of Timber Buildings under Seismic Load. Part 2: Modeling." Progress

in Structural Engineering and Materials, 6, 79-83.

81

26. Mackie, R. I. (1992). "Object-Oriented Programming of the Finite Element Method."

International Journal of Numerical Methods in Engineering, 35, 425-436.

27. McKenna, F. T. (1997). "Object-Oriented Finite Element Programming: Frameworks

for Analysis, Algorithms and Parallel Computing," PhD Thesis, University of

California at Berkeley.

28. Menetrey, P., and Zimmermann, T. (1993). "Object-Oriented Nonlinear Finite

Element Analysis: Application to J2 Plasticity." Computers and Structures, 49(5),

767-777.

29. Miller, G. R. (1988). "A LISP based Object-Oriented Approach to Structural

Analysis." Engineering with Computers, 4, 197-203.

30. Pei, S., and Lindt, J. W. v. d. (2006). "SAP WOOD: A Seismic Analysis Package for

Woodframe Structures ".

31. Peskin, R. L., and Russo, M. F. (1988). "An Object-Oriented System Environment for

Partial Differential Equation Solving." In Proceedings ASME Computations in

Engineering, 409-415.

32. Pidaparti, R. M. V., and Hudli, A. V. (1993). "Dynamic Analysis of Structures Using

Object-Oriented Techniques." Computers and Structures, 49(1), 149-156.

33. Rumbaugh, J., Blaha, M., Premerlani, W., Eddy, F., and Lorensen, W. (1991).

Object-Oriented Modeling and Design, Prentice Hall, Englewood Cliffs, New Jersey.

34. Stepanov, A., and Lee, M. (1995). The Standard Template Library, Hewlett Packard

Company.

35. Vamvatsikos, D., and Cornell, C. A. (2002a). "Incremental Dynamic Analysis."

Earthquake Engineering and Structural Dynamics, 31(3), 491-514.

36. Velivasakis, E. E., and DeScenza, R. (1983). "Design Optimization of Lateral Load

Resisting Frameworks." Proceedings of the Eight Conference on Electronic

Computation, Houston, Texas.

82

Figure 4-1: Class diagram of WoodFrameSolver program

83

Dense System: Bandwidth = 5433, Number of Equations = 48843

Sparse System: Bandwidth = 607, Number of Equations = 30651

Figure 4-2: Benchmark problems (SAP2000 view) to compare WoodFrameSolver performance

84

0

20

40

60

80

100

120

140

160

180

0 20000 40000 60000 80000 100000 120000 140000

NUMBER OF EQUATIONS

TIM

E (

sec)

WOODFRAMESOLVER

SAP VERSION 10

Figure 4-3: Speed comparison between SAP version 10 and WoodFrameSolver

(dense system of equations, linear static analysis)

0

10

20

30

40

50

60

70

80

0 50000 100000 150000 200000 250000 300000

NUMBER OF EQUATIONS

TIM

E (

sec)

WOODFRAMESOLVER

SAP VERSION 10

Figure 4-4: Speed comparison between SAP version 10 and WoodFrameSolver

(sparse system of equations, linear static analysis)

85

Figure 4-5: Two node frame element

Figure 4-6: Three, four node shell elements

U1x,F1x

U1y,F1y

U1z,F1z θ1x,M1x

θ1y,M1y

θ1z,M1z

U2x,F2x

U2y,F2y

U2z,F2z

θ2z,M2z

θ2x,M2x

x

y

z1 2

θ2y,M2y

U1x,F1x

U1y,F1y

U1z,F1z

θ1y,M1y

θ1z,M1z

θ1x,M1x U2x,F2x

U2y,F2y

U2z,F2z

θ2y,M2y

θ2z,M2z

θ2x,M2x

U3x,F3x

U3y,F3y

U3z,F3z

θ3y,M3y

θ3z,M3z

θ3x,M3x

1 2

3

x

y

z

U1x,F1x

U1y,F1y

U1z,F1z

θ1y,M1y

θ1z,M1z

θ1x,M1x U2x,F2x

U2y,F2y

U2z,F2z

θ2y,M2y

θ2z,M2z

θ2x,M2x

U3x,F3x

U3y,F3y

U3z,F3z

θ3y,M3y

θ3z,M3z

θ3x,M3x

1 2

3

U4x,F4x

U4y,F4y

U4z,F4z

θ4y,M4y

θ4z,M4z

4θ4x,M4x

x

y

z

86

Figure 4-7: Eight node solid element

Figure 4-8: One node spring element

U1x,F1x

U1y,F1y

U1z,F1z

1

23

4

5

6

7

8

U1x,F1x

U2y,F2y

U2z,F2z

U5x,F5x

U5y,F5y

U5z,F5z

U7x,F7x

U7y,F7y

U7z,F7z

U3x,F3x

U3y,F3y

U3z,F3z

U6x,F6x

U6y,F6y

U6z,F6z

U8x,F8x

U8y,F8y

U8z,F8z

U4x,F4x

U4y,F4y

U4z,F4z

x

y

z

x

y

z

U1y,F1y

θ1z,M1z

U1x,F1x

1

87

Figure 4-9: Nllink element

Figure 4-10: Gap spring behavior

1

2

x

y

z

U1x,F1x

U1y,F1y

θ1z,M1z

U2x,F2x

U2y,F2y

θ2z,M2z

88

Figure 4-11: Hook spring behavior

Figure 4-12: Trilinear spring behavior

89

Figure 4-13: Modified Stewart spring behavior

Figure 4-14: WoodFrameSolver program interface

K0

1

P0

PI

r1K01

(δu, Pult)

r2K0

1

r3K0

1

r4K0

1

Kd

1

δult

FORCE, P

DISPLACEMENT, δJ0

J1

J8

J2

J3

J5

J6

J7

J4

J9

J10

K0

1

P0

PI

r1K01

(δu, Pult)

r2K0

1

r3K0

1

r4K0

1

Kd

1

δult

FORCE, P

DISPLACEMENT, δJ0

J1

J8

J2

J3

J5

J6

J7

J4

J9

J10

90

Figure 4-15: Shear wall with two sheathing panels (Dolan 1989)

Figure 4-16: Shear wall finite element model

TOP PLATE

SILL

SIDE STUD

NAILS

SHEATHING ( 1) SHEATHING ( 2)

INTERIOR

STUD 4”

96”

96”

12”

91

0

2

4

6

8

10

0 2 4 6

DISPLACEMENT (in)

FO

RC

E (

kip

)

WOODFRAMESOLVER - NONORIENTED

WOODFRAMESOLVER - ORIENTED

EXPERIMENT 1 - DOLAN 1989



ABAQUS - JUDD 2003

Figure 4-17: Plywood sheathed shear wall response

0

2

4

6

8

10

0 2 4 6

DISPLACEMENT (in)

FO

RC

E (

kip

)

WOODFRAMESOLVER - NONORIENTED

WOODFRAMESOLVER - ORIENTED





ABAQUS - JUDD 2003

Figure 4-18: Waferboard sheathed wall response

92

Figure 4-19: A light frame wood house floor plan

Figure 4-20: Analytical model – U house

X

Y

1 2 3

4

5

6

7

8

9

10

11

12

(0,0) (480,0)

(480,480)(320,480)

(320,240)(160,240)

(160,480)(0,480)

X

Y

1 2 3

4

5

6

7

8

9

10

11

12

(0,0) (480,0)

(480,480)(320,480)

(320,240)(160,240)

(160,480)(0,480)

93

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45

TIME (sec)

AC

CE

LE

RA

TIO

N (

g)

Figure 4-21: An arbitrarily selected ground motion

-15

-10

-5

0

5

10

15

20

0 10 20 30 40 50

TIME (sec)

DE

FO

R (i

n)

SAPWOOD

WOODFRAMESOLVER

Figure 4-22: Deformation-time histories – Wall 1

94

-100

-80

-60

-40

-20

0

20

40

60

80

100

-15 -10 -5 0 5 10 15 20

DEFOR (in)

FO

RC

E (

kip

)

SAPWOOD

WOODFRAMESOLVER

Figure 4-23: Force-deformation histories – Wall 1

-4

-2

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

T IME (sec)

DE

FO

R (

in)

SAPWOOD

WOODFRAMESOLVER

Figure 4-24: Deformation-time histories – Wall 11

95

-4

-3

-2

-1

0

1

2

3

4

5

-4 -2 0 2 4 6 8

DEFOR (in)

FO

RC

E (

kip

)

SAPWOOD

WOODFRAMESOLVER

Figure 4-25: Force-deformation histories – Wall 11

Figure 4-26: A 3 story 1 bay moment frame

96

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0 5 10 15 20 25 30 35 40

T IME (sec)

DE

FO

R (

rad

)

WOODFRAMESOLVER

SAP2000

Figure 4-27: Nllink 6 deformation-time histories, 0% damping case

-16000

-11000

-6000

-1000

4000

9000

14000

19000

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

DEFOR (rad)

MO

ME

NT

(kip

-in

)

WOODFRAMESOLVER

SAP2000

Figure 4-28: Nllink 6 force-deformation histories, 0% damping case

97

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 5 10 15 20 25 30 35 40

TIME (sec)

DE

FO

R (

rad)

WOODFRAMESOLVER

SAP2000

Figure 4-29: Nllink 1 deformation-time histories, 2% damping modes 1 and 3

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

-0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 0.15

DEFOR (rad)

MO

ME

NT

(kip

-in

)

WOODFRAMESOLVER

SAP2000

Figure 4-30: Nllink 1 force-deformation histories, 2% damping modes 1 and 3

98

Figure 4-31: A garage type structure

Figure 4-32: Finite element model of garage

96”96”

96”WALL 1

WALL 2

WALL 3

FLOOR

OPENING

JOINT450

JOINT23

JOINT1322

JOINT1499

WALL

NAIL25

FLOOR

NAIL383

TOE NAIL505

99

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

TIME (sec)

DIS

PL

(in

)

WOODFRAMESOLVER

SAP 2000

Figure 4-33: Displacement response history – Joint23

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

TIME (sec)

DIS

PL

(in

)

WOODFRAMESOLVER

SAP 2000


100

-3

-2

-1

0

1

2

3

0 2 4 6 8 10

TIME (sec)

DIS

PL

(in

)

WOODFRAMESOLVER

SAP 2000


-3

-2

-1

0

1

2

3

0 2 4 6 8 10

T IME (sec)

DIS

PL

(in

)

WOODFRAMESOLVER

SAP 2000


101

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

DEFOR (in)

FO

RC

E (

kip

s)

WOODFRAMESOLVER

SAP2000

Figure 4-37: Force-deformation response history – Nail25

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

-0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25

DEFOR (in)

FO

RC

E (

kip

s)

WOODFRAMESOLVER

SAP2000


102

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

DEFOR (in)

FO

RC

E (

kip

s)

WOODFRAMESOLVER

SAP2000


-27

-24

-21

-18

-15

-12

-9

-6

-3

0

3

6

9

12

15

18

21

24

27

0 1 2 3 4 5 6 7 8 9

TIME (sec)

FO

RC

E (

Kip

s)

SAP2000

WOODFRAMESOLVER

Figure 4-40: X direction base shear response history

103

Table 4-1: Node attributes

DATA TYPE DATA COMMENT

INT m_Tag node number

DOUBLE m_AxisRotation[3] local axis rotations

DOUBLE m_Coord[3] global co-ordinates

DEQUE<DOUBLE> m_Displ Displacements

DEQUE<DOUBLE> m_Velo Velocities

DEQUE<DOUBLE> m_Accl Accelerations

DEQUE<DOUBLE> m_ElemJointMass Joint mass from elements

DEQUE<DOUBLE> m_AssignJointMass Assigned joint mass

MATRIX m_MassMatrix Joint mass matrix

Table 4-2: Static analysis steps

STEP # CALL COMMENT

1 this->GetStaticLoadCases Get static load cases

2 this->DetermineEqnNumbers Determine equation numbers

3 this->DetermineBandWidth Determine bandwidth of system of equations

4 SOE->SetDimensions Set dimensions for assigning storage

5 SOE->SetRHSStorage Assign right hand side storage

6 SOE->FormRHS Fill in right hand side

7 SOE->SetTangentStorage Set global stiffness matrix storage

8 SOE->FormTangent Fill in global stiffness matrix

9 SOE->Solve Solve Ax = B

10 SOE->GetX Get x

Table 4-3: Modified Stewart hysteresis parameters description

PARAMETER DESCRIPTION

K0 Initial stiffness

P0 The secondary stiffness force

PI Pinching force

r1 Secondary stiffness ratio

r2 Tertiary stiffness ratio

r3 Unloading path stiffness to initial stiffness ratio

r4 Pinching stiffness to initial stiffness ratio

δu Ultimate displacement corresponding to ultimate load

α Stiffness degradation parameter

β Strength degradation parameter

104

Table 4-4: Shear wall properties

COMPONENT DIMENSIONS PROPERTIES ELEMENT

USED

TOP PLATE & b x h = 3" x 3.5" E = 1400 ksi, µ = 0.3 2 node frame

SIDE STUDS

BOTTOM PLATE & b x h = 1.5" x 3.5" E = 1400 ksi, µ = 0.3 2 node frame

INTERIOR STUDS

PLYWOOD SHEATHING thickness = 0.278" E = 1800 ksi, G = 90 ksi, µ = 0.3 4 node shell

WAFERBOARD SHEATHING thickness = 0.375" E = 1800 ksi, G = 90 ksi, µ = 0.3 4 node shell

CONNECTION length = 0 K0 = 4.87 kip/in oriented nllink &

r1 = 0.04928 nonoriented nllink

r2 = -0.04928

P0 = 0.180 kip

δult = 0.5 in

δfail = 1.1 in

Table 4-5: Shear wall results with plywood sheathings

DISPLACEMENT (in) ULTIMATE

FORCE (kip)

EXPERIMENT 1 - DOLAN 1989 3.23 8.19



EXPERIMENT 1, 2, 3 - AVERAGE 3.07 7.55

NONORIENTED - WOODFRAMESOLVER 2.90 8.59

ORIENTED - WOODFRAMESOLVER 3.02 8.25

ABAQUS - JUDD 2003 3.20 8.30

Table 4-6: Shear wall results with waferboard sheathings

DISPLACEMENT (in) ULTIMATE

FORCE (kip)





EXPERIMENT 1, 2, 3, 4 - AVERAGE 2.36 7.21

NONORIENTED - WOODFRAMESOLVER 2.66 9.16

ORIENTED - WOODFRAMESOLVER 2.80 8.78

ABAQUS - JUDD 2003 3.00 8.82

105

Table 4-7: Bilinear shear wall properties

WALL # K1 K2 K3 FYP FYN

1 10 5 5 20 -20

2 13 6 6 20 -20

3 16 8 8 32 -32

6 14 1.4 1.4 42 -42

8 9 1 1 15 -15

10 30 6 6 30 -30

BILINEAR

Table 4-8: Modified Stewart shear wall properties

WALL # K0 P0 PI DU r1 r2 r3 r4 α β

4, 9, 11 5.21 2.44 0.473 3.86 0.0695 -0.087 1.29 0.0593 0.773 1.09

5, 12 32.4 7 1.67 1.53 0.0765 -0.0371 1.3 0.0694 0.571 1.1

7 15 3.1 0.734 1.53 0.0749 -0.059 1.29 0.0702 0.568 1.1

MODIFIED STEWART

Table 4-9: Moment frame properties

ELEMENT PROPERTY

2 node FRAME E = 10000 ksi

A = 100 in2

I = 10000 in4

2 node NLLINK K1 = 500,000 in-kip

(bilinear rotational spring) K2 = 500,00 in-kip

K3 = 500,00 in-kip

Fyp = 500 in-kip

Fyn = -500 in-kip

106

Table 4-10: Frame, sheathing and connection properties used in 3D house model


USED

SHEARWALL


SIDE STUDS


INTERIOR STUDS

SHEATHING thickness = 0.375" E = 1800 ksi, G = 90 ksi, µ = 0.33 4 node shell

DIAPHRAGM

JOISTS, BLOCKINGS b x h = 9.5" x 1.5" E = 1400 ksi, µ = 0.33 2 node frame


CONNECTION

NAILS length = 0 DOF = U1, DOFTYPE = LIN, nonoriented nllink

K1 = 1e-3, K2 = 1e-4, K3 = 1e-4, with bilinear spring

FYP = 1600, FYN = -1600 pair

DOF = U2, DOFTYPE = NON,

K1 = 3.2, K2 = 1.6, K3 = 1.6,

FYP = 0.32, FYN = -0.32

DOF = U3, DOFTYPE = NON,

K1 = 3.2, K2 = 1.6, K3 = 1.6,

FYP = 0.32, FYN = -0.32

107

CHAPTER 5 THE EFFECTS OF DIAPHRAGM FLEXIBILITY ON THE SEISMIC PERFORMANCE OF LIGHT FRAME WOOD

STRUCTURES, PART II: PARAMETRIC STUDY

INTRODUCTION: Practicing structural engineers rely heavily on the structural behavior

observed in experiments or the results obtained from validated analytical models. In the

case of LFWS, neither can be economically performed and thus in practice this has

resulted in the development of simplified procedures to obtain the force distribution in

the structural members. These procedures are only applicable to simple structures, and for

complex structures good engineering judgment is required. The residential structural

design guide (NAHBRC 2000) document specifies that: “Designer judgment is essential

in the early stages of design because the analytic methods and assumptions used to

evaluate the lateral resistance of light-frame buildings are not in themselves correct

representations of the problem. They are analogies that are sometimes reasonable but at

other times depart significantly from reason and actual system testing or field

experience”.

The design of wood structures follows one of the three popular approaches: (a) tributary

area, (b) total shear and (c) relative stiffness. The tributary area approach considers the

diaphragm as completely flexible and assigns forces to the shear walls based on the

proportion of the area of the diaphragm it covers. The total shear approach uses the story

shear to calculate the total wall length required to resist the force in each direction of

loading and then distributes the obtained wall length on that story based on the engineer’s

view, and hence is sometimes also referred to as an eyeball method. The relative stiffness

method assumes the horizontal diaphragm as rigid compared to the shear walls. This

results in a force distribution based on the relative stiffnesses of the shear walls. Here one

should note that the three approaches described above may give significantly different

forces in the shear walls for the same structural design.

108

The tributary area and relative stiffness are the most popular approaches used in

distributing the lateral forces to various shear walls in LFWS design. Figure 5-1 shows

the lateral force distribution in a box-like shear wall and diaphragm assembly due to these

two approaches. In this example all the walls are structurally identical and the results

presented in the figure clearly show the obvious disparity in the methods as a different

assumption may lead to underestimation or overestimation of the design forces in the

shear walls. In reality, none of the approaches may be correct, as the actual diaphragm

stiffness will most likely lie somewhere between entirely flexible and entirely rigid. Thus

an approach which considers the response of LFWS based on actual modeling of the

structural elements in the structure is more appropriate.

This chapter investigates the effect of diaphragm flexibility on the seismic response of

LFWS and compares it with in-plane rigid diaphragm response. To accomplish this, finite

element models of various LFWS have been developed and analyzed. The finite element

modeling approach used in developing these models and their structural configuration is

discussed in Chapter 3. This chapter also presents a verification study of two shear wall

models with the available experimental results. All the nonlinear response history

analysis presented in this study is performed using a high performance in-house finite

element analysis program named WoodFrameSolver discussed in Chapter 4.

NONLINEAR RESPONSE HISTORY ANALYSIS: All the finite element models

presented in this study are subjected to some earthquake loading and as aforementioned

the analysis is performed using an in-house utility designed and developed for this

purpose. The necessity of nonlinear analysis arises from the nonlinear nature of the

various connections occurring in the LFWS systems. Also, as the structure is subjected to

an earthquake, it tends to yield and degrade in strength and stiffness over the course of

loading, and hence only a direct numerical integration approach remains suitable for

solving the equations of motion. The fast nonlinear analysis approach also known as FNA

(Wilson 2004) would have been another option, but is not used here because of the large

109

number of nonlinear connections being considered in modeling the system. The program

internally formulates the equation of motion as shown in equation (1) and uses

Newmark’s constant average acceleration method for direct numerical integration over

the time domain. The Newton-Raphson iteration method as depicted in Figure 5-2 is used

within a time step to reduce the error introduced by the use of the tangent stiffness matrix

instead of the unknown secant stiffness matrix. The iteration within a time step is

terminated once the convergence criterion is satisfied. This convergence criterion

implemented inside the program is shown in equation (2) and is based on the work done

by the residual forces in each displacement increment compared to the work associated

with the total incremental force and the cumulative incremental displacement. The

numerical value of the convergence tolerance (ε) for all the models analyzed in this

chapter is set to be between 10-7

– 10-13

. The time step chosen for the dynamic analysis of

all the models is 0.005 sec which is one-fourth of the input loading time interval. A

convergence study was performed with refined time steps on a few house models

presented in this chapter. The loadings in these house models caused yielding in the nails

and the analysis was performed for three different time steps of 0.005, 0.00125 and

0.000625 seconds. The responses using a 0.005 second time step were found to be within

1% of the converged solutions. Also, the time taken for the dynamic analysis with the

refined time steps was significantly greater than what was obtained for 0.005 second.

Figure 5-3 presents the base shear response histories of Wall 1 in Type 4 model I using

three different time steps. We can see from the plot that convergence is obtained, as the

three response histories lay almost on the top of each other.

[ ]( ) [ ]( ) [ ]( ) [ ][ ]( )gUiMUKUCUM &&&&& ˆ−=++ (1)

[ ][ ]

ε<∆∆

∆∆

UP

URT

jTj )()(

(2)

110

MASS MATRIX: The mass matrix [M] in equation (1) is assembled by adding the mass

contribution of elements and nodes to each translational degree of freedom. This is a

lumped mass matrix and hence is diagonal. The mass contribution in the house models

comes from the self-weight of frames, sheathings and the superimposed nodal mass

representing other loads on the horizontal diaphragm. The mass contribution from the

nails is neglected in all the models.

DAMPING MATRIX: The damping matrix [C] in equation (1) is linear and is based on the

equivalent viscous damping mechanism distributed throughout the structure. This matrix

for all the finite element models is considered as mass proportional and is shown in

equation (3). Experiments on light frame wood structural systems have shown that at

large displacement amplitudes, the fasteners and other connections are the predominant

sources of hysteretic damping and non-viscous energy dissipation. These characteristics

of fasteners and connections are already included in their force-deformation relationships.

The hysteretic damping effect which comes into play at large displacement amplitude

also justifies the use of mass proportional damping which provides lesser damping in

higher frequency modes. The use of stiffness proportional damping or Rayleigh damping

instead of mass proportional damping to damp out the response in higher frequency

modes may inhibit the hysteretic damping from taking its full effect and hence may also

introduce errors in some problems. In this analysis we assume that no errors are getting

introduced due to low damping in higher modes. We also don’t observe any visible error

in the responses obtained for all the models, which may be attributed to low damping in

higher modes. However, we do mark this area as a potential for future detailed studies in

LFWS response history analysis. To incorporate the equivalent viscous damping in

LFWS models analyzed herein, the damping constant (α0) in equation (3) is assigned a

value based on the damping ratio, which is 2% of the critical at the X translational

frequency of vibration. The damping ratio in the range of 1% to 5% has been found

suitable for most wood structural systems (Chui and Smith 1989, Yeh et al. 1971).

111

[ ] [ ]MC 0α= (3)

STIFFNESS MATRIX: The stiffness matrix [K] in equation (1) is nonlinear and is

formulated by assembling the global stiffness matrix of all the elements in the model. The

matrix can be seen as a sum of linear and nonlinear element matrices as shown in

equation (4). The numerical values in KL are the contributions from linear frames and

shells and remain constant throughout the analysis. The numerical values in KN come

from nllinks representing fasteners and intercomponent connections in the models, and

they change with the internal deformations within the elements.

[ ] [ ] [ ]NL

KKK += (4)

The force-deformation relationship in the nllink springs of all the models is based on the

modified Stewart hysteresis. This hysteresis model was developed by Folz and Filiatrault

(2001) and is used to represent nail fasteners in all the models. This is a ten-parameter

model and its force-deformation behavior under monotonic and cyclic loading is as

shown in Figure 5-4. The monotonic loading portion of this model requires six

parameters and phenomenologically captures the crushing of wood, connector yielding

and connector removal. The cyclic response of this model is path dependent and is

capable of capturing pinched hysteretic behavior with stiffness and strength degradation.

This hysteresis model and its branches are discussed in more detail in chapter 4. The

parameters used for these springs in all the models are obtained from the work done by

other researchers in the past.

LOADING: The right hand side of equation (1) represents the dynamic loading vector

which is the product of mass matrix, influence coefficient matrix [ ]i and the ground

motion history used in the respective analysis. The ground motions used in the respective

analyses are discussed later along with the models.

112

NONLINEAR RESPONSE HISTORY ANALYSIS OF SHEAR WALLS AND

VERIFICATION WITH DOLAN (1989) EXPERIMENTS: In this section, a

verification study of the shear walls model developed using the procedure described in

Chapter 3 is presented. Two 8 ft x 8 ft shear walls experimentally tested by Dolan (1989)

are chosen for the dynamic analysis verification. One should note that the nonlinear static

pushover results of the same finite element models have been compared in Chapter 4 with

the experimental responses presented in Dolan and are found to be in the acceptable

range. The parameters for the cyclic portion of the modified Stewart hysteresis

connection could not be obtained and hence are reasonably assumed from the Stewart

hysteresis data of the other similar nails. Therefore, the results presented for these models

are checked for being in the ballpark instead of their accuracy.

WALL DESCRIPTION: Each of the walls was made of two double side studs, three

interior studs, a double top plate, a bottom sill, a mid-level blocking and two sheathings

as shown in Figure 5-5. The two walls were identical except for their sheathings which

were plywood and waferboard. The framing material had 1.5” x 3.5” cross-section

dimensions and the sheathing had 0.375” thickness. The sheathing to framing

connections were made using 8d galvanized common nails spaced at 4” center to center

on all the panel boundaries and 6” center to center in the field. The shear walls supported

a mass of 45.45 x 10-4

kip-sec2/in. at their top for the purpose of response history

analysis.

FINITE ELEMENT MODEL DESCRIPTION: The finite element model of one of the walls

is shown in Figure 5-6 and the element properties for plywood and waferboard models

are presented in Tables 5-1 and 5-2 respectively. The horizontal and vertical studs are

modeled using frame elements. The side studs and top plates are modeled using a single

frame element having double thicknesses. The vertical frames have moment releases at

their connections with the plate and sill frames. The horizontal blocking frames also have

moment releases at their connections with the vertical studs. The sheathings are modeled

113

using shell elements and are connected to framings on their perimeter and inside using

modified Stewart nllink elements with oriented springs. The edge interface interaction

between the adjacent sheathings is not considered in the modeling. The mass at the wall

top is distributed evenly at the nodes of the plate. A 3% linear viscous damping is also

assumed in both the models, which are subjected to the S69E component of the 1952

Kern County earthquake (California) with its peak ground acceleration scaled to 0.18g as

shown in Figure 5-7. This is the same ground motion used in Dolan’s study.

RESULTS COMPARISON: The shear wall models are developed using the WHFEMG

program (Appendix D) and the analyses are performed using WoodFrameSolver. The

center top node displacement response history is recorded for the comparison of peak

displacements. Figures 5-8 and 5-9 show the horizontal displacement response histories

for the top of the plywood and waferboard shear wall models. Table 5-3 presents the

comparison of the maximum and minimum displacement responses of the models with

the experiments. The plywood shear wall model underestimates the maximum and

minimum displacements by 28% and 16%, respectively. The waferboard shear wall

model overestimates the maximum and minimum displacements by 11% and 15%,

respectively.

LFWS HOUSE FE MODELS AND ANALYSIS DESCRIPTION: All the finite

element house models are created using the WHFEMG program and are classified into

seven different Types. Each Type contains up to 4 models with different shear wall

configuration and constant floor plan aspect ratio. The roof plans for all the models are

shown in Figures 5-10 to 5-16. The models 1 and 2 in each Type creates a symmetric

lateral force resisting system along both X and Y global directions. The models 3 and 4 in

each Type have one and two walls missing, respectively, compared to model 1 and this

results in an asymmetric lateral force resisting system, i.e. torsional irregularity.

114

The finite element models of each Type are viewed in SAP2000 GUI and are shown in

Figures 5-17 to 5-23. These models incorporate various structural elements which include

frame for framing, orthotropic shell for sheathing, and oriented nllink elements with

Stewart hysteresis spring pair for nails. The aspect ratio of the house plan in these models

varies from 1 to 5 and is kept constant within the same type of model. Two aspect ratios

are defined for each type and are referred to as X and Y direction aspect ratio. The X

direction aspect ratio is calculated as the ratio of the Y dimension to the X dimension of

the floor plan in a model. Similarly, the Y direction aspect ratio is calculated as the ratio

of the X dimension to the Y dimension of the floor plan in a model. These ratios are

defined because in a few models the earthquake loading is applied in both X and Y

directions simultaneously and these models have different lateral force resisting systems

along the X and Y directions. These direction aspect ratios will be used to correlate with

the direction response later in the discussion. Table 5-4 presents the direction aspect

ratios and the periods of vibration in the two translational modes and one rotational mode

about the vertical Z axis of all the flexible and rigid diaphragm house models used in the

analysis. These periods of vibration are calculated using the initial stiffness of the

elements in the structure, and they change over the course of loading because of the

yielding occurring in the connection elements.

The material, nails, sheathing and frame properties are kept constant in all the models

analyzed in this study except where model diaphragm flexibility is modified. These

properties for flexible diaphragm models are presented in Table 5-5. In all the models the

shear walls are 8’ x 8’ with studs spaced at 2’ intervals, and have two 4’ x 8’ OSB panels.

The area between the shear walls on the boundary is covered by the stud framing spaced

at 2’ intervals. All the walls have a double top plate and a mid-height blocking which also

extends on the boundary between the shear walls. In the models where the shear wall is

removed from the boundary, only the sheathing panels and their connecting nails are

removed and not the stud framing, as can be seen in Figures 5-17 to 5-23. The roof

diaphragm consists of sheathing panels, joists, blockings and nails. OSB sheathing panels

115

of sizes 4’ x 8’ and 4’ x 4’ are used in modeling the roof diaphragm of all the models in

Types 1, 3 and 5. Types 2, 4, 6 and 7 model diaphragms comprise of 4’ x 8’ OSB

sheathing panels only. In all the models, the same nails are used in connecting the

sheathing to framing and framing to framing (toe nails). The frame and panel self-mass is

calculated using the mass per unit volume of the wood by using the assumed specific

gravity of 0.55. This specific gravity results in a mass per unit volume of 5.144 x 10-8

kips-sec2/in

4. The nail mass is neglected in all the models, and in all the models a mass of

0.06 kips-sec2/in. is applied, which is uniformly distributed on the roof nodes. This

uniform distribution of the mass on the roof nodes takes care of the mass moment of

inertia occurring in the diaphragms rotating about their vertical axes.

All the 7 house types have a total of 22 house models with flexible diaphragms. These

flexible diaphragm model’s in-plane member properties are modified to get 22

corresponding rigid diaphragm models. The rigid in-plane behavior of an object refers to

no in-plane deformations occurring in the object. Theoretically, for a shell element this

may be obtained by using an extremely high membrane thickness and for a frame element

by using an extremely high cross-sectional area, in-plane bending moment of inertia, and

in-plane shear area. However, an extremely high number is found to cause numerical

difficulties practically while obtaining the solution and hence instead a number

generating negligible internal deformations is typically used. In the in-plane rigid

diaphragm models presented herein, this is achieved by increasing the in-plane thickness

of the sheathing, and cross-sectional area, in-plane bending moment of inertia and in-

plane shear area of the framing, by a factor of 1000. The out-of-plane element section

properties of the diaphragms are not changed. The mass per unit volume is appropriately

modified in order to keep the self-mass of the elements the same as for the flexible

diaphragm models. Similarly, the nail connections between the sheathing and the framing

of the diaphragm in the rigid cases are made rigid by increasing their stiffness by a factor

of 1000 and changing their behavior to linear.

116

In addition to the above 44 models, two sets of model 2 from Types 2, 3, and 4 and two

sets of model 1 from Types 5, 6 and 7 are created with the following diaphragm

flexibility:

(1) Sheathings are modeled as rigid in-plane with nails and joists as

flexible

(2) Nail connections are modeled as rigid in-plane and sheathing and

joist as flexible

All the 56 resulting house models are subjected to the first twelve seconds of at least one

of the following two earthquake records:

(1) Imperial Valley earthquake, El Centro

(2) Northridge earthquake, Sylmar county hospital

All the flexible and in-plane rigid diaphragm models of Type 1 to 4 are subjected to a

bidirectional input of each above earthquake with its peak ground acceleration (PGA)

scaled to 0.30g and 0.10g in the global X and Y directions, respectively. The flexible and

in-plane rigid diaphragm models of Types 5 to 7 are subjected to only the Northridge

earthquake in the global X direction with its PGA scaled to 0.30g. The two sets of model

2 Types 2, 3 and 4 are subject to bidirectional input of Imperial Valley with its peak

ground acceleration (PGA) scaled to 0.30g and 0.10g in the global X and Y directions,

respectively. The last two sets of model 1 Types 5, 6 and 7 are subjected to only the

Northridge earthquake in the global X direction with its PGA scaled to 0.30g. The scaling

factors used in the analysis are somewhat smaller than what would normally be used,

however, the scaled input generates yielding in various connections as would be seen in

the various force-deformation plots later. Figures 5-24 and 5-25 show the first twelve

seconds of the Imperial Valley and the Northridge earthquake records respectively. The

ground motions shown in these figures are scaled to 1g.

117

Overall, in the parametric study reported herein, a total of 88 nonlinear response history

analyses are performed. The varying parameters are (1) in-plane flexibility of roof

diaphragm in the models, (2) lateral load resisting system configuration, (3) floor plan

aspect ratio, and (4) earthquake loading input. These cases are put together in a chart and

are presented in Table 5-6. Also, all the house finite element models analyzed in this

study are based on the following assumptions:

(1) The interconnection between the adjoining shear walls is not modeled;

hence no force transfer from wall to wall is incorporated

(2) Contact between the adjacent panels is ignored

(3) Buckling of panels is not considered in the modeling

(4) The material for the stud frames is assumed to be isotropic

(5) The stud frames and panel behavior are assumed to be linear

(6) Only the nail connection between diaphragm and walls is modeled

(7) The member dimensions are not transformed for the cross-section overlap

of sheathing and framing in the models

(8) Tie down anchors are modeled as restraints and hence no uplift of walls or

base translation is modeled

(9) Anchor bolts are not modeled

RESULTS OF THE ANALYSIS: As mentioned earlier, all the analysis is performed

using the WoodFrameSolver program, and various responses are recorded. These

responses include base reaction (also referred to as base shear below) response histories

in the walls, displacement response histories at the top center node of the walls, and

maximum absolute displacements at mid-nodes on the edges of the flexible roof

diaphragms. The walls in all the models are assigned numbers from 1 to 9 for the purpose

of reference to walls in the discussion. The numbers assigned to the walls in each parent

118

house model are presented in Figure 5-26 i.e. on Type x1 Model 1. The reference number

of a wall at a location in a model within a type remains the same as shown in its parent

model. The peak base shears in all the walls of all the models are presented in various

Tables in Appendix E. The aspects of the results which are studied in detail and discussed

hereon are as follows:

(1) Comparison of flexible and rigid diaphragm response of all the models

(2) Peak in-plane base shear occurring in partition walls in Type x model 1

(3) Effects of torsional irregularity

(4) Study using the code specified measure of rigidity

FLEXIBLE AND RIGID DIAPHRAGM MODEL’S RESPONSE: In this section, a

comparison of peak in-plane base shear occurring in the walls of the flexible and rigid

diaphragm models is made. The comparison is made for all the corresponding walls in

the corresponding flexible and rigid diaphragm models. The ratios of peak in-plane base

shear occurring in the walls of rigid and flexible diaphragm models are calculated and are

presented in Tables 5-7 to 5-17. The base shear ratios presented in these tables are for

Imperial Valley and Northridge ground motions. It is noted that the flexible and rigid

diaphragm assumptions give different base shears in corresponding walls of all the

models.

The Type 1 rigid diaphragm models show a maximum over and underestimation of up to

28% and 6%, respectively, in their wall’s in-plane peak base shears. The maximum

overestimation is obtained for walls 8 and 9 in model 3 using the Imperial Valley ground

motion. The maximum underestimation is noted for wall 1 in model 3 using the

Northridge ground motion.

The Type 2 rigid diaphragm models show a maximum over and underestimation of up to


1 x refers to all of the 1, 2, 3, 4, 5, 6 and 7 house Types used in the analysis

119

overestimation is obtained for walls 1, 2, 3 and 4 in model 2 using the Imperial Valley

ground motion. The maximum underestimation is noted for wall 6 in model 3 using the

Northridge ground motion.

The Type 3 rigid diaphragm models report a maximum over and underestimation of up to


overestimation is obtained for walls 1, 2, 3 and 4 in model 1 using the Northridge ground

motion. The maximum underestimation is noted for wall 6 in model 3 using the Imperial

Valley ground motion.

The Type 4 rigid diaphragm models report a maximum over and underestimation of up to


overestimation is obtained for walls 1, 2, 3 and 4 in model 1 using the Northridge ground

motion. The maximum underestimation is noted for wall 6 in model 3 using both the

ground motions.

In Type 5, 6 and 7 models, only Northridge ground motions were used and there was not

much significant in-plane shear reported in their walls, 4, 5, 6 and 7. Thus in Tables 5-15,

5-16 and 5-17, the ratios are presented only for the walls 1, 2 and 3. The Type 5 rigid

diaphragm models report a maximum over and underestimation of up to 2% and 1%,

respectively, in their in-plane peak base shear of all the walls. The Type 6 rigid


respectively, in their in-plane peak base shear of all the walls. The Type 7 rigid


respectively, in their in-plane peak base shear of all the walls.

Overall, the maximum underestimation is found for the Type 3 Model 3 wall 6 and

maximum overestimation is found for the Type 1 Model 3 walls 8 and 9. The rigid

diaphragm assumption reports difference in wall forces ranging from 33%

120

(underestimation) to 28% (overestimation). However, one should note that these

maximum and other higher differences (see Tables 5-7 to 5-15) are obtained when the

lateral force resisting system is asymmetric. To see the effect of diaphragm flexibility in

the symmetric systems, only Models 1, and their X direction walls can be considered. It is

found that for these models the rigid diaphragm assumption reports difference in wall

forces ranging from 9% (underestimation) to 16% (overestimation). The maximum

underestimation is found for the interior wall 2 in Type 7 Model 1 and maximum

overestimation is obtained in the outer walls (1, 2, 3 and 4) of Type 1 Model 1.

INTERIOR SHEAR WALL PEAK IN-PLANE LOAD SHARING: This section presents a

comparison of peak in-plane base shear per unit wall length acting in the interior shear

wall relative to the peak in-plane base shear per unit wall length acting on the boundary

walls for the two types of lateral force resisting systems analyzed in this study. The Types

1, 2, 3 and 4 model 1 present one symmetric configuration of the lateral force resisting

system where 4 boundary walls (numbered 1, 2, 3 and 4) are present along the X

direction and one interior wall (numbered 5) is placed in the center along the X direction.

The Types 5, 6 and 7 model 1 present another symmetric configuration where 2 boundary

walls (numbered 1 and 3) are present along the X direction and one interior wall

(numbered 3) is placed in the center along the X direction. All of the Types x model 1 are

subjected to the 0.3g Northridge earthquake in the X direction, as mentioned previously.

A ratio of resulting peak in-plane base shear per unit length in the interior shear wall and

the peak in-plane base shear per unit length in the outer shear walls is calculated for all

the flexible and rigid diaphragm cases of these models. The idea is to compare the

relative variation in interior wall forces when the direction aspect ratio and the flexibility

of the model changes. Figures 5-27 and 5-28 present these results for flexible and rigid

diaphragm cases, respectively, for the two different lateral force resisting systems

combined in one plot. In the flexible diaphragm models, as the X direction aspect ratio

increases, an increase in interior wall peak base shear per unit length relative to the outer

121

walls is suggested from Figure 5-27. Figure 5-28 suggests just the opposite for the rigid

diaphragm models.

TORSIONAL IRREGULARITY: Torsional irregularity refers to the change in the stiffness

center relative to the center of mass in the structure. In Types 1, 2, 3 and 4 models 3 and

4, a significant torsional irregularity is generated due to removal of wall 2 and walls 2

and 5, respectively. Thus, when loading is applied in the X direction in these models, it

induces torsional moments in the system. This moment is shared by all the walls,

resulting in a change of their base shears, the sign and magnitude of which vary from

wall to wall in a model. A comparison2 between the Type 1 flexible diaphragm models 1

and 3 shows that the removal of wall 2 changes the force in the walls up to 56%. The

corresponding rigid diaphragm shows the force changes occurring up to 75%. The

comparison between the Type 1 flexible diaphragm models 2 and 4 shows that removal

of wall 2 changes the force in walls up to 56%. The corresponding rigid diaphragm shows

the force changes lying between 14% and 87%. A comparison between the Type 2

flexible diaphragm models 1 and 3 shows that the removal of wall 2 changes the force in

walls up to 39%. The corresponding rigid diaphragm model shows the force changes

lying between 2.3% and 65%. A comparison between the Type 2 flexible diaphragm

models 2 and 4 shows that the removal of wall 2 may change the force in the walls

between 13% and 57%. The corresponding rigid diaphragm shows the force changes

lying between 13% and 85%. A comparison between the Type 3 flexible diaphragm

models 1 and 3 shows that removal of wall 2 changes the force in the walls between 1%

and 39%. The corresponding rigid diaphragm shows the force changes lying between

1.8% and 54.3%. A comparison between the Type 3 flexible diaphragm models 2 and 4

shows that removal of wall 2 changes the force in the walls between 8.4% and 39.5%.

The corresponding rigid diaphragm shows the force changes lying between 10.3% and

63%. A comparison between the Type 4 flexible diaphragm models 1 and 3 shows that

the removal of wall 2 changes the force in the walls between 2.4% and 30.2%. The

2 Results presented in Appendix E

122

corresponding rigid diaphragm shows the force changes lying between 7.2% and 48%. A

comparison between the Type 4 flexible diaphragm models 2 and 4 shows that removal

of wall 2 changes the force in the walls between 15.4% and 37%. The corresponding

rigid diaphragm shows the force changes lying between 6.84% and 48%.

The wall in-plane base shear and its top center displacement histories are recorded during

the analysis of the models. It is found that all walls in all the Types 1, 2, 3 and 4 models

undergo hysteretic yielding due to loads applied in both X and Y directions. It was

already mentioned that this yielding occurs due to the hysteretic nature of the nail

connections in the walls. Wall 1 undergoes maximum yielding among all the X direction

walls of Type 1, 2, 3 and 4 models. This yielding in Wall 1 for all the flexible models is

compared within a type for Imperial Valley and Northridge ground motions in Figures 5-

29 to 5-36. The flexible models 3 and 4 wall 1 dissipates more energy compared to the

corresponding wall in models 1 and 2, respectively. This again is due to the introduction

of torsional moments in the models, which increases the base shear. The models 1 and 2

have no asymmetry and hence we do not see a significant difference in the force

displacement plots of their wall 1. The model 2 wall 1 dissipates more energy compared

to model 1 wall 1 due to the absence of wall 5 resulting in more forces in the X direction

walls of model 2.

In Types 5, 6 and 7 model 3, removal of wall 7 creates asymmetry in the Y direction

lateral force resisting system, however, as these models are subjected to loading only in

the X direction where the stiffness center and the center of mass are almost on the top of

each other (no change because Y shear wall has practically zero stiffness in the out-of-

plane direction). Hence, Y direction walls (numbered 4, 5 and 6) don’t pick up any

significant in-plane shear loading in these models.

Overall, it is found that there are significant in-plane peak shear forces changes occurring

in the walls due to the introduction of torsional irregularity using both the flexible and

123

rigid diaphragm modeling assumption. This suggests that torsional irregularity in the

structure must be critically analyzed for design of shear walls in LFWS. In such

problems, modeling the actual flexibility of the diaphragm for analysis would certainly

give an economical solution.

STUDY USING THE CODE SPECIFIED MEASURE OF RIGIDITY: The design code

specifies that the horizontal diaphragms can be considered to be rigid until the maximum

floor deflection exceeds twice the wall displacement. We calculate this criterion along X

and Y directions for the models discussed in this section and denote it generically as RCd,

where RC denotes rigidity criteria and the subscript d is the direction in which it is

measured. This criterion in a direction d is linked with the direction aspect ratio and is

generically referred to as ARd hereon. AR denotes aspect ratio and d is the direction in

which the rigidity is measured. These measures are presented in Figure 5-37. RCd is

calculated as the ratio of δ1dmax and δ2dmax and these are the maximum deflections in the

d direction at the points denoted by a cross (x) in the figure. This section investigates the

diaphragm rigidity criterion for the following set of models:

(1) Flexible diaphragm models 1 and 2 in Types 2, 3 and 4

(2) Flexible diaphragm model 1 in Types 5, 6 and 7

(3) Model 2 in Types 2, 3 and 4

(3.1) Only nail and joist flexibility included in the diaphragm (rigid sheathing)

(3.2) Only sheathing and joist flexibility included in the diaphragm (rigid nails)

(4) Model 1 in Types 5, 6 and 7

(4.1) Only nail and joist flexibility included in the diaphragm (rigid sheathing)

(4.2) Only sheathing and joist flexibility included in the diaphragm (rigid nails)

All the models considered for investigation are symmetric and therefore there is no

torsion coming into action. Hence, the measured deflections δ1dmax and δ2dmax are purely

124

due to loading and stiffness in the d direction of the model (neglecting the Poisson effect)

even if a bidirectional ground motion loading is applied during the analysis.

INVESTIGATION (1): The flexible diaphragm models 1 and 2 in Types 2, 3 and 4 were

subjected to two sets of bidirectional loadings and the deflections were recorded in the

models at the points and in the directions shown in Figure 5-37. The RCd ratios are

measured in both the directions for models 1 and 2 in Types 2, 3 and 4 and are presented

in Table 5-18. It is interesting to note that RCd is linked to ARd; however, an absolute

comparison can also be made on the basis of RC and AR without referring to the

direction in which they are calculated. The RCd versus ARd is plotted in Figure 5-38 for

models 1 and 2 and both the loading cases. A similar trend is obtained for both models

with the two loadings used in the analysis. The thick dark line at RCd equal to 1

represents the in-plane rigid diaphragm case response with varying aspect ratio. It is

found that if the lateral force resisting system configuration is kept the same in a direction

and if the aspect ratio of the diaphragm in that direction is increased the flexibility of the

diaphragm tends to increase. The RCX values for model 1 are lower than the

corresponding RCX values in model 2, which suggests that the presence of an interior

shear wall reduces the diaphragm flexibility.

INVESTIGATION (2): The flexible diaphragm model 1 in Types 5, 6 and 7 were subjected

to unidirectional Northridge earthquake loading and the deflections were recorded in the

loading direction (global X) at the points shown in Figure 5-37. The RCX ratios are

measured using the deflections and are presented in Table 5-19. The RCX vs ARX is

plotted in Figure 5-39, and from the trend it is found that as the aspect ratio ARX

increases, RCX also increases. However, even with the aspect ratio of 5, the RCX value

remained less than 2. This is due to the presence of an interior wall, which reduces the

flexibility of diaphragm. This is also substantiated by the fact that an RC value of 2.2 was

obtained for the corresponding aspect ratio case (Type 4 model 2), in the previous

investigation, even with a smaller ground motion scaling.

125

INVESTIGATION (3): Investigation (1) on the flexible diaphragm model 2 in the Types 2, 3

and 4 showed how the flexibility of the diaphragm varied with the aspect ratio. The

diaphragm constitutes joists, sheathing and nail connections, which are also the sources

of flexibility in its response, and investigation (1) on model 2 isn’t sufficient for

quantifying these sources. Thus, in this investigation we use model 2 from Types 2, 3 and

4 but with the following two in-plane diaphragm flexibilities:

(1) Sheathings are modeled as rigid in-plane with nail connections

and joists as flexible

(2) Nail connections are modeled as rigid in-plane and sheathing

and joists as flexible

The deflection results are obtained for these two model sets using Imperial Valley ground

motion with the same scaling as used in the corresponding completely flexible diaphragm

models. Using the results, we compute the RCd values corresponding to ARd values as

was done in the Investigation (1) and these are presented in Table 5-20 and are plotted in

Figure 5-40. It is found that nail connections are the major source of flexibility compared

to sheathings in the diaphragm throughout the range of aspect ratio studied and for the

two different lateral force resisting systems.

INVESTIGATION (4): This is a similar investigation as done above but for a different

lateral force resisting system. In this investigation model 1 from Types 5, 6 and 7 is used

with the two different sets of in-plane flexibility discussed above. The deflection results

are obtained for the resulting model sets using Northridge ground motion with the same

scaling as is used in the corresponding completely flexible diaphragm models. Using the

results we compute the RCX values corresponding to ARX values as done in the previous

investigations, and these are presented in Table 5-21. The RCX versus ARX is plotted in

Figure 5-41. It is found that nail connections are the major source of flexibility compared

to sheathings in the diaphragm throughout the range of aspect ratio studied.

126

SUMMARY AND CONCLUSIONS: In this chapter we have studied the effect of

diaphragm flexibility on the seismic performance of LFWS systems. A total of 88

nonlinear response history analyses are performed on 56 finite element house models.

The relevant aspects of the obtained responses are discussed in detail in this chapter and

overall the following conclusions are drawn from the above modeling and analysis:

1. Finite element modeling and dynamic analysis of light frame wood structural

systems is a time consuming task. This is due to the system being inherently

complex and nonlinear, resulting in several linear and nonlinear degrees of

freedom in the analytical model.

2. The shear wall model used in the verification study predicts the seismic response

in the acceptable range when compared with experiments. The maximum and

minimum displacement responses obtained from the model lie between 11% and

28% of the experimental response. This is reasonable considering the uncertainties

involved in wood properties, approximate connection behavior modeling and

experimental errors.

3. The shear walls transverse to the earthquake loading resist negligible out-of-plane

forces.

4. The flexible and rigid diaphragm assumptions give different in-plane peak base

shears in corresponding walls for all the models. The rigid diaphragm assumption

overestimates and underestimates the in-plane peak base shear in the walls when

compared with the results from corresponding models of flexible diaphragms. In

the above analysis, it is found that the rigid diaphragm assumption overestimated

and underestimated the in-plane peak base shears maximum up to 28% and 33%,

respectively. These maximum values were obtained for torsionally irregular

127

models, and for the symmetric lateral load resisting system with interior wall

analyzed herein, this range comes down to 9% (underestimation) and 16%

(overestimation). Thus, modeling and analysis of torsionally irregular systems

should certainly incorporate flexibility of diaphragm elements in their analysis.

5. The interior shear walls resist a significant amount of seismic forces in their planes

and the distribution depends upon the aspect ratio of the diaphragm, its flexibility

and the stiffness of the shear walls. In the above analysis the stiffnesses of all the

walls are kept the same, and hence only the variation of shear force in the interior

shear wall with changing aspect ratio and flexibility is captured. It is found that in

the flexible diaphragm models, as the X direction aspect ratio increases, an

increase in interior wall peak base shear per unit length relative to the outer walls

is suggested.

6. The interior wall in Types 1, 2, 3 and 4 Model 3 helps in reducing the shear forces

due to torsion when compared with Types 1, 2, 3 and 4 Model 4 for both flexible

and rigid diaphragm assumptions. This is because the presence of an interior wall

in Model 3 keeps the center of rigidity and center of mass closer to each other and

in effect reduces the moment arm.

7. The walls transverse to the loadings are critical in resisting in-plane forces induced

by torsional moments. This is evident from the change in in-plane forces of Y

direction walls in asymmetric models (3 and 4) when compared to symmetric

models (1 and 2).

8. Stiffness irregularity increases the yielding and energy dissipation in some walls,

as is evident from the force-deformation plots of wall 1 of various models shown

in Figures 5-21 to 5-28. Hence, identifying such walls in a structural system is

critical from design perspective.

128

9. The in-plane flexibility of a diaphragm is found to get significantly affected by the

presence of interior shear wall located at the geometric center in the direction of

loading. This effect due to an interior wall shall vary depending upon the spatial

location and needs further investigation.

10. Nail connections are the major source of in-plane flexibility compared to

sheathings within a diaphragm, irrespective of the aspect ratio of the diaphragm.

129

REFERENCES:

1. Chui, Y. H., and Smith, I. (1989). "Quantifying Damping in Structural Timber

Components." Proc., 2nd

Pacific Timber Engineering Conference., Institute of

Professional Engineers., Wellington, NZ, Vol. 1, 57-60.





4. NAHBRC (2000). Residential Structural Design Guide: 2000 Edition, U.S.

Department of Housing and Urban Development, Rockville, MD.

5. Wilson, E. L. (2004). "Static and Dynamic Analysis of Structures." Computers and

Structures, Inc.

6. Yeh, T. C., Hartz, B. J., and Brown, C. B. (1971). "Damping Sources in Wood

Structures." Journal of Sound and Vibration, 19(4), 411-419.

130

Figure 5-1: Lateral force distribution in a shear wall under rigid and flexible diaphragm

assumption

Shear Wall

Diaphragm

Lateral Load

W force/length

L

Rigid Diaphragm Assumption:R1=R2=R3=WL/3

Flexible Diaphragm Assumption:

R1=R3=WL/4 & R2=WL/2

R1

R2

R3Shear Wall

Diaphragm

Lateral Load

W force/length

L

Rigid Diaphragm Assumption:R1=R2=R3=WL/3

Flexible Diaphragm Assumption:

R1=R3=WL/4 & R2=WL/2

R1

R2

R3

131

Figure 5-2: Newton-Raphson within a load step

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10 12 14

TIME (sec)

BA

SE

SH

EA

R (

Kip

s)

DT=0.005

DT=0.00125

DT=0.000625

Figure 5-3: Base shear convergence test of wall 1 Type 4 model 1

U

P

∆P

∆R(2)

∆R(4)

∆R(3)

∆U(1) ∆U(2) ∆U(3) ∆U(4)

KT(1)

KT(2)

KT(3)

132

Figure 5-4: Modified Stewart spring behavior

Figure 5-5: Shear wall with two sheathing panels (Dolan 1989)

K0

1

P0

PI

r1K01

(δu, Pult)

r2K0

1

r3K0

1

r4K0

1

Kd

1

δult

FORCE, P

DISPLACEMENT, δJ0

J1

J8

J2

J3

J5

J6

J7

J4

J9

J10

TOP PLATE

SILL

SIDE STUD

NAILS

SHEATHING ( 1) SHEATHING ( 2)

INTERIOR

STUD 4”

96”

96”

12”

133

Figure 5-6: Shear wall finite element model

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 5 10 15 20 25 30

TIME (sec)

AC

CE

LE

RA

TIO

N (

g)

Figure 5-7: Kern County earthquake

134

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 5 10 15 20 25 30

TIME (sec)

DIS

PL

(in

)

PLYWOOD

Figure 5-8: Plywood wall displacement history

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

TIME (sec)

DIS

PL

(in

)

WAFERBOARD

Figure 5-9: Waferboard wall displacement history

135

Type 1, Model 1 Type 1, Model 2


Figure 5-10: Type 1 floor plans

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20’

8’

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136




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137




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138




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139





20’

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10’

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140



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141



Figure 5-17: Type 1 houses finite element models

142




143




144




145





146



-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2 4 6 8 10 12 14

TIME (sec)

AC

CE

LE

RA

TIO

N (

g)

Figure 5-24: Imperial Valley earthquake

147

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2 4 6 8 10 12 14

TIME (sec)

AC

CE

LE

RA

TIO

N (

g)

Figure 5-25: Northridge earthquake

148

Type 1 Type 2

Type 3 Type 4

Type 5 Type 6

Type 7

Figure 5-26: Wall numbering

1 2

3 4

5

6

7

8

9

1 2

3 4

56 7

1 2

3 4

56 7

1 2

3 45

6 7

3

4

5

1

2

7

6

3

1

2

4

5

7

6

1

2

3

4

5

6

7

149

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

0 1 2 3 4 5 6

X DIRECTION ASPECT RATIO

RA

TIO

TYPES 1,2,3 and 4

TYPES 5, 6 and 7

Figure 5-27: Ratio of interior and exterior shear wall in-plane peak base shear per unit

length vs the X direction aspect ratio, Flexible diaphragm model 1

0.85

0.87

0.89

0.91

0.93

0.95

0.97

0.99

0 1 2 3 4 5 6

X DIRECTION ASPECT RATIO

RA

TIO

TYPES 1,2,3 and 4

TYPES 5, 6 and 7

Figure 5-28: Ratio of interior and exterior shear wall in-plane peak base shear per unit

length vs the X direction aspect ratio, Rigid diaphragm model 1

150

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

DISPL (in)

FO

RC

E (

Kip

)

TYP1M1

TYP1M2

TYP1M3

TYP1M4

Figure 5-29: Type 1, wall 1 force-displacement response history, Imperial Valley

earthquake

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

DISPL (in)

FO

RC

E (

Kip

)

TYP1M1

TYP1M2

TYP1M3

TYP1M4

Figure 5-30: Type 1, wall 1 force-displacement response history, Northridge earthquake

151

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

DISPL (in)

FO

RC

E (

Kip

)

TYP2M1

TYP2M2

TYP2M3

TYP2M4


earthquake

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

DISPL (in)

FO

RC

E (

Kip

)

TYP2M1

TYP2M2

TYP2M3

TYP2M4


152

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

DISPL (in)

FO

RC

E (

Kip

)

TYP3M1

TYP3M2

TYP3M3

TYP3M4


earthquake

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

DISPL (in)

FO

RC

E (

Kip

)

TYP3M1

TYP3M2

TYP3M3

TYP3M4


153

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

DISPL (in)

FO

RC

E (

Kip

)

TYP4M1

TYP4M2

TYP4M3

TYP4M4


earthquake

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

DISPL (in)

FO

RC

E (

Kip

)

TYP4M1

TYP4M2

TYP4M3

TYP4M4


154

Figure 5-37: Directional rigidity criterion

FORCE

X

Y

δ2Ymax

δ1Ymax

Undeformed Diaphragm

Shear Walls

L

W

ARY = L/W

RCY = δ1Ymax/δ2Ymax

FO

RC

E

X

Y

δ2Xmax

δ1Xmax

Undeformed Diaphragm

Shear Walls

L

W

ARX = W/L

RCX = δ1Xmax/δ2Xmax

155

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 1 2 3 4 5 6

ASPECT RATIO (ARd)

RIG

IDIT

Y C

RIT

ER

ION

(R

Cd

)

M1_IMPVAL

M2_IMPVAL

M1_NRIDGE

M2_NRIDGE

RIGID

Figure 5-38: Rigidity criterion plot for Types 2, 3 and 4 models 1 and 2

1.00

1.10

1.20

1.30

1.40

1.50

1.60

0 1 2 3 4 5 6

ASPECT RATIO (ARX)

RIG

IDIT

Y C

RIT

ER

ION

(R

CX

)

M1_NRIDGE

RIGID

Figure 5-39: Rigidity criterion plot for Types 5, 6 and 7 model 1

156

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 1 2 3 4 5 6

ASPECT RATIO (ARd)

RIG

IDIT

Y C

RIT

ER

ION

(R

Cd

)

FLEXIBLE

RIGID NAILS

RIGID PANEL

RIGID

Figure 5-40: Rigidity criterion plot for Types 2, 3 and 4 model 1 with various in-plane

diaphragm flexibilities

1.00

1.10

1.20

1.30

1.40

1.50

1.60

0 1 2 3 4 5 6

ASPECT RATIO (ARd)

RIG

IDIT

Y C

RIT

ER

ION

(R

Cd

)

FLEXIBLE

RIGID NAILS

RIGID PANEL

RIGID

Figure 5-41: Rigidity criterion plot for Types 5, 6 and 7 model 1 with various in-plane

diaphragm flexibilities

157

Table 5-1: Plywood shear wall element properties used in verification analysis


USED


SIDE STUDS


INTERIOR STUDS


CONNECTION length = 0 K0 = 4.8 kip/in oriented nllink

r1 = 0.04924

r2 = -0.04924

r3 = 1.01

r4 = 0.144

P0 = 0.180 kip

PI = 0.075

δult = 0.49 in

α = 0.80

β = 1.1

Table 5-2: Waferboard shear wall element properties used in verification analysis


USED


SIDE STUDS


INTERIOR STUDS


CONNECTION length = 0 K0 = 4.72 kip/in oriented nllink

r1 = 0.0508

r2 = -0.0508

r3 = 1.01

r4 = 0.144

P0 = 0.200 kip

PI = 0.075

δult = 0.49 in

α = 0.80

β = 1.1

158

Table 5-3: Maximum and minimum displacements used in verification analysis

MAX MIN MAX MIN

MODEL 0.48 -0.49 0.70 -0.64

EXPERIMENT (DOLAN 1989) 0.67 -0.58 0.63 -0.56

DISPLACEMENT (in)

PLYWOOD WAFERBOARD

Table 5-4: Direction aspect ratios and vibration periods of all the models

X Y X Y θ X Y θ

TYPE1

M1 1.00 1.00 0.279 0.322 0.187 0.243 0.271 0.167

M2 1.00 1.00 0.323 0.322 0.188 0.271 0.271 0.168

M3 1.00 1.00 0.313 0.322 0.198 0.280 0.271 0.176

M4 1.00 1.00 0.372 0.322 0.199 0.323 0.271 0.177

TYPE2

M1 0.50 2.00 0.270 0.454 0.242 0.247 0.385 0.215

M2 0.50 2.00 0.305 0.455 0.243 0.275 0.386 0.216

M3 0.50 2.00 0.307 0.455 0.246 0.282 0.369 0.212

M4 0.50 2.00 0.356 0.457 0.248 0.327 0.386 0.222

TYPE3

M1 0.33 3.00 0.271 0.499 0.267 0.245 0.382 0.221

M2 0.33 3.00 0.305 0.500 0.268 0.273 0.382 0.222

M3 0.33 3.00 0.309 0.501 0.267 0.278 0.382 0.223

M4 0.33 3.00 0.357 0.501 0.270 0.320 0.383 0.225

TYPE4

M1 0.20 5.00 0.265 0.559 0.289 0.242 0.379 0.226

M2 0.20 5.00 0.296 0.560 0.291 0.271 0.379 0.227

M3 0.20 5.00 0.285 0.562 0.303 0.273 0.379 0.226

M4 0.20 5.00 0.345 0.562 0.291 0.315 0.380 0.228

TYPE5

M1 1.00 1.00 0.349 0.334 0.219 0.334 0.291 0.204

M3 1.00 1.00 0.344 0.381 0.229 0.330 0.346 0.214

TYPE6

M1 2.00 0.50 0.353 0.305 0.242 0.333 0.288 0.225

M3 2.00 0.50 0.356 0.353 0.247 0.339 0.339 0.231

TYPE7

M1 5.00 0.20 0.383 0.288 0.281 0.335 0.281 0.240

M3 5.00 0.20 0.385 0.334 0.153 0.335 0.326 0.241

DIRECTION ASPECT

RATIO

FLEXIBLE DIAPHRAGM RIGID DIAPHRAGM

VIBRATION PERIODS (sec) VIBRATION PERIODS (sec)

159

Table 5-5: Element properties used in the house models


USED

SHEARWALLS


SIDE STUDS


INTERIOR STUDS


DIAPHRAGM

JOISTS, BLOCKINGS b x h = 9.5" x 1.5" E = 1400 ksi, µ = 0.33 2 node frame


CONNECTION

NAILS length = 0 K0 = 3.2 kip/in nonoriented nllink

r1 = 0.0610 with modified

r2 = -0.0780 Stewart spring pair

r3 = 1.40

r4 = 0.143

P0 = 0.169 kip

PI = 0.032

δult = 0.49 in

α = 0.80

β = 1.1

160

Table 5-6: Analysis cases

ANALYSIS x y RIGID NAILS & PANELS ALL

COUNT JOISTS & JOISTS FLEXIBLE IMPVAL NRIDGE IMPVAL NRIDGE

1 to 16 1,2,3,4 1,2,3,4 X X X √ √ X √ X

17 to 32 1,2,3,4 1,2,3,4 √ X X X √ X √ X

33 to 48 1,2,3,4 1,2,3,4 X X X √ X √ X √

49 to 64 1,2,3,4 1,2,3,4 √ X X X X √ X √

65 to 70 5,6,7 1,3 X X X √ X √ X X

71 to 76 5,6,7 1,3 √ X X X X √ X X

77 to 79 2,3,4 2 X √ X X √ X √ X

80 to 82 2,3,4 2 X X √ X √ X √ X

83 to 85 5,6,7 1 X √ X X X √ X X

86 to 88 5,6,7 1 X X √ X X √ X X

X DIR (0.3g PGA) Y DIR (0.1g PGA)

(4) All the 88 analysis are performed using WoodFrameSolver program

TYPxMy

(1) x and y columns in TYPxMy should be interpreted as numbers y varied within each x i.e. TYP1M1, TYP1M2….TYP2M1….

(2) Total of 88 nonlinear response history analysis cases presented in the table

(3) X = Not included in the analysis, √ = Included in the analysis

IN-PLANE FLEXIBILITY INCLUDED EARTHQUAKE LOADING

Table 5-7: Type 1 Models, ratio of peak in-plane base shears obtained using rigid and

flexible diaphragm assumptions for Imperial Valley earthquake loading

• x indicates wall is not present in the model

• numbers in bold present maximum and minimum ratio

TYP1M1 TYP1M2 TYP1M3 TYP1M4

WALL1 1.16 1.09 1.02 0.98

WALL2 1.16 1.09 x x

WALL3 1.16 1.09 1.15 1.07

WALL4 1.16 1.09 1.15 1.07

WALL5 0.95 x 0.96 x

WALL6 1.05 1.02 1.10 1.19

WALL7 1.05 1.02 1.10 1.19

WALL8 1.05 1.02 1.28 1.24

WALL9 1.05 1.02 1.28 1.24

RIGID/FLEXIBLE

PEAK IN-PLANE BASE SHEAR RATIO

161


flexible diaphragm assumptions for Northridge earthquake loading


WALL1 1.13 1.09 0.94 0.96

WALL2 1.12 1.09 x x

WALL3 1.12 1.10 1.18 1.08

WALL4 1.12 1.09 1.19 1.08

WALL5 1.00 x 0.97 x

WALL6 1.08 1.11 1.17 1.21

WALL7 1.08 1.11 1.18 1.21

WALL8 1.08 1.08 1.09 1.05

WALL9 1.08 1.08 1.09 1.06

RIGID/FLEXIBLE







WALL1 1.01 1.07 1.04 0.99

WALL2 1.01 1.07 x x

WALL3 1.01 1.07 1.02 1.01

WALL4 1.01 1.07 1.03 1.01

WALL5 0.94 x 0.99 x

WALL6 0.92 0.90 0.89 0.78

WALL7 0.93 0.90 1.04 1.07

RIGID/FLEXIBLE




162




WALL1 1.03 1.01 0.98 0.97

WALL2 1.02 1.01 x x

WALL3 1.02 1.01 1.05 1.03

WALL4 1.03 1.02 1.06 1.04

WALL5 1.00 x 0.95 x

WALL6 0.76 0.77 0.73 0.96

WALL7 0.76 0.77 0.93 0.92

RIGID/FLEXIBLE







WALL1 0.99 1.06 1.03 0.99

WALL2 0.99 1.06 x x

WALL3 0.99 1.06 1.01 0.99

WALL4 0.99 1.06 1.02 0.99

WALL5 0.92 x 1.02 x

WALL6 0.86 0.82 0.67 0.68

WALL7 0.86 0.82 1.06 0.96

RIGID/FLEXIBLE







WALL1 1.12 1.02 0.98 1.00

WALL2 1.12 1.02 x x

WALL3 1.12 1.02 1.00 1.04

WALL4 1.12 1.02 1.00 1.04

WALL5 1.06 x 1.00 x

WALL6 0.75 0.75 0.68 0.82

WALL7 0.74 0.75 0.91 0.88

RIGID/FLEXIBLE




163




WALL1 0.97 1.05 1.03 1.00

WALL2 0.97 1.05 x x

WALL3 0.97 1.05 1.02 0.98

WALL4 0.97 1.05 1.01 0.98

WALL5 0.92 x 1.03 x

WALL6 0.81 0.76 0.74 0.76

WALL7 0.81 0.76 1.03 0.93

RIGID/FLEXIBLE







WALL1 1.08 1.01 1.01 1.01

WALL2 1.08 1.01 x x

WALL3 1.08 1.02 1.01 1.00

WALL4 1.08 1.02 1.02 1.00

WALL5 1.07 x 1.01 x

WALL6 0.75 0.81 0.74 0.94

WALL7 0.75 0.82 0.96 0.95

RIGID/FLEXIBLE






TYP5M1 TYP5M3

WALL1 1.01 1.02

WALL2 1.00 0.99

WALL3 1.01 1.02

RIGID/FLEXIBLE

PEAK IN-PLANE

BASE SHEAR RATIO

164



TYP6M1 TYP6M3

WALL1 1.05 1.05

WALL2 0.97 0.96

WALL3 1.05 1.05

PEAK IN-PLANE

BASE SHEAR RATIO

RIGID/FLEXIBLE



TYP7M1 TYP7M3

WALL1 1.07 1.06

WALL2 0.91 0.91

WALL3 1.07 1.06

RIGID/FLEXIBLE

PEAK IN-PLANE

BASE SHEAR RATIO

Table 5-18: Rigidity criteria ratios for Types 2, 3 and 4 models 1 and 2

ASPECT DIRECTION

TYPE RATIO (AR) (d) M1_IMPVAL M2_IMPVAL M1_NRIDGE M2_NRIDGE

4 5 Y 2.36 2.28 2.14 2.2

3 3 Y 1.68 1.7 1.68 1.73

2 2 Y 1.51 1.46 1.45 1.52

2 0.5 X 1.13 1.15 1.12 1.15

3 0.33 X 1.07 1.12 1.07 1.12

4 0.2 X 1.05 1.07 1.05 1.05

RIGIDITY CRITERIA (RC)

Table 5-19: Rigidity criteria ratios for Types 5, 6 and 7 model 1

ASPECT DIRECTION RC

TYPE RATIO (AR) (d) M1_NRIDGE

5 1 X 1.08

6 2 X 1.30

7 5 X 1.49

165

Table 5-20: Rigidity criteria ratios for Types 2, 3 and 4 model 2 with various in-plane

flexibilities

ASPECT DIRECTION

TYPE RATIO (AR) (d) FLEXIBLE RIGID NAILSRIGID PANEL

4 5 Y 2.28 1.17 2.17

3 3 Y 1.7 1.05 1.66

2 2 Y 1.46 1.03 1.47

2 0.5 X 1.15 1.01 1.14

3 0.33 X 1.12 1 1.11

4 0.2 X 1.07 1 1.07


Table 5-21: Rigidity criteria ratios for Types 5, 6 and 7 model 1 with various in-plane

flexibilities

ASPECT DIRECTION

TYPE RATIO (AR) (d) FLEXIBLE RIGID NAILS RIGID PANEL

5 1 X 1.08 1.01 1.08

6 2 X 1.30 1.02 1.27

7 5 X 1.49 1.09 1.53


166

CHAPTER 6

FUTURE WORK

Previous chapters in this thesis present a literature review, the development of finite

element models, analysis tools, and a parametric study of light frame wood structures

(LFWS). Each of these chapters includes a detailed summary or conclusion in the end,

and hence this chapter provides only the future work recommendations by the author.

FUTURE WORK: The finite element method is the most robust analysis procedure in

the current state of the art. The use of finite elements in analyzing LFWS is restricted by

the lack of available input information for the models, high computational time for

repeated nonlinear dynamic analysis and the time taken in creating detailed house

models. The following recommendations for future work are made by the author:

RECOMMENDATION 1: A further detailed parametric study may be performed if more

connection data, material properties and house dimensions can be obtained. The

assumptions and limitations in the finite element modeling and analysis of houses may be

reduced to get a more realistic response.

RECOMMENDATION 2: The current state of the art (SAWS and SAPWOOD) consists of

2D in-plane rigid diaphragm models with walls represented by one-dimensional nonlinear

hysteretic springs in the direction of the walls. This approach may be verified for

asymmetric plan houses modeled using finite elements with flexible diaphragms.

RECOMMENDATION 3: The current limitation of the WoodFrameSolver program is the

fact that it does not have any self graphical user interface (GUI) for the creation of

models. It is dependent upon the .S2K input format files created by SAP 2000 version 7.x

or the automated WHFEMG program. The former, however, may not always be used

because of the additional features implemented in WoodFrameSolver and different

element, hysteresis properties used in LFWS analysis. The latter program is restricted to

167

only finite element house models. Currently, the use of additional features is done by

manipulating the input files manually, which for larger models may not be trivial.

RECOMMENDATION 4: A detailed finite element model of a light wood frame structure

can easily overwhelm the current capacity of a single processor computer and hence

alternatives available in the current state of the art computing may be explored. Parallel

finite element analysis programs running on clusters are now pervasive. The programs

running on clusters are developed using a distributed memory model which is difficult to

code and requires communication between the CPUs. A shared memory model is another

option for writing a parallel program. A shared memory is a computer architecture in

which two or more similar processors are attached to a common memory. Although

shared memory programming is far simpler than distributed memory programming, much

research work has been done in the development of programs using distributed memory

models.

Today, the CPU architectures (SMPs, dual core, quad core, and 64 bit) supporting high

performance computing using shared memory programming are evolving rapidly. The 64

bit architecture theoretically provides the user with unlimited random access memory

(RAM). In this scenario, developing programs using a shared memory model seems

optimistic. The current state of the art computing provides an application programming

interface called OpenMP which can be used to write programs that can run on shared

memory systems. A serious advantage which comes with the use of OpenMP is the

incremental parallelization of various existing finite element analysis codes without

altering the existing architecture of the program. Also, when compared to MPI for

distributed memory programming, it requires much less development time. The

parallelization is done by adding the compiler directives at the beginning and end of the

portions of the code which are desired to be parallelized. OpenMP provides an easy

interface for developers and its programming standard is portable. It is very efficient for

programs where loop level parallelism is abundant. It uses a fork-join model of parallel

execution, and the program development is tidy and understandable. The PARDISO

solver in WoodFrameSolver uses OpenMP for parallel solution of SOEs. This solver may

168

be a good starting point to see the increased efficiency and speed up we get in solving

larger systems before starting incremental parallelization.

RECOMMENDATION 5: The WoodFrameSolver program is a powerful utility and its

object oriented design facilitates the enhancement of program features. The program

currently can perform linear static, linear and nonlinear dynamic, incremental dynamic

analysis and DISPAR using virtual work. As may be recalled, the WoodFrameSolver

program has been used for nonlinear static analysis of wood shear walls by performing a

dynamic analysis on the sdof mass-spring system attached adjacent to the wall. Nonlinear

static analysis is widely used in engineering practice and theory, and hence having this

feature might really prove useful and of course less time consuming compared to the

existing procedure.

The shear walls and diaphragms in house models tend to yield and displace when

subjected to dynamic loading. This displacement under the gravity loading may induce

second-order moments or P-∆ effects in the structure. The P-∆ inclusion along with

regular nonlinear dynamic analysis may provide more detailed information on the

behavior of LFWS systems. This aspect of the LFWS has not been researched much. and

if one desires to proceed in this direction, then P-∆ analysis capability needs to be added

to the WoodFrameSolver program for analysis purposes.

RECOMMENDATION 6: A large part of this dissertation is the development of

WoodFrameSolver program with advanced features, and the use of virtual work to

calculate displacement participation of various elements in the structure is one aspect of

it. This method has been extended to finite elements in the program and has been

successfully used in linear static analysis. Theoretically, this procedure may be extended

to nonlinear analysis. The architecture of the program provides a flexible framework for

the extension of this procedure from linear to nonlinear. This feature may further be used

for the identification of the potential sources of deflection in LFWS and verification of

the accuracy of the design formulas for shear walls and diaphragms. Identifying the

potential sources of deflection in a LFWS may lead to improved understanding of its

169

behavior, and may help in reducing the inter-story drift and costly structural and

nonstructural damage. Quantifying axial, shear, flexure and torsional deformations of

structural elements in a subassemblage under a given load may lead to improved design

formulas and optimization. No such research identified in this area which tries to

calculate displacement contributions of structural elements to the displacement at a

particular point and direction in the structure.

RECOMMENDATION 7: Hysteretic response of the connections is the major source of

energy dissipation or damping in LFWS. Not much research has been done on which

inherent classical damping model one should choose in the response history analysis of

LFWS systems. An initial selection of damping ratios in some modes may result in

different damping in other modes than what is intended initially. This would occur

primarily because of the frequency shifts of the structural system, and hence it must be

investigated before any classical damping model is used in the LFWS response history

analysis.

RECOMMENDATION 8: A side project to automate the generation of LFWS models led

to the development of an automatic wood house finite element model generator

(WHFEMG). This utility has been found to really produce time saving in the generation

of house models and their subassemblages compared to developing models manually.

However, this program too has its limitations (see the manual for details) and needs

further work if, for example, non-rectangular plan houses or multi-story houses are to be

analyzed.

Overall, the WoodFrameSolver program is a general purpose finite element analysis tool

and may be used (obviously within limitations) and extended in several ways. The most

immediate and important need in the author’s opinion is the creation of a GUI for the

most fruitful use of the program. The finite element analyses of LFWS also need to

include more parameters in the parametric study so that further detailed understanding of

responses is obtained.

Version 1.0 Beta

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170

APPENDIX A WOODFRAMESOLVER PROGRAM ARCHITECTURE

Version 1.0 Beta

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171

WOODFRAMESOLVER

General Purpose Finite Element Analysis of Structures

PROGRAM ARCHITECTURE

Written by: Rakesh Pathak ([email protected])

Version 1.0 Beta

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172

PREFACE

Before reading this document one must read through the following two

documents in detail:

(1) KeySolver Architecture Commentary by Paul W. Spears

(2) Chapter 6 in “Development of Finite Element Modeling Mesh

Generation and Analysis Software of Light Wood Frame Houses”

M.S. Thesis by Rakesh Pathak

The program development is based on the architecture presented in these

two documents. This document presents the current class structure of the

program. Also, WoodFrameSolver was initially named as KeySolver and

hence at several places in the program documents user may find program

referred to as KeySolver.

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WOODFRAMESOLVER PROGRAM

ARCHITECTURE

INTRODUCTION: This document presents the current class structure of the

WoodFrameSolver program. WoodFrameSolver is a finite element analysis

program capable of static and dynamic analysis. The program has been

designed to read S2K files which may be created either from SAP2000

version 7.4 or from any other processor generating input in the format

described in the users manual document (Appendix B). This program has

been written on a mixed language Visual Studio 2005 .Net platform using

object oriented C++, FORTRAN and C.

WoodFrameSolver performs various operations to analyze a finite element

model however this entire process can be broken down into three key major

operations which are as follows: (1) reading the input file and generating the

analytical model, (2) analysis of the generated model, and (3) printing the

required output. This program is designed using object oriented philosophy

and structured such that it provides a flexible platform for extension. Table 1

presents the list of all the classes implemented1 in the WoodFrameSolver

program.

Table 1. WoodFrameSolver class structure

1. Classes involved in the generation of an analytical model from the input data COMMENTS

1.1 KSS2KFileReader ………… Input file reader

1.2 KSModelBuilder ………… Interface between

1 Classes shown in red color either needs further work or some debugging to become fully functional. Also,

one should refer to their footnotes for the discussion of the problem.

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174

the file reader and

the container

1.3 KSModel ………… The container class

of various entities

constituting a finite

element model.

1.3.1 KSNode ………… Node class

1.3.2 KSRestraint ………… Restraint class

1.3.3 KSConstraint ………… Constraint base

class

1.3.3.1 KSEqualConstraint ………… Equal Constraint

class

1.3.4 KSMaterial ………… Material base class

1.3.4.1 KSElasticMaterial ………… Elastic material

class

1.3.5 KSSection ………… Section base class

1.3.5.1 KSShellSection ………… Shell section class

1.3.5.2 KSFrameSection ………… Frame section class

1.3.5.3 KSNonLinearLinkSection ………… Nonlinear link

section class

1.3.5.3.1 KSStewartHysteresisElementSection ………… Stewart hysteresis

element section

class

1.3.5.3.2 KSIMKHysteresisElementSection ………… Ibarra, Medina,

Krawinkler

hysteresis element

section class

1.3.5.3.3 KSNonlinearGapOrHookSection ………… Gap or Hook section

class

1.3.5.3.4 KSBilinearElementSection ………… Bilinear element

section class

1.3.6 KSElement ………… Element base class

1.3.6.1 KSFrameElement ………… Frame element class

1.3.6.2 KSEightNodeSolidElement ………… Solid element class

1.3.6.3 KSNonLinearLinkElement ………… Nllink element class

1.3.6.4 KSShellElement ………… Shell element class

1.3.6.5 KSSpringElement ………… Spring element class

1.3.7 KSLoadCase ………… Load case base class

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1.3.7.1 KSStaticForceLC ………… Static force load

case class

1.3.7.2 KSDynamicForceLC ………… Dynamic force load

case class

1.3.8 KSFunction ………… Input loading

functions base class

1.3.8.1 KSTimeHistoryFunction ………… Time history

loading functions

class

1.3.8.2 KSResponseHistoryFunction

1.3.9 KSDampingModel ………… Damping model

class

1.3.10 KSIDAData ………… Incremental

dynamic analysis

data

1.3.11 KSModalAnalysisData ………… Modal analysis data

base class

1.3.11.1 KSEigenAnalysisData ………… Eigen analysis data

class

1.3.11.2 KSRitzAnalysisData

1.4 KSEcho ………… Echo class which is

used to echo all the

input read using the

file reader

2. Classes involved in the analysis of the model

2.1 KSAnalysisCase ………… Analysis base class

2.1.1 KSStaticAnalysisCase ………… Static analysis class

2.1.2 KSDynamicAnalysisCase ………… Dynamic analysis

class

2.2 KSDOFNumberer ………… Degree of freedom

numberer base class

2.2.1 KSPlainNumberer ………… Plain degree of

freedom numberer

class

2.2.2 KSPacManReNumberer ………… PacMan degree of

freedom numberer

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class

2.2.3 KSRCMReNumberer ………… Reverse Cuthill and

McKee numberer

class calling a C

method

2.2.4 KSPFMNumberer2 ………… Profile front

minimization

numberer class

calling a FORTRAN

routine

2.2.5 KSFortranPacManReNumberer2 ………… PacMan degree of

freedom numberer

class calling a

FORTRAN routine

2.2.6 KSAdvancedPacManReNumberer2 ………… Advanced more

effective PacMan

degree of freedom

numberer class

calling a FORTRAN

routine

2.3 KSSOEandSolver ………… Base class for

system of equation

solvers

2.3.1 KSBandedSOEnSolver ………… Banded system of

equation solver class

2.3.2 KSCGSOEnSolver ………… Conjugate gradient

system of equation

solver base class

2.3.2.1 KSPCGSOEnSolver3 ………… Parallel conjugate

gradient system of


2.3.2.2 KSSCGSOEnSolver ………… Serial conjugate

gradient system of

2 Need compatible INTEL FORTRAN compiler for visual studio 2005 for the FORTRAN routines. These

classes call FORTRAN methods for renumbering. The FORTRAN methods were tested in visual studio 2003 having a compatible INTEL FORTRAN compiler attached to it. There was no compatible INTEL

FORTRAN compiler when we shifted to visual studio 2005. Also, one might also ever skip using these

renumberers because for solving bigger problems we recommend using DSS or PARDISO equation solvers

which don’t require these renumbering schemes. 3 Needs to be debugged and some coding is required for this class to become fully functional

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2.3.3 KSSparseSOEnSolver ………… Sparse system of

equation solver base

class

2.3.4 KSDSSSOEnSolver ………… Direct sparse system

of equation solver

class

2.3.5 KSPardisoSOEnSolver ………… Parallel direct sparse

system of equation

solver class

2.4 KSEigenValueSolver ………… Eigen value solver

base class

2.4.1 KSdsbgvxSolver ………… Banded eigen value

solver class

2.5 KSNumericalIntegrator ………… Numerical

Integrator base class

2.5.1 KSNewmarkIntegrator ………… Newmark integrator

class

2.5.2 KSLinearIntlIntegrator ………… Linear interpolation

Integrator class

2.5.3 KSCDMIntegrator ………… Central difference

method class

2.5.4 KS4thOrderRungeKutta ………… 4th order Runge

Kutta method class

3. Classes involved in printing output to file

3.1 KSResponse ………… Response class for

all kinds of response

4. Other Classes

4.1 KSMatrixBase ………… Matrix base class

4.1.1 KSFullMatrix ………… Matrix child class

4.2 KSAxesTransformation ………… Axes transformation

class for various

elements

4.3 KSString ………… String class

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currently used while

reading the input file

4.4 KSUtilitySTL ………… It’s a purge utility

class

The objects and the methods of these classes are used in performing

operations required in the above three steps. The following sections discuss

these three steps:

READING THE INPUT FILE AND GENERATING THE ANALYTICAL MODEL:

The program reads the input file by calling on the methods of

KSS2KFileReader object. Each method of KSS2KFileReader reads one

block of the input from the file and passes it to the corresponding method

inside KSModelBuilder object. The corresponding KSModelBuilder object

method checks the input for consistency and passes the block to the

corresponding KSEcho object methods for echoing the block inside .KSECH

file. If there are any errors in the input then an error count is also done inside

the KSEcho object which prints the count and the list of errors at the end of

the .KSECH file. One should note that KSEcho object will not capture errors

related to modeling of the problem. If there is no error in the block read by

the KSS2KFileReader method then the input data is saved in the

corresponding map container inside KSModel object. This method for saving

the data inside the container is called inside the KSModelBuilder object. This

way every time a block is read by the KSS2KFileReader object, it is passed

to the corresponding KSModelBuilder object method, where it is checked for

errors and passed to the corresponding KSEcho method for echoing and if

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the data is correct then it is also passed to the corresponding method of

KSModel for storage inside the map container.

ANALYSIS OF THE GENERATED MODEL: When the KSModel object’s

container has been fully populated it means that model has been generated

for the analysis. The analysis is performed inside the respective analysis

class objects. The linear static analysis is the default analysis even when the

loads are not applied on the structure. This is done by assuming zero loads.

PRINTING THE REQUIRED OUTPUT: The output printing is facilitated by the

methods inside the KSResponse class object. The relevant methods are called

inside the analysis object of the analysis being performed i.e. static or

dynamic. Various output files may be generated depending upon what type

of analysis is being performed. In the static analysis, the output data is in the

form of nodal displacements, frame element forces, spring forces, nllink

forces, solid stresses, in-plane shell stresses. In the dynamic analysis, the

output data is in the form of maximum/minimum nodal displacements,

velocities, accelerations, nodal displacement response histories at the

requested nodes, base reaction response histories for all the restraints and

nllink force deformation response histories at the requested nllinks.

CLASS INTERFACE AND PROGRAM FLOW: The program comes

along with the source code and contains several thousand lines of code. The

public, protected and private interface of classes may be seen by opening the

respective *.h or *.hpp file. To understand the program flow the author

recommends opening the project file in Visual Studio .Net 2005 and

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executing it in “DEBUG” mode with break points placed at several locations

starting from the driver program KeySolver.cpp. The program shall take one

through various branches depending upon the input being provided and the

options being selected. Currently, this is perceived as the best mean to

understand the flow of the program.

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APPENDIX B WOODFRAMESOLVER USERS MANUAL AND INPUT FILE FORMAT

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WOODFRAMESOLVER


INPUT FILE FORMAT

and

USER’S MANUAL


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DISCLAIMER

The first version of this program was released in May 2004 and this program

has come a long way since then. It has been thoroughly tested over these

years and has been used in various research applications. However, we do

not give any warranty to the users of this program. The user must understand

the assumptions involved in the program and the results obtained must be

verified by the user through other means.

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ACKNOWLEDGEMENT

Thanks are due to monetary grant of Simpson Strong Tie. and initial grant of

Keymark Inc. which helped in the development of WoodFrameSolver

program.

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TABLE OF CONTENTS

DISCLAIMER

ACKNOWLEDGEMENT

1. INTRODUCTION

2. INPUT FILE FORMAT 2.1 SYSTEM

2.2 JOINT

2.3 LOCAL

2.4 RESTRAINT

2.5 CONSTRAINT

2.6 SPRING

2.7 MASS

2.8 MATERIAL

2.9 FRAME SECTION

2.10 SHELL SECTION

2.11 NLPROP

2.12 FRAME

2.13 SHELL

2.14 SOLID

2.15 NLLINK

2.16 LOAD

2.17 MODE

2.18 FUNCTION

2.19 DAMPING MODEL

2.20 HISTORY

2.21 INCREMENTAL DYNAMIC ANALYSIS

2.22 VIRTUAL WORK ANALYSIS

2.23 COMBO

2.24 END

3. PROGRAM INTERFACE

4. OUTPUT FILES

5. DLL’S NEEDED

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1. INTRODUCTION

This document describes the text input file format for the WoodFrameSolver

program. The program can only read certain files with .S2K extension

developed using SAP2000 version 7.x and all the .S2K files developed using

WHFEMG program. The WoodFrameSolver program currently does not

provide any self GUI and hence most of the model input is provided to it via

a text formatted file having .S2K extension. These files must follow a fixed

format to describe the input model which is mostly adopted from SAP2000

version 7.x input file format. Thus, SAP2000 may serve as a GUI tool for

complete or partial model input file development for WoodFrameSolver.

However, an input model file created for WoodFrameSolver independently

in .S2K format may not be viewed by the SAP program due to the use of

additional/different program features present in WoodFrameSolver.

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2. INPUT FILE FORMAT

The text input data written inside the .S2K input file to describe the input

model is divided into various blocks the starting keywords of which are

listed in Table 1. Some of the blocks in the Table may or may not be present

depending upon the problem. A block consists of the block keyword which

is followed by the detailed input data for that block, for example, “JOINT” is

followed by the joint co-ordinates and “FRAME” is followed by the list of

frame elements. Any two blocks must be separated with each other by two

lines (press ENTER) containing no blank characters anywhere in them. One

should note that the presence of the blocks with input in the right format,

however does not guarantee the accurate solution. To obtain an accurate

solution one must have a correct model and must understand the

assumptions involved in the program. Also, the order of these blocks is

important as is the format which is discussed in the following sections. The

information stored in each block is different and thus each block follows a

different input format.

All the blocks described in the following sections contain a block definition

and a block example. The block definition presents how the block should

look like when used in the input file. Each block definition also presents a

table which describes the text to be used in the block and their type i.e.

integer, string etc. The type could either be variable or fixed. By variable it

means that the text could take any other value apart from what is presented.

By fixed it means that the text may only take that value which is presented in

the table. The definition column in these tables also gives some possible

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values of the variable text. When a block is required the user needs to write

and define all the text which is presented in bold red color in all the block

descriptions. The text pattern and its ordering must be strictly followed for

the program to read the data correctly.

Table 2.1: Keywords for WoodFrameSolver input Blocks

ORDER # BLOCK KEYWORD

1 SYSTEM

2 JOINT

3 LOCAL

4 RESTRAINT

5 CONSTRAINT

6 SPRING

7 MASS

8 MATERIAL

9 FRAME SECTION

10 SHELL SECTION

11 NLPROP

12 FRAME

13 SHELL

14 SOLID

15 NLLINK

16 LOAD

17 COMBO

18 MODE

19 FUNCTION

20 DAMPING MODEL

21 HISTORY

22 INCREMENTAL DYNAMIC ANALYSIS

23 VIRTUAL WORK ANALYSIS

24 END

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2.1 SYSTEM: This is the first keyword read by the program and it

defines the first block which is mandatory to be defined for any model. It

contains force and displacement units to be defined by the user.

2.1.1 BLOCK DEFINITION: SYSTEM DOF=UX,UY,UZ,RX,RY,RZ LENGTH=LU FORCE=FU PAGE=SECTIONS

TEXT DEFINITION TYPE

LU LENGTH UNIT e.g. mm,

in, ft etc.

STRING

VARIABLE

FU FORCE UNIT e.g. KN,

Kip etc.

STRING

VARIABLE

2.1.2 BLOCK EXAMPLE: SYSTEM DOF=UX,UY,UZ,RX,RY,RZ LENGTH=mm FORCE=KN PAGE=SECTIONS

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2.2 JOINT: The data block following this keyword defines the joints

occurring in the model. This block contains joint identification numbers

(id’s) and co-ordinates as shown below to be defined by the user. A joint

identification number has to be a unique however the co-ordinates may not

be so.

2.2.1 BLOCK DEFINITION: JOINT ID1 X=XCOORDINATE1 Y=YCOORDINATE1 Z=ZCOORDINATE1 ID2 X=XCOORDINATE2 Y=YCOORDINATE2 Z=ZCOORDINATE2 ID3 X=XCOORDINATE1 Y=YCOORDINATE1 Z=ZCOORDINATE1 … …


ID JOINT

IDENTIFICATION

NUMBER (UNIQUE)

INTEGER

VARIABLE

XCOORDINATE X COORDINATE OF

THE JOINT

DOUBLE

VARIABLE

YCOORDINATE Y COORDINATE OF

THE JOINT

DOUBLE

VARIABLE

ZCOORDINATE Z COORDINATE OF

THE JOINT

DOUBLE

VARIABLE

2.2.2 BLOCK EXAMPLE: JOINT 1 X=0 Y=0 Z=0 2 X=50 Y=0 Z=0 3 X=0 Y=0 Z=0 … …

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2.3 LOCAL: The data block following this keyword defines the joints

local axis.

2.3.1 BLOCK DEFINITION: LOCAL ADD=ID1 ANG=rZ1,rY’1,rX’’1 ADD=ID2 ANG=rZ2,rY’2,rX’’2 … …


ID JOINT

IDENTIFICATION

NUMBER (UNIQUE)

INTEGER

VARIABLE

rZ ROTATION IN

DEGREES ABOUT

GLOBAL Z AXIS

DOUBLE

VARIABLE

rY’ ROTATION IN

DEGREES ABOUT

NEW Y AXIS

DOUBLE

VARIABLE

rX’’ ROTATION IN

DEGREES ABOUT

NEW X AXIS

DOUBLE

VARIABLE

2.2.2 BLOCK EXAMPLE: LOCAL ADD=2 ANG=10.2,0,5 ADD=9 ANG=1,100,30 … …

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2.4 RESTRAINT: The data block following this keyword defines the

joint restraints block. A joint may have a single restraint or a set of

restraints. These restraints at a joint are defined shown as follows:

2.4.1 BLOCK DEFINITION:

RESTRAINT ADD=ID1 DOF=U1,U2,U3,R1,R2,R3 ADD=ID2 DOF=U2,U3,R1,R2 … …

2.4.2 BLOCK EXAMPLE:

RESTRAINT ADD=1 DOF=U1,U2,U3,R1,R2,R3 ADD=86 DOF=U2,U3,R1,R2 … …


ID JOINT IDENTIFICATION

NUMBER AT WHICH

RESTRAINTS ARE APPLIED

INTEGER VARIABLE

U1 X TRANSLATIONAL RESTRAINT STRING FIXED

U2 Y TRANSLATIONAL RESTRAINT STRING FIXED

U3 Z TRANSLATIONAL RESTRAINT STRING FIXED

R1 X ROTATIONAL RESTRAINT STRING FIXED

R2 Y ROTATIONAL RESTRAINT STRING FIXED

R3 Z ROTATIONAL RESTRAINT STRING FIXED

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2.5 CONSTRAINT: The data block following this keyword defines the

joint constraints block. Currently, the program can only handle equal

constraint type and hence that is the only type which may be defined. This

block contains joint constraints shown as follows:


CONSTRAINT NAME=CONSNAME1 TYPE=EQUAL DOF=UX,UY,UZ,RX,RY,RZ CSYS=0 ADD=MJOINTID1 ADD=SJOINTID1 NAME=CONSNAME2 TYPE= EQUAL DOF=UX,UY,UZ,RX CSYS=0 ADD=MJOINTID2 ADD=SJOINTID2 ADD=SJOINTID3 ….


CONSTRAINT NAME=EQUAL1 TYPE=EQUAL DOF=UX,UY,UZ,RX,RY,RZ CSYS=0 ADD=1 ADD=2 ADD=3


CONSNAME CONSTRAINT NAME STRING VARIABLE

EQUAL CONSTRAINT TYPE STRING FIXED

UX X TRANSLATIONAL

CONSTRAINT

STRING FIXED

UY Y TRANSLATIONAL

CONSTRAINT

STRING FIXED

UZ Z TRANSLATIONAL

CONSTRAINT

STRING FIXED

RX X ROTATIONAL CONSTRAINT STRING FIXED

RY Y ROTATIONAL CONSTRAINT STRING FIXED

RZ Z ROTATIONAL CONSTRAINT STRING FIXED

MJOINTID MASTER JOINT ID NUMBER.

ALWAYS FIRST IN THE LIST

AND ONLY ONE MASTER IN A

CONSTRAINT

INTEGER

VARIABLE

SJOINTID SLAVE JOINT ID NUMBER.

CAN HAVE MULTIPLE SLAVES

ATTACHED TO A MASTER

INTEGER

VARIABLE

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2.6 SPRING: The data block following this keyword defines the joint

springs block. Each joint spring may have up to six springs attached along

each degree of freedom at that joint. This block contains joint springs

defined as follows:


SPRING ADD=ID1 U1=XTSPRS U2=YTSPRS U3=ZTSPRS R1=XRSPRS R2=YRSPRS R3=ZRSPRS ADD=ID2 U2=YTSPRS R1=XRSPRS R2=YRSPRS … …


SPRING ADD=1 U1=100 U2=200 U3=300 R1=8000 R2=9000 R3=10000 ADD=8 U2=111 R1=432 R2=334 … …

NOTE: If a spring stiffness (U1, U2 etc) is not defined then its value is assumed to be zero.


ID JOINT ID AT WHICH SPRING IS

TO BE ATTACHED

INTEGER

VARIABLE

XTSPRS X TRANSLATIONAL SPRING

STIFFNESS

DOUBLE

VARIABLE

YTSPRS Y TRANSLATIONAL SPRING

STIFFNESS

DOUBLE

VARIABLE

ZTSPRS Z TRANSLATIONAL SPRING

STIFFNESS

DOUBLE

VARIABLE

XRSPRS X ROTATIONAL SPRING

STIFFNESS

DOUBLE

VARIABLE

YRSPRS Y ROTATIONAL SPRING

STIFFNESS

DOUBLE

VARIABLE

ZRSPRS Z ROTATIONAL SPRING

STIFFNESS

DOUBLE

VARIABLE

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2.7 MASS: The data block following this keyword defines the joint

masses block. The masses may be defined for three translational and three

rotational degrees of freedoms. This block contains joint masses shown as

follows:


MASS ADD=ID1 U1=XTMASS U2=YTMASS U3=ZTMASS R1=XRMASS R2=YRMASS R3=ZRMASS ADD=ID2 U2=YTMASS R1=XRMASS R2=YRMASS … …

2.7.2 BLOCK EXAMPLE: MASS ADD=14 U1=9.2755E-05 U2=9.2755E-05 U3=9.2755E-05 ADD=15 U1=9.2755E-05 U2=9.2755E-05 U3=9.2755E-05 … …

NOTE: If a joint mass (R1, R2 etc) is not defined then its value is assumed to be zero.


ID JOINT ID INTEGER

VARIABLE

XTMASS X TRANSLATIONAL JOINT MASS DOUBLE

VARIABLE

YTMASS Y TRANSLATIONAL JOINT MASS DOUBLE

VARIABLE

ZTMASS Z TRANSLATIONAL JOINT MASS DOUBLE

VARIABLE

XRMASS X ROTATIONAL JOINT MASS DOUBLE

VARIABLE

YRMASS Y ROTATIONAL JOINT MASS DOUBLE

VARIABLE

ZRMASS Z ROTATIONAL JOINT MASS DOUBLE

VARIABLE

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2.8 MATERIAL: The data block following this keyword defines the

materials block. This block contains material properties shown as follows:

2.8.1 BLOCK DEFINITION: MATERIAL NAME=MNAME1 IDES=N TYPE=ISO M=MPUV W=WPUV T=0 E=EMOD U=PR A=0 NAME=MNAME2 IDES=N TYPE=ISO M=MPUV W=WPUV T=0 E=EMOD U=PR A=0 NAME=MNAME3 IDES=N TYPE=ORTHO M=MPUV W=WPUV T=0 E=EMOD1,EMOD2,EMOD3 G=SMOD1,SMOD2,SMOD3 U=PR1,PR2,PR3 A=0,0,0

….

2.8.2 BLOCK EXAMPLE: MATERIAL NAME=STEEL IDES=N TYPE=ISO M=0.0001 W=0 T=0 E=29000 U=0.29 A=0 NAME=OTHER IDES=N TYPE=ISO M=0 W=0 T=0 E=15000 U=0.23 A=0 NAME=PLYWOOD IDES=N TYPE=ORTHO M=0.00023 W=0.0021 T=0 E=12.3,10,9.8 G=0.43,1.2,1.9 U=0.2,0.13,0.04 A=0,0,0

….


MNAME MATERIAL NAME STRING

VARIABLE

ISO ISOTROPIC MATERIAL STRING FIXED

ORTHO ORTHOTROPIC MATERIAL STRING FIXED

MPUV MASS PER UNIT VOLUME DOUBLE

VARIABLE

WPUV WEIGHT PER UNIT VOLUME DOUBLE

VARIABLE

EMOD MODULUS OF ELASTICITY DOUBLE

VARIABLE

PR POISSONS RATIO DOUBLE

VARIABLE

SMOD SHEAR MODULUS DOUBLE

VARIABLE

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2.9 FRAME SECTION: The data block following this keyword defines

the frame section block. This block contains frame section properties shown

as follows:

2.9.1 BLOCK DEFINITION: FRAME SECTION NAME=FSNAME1 MAT=MNAME1 A=AREA1 J=TC1 I=IXX1,IYY1 AS=SAXX1,SAYY1 NAME=FSNAME2 MAT=MNAME2 A=AREA2 J=TC2 I=IXX2,IYY2 AS=SAXX2,SAYY2 ….

2.9.2 BLOCK EXAMPLE: FRAME SECTION NAME=PLATE MAT=STEEL A=67 J=63 I=32,44 AS=5,5 NAME=STUDS MAT=PLYWOOD A=31 J=11 I=40,22 AS=28,21 ….


FSNAME FRAME SECTION NAME STRING

VARIABLE


VARIABLE

AREA CROSS-SECTIONAL AREA DOUBLE

VARIABLE

TC TORSIONAL CONSTANT DOUBLE

VARIABLE

IXX MOMENT OF INERTIA ABOUT 3

AXIS

DOUBLE

VARIABLE

IYY MOMENT OF INERTIA ABOUT 2

AXIS

DOUBLE

VARIABLE

SAXX SHEAR AREA IN 2 DIRECTION DOUBLE

VARIABLE

SAYY SHEAR AREA IN 3 DIRECTION DOUBLE

VARIABLE

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2.10 SHELL SECTION: The data block following this keyword defines

the shell section block. This block contains shell section properties shown as

follows:

2.10.1 BLOCK DEFINITION: SHELL SECTION NAME=SSNAME1 MAT=MNAME1 TYPE=Shell,Thin TH=MT1 THB=BT1 NAME=SSNAME2 MAT=MNAME2 TYPE=Plate,Thin THB=BT2 NAME=SSNAME3 MAT=MNAME3 TYPE=Membrane,Thin TH=MT3

…

2.10.2 BLOCK EXAMPLE: SHELL SECTION NAME=PLYWOOD MAT=WOOD TYPE=Shell,Thin TH=1 THB=2 NAME=OSB MAT=OTHER TYPE=Plate,Thin THB=2.5 NAME=WAFER MAT=WOOD TYPE=Membrane,Thin TH=3

…


SSNAME SHELL SECTION NAME STRING

VARIABLE


VARIABLE

Shell SHELL SECTION BEHAVIOR STRING FIXED

Plate PLATE SECTION BEHAVIOR STRING FIXED

Membrane MEMBRANE SECTION BEHAVIOR STRING FIXED

MT MEMBRANE THICKNESS DOUBLE

VARIABLE

BT BENDING THICKNESS DOUBLE

VARIABLE

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2.11 NLPROP: The data block following this keyword defines the node

link block. The nllink spring properties may be of type gap, hook, simple

bilinear, and modified Stewart. Each sub-block to define these properties is

different and is separated by a single line if more than one property is being

defined. This block contains node link properties shown as follows:

2.11.1 BLOCK DEFINITION: NLPROP NAME=NLNAME1 TYPE=STW_HYSTERESIS M=m W=w MR1=mr1 MR2=mr2 MR3=mr3 ->

1 IMODE=imode

DOF=U1 DOFTYPE=LIN TYPE=STEWART KE=ke F0=f0 FI=fi DU=du S0=s0 R1=d R2=r2 -> R3=r3 R4=r4 ALPHA=α BETA=β DOF=U2 DOFTYPE=LIN TYPE=STEWART KE=ke F0=f0 FI=fi DU=du S0=s0 R1=d R2=r2 -> R3=r3 R4=r4 ALPHA=α BETA=β DJ2=dj2 DOF=U3 DOFTYPE=LIN TYPE=STEWART KE=ke F0=f0 FI=fi DU=du S0=s0 R1=d R2=r2 -> R3=r3 R4=r4 ALPHA=α BETA=β DJ3=dj3 DOF=R1 DOFTYPE=LIN TYPE=STEWART KE=ke F0=f0 FI=fi DU=du S0=s0 R1=d R2=r2 -> R3=r3 R4=r4 ALPHA=α BETA=β DOF=R2 DOFTYPE=LIN TYPE=STEWART KE=ke F0=f0 FI=fi DU=du S0=s0 R1=d R2=r2 -> R3=r3 R4=r4 ALPHA=α BETA=β DOF=R3 DOFTYPE=LIN TYPE=STEWART KE=ke F0=f0 FI=fi DU=du S0=s0 R1=d R2=r2 -> R3=r3 R4=r4 ALPHA=α BETA=β

NAME=NLNAME2 TYPE=SIMPLE_BILINEAR M=m W=w MR1=mr1 MR2=mr2 -> MR3=mr3 DOF=U1 DOFTYPE=LIN K1=k11 K2=k21 K3=k31 FYP=fyp1 FYN=-fyn1 DOF=U2 DOFTYPE=NON K1=k12 K2=k22 K3=k32 FYP=fyp2 FYN=-fyn2 DOF=U3 DOFTYPE=NON K1=k13 K2=k23 K3=k33 FYP=fyp3 FYN=-fyn3 DOF=R1 DOFTYPE=LIN K1=k14 K2=k24 K3=k34 FYP=fyp4 FYN=-fyn4 DOF=R2 DOFTYPE=LIN K1=k15 K2=k25 K3=k35 FYP=fyp5 FYN=-fyn5 DOF=R3 DOFTYPE=LIN K1=k16 K2=k26 K3=k36 FYP=fyp6 FYN=-fyn6

NAME=NLNAME3 TYPE=Gap M=m W=w MR1=mr1 MR2=mr2 MR3=mr3 DOF=U1 KE=KE1 CE=0 K=KN1 OPEN=O1 DOF=U2 KE=KE2 CE=0 DJ=DJ2 K=KN2 OPEN=O2 DOF=U3 KE=KE3 CE=0 DJ=DJ3 K=KN3 OPEN=O3 DOF=R1 KE=KE4 CE=0 K=KN4 OPEN=O4 DOF=R2 KE=KE5 CE=0 K=KN5 OPEN=O5 DOF=R3 KE=KE6 CE=0 K=KN6 OPEN=O6 NAME=NLNAME4 TYPE=Hook M=m W=w MR1=mr1 MR2=mr2 MR3=mr3 DOF=U1 KE=KE1 CE=0 K=KN1 OPEN=O1 DOF=U2 KE=KE2 CE=0 DJ=DJ2 K=KN2 OPEN=O2 DOF=U3 KE=KE3 CE=0 DJ=DJ3 K=KN3 OPEN=O3 DOF=R1 KE=KE4 CE=0 K=KN4 OPEN=O4 DOF=R2 KE=KE5 CE=0 K=KN5 OPEN=O5 DOF=R3 KE=KE6 CE=0 K=KN6 OPEN=O6

1 Presents continuation from previous line. When writing in the input file it all should appear in one line.

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NLNAME NODE LINK NAME STRING

VARIABLE

m NODE LINK MASS DOUBLE

VARIABLE

w NODE LINK WEIGHT DOUBLE

VARIABLE

mr1 ROTATIONAL INERTIA 1 DOUBLE

VARIABLE


VARIABLE


VARIABLE

imode ANALYSIS MODE, USE 2 FOR

STATIC ANALYSIS, USE 0 FOR

CYCLIC ANALYSIS

INTEGER

VARIABLE

U1 X TRANSLATIONAL DEGREE OF

FREEDOM

STRING FIXED

U2 Y TRANSLATIONAL DEGREE OF

FREEDOM

STRING FIXED

U3 Z TRANSLATIONAL DEGREE OF

FREEDOM

STRING FIXED

R1 X ROTATIONAL DEGREE OF

FREEDOM

STRING FIXED

R2 Y ROTATIONAL DEGREE OF

FREEDOM

STRING FIXED

R3 Z ROTATIONAL DEGREE OF

FREEDOM

STRING FIXED

LIN FOR DEGREE OF FREEDOM TO

BEHAVE AS LINEAR

STRING FIXED

NON FOR DEGREE OF FREEDOM TO

BEHAVE AS NONLINEAR

STRING FIXED

KE LINEAR ELASTI STIFFNESS DOUBLE

VARIABLE

f0 SECONDARY STIFFNESS FORCE DOUBLE

VARIABLE

fI PINCHING FORCE DOUBLE

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VARIABLE

r1 SECONDARY STIFFNESS RATIO DOUBLE

VARIABLE

r2 TERTIARY STIFFNESS RATIO DOUBLE

VARIABLE

r3 UNLOADING PATH STIFFNESS TO

INITIAL STIFFNESS RATIO

DOUBLE

VARIABLE

r4 PINCHING STIFFNESS TO INITIAL

STIFFNESS RATIO

DOUBLE

VARIABLE

du ULTIMATE DISPLACEMENT

CORRESPONDING TO ULTIMATE

LOAD

DOUBLE

VARIABLE

α STIFFNESS DEGRADATION

PARAMETER

DOUBLE

VARIABLE

β STRENGTH DEGRADATION

PARAMETER

DOUBLE

VARIABLE

k1 INITIAL STIFFNESS IN BILINEAR

FORCE DEFORMATION CURVE

DOUBLE

VARIABLE

k2 SECONDARY STIFFNESS ABOVE

X AXIS IN BILINEAR FORCE

DEFORMATION CURVE

DOUBLE

VARIABLE

k3 SECONDARY STIFFNESS BELOW

X AXIS IN BILINEAR FORCE

DEFORMATION CURVE

DOUBLE

VARIABLE

fyp POSITIVE YIELD FORCE DOUBLE

VARIABLE

fyn NEGATIVE YIELD FORCE DOUBLE

VARIABLE

KN NONLINEAR STIFFNESS TO BE

USED IN NONLINEAR ANALYSIS

DOUBLE

VARIABLE

DJ DISTANCE OF SHEAR SPRINGS

FROM J END OF THE ELEMENT

DOUBLE

VARIABLE

O INITIAL OPENING BETWEEN TWO

NODES OF GAP OR HOOK

ELEMENT

DOUBLE

VARIABLE

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2.11.2 BLOCK EXAMPLE: NLPROP

NAME=NL1 TYPE=STW_HYSTERESIS M=0 MR1=0 MR2=0 MR3=0 IMODE=2 DOF=U1 DOFTYPE=LIN TYPE=STEWART KE=1e-3 F0=0 FI=0 DU=0 S0=0 R1=0 -> R2=0 R3=0 R4=0 ALPHA=0 BETA=0 DOF=U2 DOFTYPE=NON TYPE=STEWART KE=0.827 F0=0.890 FI=0.334 DU=12.70 -> S0=0.827 R1=0.0508 R2=-0.0508 R3=1.01 R4=0.144 ALPHA=0.80 BETA=1.1 DJ2=0 DOF=U3 DOFTYPE=NON TYPE=STEWART KE=0.827 F0=0.890 FI=0.334 DU=12.70 -> S0=0.827 R1=0.0508 R2=-0.0508 R3=1.01 R4=0.144 ALPHA=0.80 BETA=1.1 DJ3=0

NAME=NL2 TYPE=SIMPLE_BILINEAR M=0 W=0 MR1=0 MR2=0 MR3=0 DOF=U1 DOFTYPE=LIN K1=1e-3 K2=1e-4 K3=1e-4 FYP=1600 FYN=-1600 DOF=U2 DOFTYPE=NON K1=3.2 K2=1.6 K3=1.6 FYP=0.32 FYN=-0.32 DOF=U3 DOFTYPE=NON K1=3.2 K2=1.6 K3=1.6 FYP=0.32 FYN=-0.32 DOF=R1 DOFTYPE=LIN K1=3.2 K2=3.2 K3=3.2 FYP=1600 FYN=-1600 DOF=R2 DOFTYPE=LIN K1=3.2 K2=1.6 K3=1.6 FYP=0.32 FYN=-0.32 DOF=R3 DOFTYPE=LIN K1=3.2 K2=1.6 K3=1.6 FYP=0.32 FYN=-0.32

NAME=NL3 TYPE=Gap M=.1 MR1=1 MR2=2 MR3=3 W=.2 DOF=U1 KE=100 CE=0 K=100 OPEN=.1 DOF=U2 KE=11 CE=0 DJ=111 K=100 OPEN=.1 DOF=U3 KE=111 CE=0 DJ=.22 K=100 OPEN=.1 DOF=R1 KE=1111 CE=0 K=100 OPEN=.1 DOF=R2 KE=1232 CE=0 K=100 OPEN=.1 DOF=R3 KE=111 CE=0 K=100 OPEN=.1 NAME=NL4 TYPE=Hook M=.1 MR1=1 MR2=2 MR3=3 W=.2 DOF=U1 KE=100 CE=0 K=100 OPEN=.1 DOF=U2 KE=11 CE=0 DJ=111 K=100 OPEN=.1 DOF=U3 KE=111 CE=0 DJ=.22 K=100 OPEN=.1 DOF=R1 KE=1111 CE=0 K=100 OPEN=.1 DOF=R2 KE=1232 CE=0 K=100 OPEN=.1 DOF=R3 KE=111 CE=0 K=100 OPEN=.1

NOTE: For details on these properties one may refer to Chapter 4 of this dissertation.

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2.12 FRAME: The data block following this keyword defines the frame

element block. This block contains frame elements shown as follows:

2.12.1 BLOCK DEFINITION: FRAME 1 J=ID1,ID2 SEC=FSNAME1 NSEG=2 ANG=A1 IOFF=IO JOFF=JO RIGID=RF IREL=U1,U2,U3,R1,R2,R3 2 J=ID3,ID4 SEC=FSNAME2 NSEG=2 ANG=A2 JREL=U1,U2,U3,R1,R2,R3 3 J=ID5,ID6 SEC=FSNAME3 NSEG=2 ANG=A3 …

2.12.2 BLOCK EXAMPLE: FRAME 1 J=1,20 SEC=FSEC1 NSEG=2 ANG=0 IOFF=10 JOFF=10 RIGID=0.1 IREL=U1,U2,U3,R1,R2,R3 2 J=20,3 SEC=FSEC2 NSEG=2 ANG=20 JREL=U1,U2,U3,R1,R2,R3 3 J=10,11 SEC=FSECN1 NSEG=2 ANG=-30 …

NOTE: Input up to ANG= is minimum necessary to define a frame element. Also, all the input for a frame

element has to be defined in one single line.


ID JOINT ID INTEGER

VARIABLE

FSNAME FRAME SECTION NAME FROM

THE FRAME SECTIONS LIST

STRING

VARIABLE

A FRAME ELEMENT LOCAL AXIS

ANGLE

DOUBLE

VARIABLE

IO I END OFFSET DOUBLE

VARIABLE

JO J END OFFSET DOUBLE

VARIABLE

RF RIGID END FACTOR (may take value

only between 0 and 1)

DOUBLE

VARIABLE

U1 FOR U1 RELEASE STRING FIXED



R1 FOR R1 RELEASE STRING FIXED



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2.13 SHELL: The data block following this keyword defines the shell

element block. This block contains shell elements shown as follows:


SHELL 1 J=ID1,ID2,ID3,ID4 SEC=SSNAME1 2 J=ID5,ID6,ID7,ID8 SEC=SSNAME2 3 J=ID7,ID8,ID9 SEC=SSNAME3 …


SHELL 1 J=1,2,3,4 SEC=SEC1 2 J=5,6,7,8 SEC=SEC2 3 J=22,2,9 SEC=SEC1


ID JOINT ID INTEGER

VARIABLE

SSNAME SHELL SECTION NAME

FROM THE SHELL

SECTION LIST

STRING

VARIABLE

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2.14 SOLID: The data block following this keyword defines the solid

element block. This block contains solid elements shown as follows:


SOLID 1 J=ID1,ID2,ID3,ID4,ID5,ID6,ID7,ID8 MAT=MNAME1 I=Y 2 J=ID5,ID6,ID7,ID8,ID9,ID10,ID11,ID12 MAT=MNAME2 I=Y …


SOLID 1 J=1,2,3,4,5,6,7,8 MAT=MNAME1 I=Y 2 J=5,6,7,8,9,10,11,12 MAT=MNAME2 I=Y …


ID JOINT ID INTEGER

VARIABLE

MNAME MATERIAL NAME

FROM MATERIALS

LIST

STRING

VARIABLE

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2.15 NLLINK: The data block following this keyword defines the node

link element block. This block contains node link elements shown as

follows:

2.15.1 BLOCK DEFINITION: NLLINK 1 J=ID1,ID2 NLP=NSNAME1 ANG=A1 AXDIR=DIR1 SPR_BEH=ORIENTED 2 J=ID3 NLP=NSNAME2 ANG=A2 AXDIR=DIR2 SPR_BEH=GEN … …

2.15.2 BLOCK EXAMPLE: NLLINK 1 J=2,33 NLP=NLPROP1 ANG=10 AXDIR=+X SPR_BEH=ORIENTED 2 J=1 NLP=NLPROP2 ANG=21 AXDIR=+Z SPR_BEH=GEN

…


ID JOINT ID INTEGER

VARIABLE

NSNAME NLPROP NAME STRING

VARIABLE

A ANGLE DOUBLE

VARIABLE

DIR AXIS DIRECTION MAY ONLY

TAKE VALUES I.E. “+X”, “-X”,

“+Y”, “-Y”, “+Z”, “-Z”

STRING

VARIABLE

ORIENTED ORIENTED SPRING IN THE PLANE

PERPENDICULAR TO THE AXES

DIRECTION

STRING FIXED

GEN NO ORIENTATION STRING FIXED

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2.16 LOAD: The data block following this keyword defines the static load

block. Currently, one can define nodal loads, frame point load, frame

distributed load and shell pressure loading.

2.16.1 BLOCK DEFINITION: LOAD NAME=LNAME1 SW=SF1 CSYS=0 TYPE=FORCE ADD=ID1 UX=xF1 UY=yF1 UZ=zF1 RX=xM1 RY=yM1 RZ=zM1 ADD=ID2 UX=xF2 UY=yF2 UZ=zF2 ADD=ID3 UX=xF3 UY=yF3 UZ=zF3 RY=yM3 RZ=zM3 NAME=LNAME2 SW=SF2 CSYS=0 TYPE=CONCENTRATED SPAN ADD=FELNO1 RD=rd1 UX=xF1 ADD=FELNO1 RD=rd2 UY=yF2 ADD=FELNO2 RD=rd3 UZ=zF3 ADD=FELNO3 RD=rd4 RX=xM1 TYPE=DISTRIBUTED SPAN ADD=FELNO2 RD=0,1 UX=fXVal1,fXVal1 ADD=FELNO3 RD=0,1 UY=fYVal2,fYVal2 ADD=FELNO4 RD=0,1 UZ=fZVal3,fYVal3 ADD=FELNO5 RD=0,1 RX=mXVal4,mXVal4 ADD=FELNO3 RD=0,1 RY=mYVal5,mYVal5 ADD=FELNO2 RD=0,1 RZ=mZVal6,mZVal6 TYPE=SURFACE PRESSURE ELEM=SHELL FACE=FNO1 ADD=SELNO1 P=pVal1 TYPE=SURFACE PRESSURE ELEM=SHELL FACE=FNO2 ADD=SELNO2 P=pVal2


LNAME1 LOAD CASE NAME STRING

VARIABLE

SF SCALE FACTOR FOR THE

LOAD CASE

DOUBLE

VARIABLE

FORCE USED TO DEFINE NODAL

FORCE

STRING FIXED

ID JOINT ID INTEGER

VARIABLE

xF X DIRECTION FORCE AT THE

JOINT

DOUBLE

VARIABLE

yF Y DIRECTION FORCE AT THE

JOINT

DOUBLE

VARIABLE

zF Z DIRECTION FORCE AT THE

JOINT

DOUBLE

VARIABLE

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xM MOMENT ABOUT X AXIS AT

THE JOINT

DOUBLE

VARIABLE

yM MOMENT ABOUT Y AXIS AT

THE JOINT

DOUBLE

VARIABLE

zM MOMENT ABOUT Z AXIS AT

THE JOINT

DOUBLE

VARIABLE

CONCENTRATED

SPAN

USED TO DEFINE FRAME

POINT LOAD

STRING FIXED

DISTRIBUTED

SPAN

USED TO DEFINE FRAME

UNIFORMLY DISTRIBUTED

LOAD

STRING FIXED

FELNO FRAME ELEMENT NUMBER

ON WHICH POINT OR

UNIFORM LOAD IS APPLIED

INTEGER

VARIABLE

rd RELATIVE DISTANCE OF THE

POINT LOAD FROM I END OF

THE FRAME ELEMENT

DOUBLE

VARIABLE

fXVal THE VALUE OF UNIFORMLY

DISTRIBUTED FORCE IN THE

X DIRECTION

DOUBLE

VARIABLE

fYVal THE VALUE OF UNIFORMLY


Y DIRECTION

DOUBLE

VARIABLE

fZVal THE VALUE OF UNIFORMLY


Z DIRECTION

DOUBLE

VARIABLE

mXVal THE VALUE OF UNIFORMLY

DISTRIBUTED MOMENT IN

THE X DIRECTION

DOUBLE

VARIABLE

mYVal THE VALUE OF UNIFORMLY


Y DIRECTION

DOUBLE

VARIABLE

mZVal THE VALUE OF UNIFORMLY


Z DIRECTION

DOUBLE

VARIABLE

SURFACE

PRESSURE

USED TO DEFINE SURFACE

PRESSURE LOADING

STRING FIXED

FNO FACE NO ON THE SHELL INTEGER

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2.16.2 BLOCK EXAMPLE: LOAD NAME=LOAD1 SW=1 CSYS=0 TYPE=FORCE ADD=6 UX=10 UY=10 UZ=111 ADD=9 UX=10 UY=10 UZ=111 ADD=2 UX=10 UY=10 UZ=10 RY=10 RZ=10 NAME=LOAD2 SW=1.1 CSYS=0 TYPE=CONCENTRATED SPAN ADD=9 RD=0 UZ=-1 ADD=9 RD=.25 UZ=-2 ADD=9 RD=.75 UZ=-4 ADD=9 RD=1 UZ=-5

TYPE=DISTRIBUTED SPAN ADD=7 RD=0,1 UX=-0.33,-0.33 ADD=3 RD=0,1 UY=-0.24,-0.24 ADD=7 RD=0,1 UZ=-0.22,-0.22 ADD=4 RD=0,1 RX=1.0,1.0 ADD=9 RD=0,1 RY=-0.22,-0.22 ADD=9 RD=0,1 RZ=-0.32,-0.32 TYPE=SURFACE PRESSURE ELEM=SHELL FACE=5 ADD=1 P=0.22 NAME=LOAD3 SW=0.9 CSYS=0 TYPE=SURFACE PRESSURE ELEM=SHELL FACE=3 ADD=8 P=0.99

ELEMENT (may take value

between 1 to 6)

VARIABLE

SELNO SHELL ELEMENT NUMBER INTEGER

VARIABLE

pVal PRESSURE VALUE ON THE

FACE

DOUBLE

VARIABLE

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2.17 MODE: The data block following this keyword defines the mode

block. Currently one can only define single modal analysis case in the

program. This block contains modal analysis input parameters shown as

follows:

2.17.1 BLOCK DEFINITION: MODE

TYPE=EIGEN N=NMODES TOL=ε

2.17.2 BLOCK EXAMPLE: MODE

TYPE=EIGEN N=5 TOL=.00001


EIGEN EIGEN VALUES STRING FIXED

NMODES NUMBER OF MODES INTEGER

VARIABLE

ε TOLERANCE DOUBLE

VARIABLE

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2.18 FUNCTION: The data block following this keyword defines the

function block. This block contains input functions details to be used in the

dynamic analysis shown as follows:

2.18.1 BLOCK DEFINITION: FUNCTION NAME=FUNAME1 DT=dT NPL=npl PRINT=Y FILE=FILENAME1 NAME=FUNAME2 DT=dT NPL=npl PRINT=Y FILE=FILENAME2 ….


FUNCTION NAME=ELCENTRO DT=0.02 NPL=2 PRINT=Y FILE=ELCENTRO.DAT NAME=NRIDGE DT=0.02 NPL=2 PRINT=Y FILE=NRIDGE.DAT …..


FUNAME FUNCTION NAME STRING

VARIABLE

dT TIME INTERVAL AT WHICH DATA

IS PRESENT IN THE FILE

DOUBLE

VARIABLE

npl NUMBER OF POINTS PER LINE IN

THE FILE

INTEGER

VARIABLE

FILENAME FILE NAME WITH EXTENSION STRING

VARIABLE

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2.19 DAMPING_MODEL: The data block following this keyword

defines the damping models block. These block definition shows 4 types of

models currently available in the program. This block may contain only

linear viscous damping parameters to be used in the dynamic analysis shown

as follows:

2.19.1 BLOCK DEFINITION: DAMPING_MODEL NAME=DMNAME1 TYPE=MASS_PROP KCONST=YES ACONST=YES MODE=MNO1 DAMP=DR1 NAME=DMNAME2 TYPE=STIF_PROP KCONST=YES ACONST=YES MODE=MNO2 DAMP=DR2 NAME= DMNAME3 TYPE=RAYL_PROP KCONST=NO ACONST=YES MODE=MNO3,MNO4 DAMP=DR3,DR4 NAME= DMNAME4 TYPE=USR_PROP KCONST=YES ACONST=YES AZERO=A0 AONE=A1

2.19.2 BLOCK EXAMPLE: DAMPING_MODEL NAME=DM1 TYPE=MASS_PROP KCONST=YES ACONST=YES MODE=1 DAMP=0.05 NAME=DM2 TYPE=STIF_PROP KCONST=YES ACONST=YES MODE=2 DAMP=0.10 NAME= DM3 TYPE=RAYL_PROP KCONST=NO ACONST=YES MODE=3,4 DAMP=0.32,0.32 NAME= DM4 TYPE=USR_PROP KCONST=YES ACONST=NO AZERO=0.0001 AONE=0.00002


DMNAME DAMPING MODEL NAME STRING

VARIABLE

YES THE LEFT HAND SIDE OF =YES

REMAINS CONSTANT DURING

THE ANALYSIS

STRING FIXED

NO THE LEFT HAND SIDE OF =NO

CHANGES DURING THE

ANALYSIS

STRING FIXED

MNO MODE NUMBER INTEGER

VARIABLE

DR DAMPING RATIO DOUBLE

VARIABLE

A0 MASS PROPORTIONAL

CONSTANT

DOUBLE

VARIABLE

A1 STIFFNESS PROPORTIONAL

CONSTANT

DOUBLE

VARIABLE

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2.20 HISTORY: The data block following this keyword defines the

history block. This block contains dynamic analysis load cases and

parameters shown as follows:


HISTORY NAME=HNAME1 TYPE=NON NSTEP=NSTEP DT=dT DAMP=DMODEL1 MITER=MAXITER TOL=ε ITYPE=ITERTYPE SAVE=NSTEPS ACC=U1 ANG=0 FUNC=FUNAME1 SF=SFACTOR1 AT=0 ACC=U2 ANG=0 FUNC=FUNAME2 SF=SFACTOR2 AT=0 … … NAME=HNAME2 TYPE=LIN NSTEP=NSTEP DT=dT DAMP=DMODEL2 MITER=MAXITER TOL=ε ITYPE=ITERTYPE SAVE=NSTEPS ACC=U1 ANG=0 FUNC=FUNAME1 SF=SFACTOR1 AT=0 ACC=U3 ANG=0 FUNC=FUNAME2 SF=SFACTOR2 AT=0 … …


HNAME HISTORY CASE NAME STRING

VARIABLE

NON NONLINEAR ANALYSIS STRING FIXED

LIN LINEAR ANALYSIS STRING FIXED

NSTEP NUMBER OF LOAD STEPS INTEGER

VARIABLE

dT TIME STEP DOUBLE

VARIABLE

DMODEL DAMPING MODEL TO BE USED STRING

VARIABLE

MAXITER MAXIMUM NUMBER OF

ITERATIONS WITHIN A TIME

STEP

INTEGER

VARIABLE

ε TOLERANCE FOR ENERGY

BALANCE WITHIN A LOAD STEP

DOUBLE

VARIABLE

ITERTYPE ITERATION TYPE WITHIN A

LOAD STEP, 0 FOR MODIFIED

NEWTON RAPHSON AND 1 FOR

FULL NEWTON RAPHSON

INTEGER

VARIABLE

NSTEPS EVERY STEPS AT WHICH

RESPONSE HAS TO BE SAVED

INTEGER

VARIABLE

U1,U2,U3 DIRECTION OF EARTHQUAKE STRING

FUNAME FUNCTION NAME TO BE USED STRING

VARIABLE

SFACTOR FACTOR BY WHICH FUNCTION DOUBLE

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HISTORY NAME=HIST1 TYPE=NON NSTEP=1000 DT=0.005 DAMP=DM1 MITER=100 TOL=1e-7 ITYPE=0 SAVE=4 ACC=U1 ANG=0 FUNC=FUNAME1 SF=100 AT=0 ACC=U2 ANG=0 FUNC=FUNAME2 SF=386 AT=0 … … NAME=HIST2 TYPE=LIN NSTEP=2000 DT=0.0025 DAMP=DM2 MITER=50 TOL=1e-5 ITYPE=1 SAVE=8 ACC=U1 ANG=0 FUNC=FUNAME1 SF=23.5 AT=0 ACC=U3 ANG=0 FUNC=FUNAME2 SF=38.1 AT=0 … …

LOADING IS TO BE SCALED VARIABLE

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2.21 INCREMENTAL DYNAMIC ANALYSIS: The data block

following this keyword defines the incremental dynamic analysis (IDA)

block. The IDA feature of the program is not fully developed and currently

works just by analyzing the response history cases by scaling their ground

motions in the requested increments. This block contains incremental

dynamic analysis load cases and parameters shown as follows:

2.21.1 BLOCK DEFINITION: INCREMENTAL_DYNAMIC_ANALYSIS NAME=IDANAME1 TA=0.4 TP=0.4 TD=0.9 TM=2.0 INCR=INSTEPS NOHIS=noHIS HIST=HISTNAME1 HIST=HISTNAME2 …. …. upto noHIS cases

2.21.2 BLOCK EXAMPLE: INCREMENTAL_DYNAMIC_ANALYSIS NAME=IDA1 TA=0.4 TP=0.4 TD=0.9 TM=2.0 INCR=10 NOHIS=3 HIST=HIST1 HIST=HIST2 HIST=HIST3


IDANAME INCREMENTAL DYNAMIC

ANALYSIS LOAD CASE NAME

STRING

VARIABLE

INSTEPS NUMBER OF STEPS FOR

INCREMENT

INTEGER

VARIABLE

HISTNAME RESPONSE HISTORY CASE NAME

DEFINED IN HISTORY CASES

STRING

VARIABLE

noHIS NUMBER OF HISTORY CASES TO

BE ANALYZED WITH THE GIVEN

PARAMETERS

INTEGER

VARIABLE

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2.22 VIRTUAL WORK ANALYSIS: The data block following this

keyword defines the virtual work analysis load cases. The virtual work

analysis currently can only be performed using nodal loads on the structures

and is only applicable to frame, spring, solid and nllink elements.

2.22.1 BLOCK DEFINITION: VIRTUAL_WORK_ANALYSIS RCASE1,VWCASE1 RCASE1,VWCASE2 ….. …..

2.22.2 BLOCK EXAMPLE: VIRTUAL_WORK_ANALYSIS WINDNS,VWLOADNS WINDEW,VWLOADEW ….. …..


RCASE REAL LOAD CASE NAME (ONLY

NODAL)

STRING

VARIABLE

VWCASE VIRTUAL WORK LOAD CASE

NAME (ONLY NODAL)

STRING

VARIABLE

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2.23 COMBO: The data block following this keyword defines the load

combination block. Currently only static load cases may be combined. The

resulting load case is treated as a new load case.

2.23.1 BLOCK DEFINITION: COMBO NAME=CONAME1 LOAD=SLCNAME1 SF=sf1 LOAD=SLCNAME2 SF=sf2 NAME=CONAME2 LOAD=SLCNAME3 SF=sf3 LOAD=SLCNAME4 SF=sf4 LOAD=SLCNAME1 SF=sf5

2.23.2 BLOCK EXAMPLE: COMBO NAME=COMBO1 LOAD=LOAD1 SF=1 LOAD=LOAD2 SF=0.87 LOAD=LOAD3 SF=0.22 NAME=COMB2 LOAD=LOAD1 SF=1.6 LOAD=LOAD2 SF=1.2 LOAD=LOAD3 SF=0.67


CONAME LOAD COMBINATION CASE

NAME

STRING

VARIABLE

SLCNAME STATIC LOAD CASE NAME STRING

VARIABLE

sf SCALE FACTOR BY WHICH THE

CORRESPONDING STATIC LOAD

CASE HAS TO BE MULTIPLIED

BEFORE ADDING IN THE COMBO

DOUBLE

VARIABLE

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2.24 END: The data block following this keyword defines the end block.

This block defines the end of the input file and program doesn’t read

anything after it encounters this keyword.

2.24.1 BLOCK DEFINITION: END

2.24.2 BLOCK EXAMPLE: END

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3. PROGRAM INTERFACE

The program has a DOS interface and it prompts the user for a few inputs

when the analysis is being performed. The input file must be present in the

folder where the program executable is present. The following steps describe

on how to use the program for static and dynamic analysis.

STEP 1: Click on the executable and the following window must appear.

STEP 2: Write the input file name without any extension and press enter.

The user is now prompted to select the renumbering schemes. Select one of

the renumbering schemes (prefer plain renumberer) and press enter.

STEP 3: The program now prompts the user to select one of the equation

solvers. Select one of the renumbering schemes (prefer DSS or PARDISO)

and press enter.

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STEP 4: Program tells the user if the input is correct or not and prompts the

user to press any key to continue. Once any key is pressed the program

proceeds to solve the linear system of equations.

STEP 5: Program now prompts the user to input 0 if dynamic analysis is to

be performed or input 1 if analysis has to be discontinued. If user selects 1

then the program leads to STEP 9 else to STEP 6. The user must have

dynamic analysis parameters defined inside the file if he/she proceeds to

perform dynamic analysis.

STEP 6: Program now prompts the user to select a numerical integrator for

analysis. Select one of the integrators (prefer Newmark alpha as it works for

all) and press enter.

STEP 7: Program now prompts the user to select the analysis method. Select

one of the methods and press enter.

STEP 8: Program now prompts the user to select the solver to be used in the

analysis. Select one of the solvers and press enter. Program now proceeds to

perform the dynamic analysis.

STEP 9: Once the analysis is performed the program prompts the user to

press enter to complete the analysis. Now the user may proceed to see the

output files created by the program.

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4. FILES GENERATED

The program generates the following files during the analysis when an input

file name named “INPUTFILENAME.S2K” is being executed:

1.) INPUTFILENAME.KSECH: This file contains the echoed input

information. It shall contain the list of errors in the end if the input

data isn’t consistent.

2.) INPUTFILENAME.KSERR: This file contains the number of

errors found during the reading of file.

3.) INPUTFILENAME.KSOUT: This file contains the output for

nodes displacements, elements forces, modes responses, and

maximum and minimum of displacement, velocity and acceleration

response histories.

4.) INPUTFILENAME.DISPAR: This file contains the element

displacement participation results if the virtual work analysis is

performed for any load cases.

5.) INPUTFILENAME.KSNRESHIS: This file contains the nodal

displacements, velocity, and acceleration response histories at the

requested nodes.

6.) INPUTFILENAME.KSNRKTHIS: This file contains the base

reaction response histories at the restrained nodes.

7.) INPUTFILENAME.KSDEFOR: This file contains the force-

deformation response histories for the selected nllink elements.

8.) KeySolverArchiveEData and KeySolverArchiveEData: These are

two binary files created by the program during the analysis. These

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files are not readable and contain intermediate information

produced during the analysis.

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5. DLL’S NEEDED

The program may require the following dll’s while performing static and

dynamic analysis. These dll’s come along with the installation package.

BILIN2001.dll

CLOUGH_MOD.dll

CLOUGH_STIF.dll

PINCHSTIF.dll

STW_HYSTR.dll

MSKPINCH2.dll

libguide40.dll

mkl_def.dll

mkl_lapack32.dll

mkl_lapack64.dll

msvcm80.dll

msvcp80.dll

msvcr80.dll

vcomp.dll

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APPENDIX C WOODFRAMESOLVER VERIFICATION MANUAL

225

WOODFRAMESOLVER


VERIFICATION MANUAL 2.0


226

WOODFRAMESOLVER

VERIFICATION MANUAL 2.0

This verification manual consists of various examples verified with

SAP2000 and SAPWOOD programs. The input files for the problems

described in this manual are contained in the folder named “WFS

VERIFIED PROBLEMS” which come along with the executable and the

source code of WoodFrameSolver program. This verification manual is part

two of previous manual printed in 2005. The previous manual and its

example problems are present in the other folder named “KS VERIFIED

PROBLEMS”. Also, WoodFrameSolver was initially named as KeySolver

and hence at several places in the program documentation/manual user may

find program referred to as KeySolver.

227

DISCLAIMER

The author does not give any warranty to the users of this program. The

examples presented should be independently verified before they are set as

the basis for other analysis or interpretation. Also, the users must understand

the assumptions involved in the program and the results obtained must be

verified by the users through other means.

Rakesh Pathak

228

TABLE OF CONTENTS

CONVENTION FOR ORIENTATION OF AXES 1

Example1: A 2 bay, 2 story plane frame with nodal loads. The frame

elements are W sections with rigid end zone factor of 1.0

2

Example1Big: A 20 bay, 20 story plane frame with nodal loads. The frame


3

Example2: A 2x2 bay, 2 story space frame with nodal loads. The frame


4

Example2Big: A 5x4 bay, 20 story space frame with nodal loads. The

frame elements are W sections with rigid end zone factor as

0.5

5

Example3: A 2x2 bay, 2 story space frame with only self-weight. 6

Example3Big: A 4x5 bay, 10 story space frame with only self-weight. 7

Example4: A 2x2 bay, 2 story space frame with nodal loads. Four load

cases are defined one real and other three virtual (along UX,

UY and RZ respectively). The virtual loads are applied at

joint id 20. The nodal displacements for UX, UY and RZ are

obtained from the .DISPAR output file generated by

WoodFrameSolver.

8

Example4Big: A 10 story space grid with nodal loads. Four load cases are

defined one real and other three virtual (along UX, UY and

RZ respectively). The virtual loads are applied at joint id 26.

The nodal displacements for UX, UY and RZ are obtained

from the .DISPAR output file generated by

WoodFrameSolver.

9



UZ and RY respectively). In this model the vertical supports

are springs in UX, UY, UZ, RX, RY and RZ directions. The

virtual loads are applied at joint id 2. The nodal

displacements for UX, UZ and RY are obtained from the

.DISPAR output file generated by WoodFrameSolver.

10

Example5Big: A 5x5 bay, 10 story space frame with nodal loads. Four load


UY and UZ respectively). In this model the vertical supports

are springs in UX, UY, UZ, RX, RY and RZ directions. The

virtual loads are applied at joint id 396. The nodal

displacements for UX, UZ and RY are obtained from the

.DISPAR output file generated by WoodFrameSolver.

11



UY and UZ respectively). In this model the vertical supports

are zero length linear link element with stiffnesses in UX,

UY, UZ, RX, RY and RZ directions. The virtual loads are

229

applied at joint id 24. The nodal displacements for UX, UY

and UZ are obtained from the .DISPAR output file

generated by WoodFrameSolver.

12

Example7: A cantilever beam modeled with solid elements. 13

Example7VWx: A cantilever beam modeled with solid elements and axial

loadings on the free end. Two load cases are defined: one

with real load and another with virtual load.

14

Example7VWz: A cantilever beam modeled with solid elements and vertical

loading. Two load cases are defined: one with real load and

another with virtual load.

15

Example8: A cantilever beam with solid, frame and spring elements. 16

Example9: A beam column sub assemblage modeled with solid

elements

17

Example10: A 2 bay x 2 story plane frame analyzed for Eigen Values 18

Example11: A 2 x 2 bay and 2 story 3D space frame analyzed for Eigen

values

19

Example12: A cantilever beam modeled using quadrilateral shell

elements.

20

Example13: A cantilever beam modeled using solid elements. 21

Example14: A cantilever beam modeled using frame element. 22

Example15: A 2 story 2 bay frame structure. 23

Example16: A 10 story 2 x 2 bay frame structure 24

Example17: A SDOF mass spring damper system 25

Example18: A beam on elastic foundation is modeled using frame and

spring elements. The beam is subjected to three different

transient load cases. Each load case has a different damping

model. The eigen values in each mode and the

corresponding damping ratios are also generated based on

the values initially provided to the program. The mass and

stiffness coefficients (a0 and a1) for analysis using direct

integration are based on these values.

26

Example19: A simple 2D frame structure. The structure is subjected to

three different transient load cases. The eigen values in each

mode and the corresponding damping ratios are also

generated based on the values initially provided to the

program. The mass and stiffness coefficients (a0 and a1) for

analysis using direct integration are based on these values.

30

Example20: A simple 2D frame structure with a brace connected using

linear hook. The frame is subjected to three different

transient load cases. The eigen values in each mode and the





34

Example21: A simple 3D frame structure is subjected to three different

transient load cases. The eigen values in each mode and the

230





38

Example22: This model is a fixed end beam with a two node nonlinear

gap element at the center. The model is subjected to one

transient load case with three acceleration loads in UX, UY

and UZ directions. A Rayleigh damping model is used for

the analysis of this load case.

42

Example23: A 3 story 1 bay frame with bilinear rotational springs as

shown in Figure 1 is subjected to Loma Preita earthquake

(1989). The bilinear moment rotation relationship for

springs is shown in Figure 2. The model is executed both on

WoodFrameSolver and SAP2000. The results are presented

for two cases one with 0% damping and other with 2%

damping in modes 1 and 3.

47

Example24: Rectangular floor plan with 3 linear and 1 Stewart hysteresis

walls on its perimeter

57

Example25: Rectangular floor plan with 4 bilinear walls on its perimeter 58

Example26: Rectangular floor plan with 2 bilinear and 2 Stewart

hysteresis walls on its perimeter

63

Example27: L shaped floor plan with 8 Stewart hysteresis walls on its

perimeter

68

Example28: U shaped floor plan with 6 bilinear and 6 Stewart hysteresis

walls on its perimeter

78

VERIFICATION MANUAL 2.0: General problem verification with SAP2000

___________________________________________________________________________________________________________________

-231-

Convention for Orientation of Axes

GLOBAL axes orientation LOCAL axes orientation

Z

Y

X

x y

z

i

j


___________________________________________________________________________________________________________________

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Verification Examples

Input Filename: Example1.s2k Date tested: 12/05/05

Program Version: Release 5

Tested by: Rakesh Pathak

Description: A 2 bay, 2 story plane frame with nodal loads. The frame elements are W sections with rigid end zone factor of 1.0

Purpose: Verify the accuracy of WoodFrameSolver rigid end zone for frame elements.

Results:

Nodal Displacements: WoodFrameSolver SAP2000 Node

UX UY RZ UX UY RZ

2 0.103534 0.000000 0.001320 0.103534 1.08E-20 0.001320

3 0.146738 0.000000 0.001699 0.146738 3.80E-19 0.001699

Element Force: WoodFrameSolver SAP2000 Element Node

FX FY MZ FX FY MZ

1 2 7.6165 5.9013 770.3608 7.616469 5.901339 -770.3608

2 3 2.1894 2.1604 307.2191 2.189411 2.160434 -307.2191


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Input Filename: Example1Big.s2k

Date tested: 12/05/05



Description: A 20 bay, 20 story plane frame with nodal loads. The frame elements are W sections with rigid end zone factor of 0.8


Results:


UX UZ RY UX UZ RY

28 0.049179 0.002854 3.297e-05 0.049179 0.002854 3.30E-05

58 0.103361 0.002523 2.005e-05 0.103361 0.002523 2.01E-05

100 0.103473 0.000420 1.669e-05 0.103473 0.000420 1.67E-05


FX FY MZ FX FY MZ

100 104 -0.0086 -0.4713 15.7577 0.008573 0.471286 15.757747

120 125 0.0042 -0.4822 15.8223 -0.004152 0.482208 15.822294

250 262 -0.0114 -4.0494 206.2489 0.011368 4.049438 206.248909


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Input Filename: Example2.s2k Date tested: 12/05/05



Description: A 2x2 bay, 2 story space frame with nodal loads. The frame elements are W sections with rigid end zone factor of 1.0


Results:


UX UY RZ UX UY RZ

2 0.004971 0.042263 -4.70e-005 0.004971 0.042263 -4.70E-05

20 0.004739 0.029393 2.96e-005 0.004739 0.029393 2.96E-05


FX FY MZ FX FY MZ

1 2 13.5779 -7.3526 904.4849 13.577856 7.3526 - 904.4849

13 20 0.9342 -4.8629 598.2118 0.934236 4.8629 -598.2118


___________________________________________________________________________________________________________________

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Input Filename: Example2Big.s2k Date tested: 12/05/05



Description: A 5x4 bay, 20 story space frame with nodal loads. The frame elements are W sections with rigid end zone factor as 0.5


Results:


UX UY RY UX UY RY

50 0.991555 7.364326 0.000777 0.991555 7.364326 0.000777

151 0.488960 4.401416 0.000562 0.488960 4.401416 0.000562

302 0.991637 7.364292 0.000604 0.991637 7.364292 0.000604


FX FY MZ FX FY MZ

1016 101 -0.0181 -4.8116 675.5996 0.018137 4.811610 675.599597

105 110 -121.6336 -11.5938 705.8913 121.633557 11.593846 705.891261

1106 7 -0.0585 6.7553 -1207.5527 0.058496 6.755348 1207.553


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Input Filename: Example3.s2k




Description: A 2x2 bay, 2 story space frame with only self-weight.

Purpose: Verify the accuracy of WoodFrameSolver for frame elements with self weight.

Results:


UX UY RZ UX UY RZ

8 -0.013509 0.008944 0.00 -0.013509 0.008944 0.00

20 0.013509 -0.008944 0.00 0.013509 -0.008944 0.00


FX FY MY FX FY MY

27 8 -253.2931 705.7556 -29700.075 253.2931 -705.7556 -29700.075

39 20 -167.7071 -662.6336 21370.287 167.7071 -662.6336 -21370.287


___________________________________________________________________________________________________________________

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Description: A 4x5 bay, 10 story space frame with only self-weight.

Purpose: Verify the accuracy of WoodFrameSolver for frame elements with self weight.

Results:


UX UY UZ UX UY UZ

220 -0.000219 0.000214 -0.054655 -0.000219 0.000214 -0.054655

285 0.000120 -3.681e-005 -0.043371 0.000120 -3.68E-05 -0.043371

297 -0.000423 7.102e-005 -0.043813 -0.000423 7.10E-05 -0.043813


FX FY FZ FX FY FZ

11 12 297.2766 0.7860 0.0583 -297.276604 -0.785969 0.058336

119 130 57.7285 0.4430 2.0181 -57.728550 -0.442965 2.018117

157 172 147.0653 -0.0000 -0.0198 -147.065339 7.21E-15 -0.019822


___________________________________________________________________________________________________________________

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Description: A 2x2 bay, 2 story space frame with nodal loads. Four load cases are defined one real and other three virtual (along UX, UY and

RZ respectively). The virtual loads are applied at joint id 20. The nodal displacements for UX, UY and RZ are obtained from the .DISPAR

output file generated by WoodFrameSolver.

Purpose: Verify the accuracy of WoodFrameSolver for virtual work on frame elements.

Results:

Nodal Displacements: DISPAR SAP2000 Node

UX UY RZ UX UY RZ

20 0.116642 0.282358 0.000179 0.11664 0.282358 0.000179


___________________________________________________________________________________________________________________

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Description: A 10 story space grid with nodal loads. Four load cases are defined one real and other three virtual (along UX, UY and RZ

respectively). The virtual loads are applied at joint id 26. The nodal displacements for UX, UY and RZ are obtained from the .DISPAR output

file generated by WoodFrameSolver.

Purpose: Verify the accuracy of WoodFrameSolver for virtual work on frame elements.

Results:


UX UY RZ UX UY RZ

26 0.133670 0.134367 -0.000047 0.133670 0.134367 -4.658e-005


___________________________________________________________________________________________________________________

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Description: A 2x2 bay, 2 story space frame with nodal loads. Four load cases are defined one real and other three virtual (along UX, UZ and

RY respectively). In this model the vertical supports are springs in UX, UY, UZ, RX, RY and RZ directions. The virtual loads are applied at

joint id 2. The nodal displacements for UX, UZ and RY are obtained from the .DISPAR output file generated by WoodFrameSolver.

Purpose: Verify the accuracy of WoodFrameSolver for virtual work on a model with frame and spring elements.

Results:


UX UZ RY UX UZ RY

2 0.029278 0.000310 0.000231 0.029278 0.000310 0.000231


___________________________________________________________________________________________________________________

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UZ respectively). In this model the vertical supports are springs in UX, UY, UZ, RX, RY and RZ directions. The virtual loads are applied at

joint id 396. The nodal displacements for UX, UZ and RY are obtained from the .DISPAR output file generated by WoodFrameSolver.

Purpose: Verify the accuracy of WoodFrameSolver for virtual work on a model with frame and spring elements.

Results:


UX UY UZ UX UY UZ

396 0.323455 0.533113 -0.008807 0.323455 0.533113 -0.008807


___________________________________________________________________________________________________________________

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UZ respectively). In this model the vertical supports are zero length linear link element with stiffnesses in UX, UY, UZ, RX, RY and RZ

directions. The virtual loads are applied at joint id 24. The nodal displacements for UX, UY and UZ are obtained from the .DISPAR output file

generated by WoodFrameSolver.

Purpose: Verify the accuracy of WoodFrameSolver for virtual work on a model with frame and linear link elements.

Results:


UX UY UZ UX UY UZ

24 1.159530 -1.231471 0.008043 1.159530 -1.231471 0.008043


___________________________________________________________________________________________________________________

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Description: A cantilever beam modeled with solid elements

Purpose: Verify the accuracy of WoodFrameSolver for solid elements.

Results:


UX UY UZ UX UY UZ

8 0.003898 0.001944 0.001944 0.003898 0.001944 0.001944

20 0.002600 0.001697 0.001233 0.002600 0.001697 0.001233

30 0.002905 0.001853 0.000691 0.002905 0.001853 0.000691

40 0.000760 0.001024 0.000557 0.000760 0.001024 0.000557

100 0.000988 0.001246 0.001202 0.000988 0.001246 0.001202

Solid Element Stresses: WoodFrameSolver SAP2000 Element Node

σx σy σz σx σy σz

270 318 -30.6106 -4.0207 0.2741 -30.6106 -4.0207 0.2741

270 330 -50.4629 -5.2543 0.2956 -50.4629 -5.2543 0.2956

567 678 9.9450 0.6730 0.6730 9.9450 0.6730 0.6730

567 800 4.1557 -0.5155 -0.3276 4.1557 -0.5155 -0.3276

999 1197 15.2077 2.3329 4.6574 15.2077 2.3329 4.6574

999 1198 11.8060 2.1840 5.5199 11.8060 2.1840 5.5199


___________________________________________________________________________________________________________________

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Input Filename: Example7VWx.s2k




Description: A cantilever beam modeled with solid elements and axial loadings on the free end. Two load cases are defined: one with real load

and another with virtual load.

Purpose: Verify the accuracy of WoodFrameSolver for solid elements virtual work.

Results:

Nodal Displacements: DISPAR SAP2000 Theory

UX UX UX

Free End 0.00033333 3.333e-005 0.000333


___________________________________________________________________________________________________________________

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Input Filename: Example7VWz.s2k




Description: A cantilever beam modeled with solid elements and vertical loading. Two load cases are defined: one with real load and another

with virtual load.

Purpose: Verify the accuracy of WoodFrameSolver for solid elements virtual work.

Results:

Nodal Displacements: DISPAR SAP2000 Theory

UZ UZ UZ

Free End 0.00281978 0.002817 0.002533


___________________________________________________________________________________________________________________

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Description: A cantilever beam with solid, frame and spring elements.

Purpose: Verify the accuracy of WoodFrameSolver for mixed element models.

Results:


UX UY UZ UX UY UZ

220 -0.001442 -0.005511 -0.002161 -0.001442 -0.005511 -0.002161

300 0.042807 0.000745 -0.001104 0.042807 0.000745 -0.001104

400 0.050236 0.005087 0.001808 0.050236 0.005087 0.001808

800 0.022058 0.017122 0.015336 0.022058 0.017122 0.015336

870 0.072159 0.025651 0.019374 0.072159 0.025651 0.019374


τxy τyz τyz τxy Τyz τyz

355 423 -2.4270 4.4230 1.7989 -2.426976 4.423018 1.798886

357 547 1.5585 1.1488 -0.8046 1.558515 1.148809 -0.804624

45 97 -1.2803 10.2518 -1.9700 -1.280342 10.251843 -1.969997

467 558 2.1442 0.8646 1.6293 2.144163 0.864596 1.629261

623 763 0.0842 0.8166 4.6046 0.084211 0.816576 4.604587

673 806 -6.1336 -2.1707 5.2973 -6.133614 -2.170654 5.297290

Frame Element Forces: WoodFrameSolver SAP2000 Element Node

FX FY MY FX FY MY

7 1331 93.7141 191.092 1017.3921 -93.7141 -191.0920 1017.392

9 626 -70.0094 -37.396 -763.4629 70.009390 -37.396 763.4629


___________________________________________________________________________________________________________________

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Description: A beam column sub assemblage modeled with solid elements

Purpose: Verify the accuracy of WoodFrameSolver for solid elements.

Results:


UX UY UZ UX UY UZ

3110 0.158470 -1.534e-005 0.023876 0.158470 -1.53E-05 0.023876

5023 0.260111 4.791e-005 -0.036300 0.260111 4.79E-05 -0.036300

6207 0.003257 0.001182 -0.025059 0.003257 0.001182 -0.025059

6894 0.200223 0.000108 0.035454 0.200223 0.000108 0.035454

7000 0.044290 0.000448 0.028211 0.044290 0.000448 0.028211


σx σy τxy σx σy τxy

10 37 -0.0233 -0.0210 -0.0001 -0.023317 -0.021025 -0.000106

100 8 -0.0692 0.1860 -0.0146 -0.069152 0.186001 -0.014597

1006 1742 -0.0888 2.2668 -0.2299 -0.088816 2.266819 -0.229934

1009 1750 0.8386 7.4175 -0.9507 0.838604 7.417501 -0.950729


___________________________________________________________________________________________________________________

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Description: A 2 bay x 2 story plane frame analyzed for Eigen values

Purpose: Verify the accuracy of WoodFrameSolver for Eigen value generation.

Results:

Frequencies and Eigen Values

WoodFrameSolver SAP 2000

MODE PERIOD FREQUENCY FREQUENCY EIGENVALUE PERIOD FREQUENCY FREQUENCY EIGENVALUE

(TIME) (CYC/TIME) (RAD/TIME) (RAD/TIME)**2 (TIME) (CYC/TIME) (RAD/TIME) (RAD/TIME)**2

1 0.535871 1.866122 11.738654 137.795996 0.535245 1.868302 11.738889 137.801509

2 0.311398 3.211329 20.200547 408.062082 0.311033 3.215096 20.201045 408.08222

3 0.221252 4.519728 28.430895 808.315816 0.220997 4.524942 28.43105 808.324626

4 0.194222 5.148737 32.387613 1048.957447 0.193993 5.154814 32.388649 1049.025

5 0.092388 10.823923 68.086808 4635.81336 0.092275 10.837189 68.092067 4636.53


___________________________________________________________________________________________________________________

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Description: A 2 x 2 bay and 2 story 3D space frame analyzed for Eigen values


Results:


WoodFrameSolver SAP2000



1 0.417094 2.397542 15.081497 227.451546 0.416614 2.400303 15.081546 227.453031

2 0.335592 2.979806 18.744171 351.343943 0.335205 2.983253 18.744329 351.349882

3 0.287416 3.479281 21.886068 478.999983 0.287085 3.48329 21.886158 479.003906

4 0.221582 4.513002 28.388589 805.911997 0.221327 4.518209 28.388741 805.92063

5 0.208045 4.806654 30.235777 914.202193 0.207805 4.812213 30.236024 914.217122

6 0.182627 5.47565 34.444029 1186.391158 0.182416 5.481986 34.444331 1186.412

7 0.100222 9.977819 62.764475 3939.379348 0.100106 9.98941 62.765316 3939.485

8 0.089071 11.227034 70.622537 4987.542741 0.088967 11.2401 70.623634 4987.698


___________________________________________________________________________________________________________________

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Description: A cantilever beam modeled using quadrilateral shell elements

Purpose: Verify the accuracy of WoodFrameSolver for Eigen Value generation.

Results:





1 11.162176 0.089588 0.563546 0.317584 11.150975 0.089678 0.563465 0.317493

2 4.388321 0.227878 1.433441 2.054754 4.399767 0.227285 1.428072 2.039391

3 1.820031 0.549441 3.456205 11.945355 1.81755 0.550191 3.456954 11.950528

4 1.44952 0.689883 4.339643 18.832498 1.453045 0.68821 4.32415 18.698275

5 1.235013 0.809708 5.093389 25.942616 1.241169 0.805692 5.062313 25.627014

6 0.714907 1.398784 8.798908 77.420782 0.724135 1.380958 8.676813 75.287085

7 0.649281 1.540165 9.688256 93.862301 0.647231 1.545044 9.707795 94.241289

8 0.626437 1.596331 10.04156 100.832927 0.627396 1.59389 10.014707 100.294357


___________________________________________________________________________________________________________________

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Description: A cantilever beam modeled using solid elements


Results:





1 0.038159 26.206453 164.849074 27175.21736 0.038115 26.236545 164.849075 27175.218

2 0.030698 32.57563 204.913745 41989.64286 0.030663 32.613036 204.913746 41989.643

3 0.030544 32.740149 205.948636 42414.84056 0.030509 32.777743 205.948636 42414.841

4 0.019194 52.100293 327.731681 107408.0547 0.019172 52.16012 327.731702 107408.068

5 0.016957 58.974261 370.971694 137619.9977 0.016937 59.042045 370.972109 137620.306

6 0.015913 62.840931 395.29459 156257.8131 0.015895 62.913134 395.294881 156258.043

7 0.015305 65.340139 411.015614 168933.8345 0.015287 65.415174 411.015663 168933.875

8 0.015005 66.64388 419.216664 175742.611 0.014988 66.720429 419.216822 175742.744

9 0.014501 68.95996 433.785734 188170.0626 0.014485 69.039281 433.786596 188170.811

10 0.012751 78.425458 493.327498 243372.0203 0.012736 78.515641 493.328323 243372.835


___________________________________________________________________________________________________________________

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Description: A cantilever beam modeled using frame element

Purpose: Verify the accuracy of WoodFrameSolver for time history analysis for El Centro ground motion using mode superposition.

Results:

Maximum Nodal Displacements: WoodFrameSolver SAP2000 Node

UX UY UZ UX UY UZ

1 0 0 0 0 0 0

2 0.000032 0.00E+00 0.00E+00 3.23E-05 4.65E-20 3.37E-18

Minimum Nodal Displacements: WoodFrameSolver SAP2000 Node

UX UY UZ UX UY UZ

1 0 0 0 0 0 0

2 -0.00003 0.00E+00 0.00E+00 -3.02E-05 -7.55E-20 -2.39E-18


___________________________________________________________________________________________________________________

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Description: A 2 story 2 bay frame structure.


Results:


UX UY UZ UX UY UZ

1 0 0 0.00363 0 0 0.00363

2 0.422622 0.001002 0.001493 0.422625 0.001002 0.001493

3 0.587454 0.001284 0.000576 0.587458 0.001284 0.000576


UX UY UZ UX UY UZ

1 0 0 -0.00405 0 0 -0.00405

2 -0.47083 -0.00111 -0.00166 -0.47083 -0.00111 -0.00166

3 -0.65292 -0.00142 -0.00063 -0.65293 -0.00142 -0.00063


___________________________________________________________________________________________________________________

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Description: A 10 story 2 x 2 bay frame structure


Results:


UX UY UZ UX UY UZ

1 0 0 0.011116 0 0 0.011116

2 1.331111 0.012401 0.00536 1.331122 0.012401 0.00536

3 2.099653 0.022345 0.003808 2.099673 0.022344 0.003808

4 2.742385 0.030475 0.003504 2.74241 0.030475 0.003504


UX UY UZ UX UY UZ

1 0 0 -0.00931 0 0 -0.00931

2 -1.11124 -0.0101 -0.00443 -1.11126 -0.0101 -0.00443

3 -1.73774 -0.01811 -0.00322 -1.73776 -0.01811 -0.00322

4 -2.28793 -0.02453 -0.00301 -2.28795 -0.02453 -0.00301


___________________________________________________________________________________________________________________

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Input Filename: Example17 (CHOPRA chapter 5.1 problem)




Description: A SDOF mass spring damper system

Purpose: Verify the accuracy of different SDOF numerical integration schemes implemented in WoodFrameSolver

1 LINEAR INTERPOLATION

2 CENTRAL DIFFERENCE METHOD

3 NEWMARK AVERAGE ACCELERATION METHOD

4 NEWMARK LINEAR ACCELERATION METHOD

5 4TH

ORDER RUNGE KUTTA METHOD

Results:

CHOPRA (CHAPTER 5) WOODFRAMESOLVER

ti 1 2 3 4 5 1 2 3 4 5

0 0 0 0 0 x 0 0 0 0 0

0.1 0.0318 0 0.0437 0.03 x 0.03176 0 0.04367 0.02998 0.03238

0.2 0.2274 0.1914 0.2326 0.2193 x 0.22742 0.19138 0.23262 0.21934 0.22804

0.3 0.6336 0.6293 0.6121 0.6166 x 0.63357 0.62934 0.61208 0.61662 0.63319

0.4 1.1339 1.1825 1.0825 1.113 x 1.13392 1.18251 1.08258 1.11305 1.13193

0.5 1.4896 1.5808 1.4309 1.4782 x 1.48964 1.58088 1.43102 1.47828 1.48657

0.6 1.448 1.5412 1.4231 1.4625 x 1.44812 1.5413 1.42319 1.4626 1.44561

0.7 0.9037 0.9141 0.9622 0.9514 x 0.90381 0.91422 0.96232 0.95158 0.90454

0.8 0.0579 -0.0247 0.1908 0.1273 x 0.05807 -0.02457 0.19092 0.12746 0.06267

0.9 -0.7577 -0.8968 -0.6044 -0.6954 x -0.75767 -0.89677 -0.60428 -0.69533 -0.75071

1 -1.2432 -1.3726 -1.1442 -1.2208 x -1.24325 -1.37261 -1.14419 -1.22083 -1.23704


___________________________________________________________________________________________________________________

-256-

Input Filename: Example18 (BEAM_ONEF_TH.S2K)




Description: A beam on elastic foundation is modeled using frame and spring elements. The beam is subjected to three different transient load

cases. Each load case has a different damping model. The eigen values in each mode and the corresponding damping ratios are also generated

based on the values initially provided to the program. The mass and stiffness coefficients (a0 and a1) for analysis using direct integration are

based on these values.

Purpose: Verify the accuracy of WoodFrameSolver for linear dynamic analysis using mode superposition and direct integration using mass,

stiffness and Rayleigh proportional damping.

Results:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2000 4000 6000

OMEGA (rad/sec)

ξ

ξ-MASS_PROP

ξ-STIF_PROP

ξ-RAYL_PROP

MASS_PROP STIF_PROP RAYL_PROP

a0 60.091 0 52.394


___________________________________________________________________________________________________________________

-257-

a1 0 2.44E-05 2.13E-05

MODAL PARTICIPATING MASS RATIOS

WOODFRAMESOLVER SAP 2000

MODE PERIOD INDIVIDUAL MODE % PERIOD INDIVIDUAL MODE %

UX UY UZ UX UY UZ

1 0.1798 0.00 62.87 0.00 0.1796 0.00 62.87 0.00

3 0.0289 0.00 19.43 0.00 0.0289 0.00 19.43 0.00

8 0.0048 83.05 0.00 0.00 0.0048 83.05 0.00 0.00

12 0.0022 0.00 0.00 1.07 0.0022 0.00 0.00 1.07

18 0.0011 0.00 0.57 0.00 0.0011 0.00 0.57 0.00

LINEAR TIME HISTORY - MODAL SUPERPOSITION LINEAR TIME HISTORY - MODAL SUPERPOSITION

HIST1---MAXIMUM DISPLACEMENTS HIST1---MINIMUM DISPLACEMENTS

WOODFRAMESOLVER SAP 2000 WOODFRAMESOLVER SAP 2000

NODE UX UY UZ UX UY UZ NODE UX UY UZ UX UY UZ

7 0.00002 0.00000 0.00000 0.00002 0.00000 0.00000 7 -0.00002 0.00000 0.00000 -0.00002 0.00000 0.00000

12 0.00006 0.00000 0.00000 0.00006 0.00000 0.00000 12 -0.00005 0.00000 0.00000 -0.00005 0.00000 0.00000

15 0.00007 0.00000 0.00000 0.00007 0.00000 0.00000 15 -0.00007 0.00000 0.00000 -0.00007 0.00000 0.00000

20 0.00009 0.00000 0.00000 0.00009 0.00000 0.00000 20 -0.00008 0.00000 0.00000 -0.00008 0.00000 0.00000

23 0.00009 0.00000 0.00000 0.00009 0.00000 0.00000 23 -0.00008 0.00000 0.00000 -0.00008 0.00000 0.00000





7 0.00000 0.02890 0.00000 0.00000 0.02888 0.00000 7 0.00000 -0.03037 0.00000 0.00000 -0.03037 0.00000

12 0.00000 0.17980 0.00000 0.00000 0.17984 0.00000 12 0.00000 -0.18845 0.00000 0.00000 -0.18846 0.00000

15 0.00000 0.31240 0.00000 0.00000 0.31239 0.00000 15 0.00000 -0.32672 0.00000 0.00000 -0.32673 0.00000

20 0.00000 0.56850 0.00000 0.00000 0.56853 0.00000 20 0.00000 -0.59308 0.00000 0.00000 -0.59310 0.00000

23 0.00000 0.73000 0.00000 0.00000 0.73007 0.00000 23 0.00000 -0.76082 0.00000 0.00000 -0.76085 0.00000




NODE UX UZ RY UX UZ RY NODE UX UZ RY UX UZ RY

7 0.00000 0.00459 0.00015 0.00000 0.00459 0.00015 7 0.00000 -0.00547 -0.00013 0.00000 -0.00547 -0.00013


___________________________________________________________________________________________________________________

-258-

12 0.00000 0.01170 0.00014 0.00000 0.01170 0.00014 12 0.00000 -0.01418 -0.00011 0.00000 -0.01418 -0.00011

15 0.00000 0.01556 0.00013 0.00000 0.01556 0.00013 15 0.00000 -0.01908 -0.00010 0.00000 -0.01908 -0.00010

20 0.00000 0.02149 0.00013 0.00000 0.02149 0.00013 20 0.00000 -0.02681 -0.00010 0.00000 -0.02681 -0.00010

23 0.00000 0.02488 0.00012 0.00000 0.02488 0.00012 23 0.00000 -0.03131 -0.00009 0.00000 -0.03131 -0.00009

LINEAR TIME HISTORY - DIRECT INTEGRATION LINEAR TIME HISTORY - DIRECT INTEGRATION




7 0.00002 0.00000 0.00000 0.00003 0.00000 0.00000 7 -0.00002 0.00000 0.00000 -0.00002 0.00000 0.00000

12 0.00006 0.00000 0.00000 0.00006 0.00000 0.00000 12 -0.00005 0.00000 0.00000 -0.00006 0.00000 0.00000

15 0.00007 0.00000 0.00000 0.00007 0.00000 0.00000 15 -0.00007 0.00000 0.00000 -0.00007 0.00000 0.00000

20 0.00008 0.00000 0.00000 0.00009 0.00000 0.00000 20 -0.00008 0.00000 0.00000 -0.00008 0.00000 0.00000

23 0.00009 0.00000 0.00000 0.00009 0.00000 0.00000 23 -0.00009 0.00000 0.00000 -0.00009 0.00000 0.00000





7 0.00000 0.02687 0.00000 0.00000 0.02688 0.00000 7 0.00000 -0.02843 0.00000 0.00000 -0.02839 0.00000

12 0.00000 0.16729 0.00000 0.00000 0.16735 0.00000 12 0.00000 -0.17668 0.00000 0.00000 -0.17642 0.00000

15 0.00000 0.29058 0.00000 0.00000 0.29067 0.00000 15 0.00000 -0.30662 0.00000 0.00000 -0.30617 0.00000

20 0.00000 0.52879 0.00000 0.00000 0.52892 0.00000 20 0.00000 -0.55736 0.00000 0.00000 -0.55655 0.00000

23 0.00000 0.67901 0.00000 0.00000 0.67917 0.00000 23 0.00000 -0.71539 0.00000 0.00000 -0.71435 0.00000





7 0.00000 0.00457 0.00015 0.00000 0.00459 0.00015 7 0.00000 -0.00559 -0.00012 0.00000 -0.00557 -0.00013

12 0.00000 0.01164 0.00014 0.00000 0.01169 0.00014 12 0.00000 -0.01446 -0.00011 0.00000 -0.01441 -0.00011

15 0.00000 0.01546 0.00013 0.00000 0.01553 0.00013 15 0.00000 -0.01942 -0.00010 0.00000 -0.01935 -0.00010

20 0.00000 0.02131 0.00013 0.00000 0.02140 0.00013 20 0.00000 -0.02721 -0.00010 0.00000 -0.02712 -0.00010

23 0.00000 0.02465 0.00013 0.00000 0.02476 0.00013 23 0.00000 -0.03172 -0.00009 0.00000 -0.03162 -0.00009


HIST2---MAXIMUM VELOCITIES HIST2---MINIMUM VELOCITIES




___________________________________________________________________________________________________________________

-259-

5 0.00000 0.12458 0.00000 0.00000 0.12000 0.00000 5 0.00000 -0.11931 0.00000 0.00000 -0.12000 0.00000

13 0.00000 7.95334 0.00000 0.00000 7.96000 0.00000 13 0.00000 -7.68585 0.00000 0.00000 -7.69000 0.00000

17 0.00000 14.68662 0.00000 0.00000 14.69000 0.00000 17 0.00000 -14.25173 0.00000 0.00000 -14.25000 0.00000

21 0.00000 22.11501 0.00000 0.00000 22.12000 0.00000 21 0.00000 -21.53326 0.00000 0.00000 -21.53000 0.00000

22 0.00000 24.01014 0.00000 0.00000 24.02000 0.00000 22 0.00000 -23.39444 0.00000 0.00000 -23.39000 0.00000





5 0.00000 0.11279 0.00000 0.00000 0.11000 0.00000 5 0.00000 -0.11304 0.00000 0.00000 -0.11000 0.00000

13 0.00000 7.43352 0.00000 0.00000 7.43000 0.00000 13 0.00000 -7.27901 0.00000 0.00000 -7.28000 0.00000

17 0.00000 13.90589 0.00000 0.00000 13.90000 0.00000 17 0.00000 -13.60474 0.00000 0.00000 -13.60000 0.00000

21 0.00000 21.13308 0.00000 0.00000 21.12000 0.00000 21 0.00000 -20.63724 0.00000 0.00000 -20.62000 0.00000

22 0.00000 22.98516 0.00000 0.00000 22.97000 0.00000 22 0.00000 -22.43192 0.00000 0.00000 -22.41000 0.00000


HIST3---MAXIMUM ACCELERATIONS HIST3---MINIMUM ACCELERATIONS



5 0.00000 0.00000 5.15041 0.00000 0.00000 5.15000 5 0.00000 0.00000 -4.38265 0.00000 0.00000 -4.38300

13 0.00000 0.00000 47.44657 0.00000 0.00000 47.44600 13 0.00000 0.00000 -35.89688 0.00000 0.00000 -35.89700

17 0.00000 0.00000 70.74769 0.00000 0.00000 70.74800 17 0.00000 0.00000 -51.33674 0.00000 0.00000 -51.33700

21 0.00000 0.00000 91.86813 0.00000 0.00000 91.86900 21 0.00000 0.00000 -69.62636 0.00000 0.00000 -69.62700

22 0.00000 0.00000 96.86608 0.00000 0.00000 96.86700 22 0.00000 0.00000 -74.07437 0.00000 0.00000 -74.07500





5 0.00000 0.00000 6.50963 0.00000 0.00000 6.47700 5 0.00000 0.00000 -6.10602 0.00000 0.00000 -6.10700

13 0.00000 0.00000 57.09429 0.00000 0.00000 57.02300 13 0.00000 0.00000 -42.89395 0.00000 0.00000 -42.95800

17 0.00000 0.00000 77.73771 0.00000 0.00000 77.83200 17 0.00000 0.00000 -53.00894 0.00000 0.00000 -52.95100

21 0.00000 0.00000 100.50203 0.00000 0.00000 100.42400 21 0.00000 0.00000 -75.68891 0.00000 0.00000 -75.70200

22 0.00000 0.00000 105.76891 0.00000 0.00000 105.81200 22 0.00000 0.00000 -81.81049 0.00000 0.00000 -81.77600


___________________________________________________________________________________________________________________

-260-

Input Filename: Example19 (2DFRAME_TH.S2K)




Description: A simple 2D frame structure. The structure is subjected to three different transient load cases. The eigen values in each mode and

the corresponding damping ratios are also generated based on the values initially provided to the program. The mass and stiffness coefficients

(a0 and a1) for analysis using direct integration are based on these values.

Purpose: Verify the accuracy of WoodFrameSolver for linear dynamic analysis using mode superposition and direct integration.

Results:

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 500 1000 1500 2000

OMEGA (rad/sec)

ξ

ξ-MASS_PROP

ξ-STIF_PROP

ξ-RAYL_PROP


a0 6.80919 0 6.43926

a1 0 8.44E-05 7.98E-05


___________________________________________________________________________________________________________________

-261-




UX UY UZ UX UY UZ

1 0.5358 0.00 77.22 0.5352 0.00 77.22 0.00

3 0.2212 97.27 0.00 0.00 0.2210 97.27 0.00 0.00

8 0.0485 2.73 0.00 0.00 0.0485 2.73 0.00 0.00

12 0.0090 0.00 0.00 0.05 0.0090 0.00 0.00 0.05

18 0.0038 0.00 0.00 2.48 0.0038 0.00 0.00 2.48





1 0.00000 0.00000 0.00142 0.00000 0.00000 0.00142 1 0.00000 0.00000 -0.00181 0.00000 0.00000 -0.00181

3 0.22893 0.00049 0.00022 0.22893 0.00049 0.00022 3 -0.29580 -0.00065 -0.00030 -0.29581 -0.00065 -0.00030

5 0.16572 0.00000 0.00042 0.16572 0.00000 0.00042 5 -0.21147 0.00000 -0.00054 -0.21148 0.00000 -0.00054

7 0.00000 0.00000 0.00142 0.00000 0.00000 0.00142 7 0.00000 0.00000 -0.00181 0.00000 0.00000 -0.00181

9 0.22893 0.00065 0.00022 0.22893 0.00065 0.00022 9 -0.29580 -0.00049 -0.00030 -0.29581 -0.00049 -0.00030




NODE UY RX RZ UY RX RZ NODE UY RX RZ UY RX RZ

1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

3 6.46666 0.03324 0.00210 6.46549 0.03322 0.00210 3 -6.47713 -0.03321 -0.00208 -6.47547 -0.03320 -0.00208

5 2.24029 0.02728 0.00000 2.23905 0.02726 0.00000 5 -2.27851 -0.02699 0.00000 -2.27830 -0.02697 0.00000

7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

9 6.46671 0.03324 0.00208 6.46549 0.03322 0.00208 9 -6.47715 -0.03321 -0.00210 -6.47547 -0.03320 -0.00210





1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

3 0.00000 0.00000 0.00032 0.00000 0.00000 0.00032 3 0.00000 0.00000 -0.00021 0.00000 0.00000 -0.00030

5 0.00000 0.00000 0.00035 0.00000 0.00000 0.00035 5 0.00000 0.00000 -0.00033 0.00000 0.00000 -0.00033

7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000


___________________________________________________________________________________________________________________

-262-

9 0.00000 0.00000 0.00032 0.00000 0.00000 0.00032 9 0.00000 0.00000 -0.00030 0.00000 0.00000 -0.00030





1 0.00000 0.00000 0.00143 0.00000 0.00000 0.00143 1 0.00000 0.00000 -0.00181 0.00000 0.00000 -0.00181

3 0.22951 0.00049 0.00022 0.23024 0.00049 0.00022 3 -0.29574 -0.00065 -0.00030 -0.29501 -0.00065 -0.00030

5 0.16624 0.00000 0.00042 0.16677 0.00000 0.00042 5 -0.21165 0.00000 -0.00054 -0.21112 0.00000 -0.00054

7 0.00000 0.00000 0.00143 0.00000 0.00000 0.00143 7 0.00000 0.00000 -0.00181 0.00000 0.00000 -0.00181

9 0.22951 0.00065 0.00022 0.23024 0.00065 0.00022 9 -0.29574 -0.00049 -0.00030 -0.29501 -0.00049 -0.00030





1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

3 6.30301 0.03275 0.00206 6.30388 0.03271 0.00206 3 -6.36945 -0.03216 -0.00213 -6.36157 -0.03217 -0.00213

5 2.25577 0.02694 0.00000 2.25622 0.02691 0.00000 5 -2.23482 -0.02673 0.00000 -2.23198 -0.02674 0.00000

7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

9 6.30301 0.03275 0.00213 6.30388 0.03271 0.00213 9 -6.36945 -0.03216 -0.00206 -6.36157 -0.03217 -0.00206





1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

3 0.00000 0.00000 0.00033 0.00000 0.00000 0.00033 3 0.00000 0.00000 -0.00031 0.00000 0.00000 -0.00031

5 0.00000 0.00000 0.00035 0.00000 0.00000 0.00036 5 0.00000 0.00000 -0.00034 0.00000 0.00000 -0.00034

7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

9 0.00000 0.00000 0.00033 0.00000 0.00000 0.00033 9 0.00000 0.00000 -0.00031 0.00000 0.00000 -0.00031





1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

3 76.14983 0.39948 0.03068 76.10000 0.39900 0.03100 3 -75.91721 -0.38749 -0.03101 -75.89000 -0.38700 -0.03100

5 27.79585 0.32097 0.00000 27.79000 0.32100 0.00000 5 -27.03216 -0.31792 0.00000 -27.04000 -0.31800 0.00000


___________________________________________________________________________________________________________________

-263-

7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

9 76.14996 0.39948 0.03101 76.10000 0.39900 0.03100 9 -75.91761 -0.38749 -0.03068 -75.89000 -0.38700 -0.03100





1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

3 74.83390 0.39598 0.02779 74.80000 0.39600 0.02800 3 -76.04557 -0.39629 -0.02938 -76.00000 -0.39600 -0.02900

5 26.92389 0.31291 0.00000 26.91000 0.31300 0.00000 5 -27.00384 -0.30834 0.00000 -26.99000 -0.30800 0.00000

7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 7 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

9 74.83390 0.39598 0.02938 74.80000 0.39600 0.02900 9 -76.04557 -0.39629 -0.02779 -76.00000 -0.39600 -0.02800





1 0.00000 0.00000 0.00417 0.00000 0.00000 0.00417 1 0.00000 0.00000 -0.00488 0.00000 0.00000 -0.00488

3 0.14999 5.37523 0.03131 0.15000 5.37500 0.03100 3 -0.17301 -5.99973 -0.02804 -0.17300 -6.00000 -0.02800

5 0.00000 7.53835 0.00000 0.00000 7.53800 0.00000 5 0.00000 -11.62328 0.00000 0.00000 -11.62300 0.00000

7 0.00000 0.00000 0.00488 0.00000 0.00000 0.00488 7 0.00000 0.00000 -0.00417 0.00000 0.00000 -0.00417

9 0.17301 5.37523 0.02804 0.17300 5.37500 0.02800 9 -0.14999 -5.99973 -0.03131 -0.15000 -6.00000 -0.03100





1 0.00000 0.00000 0.01226 0.00000 0.00000 0.01200 1 0.00000 0.00000 -0.01143 0.00000 0.00000 -0.01100

3 0.39928 24.51489 0.05995 0.39900 24.49200 0.06000 3 -0.42660 -22.16298 -0.06119 -0.42600 -22.14200 -0.06100

5 0.00000 15.55292 0.00000 0.00000 15.55500 0.00000 5 0.00000 -17.62298 0.00000 0.00000 -17.63400 0.00000

7 0.00000 15.67432 0.01143 0.00000 0.00000 0.01100 7 0.00000 0.00000 -0.01226 0.00000 0.00000 -0.01200

9 0.42660 24.51489 0.06119 0.42600 24.49200 0.06100 9 -0.39928 -22.16298 -0.05995 -0.39900 -22.14200 -0.06000


___________________________________________________________________________________________________________________

-264-

Input Filename: Example20 (2DFRAME_HOOK.S2K)




Description: A simple 2D frame structure with a brace connected using linear hook. The frame is subjected to three different transient load

cases. The eigen values in each mode and the corresponding damping ratios are also generated based on the values initially provided to the

program. The mass and stiffness coefficients (a0 and a1) for analysis using direct integration are based on these values.


Results:

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 100 200 300 400

OMEGA (rad/sec)

ξ

ξ-MASS_PROP

ξ-STIF_PROP

ξ-RAYL_PROP


a0 1.34824 0 1.28324

a1 0 0.000375685 0.000357573


___________________________________________________________________________________________________________________

-265-




UX UY UZ UX UY UZ

1 2.9864 0.00 72.28 0.00 2.9830 0.00 72.28 0.00

4 0.5685 0.00 22.27 0.00 0.5678 0.00 22.27 0.00

8 0.3083 0.00 4.04 0.00 0.3079 0.00 4.04 0.00

14 0.0399 0.00 0.00 11.67 0.0399 0.00 0.00 11.67

17 0.0198 1.44 0.00 0.00 0.0198 1.44 0.00 0.00





2 0.00110 0.00041 0.00001 0.00110 0.00041 0.00001 2 -0.00127 -0.00047 -0.00001 -0.00127 -0.00047 -0.00001

3 0.00177 0.00051 0.00000 0.00177 0.00051 0.00000 3 -0.00020 -0.00059 0.00000 -0.00205 -0.00059 0.00000

5 0.00103 0.00000 0.00001 0.00103 0.00000 0.00001 5 -0.00119 0.00000 -0.00001 -0.00119 0.00000 -0.00001

10 0.00211 0.00125 0.00000 0.00211 0.00125 0.00000 10 -0.00247 -0.00102 0.00000 -0.00247 -0.00102 0.00000

15 0.00087 0.00079 0.00000 0.00087 0.00079 0.00000 15 -0.00098 -0.00085 0.00000 -0.00098 -0.00085 0.00000





2 11.50285 0.12043 0.01192 11.50283 0.12043 0.01192 2 -11.02380 -0.11547 -0.01199 -11.02370 -0.11547 -0.01199

3 29.77425 0.17291 0.01254 29.77415 0.17291 0.01254 3 -30.07051 -0.17640 -0.01243 -30.07047 -0.17640 -0.01243

5 13.81980 0.12247 0.00000 13.81978 0.12247 0.00000 5 -13.04643 -0.11945 0.00000 -13.04638 -0.11945 0.00000

10 21.15611 0.15587 0.01440 21.15605 0.15587 0.01440 10 -21.60708 -0.15939 -0.01417 -21.60696 -0.15939 -0.01417

15 5.30566 0.06763 0.01454 5.30565 0.06763 0.01454 15 -4.97556 -0.06686 -0.01551 -4.97557 -0.06686 -0.01551





2 -0.00005 0.00010 0.00002 0.00005 0.00010 0.00002 2 -0.00005 -0.00010 -0.00002 -0.00005 -0.00010 -0.00002

3 0.00009 0.00013 0.00002 0.00009 0.00013 0.00002 3 -0.00009 -0.00013 -0.00002 -0.00009 -0.00013 -0.00002

5 0.00000 0.00020 0.00000 0.00000 0.00020 0.00000 5 0.00000 -0.00019 0.00000 0.00000 -0.00019 0.00000

10 0.00216 0.00474 0.00000 0.00216 0.00474 0.00000 10 -0.00216 -0.00473 0.00000 -0.00216 -0.00473 0.00000


___________________________________________________________________________________________________________________

-266-

15 0.00202 0.00421 0.00000 0.00202 0.00421 0.00000 15 -0.00203 -0.00423 0.00000 -0.00203 -0.00423 0.00000





2 0.00131 0.00036 0.00001 0.00131 0.00036 0.00001 2 -0.00120 -0.00038 -0.00001 -0.00120 -0.00038 -0.00001

3 0.00197 0.00043 0.00000 0.00197 0.00044 0.00000 3 -0.00183 -0.00047 0.00000 -0.00182 -0.00047 0.00000

5 0.00125 0.00000 0.00000 0.00125 0.00000 0.00000 5 -0.00115 0.00000 0.00000 -0.00114 0.00000 0.00000

10 0.00208 0.00070 0.00000 0.00207 0.00069 0.00000 10 -0.00187 -0.00077 0.00000 -0.00187 -0.00076 0.00000

15 0.00086 0.00054 0.00000 0.00087 0.00054 0.00000 15 -0.00084 -0.00057 0.00000 -0.00083 -0.00057 0.00000





2 11.31230 0.12087 0.01195 11.41522 0.12086 0.01202 2 -11.07701 -0.11381 -0.01237 -11.07169 -0.11500 -0.01236

3 29.40127 0.17193 0.01228 29.70342 0.17195 0.01237 3 -30.00591 -0.17458 -0.01239 -30.00432 -0.17606 -0.01240

5 13.64319 0.12318 0.00000 13.75636 0.12317 0.00000 5 -13.07903 -0.11715 0.00000 -13.07654 -0.11839 0.00000

10 21.00644 0.15508 0.01414 21.21619 0.15510 0.01423 10 -21.73716 -0.15734 -0.01395 -21.73366 -0.15873 -0.01396

15 5.17425 0.06783 0.01441 5.20895 0.06782 0.01440 15 -4.95518 -0.06615 -0.01507 -4.95340 -0.06680 -0.01515





2 0.00014 0.00043 0.00002 0.00014 0.00044 0.00002 2 -0.00015 -0.00042 -0.00002 -0.00015 -0.00042 -0.00002

3 0.00011 0.00059 0.00002 0.00011 0.00059 0.00002 3 -0.00010 -0.00057 -0.00002 -0.00010 -0.00057 -0.00002

5 0.00000 0.00065 0.00000 0.00000 0.00065 0.00000 5 0.00000 -0.00065 0.00000 0.00000 -0.00065 0.00000

10 0.00218 0.00489 0.00000 0.00217 0.00491 0.00000 10 -0.00194 -0.00536 0.00000 -0.00194 -0.00534 0.00000

15 0.00199 0.00426 0.00001 0.00200 0.00428 0.00001 15 -0.00219 -0.00468 -0.00001 -0.00218 -0.00466 -0.00001





2 48.69310 0.27039 0.09749 48.69000 0.27000 0.09700 2 -47.50336 -0.26830 -0.09704 -47.50000 -0.26800 -0.09700

3 71.94298 0.74088 0.08537 71.94000 0.74100 0.08500 3 -73.77619 -0.71026 -0.07040 -73.78000 -0.71000 -0.07000

5 55.23859 0.28618 0.00000 55.24000 0.28600 0.00000 5 -53.92181 -0.28313 0.00000 -53.92000 -0.28300 0.00000


___________________________________________________________________________________________________________________

-267-

10 62.01265 0.64973 0.08372 62.01000 0.65000 0.08400 10 -59.23862 -0.60866 -0.08971 -59.24000 -0.60900 -0.09000

15 32.49276 0.18645 0.13583 32.49000 0.18600 0.13600 15 -36.27475 -0.19334 -0.13497 -36.27000 -0.19300 -0.13500





2 48.32213 0.27259 0.10005 48.43000 0.27400 0.10000 2 -47.57876 -0.26685 -0.10268 -47.69000 -0.26800 -0.10300

3 71.34031 0.74249 0.08925 71.67000 0.74400 0.08900 3 -73.07777 -0.69631 -0.07014 -73.40000 -0.69800 -0.07000

5 54.20994 0.28012 0.00000 54.34000 0.28100 0.00000 5 -53.63321 -0.27463 0.00000 -53.75000 -0.27600 0.00000

10 60.66771 0.64794 0.08206 60.90000 0.64900 0.08200 10 -60.05751 -0.60471 -0.08438 -60.28000 -0.60600 -0.08900

15 31.79003 0.18761 0.13572 31.82000 0.18800 0.13600 15 -35.85494 -0.19327 -0.13152 -35.90000 -0.19400 -0.13200





2 0.38018 0.88393 0.16072 0.38000 0.88400 0.16100 2 -0.42112 -0.98003 -0.14504 -0.42100 -0.98000 -0.14500

3 0.92379 1.11366 0.15582 0.92400 1.11400 0.15600 3 -0.83241 -1.23494 -0.14055 -0.83200 -1.23500 -0.14100

5 0.00000 1.72513 0.00000 0.00004 1.72500 0.00000 5 0.00000 -1.91205 0.00000 -0.00003 -1.91200 0.00000

10 21.34452 42.04984 0.02954 21.34400 42.05000 0.03000 10 -19.22552 -46.67780 -0.03273 -19.22500 -46.67800 -0.03300

15 18.39378 38.37293 0.04072 18.39400 38.37300 0.04100 15 -20.34637 -42.45067 -0.04513 -20.34600 -42.45100 -0.04500





2 7.07350 11.53603 0.13801 7.07400 11.53600 0.13800 2 -7.09165 -13.96444 -0.12955 -7.09100 -13.96400 -0.12900

3 4.70334 17.37863 0.15556 4.70300 17.37900 0.15600 3 -3.60928 -21.15986 -0.13027 -3.60900 -21.15900 -0.13000

5 0.00000 14.49651 0.00000 0.00000 14.49600 0.00000 5 0.00000 -19.03187 0.00000 0.00000 -19.03100 0.00000

10 19.59291 37.87784 0.11522 19.59300 37.87800 0.11500 10 -20.09141 -39.88721 -0.08481 -20.09000 -39.88800 -0.08500

15 16.93053 34.82241 0.07069 16.93000 34.82400 0.06100 15 -16.09931 -33.05294 -0.07580 -16.09900 -33.05100 -0.06600


___________________________________________________________________________________________________________________

-268-

Input Filename: Example21 (3DFRAME.S2K)




Description: A simple 3D frame structure is subjected to three different transient load cases. The eigen values in each mode and the

corresponding damping ratios are also generated based on the values initially provided to the program. The mass and stiffness coefficients (a0

and a1) for analysis using direct integration are based on these values.


Results:

0

0.05

0.1

0.15

0.2

0.25

0.3

0 200 400 600

OMEGA (rad/sec)

ξ

ξ-MASS_PROP

ξ-STIF_PROP

ξ-RAYL_PROP


a0 7.12707 0 5.44528

a1 0 0.000222306 0.000180934


___________________________________________________________________________________________________________________

-269-




UX UY UZ UX UY UZ

1 0.4284 0.00 98.13 0.00 0.4279 0.00 98.13 0.00

3 0.2860 96.58 0.00 0.00 0.2857 96.58 0.00 0.00

17 0.0158 0.00 0.00 16.69 0.0157 0.00 0.00 16.69

24 0.0140 0.00 0.00 18.50 0.0140 0.00 0.00 18.50

36 0.0119 0.00 0.00 23.15 0.0119 0.00 0.00 23.15





5 0.36118 0.00085 0.00127 0.36116 0.00085 0.00127 5 -0.33772 -0.00080 -0.00119 -0.33771 -0.00080 -0.00119

9 0.43247 0.00095 0.00043 0.43245 0.00095 0.00043 9 -0.40423 -0.00087 -0.00039 -0.40421 -0.00087 -0.00039

15 0.50149 0.00014 0.00038 0.50146 0.00014 0.00038 15 -0.47059 -0.00147 -0.00036 -0.47058 -0.00015 -0.00036

29 0.31078 0.00068 0.00110 0.31076 0.00068 0.00110 29 -0.29251 -0.00074 -0.00103 -0.29249 -0.00074 -0.00103

33 0.50184 0.00103 0.00050 0.50181 0.00103 0.00050 33 -0.47091 -0.00109 -0.00046 -0.47091 -0.00109 -0.00046





5 1.71229 0.00159 0.00036 1.71276 0.00159 0.00036 5 -1.78908 -0.00153 -0.00034 -1.78916 -0.00153 -0.00034

9 2.15073 0.00094 0.00092 2.15132 0.00094 0.00092 9 -2.24244 -0.00091 -0.00088 -2.24255 -0.00091 -0.00088

15 2.57075 0.00065 0.00044 2.57122 0.00065 0.00044 15 -2.58774 -0.00066 -0.00042 -2.58841 -0.00066 -0.00042

29 1.71174 0.00368 0.00076 1.71231 0.00368 0.00076 29 -1.78849 -0.00353 -0.00080 -1.78868 -0.00353 -0.00080

33 2.15028 0.00058 0.00041 2.15098 0.00058 0.00041 33 -2.24197 -0.00056 -0.00043 -2.24219 -0.00056 -0.00043





5 0.00000 0.00046 0.00000 0.00000 0.00046 0.00000 5 -0.00001 -0.00044 0.00000 -0.00001 -0.00044 0.00000

9 0.00000 0.00053 0.00000 0.00000 0.00053 0.00000 9 0.00000 -0.00050 0.00000 0.00000 -0.00050 0.00000

15 0.00000 0.00092 0.00000 0.00000 0.00092 0.00000 15 0.00000 -0.00086 0.00000 0.00000 -0.00086 0.00000

29 0.00000 0.00034 0.00000 0.00000 0.00034 0.00000 29 0.00000 -0.00032 0.00000 0.00000 -0.00032 0.00000


___________________________________________________________________________________________________________________

-270-

33 0.00001 0.00073 0.00000 0.00001 0.00073 0.00000 33 -0.00001 -0.00069 0.00000 -0.00001 -0.00069 0.00000





5 0.36080 0.00084 0.00126 0.36174 0.00085 0.00126 5 -0.33756 -0.00081 -0.00119 -0.33662 -0.00081 -0.00119

9 0.43074 0.00094 0.00043 0.43189 0.00094 0.00043 9 -0.40529 -0.00087 -0.00039 -0.40414 -0.00087 -0.00039

15 0.49926 0.00014 0.00038 0.50054 0.00014 0.00038 15 -0.47154 -0.00015 -0.00037 -0.47027 -0.00015 -0.00037

29 0.30990 0.00069 0.00109 0.31074 0.00068 0.00110 29 -0.29340 -0.00073 -0.00103 -0.29255 -0.00074 -0.00103

33 0.49961 0.00104 0.00050 0.50089 0.00104 0.00050 33 -0.47187 -0.00109 -0.00048 -0.47059 -0.00109 -0.00047





5 1.70483 0.00159 0.00036 1.70703 0.00159 0.00036 5 -1.79460 -0.00151 -0.00035 -1.79304 -0.00152 -0.00035

9 2.13749 0.00093 0.00092 2.14022 0.00093 0.00092 9 -2.24604 -0.00090 -0.00092 -2.24448 -0.00090 -0.00092

15 2.56257 0.00066 0.00044 2.56564 0.00066 0.00044 15 -2.58598 -0.00065 -0.00043 -2.58380 -0.00065 -0.00043

29 1.70437 0.00368 0.00079 1.70657 0.00368 0.00079 29 -1.79412 -0.00351 -0.00080 -1.79256 -0.00351 -0.00080

33 2.13716 0.00058 0.00043 2.13989 0.00058 0.00043 33 -2.24604 -0.00055 -0.00043 -2.24412 -0.00055 -0.00043





5 0.00001 0.00048 0.00000 0.00001 0.00048 0.00000 5 -0.00001 -0.00049 0.00000 -0.00001 -0.00049 0.00000

9 0.00001 0.00052 0.00000 0.00001 0.00052 0.00000 9 -0.00001 -0.00049 0.00000 -0.00001 -0.00049 0.00000

15 0.00000 0.00091 0.00000 0.00000 0.00091 0.00000 15 0.00000 -0.00096 0.00000 0.00000 -0.00096 0.00000

29 0.00000 0.00036 0.00000 0.00001 0.00036 0.00000 29 0.00000 -0.00034 0.00000 0.00000 -0.00034 0.00000

33 0.00001 0.00070 0.00000 0.00001 0.00070 0.00000 33 -0.00001 -0.00073 0.00000 -0.00001 -0.00073 0.00000





5 25.35698 0.02445 0.00610 25.36000 0.02400 0.00610 5 -27.20544 -0.02227 -0.00568 -27.20000 -0.02200 -0.00569

9 31.52177 0.01494 0.01533 31.53000 0.01500 0.01500 9 -34.34528 -0.01391 -0.01474 -34.34000 -0.01400 -0.01500

15 38.09388 0.00940 0.00722 38.09000 0.00940 0.00722 15 -36.64222 -0.00996 -0.00696 -36.64000 -0.00995 -0.00696


___________________________________________________________________________________________________________________

-271-

29 25.34933 0.05635 0.01274 25.36000 0.05600 0.01300 29 -27.19681 -0.05147 -0.01360 -27.20000 -0.05100 -0.01400

33 31.51601 0.00907 0.00692 31.52000 0.00907 0.00692 33 -34.33837 -0.00827 -0.00715 -34.34000 -0.00827 -0.00716





5 25.34073 0.02408 0.00612 25.34000 0.02400 0.00612 5 -27.05488 -0.02237 -0.00582 -27.06000 -0.02200 -0.00582

9 31.65659 0,014376 0.01545 31.66000 0.01400 0.01500 9 -33.95498 -0.01306 -0.01536 -33.97000 -0.01300 -0.01500

15 37.61942 0.00985 0.00728 37.62000 0.00986 0.00728 15 -36.84651 -0.00933 -0.00726 -36.86000 -0.00933 -0.00726

29 25.33405 0.05561 0.01307 25.34000 0.05600 0.01300 29 -27.04770 -0.05169 -0.01365 -27.06000 -0.05200 -0.01400

33 31.65163 0.00877 0.00721 31.65000 0.00877 0.00721 33 -33.94964 -0.00813 -0.00721 -33.96000 -0.00813 -0.00721





5 0.75195 9.18321 0.00937 0.75200 9.18400 0.00938 5 -0.78199 -7.43327 -0.00767 -0.77900 -7.43500 -0.00770

9 0.45258 10.22300 0.02002 0.45300 10.22300 0.02000 9 -0.39271 -13.69858 -0.01580 -0.39300 -13.69900 -0.01600

15 0.49759 22.57290 0.02168 0.49700 22.57300 0.02200 15 -0.54246 -21.02383 -0.02490 -0.54300 -21.02200 -0.02500

29 0.35899 6.52733 0.00497 0.35700 6.52900 0.00497 29 -0.34425 -8.74653 -0.00571 -0.34400 -8.75100 -0.00569

33 1.23832 14.51288 0.02777 1.23800 14.51300 0.02800 33 -1.14106 -11.74643 -0.02919 -1.14000 -11.74700 -0.02900





5 1.02838 17.80894 0.01376 1.02800 17.70800 0.01400 5 -1.09672 -22.00304 -0.01446 -1.09700 -22.00300 -0.01400

9 1.41728 21.93378 0.04602 1.41700 21.93500 0.04600 9 -1.42651 -25.75809 -0.04451 -1.42600 -25.75900 -0.04500

15 0.72351 35.92850 0.04291 0.72400 35.93000 0.04300 15 -0.76078 -31.02953 -0.04095 -0.76100 -31.03200 -0.04100

29 0.91256 14.24521 0.01453 0.91200 14.24600 0.01500 29 -0.94450 -16.12502 -0.01319 -0.94400 -16.12500 -0.01300

33 1.74328 29.59551 0.04501 1.74400 29.59500 0.04500 33 -1.65889 -34.85608 -0.04889 -1.65900 -34.85600 -0.04900


___________________________________________________________________________________________________________________

-272-

Input Filename: Example22 (FIXED_FIXED_BEAM_GAP.S2K)




Description: This model is a fixed end beam with a two node nonlinear gap element at the center. The model is subjected to one transient load

case with three acceleration loads in UX, UY and UZ directions. A Rayleigh damping model is used for the analysis of this load case.

Purpose: Verify the accuracy of WoodFrameSolver for non-linear dynamic analysis using direct integration.

Results: NON-LINEAR TIME HISTORY NON-LINEAR TIME HISTORY

HIST1---MAXIMUM DISPLACEMENTS HIST1---MAXIMUM DISPLACEMENTS

WOODFRAMESOLVER (DI) WOODFRAMESOLVER (DI)

NODE UX UY UZ RX RY RZ NODE UX UY UZ RX RY RZ

2 0.00131 2.74351 1.78458 0.00000 0.01816 0.03426 2 -0.00126 -2.23501 -1.45381 0.00000 -0.02229 -0.02791

3 0.00131 2.74351 1.78458 0.00000 0.02229 0.02791 3 -0.00126 -2.23501 -1.45381 0.00000 -0.01816 -0.03426

SAP (DI) SAP (DI)

2 0.00129 2.74369 1.78469 0.00000 0.01814 0.03427 2 0.00129 2.74369 1.78469 0.00000 0.01814 0.03427

3 0.00129 2.74369 1.78469 0.00000 0.02229 0.02788 3 -0.00133 -2.23261 -1.45225 0.00000 -0.01814 -0.03427

SAP (FNA) SAP (FNA)

2 0.00130 2.74810 1.78890 0.00000 0.01818 0.03432 2 0.00130 2.74810 1.78890 0.00000 0.01818 0.03432

3 0.00130 2.74810 1.78890 0.00000 0.02234 0.02793 3 -0.00122 -2.23603 -1.45557 0.00000 -0.01818 -0.03432

NON-LINEAR TIME HISTORY NON-LINEAR TIME HISTORY

HIST1---MAXIMUM VELCITIES HIST1---MAXIMUM VELCITIES



2 0.05464 14.70557 9.56556 0.00000 0.10449 0.18366 2 -0.12312 -12.86231 -8.36657 0.00000 -0.11947 -0.16064

3 0.05464 14.70557 9.56556 0.00000 0.11947 0.16064 3 -0.12312 -12.86231 -8.36657 0.00000 -0.10449 -0.18366

SAP (DI) SAP (DI)

2 0.08372 14.70000 9.56000 0.00000 0.10400 0.18400 2 -0.13000 -12.85000 -8.36000 0.00000 -0.11900 -0.16100

3 0.08372 14.70000 9.56000 0.00000 0.11900 0.16100 3 -0.13000 -12.85000 -8.36000 0.00000 -0.10400 -0.18400

SAP (FNA) SAP (FNA)

2 0.05462 14.72000 9.58000 0.00000 0.10500 0.18400 2 -0.12000 -12.88000 -8.38000 0.00000 -0.12000 -0.16100

3 0.05462 14.72000 9.58000 0.00000 0.12000 0.16100 3 -0.12000 -12.88000 -8.38000 0.00000 -0.10500 -0.18400


___________________________________________________________________________________________________________________

-273-

NON-LINEAR TIME HISTORY NON-LINEAR TIME HISTORY




2 18.00767 143.45085 93.31074 0.00000 0.91879 1.79641 2 -21.05330 -112.56857 -73.22269 0.00000 -1.16852 -1.41250

3 18.00767 143.45085 93.31074 0.00000 1.16852 1.41250 3 -21.05330 -112.56857 -73.22269 0.00000 -0.91879 -1.79641

SAP (DI) SAP (DI)

2 15.13000 143.44800 93.30900 0.00000 96.61100 7845.62200 2 -14.04000 -112.53100 -73.19800 0.00000 -96.77700 -7845.36700

3 15.13000 143.44800 93.30900 0.00000 81.16100 7112.42900 3 -14.04000 -112.53100 -73.19800 0.00000 -80.91000 -7112.81600

SAP (FNA) SAP (FNA)

2 2.93500 143.63400 93.50000 0.00000 0.91700 1.79400 2 -2.50600 -112.73200 -73.38400 0.00000 -1.16800 -1.40800

3 2.93500 143.63400 93.50000 0.00000 1.16800 1.40800 3 -2.50600 -112.73200 -73.38400 0.00000 -0.91700 -1.79400


___________________________________________________________________________________________________________________

-274-

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 5 10 15 20 25 30 35

TIME (sec)

RO

TA

TIO

N (ra

d)

SAP-RY

KEYSOLVER-RY

Internal rotational deformation (RY) response history


___________________________________________________________________________________________________________________

-275-

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 5 10 15 20 25 30 35

TIME (sec)

RO

TA

TIO

N (

rad

)

SAP-RZ

KEYSOLVER-RZ

Internal rotational deformation (RZ) response history


___________________________________________________________________________________________________________________

-276-

-30

-25

-20

-15

-10

-5

0

5

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

DEFORMATION (rad)

FO

RC

E (

Kip

s)

Force-Deformation curve for the NLLINK gap element at the center

VERIFICATION MANUAL 2.0: WoodFrameSolver and SAP2000 Nonlinear Time

History Analysis

________________________________________________________________________

- 277 -

Example23: A 3 story 1 bay frame with bilinear rotational springs as shown in Figure 1 is subjected to Loma Preita earthquake (1989). The bilinear moment rotation relationship for springs is shown in Figure 2. The model is executed both on WoodFrameSolver and SAP2000.

The results are presented for two cases one with 0% damping and other with 2% damping in modes 1 and 3.

Figure 1


History Analysis

________________________________________________________________________

- 278 -

Figure 2


History Analysis

________________________________________________________________________

- 279 -

-0.150

-0.100

-0.050

0.000

0.050

0.100

0.150

0 5 10 15 20 25 30 35 40

TIME (sec)

DEFOR (rad)

KEYSOLVER

SAP2000

Figure 3: Spring 1, 0 % damping

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

-0.150 -0.100 -0.050 0.000 0.050 0.100 0.150

DEFOR (rad)

MOMENT (kips-in)

KEYSOLVER

SAP2000

Figure 4: Spring 1, 0 % damping


History Analysis

________________________________________________________________________

- 280 -

-0.250

-0.200

-0.150

-0.100

-0.050

0.000

0.050

0.100

0.150

0.200

0.250

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00

TIME (sec)

DEFOR (rad)

KEYSOLVER

SAP2000

Figure 5: Spring 3, 0% damping

-12000

-9000

-6000

-3000

0

3000

6000

9000

12000

-0.220 -0.120 -0.020 0.080 0.180

DEFOR (rad)

MOMENT (kips-in)

KEYSOLVER

SAP2000



History Analysis

________________________________________________________________________

- 281 -

-0.400

-0.300

-0.200

-0.100

0.000

0.100

0.200

0.300

0.400

0 5 10 15 20 25 30 35 40

TIME (sec)

DEFOR (ra

d)

KEYSOLVER

SAP2000


-16000

-11000

-6000

-1000

4000

9000

14000

19000

-0.400 -0.300 -0.200 -0.100 0.000 0.100 0.200 0.300 0.400

DEFOR (rad)

FORCE (kips-in)

KEYSOLVER

SAP2000



History Analysis

________________________________________________________________________

- 282 -

-0.150

-0.100

-0.050

0.000

0.050

0.100

0.150

0 5 10 15 20 25 30 35 40

TIME (sec)

DEFOR (rad)

KEYSOLVER

SAP2000

Figure 9: Spring 1, 2% damping in modes 1 and 3

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

-0.150 -0.100 -0.050 0.000 0.050 0.100 0.150

DEFOR (rad)

MOMENT (kips-in)

KEYSOLVER

SAP2000



History Analysis

________________________________________________________________________

- 283 -

-0.250

-0.200

-0.150

-0.100

-0.050

0.000

0.050

0.100

0.150

0.200

0.250

0 10 20 30 40

TIME (sec)

DEFOR (rad)

KEYSOLVER

SAP2000


-12000

-9000

-6000

-3000

0

3000

6000

9000

12000

-0.220 -0.170 -0.120 -0.070 -0.020 0.030 0.080 0.130 0.180

DEFOR (rad)

MOMENT (kips-in)

KEYSOLVER

SAP2000



History Analysis

________________________________________________________________________

- 284 -

-0.400

-0.300

-0.200

-0.100

0.000

0.100

0.200

0.300

0.400

0 10 20 30 40

TIME (sec)

DEFOR (rad)

KEYSOLVER

SAP2000


-20000

-15000

-10000

-5000

0

5000

10000

15000

20000

-0.400 -0.300 -0.200 -0.100 0.000 0.100 0.200 0.300 0.400

DEFOR (rad)

FORCE (kips-in)

KEYSOLVER

SAP2000



History Analysis

________________________________________________________________________

- 285 -

VERIFICATION MANUAL 2.0: WoodFrameSolver and SAPWOOD Nonlinear

Earthquake Analysis

________________________________________________________________________

- 286 -

The following examples compare accuracy of WoodFrameSolver with SAPWOOD. The house models are planar three degree of freedom systems with shear walls having a combination of linear, simple bilinear and Stewart hysteresis type properties. The geometric shapes for the floors

include simple rectangular, L and U shapes. A nonlinear dynamic analysis is performed with bidirectional ACM1 earthquake ground motions as input. The deformation history and force-deformation response compares quite well as can be seen in the following examples. The results

are compared for all the walls in a particular model however in some cases the results for only a few walls are presented.

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 10 20 30 40 50

ACM1 earthquake input


Earthquake Analysis

________________________________________________________________________

- 287 -

Example24: HOUSE 1: Rectangular floor plan with 3 linear and 1 Stewart hysteresis walls on its perimeter MODEL NAME: NHO1, Ux=0.1g & Uy=0.1g

STEWART PARAMETERS: F0=4.86 FI=1.01 DU=2.43 S0=14.0 R1=0.0802 R2=-0.0810 R3=1.26 R4=0.0704 ALPHA=0.740 BETA=1.09

Figure 1

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 10 20 30 40 50

TIME (sec)

DE

FO

R (in

)

SAPWOOD

KEYSOLVER

Figure 2: WALL 1


Earthquake Analysis

________________________________________________________________________

- 288 -

-8

-6

-4

-2

0

2

4

6

8

-2 -1 -1 0 1 1 2 2

DEFOR (in)

FO

RC

E (kip

s)

SAPWOOD

KEYSOLVER

Figure 3: WALL 1

Example25:

HOUSE 2: Rectangular floor plan with 4 bilinear walls on its perimeter MODEL NAME: NHO3, Ux=1g & Uy=1g BILINEAR PARAMETERS:

NAME=XWALL1 K1=20 K2=4 K3=4 FYP=50 FYN=-50 NAME=YWALL1 K1=20 K2=4 K3=4 FYP=30 FYN=-30 NAME=XWALL2 K1=15 K2=3 K3=3 FYP=45 FYN=-45 NAME=YWALL2 K1=10 K2=0.1 K3=0.1 FYP=13 FYN=-13

Figure 4


Earthquake Analysis

________________________________________________________________________

- 289 -

-10.0

-5.0

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TIME (sec)

DE

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R (

in)

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Figure 5: WALL 1

-80

-60

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20

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100

120

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DEFOR (in)

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Figure 6: WALL 1


Earthquake Analysis

________________________________________________________________________

- 290 -

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Figure 7: WALL 2

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Figure 8: WALL 2


Earthquake Analysis

________________________________________________________________________

- 291 -

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Figure 9: WALL 3

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Figure 10: WALL 3


Earthquake Analysis

________________________________________________________________________

- 292 -

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)

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Fgure 11: WALL 4

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Figure 12: WALL 4


Earthquake Analysis

________________________________________________________________________

- 293 -

Example26: HOUSE 3: Rectangular floor plan with 2 bilinear and 2 Stewart hysteresis walls on its perimeter MODEL NAME: NHO4, Ux=0.4g & Uy=0.4g

BILINEAR PARAMETERS: NAME=XWALL1 K1=20 K2=4 K3=4 FYP=50 FYN=-50 NAME=YWALL1 K1=14 K2=2 K3=2 FYP=30 FYN=-30

STEWART PARAMETERS: NAME=XWALL2 F0=4.86 FI=1.01 DU=2.43 S0=14.0 R1=0.0802 R2=-0.0810 R3=1.26 R4=0.0704 ALPHA=0.740 BETA=1.09

NAME=YWALL2 F0=7.00 FI=1.67 DU=1.53 S0=32.4 R1=0.0765 R2=-0.0371 R3=1.30 R4=0.0694 ALPHA=0.571 BETA=1.10

Figure 13


Earthquake Analysis

________________________________________________________________________

- 294 -

-4.0

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Figure 14: WALL 1

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Figure 15: WALL 1


Earthquake Analysis

________________________________________________________________________

- 295 -

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Figure 16: WALL 2

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Figure 17: WALL 2


Earthquake Analysis

________________________________________________________________________

- 296 -

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Figure 18: WALL 3

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Figure 19: WALL 3


Earthquake Analysis

________________________________________________________________________

- 297 -

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Figure 20: WALL 4


Earthquake Analysis

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- 298 -

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kip

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KEYSOLVER

Figure 21: WALL 4

Example27: HOUSE 4: L shaped floor plan with 8 Stewart hysteresis walls on its perimeter MODEL NAME: NHO5, Ux=0.1g & Uy=0.1g STEWART PARAMETERS:

NAME=XWALL1 F0=7.00 FI=1.67 DU=1.53 S0=32.4 R1=0.0765 R2=-0.0371 R3=1.30 R4=0.0694 ALPHA=0.571 BETA=1.10 NAME=YWALL1 F0=3.10 FI=0.734 DU=1.53 S0=15.0 R1=0.0749 R2=-0.0590 R3=1.29

R4=0.0702 ALPHA=0.568 BETA=1.10 NAME=XWALL2 F0=4.86 FI=1.01 DU=2.43 S0=14.0 R1=0.0802 R2=-0.0810 R3=1.26 R4=0.0704 ALPHA=0.740 BETA=1.09 NAME=YWALL2 F0=2.44 FI=0.473 DU=3.86 S0=5.21 R1=0.0695 R2=-0.0870 R3=1.29

R4=0.0593 ALPHA=0.773 BETA=1.09


Earthquake Analysis

________________________________________________________________________

- 299 -

Figure 22


Earthquake Analysis

________________________________________________________________________

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Figure 23: WALL 1

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Figure 24: WALL 1


Earthquake Analysis

________________________________________________________________________

- 301 -

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Figure 25: WALL 2

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Figure 26: WALL 2


Earthquake Analysis

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- 302 -

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Figure 27: WALL 3

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Figure 28: WALL 3


Earthquake Analysis

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- 303 -

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Figure 29: WALL 4

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Figure 30: WALL 4


Earthquake Analysis

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Figure 31: WALL 5

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Figure 32: WALL 5


Earthquake Analysis

________________________________________________________________________

- 305 -

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Figure 33: WALL 6

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Figure 34: WALL 6


Earthquake Analysis

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- 306 -

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Figure 35: WALL 7

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Figure 36: WALL 7


Earthquake Analysis

________________________________________________________________________

- 307 -

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Figure 37: WALL 8

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Figure 38: WALL 8


Earthquake Analysis

________________________________________________________________________

- 308 -

Example28: HOUSE 5: U shaped floor plan with 6 bilinear and 6 Stewart hysteresis walls on its perimeter MODEL NAME: NHO6, Ux=1g & Uy=0.1g

BILINEAR PARAMETERS: NAME=XWALL1 K1=10 K2=5 K3=5 FYP=20 FYN=-20 NAME=XWALL2 K1=13 K2=6 K3=6 FYP=20 FYN=-20

NAME=XWALL3 K1=16 K2=8 K3=8 FYP=32 FYN=-32 NAME=XWALL6 K1=14 K2=1.4 K3=1.4 FYP=42 FYN=-42 NAME=XWALL8 K1=9 K2=1 K3=1 FYP=15 FYN=-15

NAME=XWALL10 K1=30 K2=6 K3=6 FYP=30 FYN=-30 STEWART PARAMETERS: NAME=YWALL4911 F0=2.44 FI=0.473 DU=3.86 S0=5.21 R1=0.0695 R2=-0.0870 R3=1.29 R4=0.0593 ALPHA=0.773 BETA=1.09

NAME=YWALL512 F0=7.00 FI=1.67 DU=1.53 S0=32.4 R1=0.0765 R2=-0.0371 R3=1.30 R4=0.0694 ALPHA=0.571 BETA=1.10 NAME=YWALL7 F0=3.10 FI=0.734 DU=1.53 S0=15.0 R1=0.0749 R2=-0.0590 R3=1.29

R4=0.0702 ALPHA=0.568 BETA=1.10

Figure 39


Earthquake Analysis

________________________________________________________________________

- 309 -

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Figure 40: WALL 1

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Figure 41: WALL 1


Earthquake Analysis

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- 310 -

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Figure 43: WALL 2

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Figure 44: WALL 2


Earthquake Analysis

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- 311 -

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Figure 45: WALL 3

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Figure 46: WALL 3


Earthquake Analysis

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- 312 -

-6

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Figure 47: WALL 4

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Figure 48: WALL 4


Earthquake Analysis

________________________________________________________________________

- 313 -

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Figure 49: WALL 5

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Figure 50: WALL 5


Earthquake Analysis

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- 314 -

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Figure 52: WALL 6

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Figure 53: WALL 6


Earthquake Analysis

________________________________________________________________________

- 315 -

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KEYSOLVER

Figure 54: WALL 7

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Figure 55: WALL 7


Earthquake Analysis

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- 316 -

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Figure 56: WALL 8

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Figure 57: WALL 8


Earthquake Analysis

________________________________________________________________________

- 317 -

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KEYSOLVER

Figure 58: WALL 9

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kip

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Figure 59: WALL 9


Earthquake Analysis

________________________________________________________________________

- 318 -

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KEYSOLVER

Figure 60: WALL 10

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Figure 61: WALL 10


Earthquake Analysis

________________________________________________________________________

- 319 -

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Figure 62: WALL 11

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5

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DEFOR (in)

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KEYSOLVER

Figure 63: WALL 11


Earthquake Analysis

________________________________________________________________________

- 320 -

-4

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7

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)

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KEYSOLVER

Figure 64: WALL 12

-10

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0

5

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15

-1 0 1 2 3 4

DEFOR (in)

FO

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E (

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KEYSOLVER

Figure 65: WALL 12

Wood House Finite Element Model Generator

Version 1.0 Beta, April 2008

321

APPENDIX D WHFEMG PROGRAM USERS MANUAL



322

Wood House Finite Element Model Generator Program (WHFEMG)

An Automated Finite Element Model Generator for Shear Walls, Diaphragms, and

Light Frame Wood Houses

USERS MANUAL

by

Rakesh Pathak ([email protected])



323

COPYRIGHT

Wood House Finite Element Model Generator program and its related

documents are copyrighted product owned by the author. The use of the

program or its reproduction requires the written authorization of the owner.

Rakesh Pathak

Graduate Student

Virginia Tech



324

1. INTRODUCTION

This document describes the beta version 1.0 of Wood House Finite Element

Model Generator (WHFEMG) program. WHFEMG is developed to create

finite element model input files for shear walls, diaphragms and full light

frame wood houses. The input files using this program are generated for

analysis using WoodFrameSolver and only in a few cases may be viewed

and analyzed in SAP2000. The program is written in Visual Basic 6.0 and

currently implements a fixed modeling methodology which is discussed in

Chapter 3. The following sections describe how one may use the program to

develop finite element models of individual subassemblages or the complete

house.

1.1 MODEL DEVELOPMENT

The model development involves various steps. The user first needs to break

down the problem into diaphragm objects. A diaphragm object is a shear

wall or a roof having physical dimensions. These diaphragms have to be

made of framing, sheathing and nails. Following this, the user needs to

identify different materials, frame sections, sheathing sections, and nail

properties to be used in the model. The following list gives the program’s

current modeling limitations:

a. The model can only have one-story rectangular box type

geometry with the diaphragms only in the XY, XZ and YZ

plane. Currently, an individual diaphragm can only be square or

rectangular shaped.

b. Sheathing bearings cannot be modeled.



325

c. The shear wall to shear wall connection at the edges cannot be

modeled.

d. The shear wall tie-downs are modeled as restraints.

e. The anchor bolts are not modeled.

f. The spring pair stiffness for the nail is the same in both the

directions.

g. Frame sections may only be rectangular.

h. Sheathing panels are always rectangular.

i. The mesh compatibility has to be maintained between the

diaphragm objects. The mesh spacing in a diaphragm object is a

function of perimeter nail spacing, internal nail spacing and the

intermediate stud spacing. Thus, the dimension of a model, stud

spacing, and nail spacing has to be chosen in such a manner that

mesh compatibility is obtained. This may require slight

deviation from the practical input but is necessary for the

creation of a valid model.

j. The data being added to the form cannot be saved for future

reference.

k. The program currently does not add any load cases to the

model.

The model development involves up to 23 steps including the starting of the

program. These steps are presented with the creation of an example house

model named LFWS1 as follows:



326

STEP 1: Click on the program icon and you will see the program interface

as shown in Figure 1.1

Figure 1.1

STEP 2: Click on the File menu and an interface appears as shown in the

Figure 1.2

Figure 1.2

STEP 3: Click on the House button and an interface appears as shown in

Figure 1.3

Figure 1.3



327

STEP 4: Fill in the basic information box. The model name corresponds to

the model file name. The force and length units tell the program what units

the input information is going to be filled in. Once the basic information is

filled click on the Set button and the box besides the Set button turns green

from red. This shown in Figure 1.4

Figure 1.4

STEP 5: Fill in the structural information box. Click on the Set Material

Properties button and a form window appears as shown in Figure 1.5



328

Figure 1.5

STEP 6: Fill in the Material Properties form. The form gives the option of

defining mass and weight per unit volume of each material. The user also

needs to select the material type which could be either isotropic or

orthotropic. The user also needs to set the elasticity parameters

corresponding to each material. One may add multiple materials and define

up to 100 material properties to be used in the model. Once few material



329

properties are filled in the Material Properties form may look as shown in

Figure 1.6

Figure 1.6

STEP 7: Click the Close button if all the required material properties are

filled in. Now we go back to the structural information box where we see the

previous red box next to Set Material Properties button has now turned to

green as shown in Figure 1.7



330

Figure 1.7

STEP 8: Click on the Set Frame Sections button and a form window appears


Figure 1.8



331

STEP 9: Fill in the Frame Section form. Currently, the form gives the

option of having only rectangular section type. The user needs to define the

cross-section dimensions in the dimensions boxes. The area, shear areas,

moments of inertia and torsional constant are calculated internally for a

section. The material property name defined in the steps 5 to 7 may be seen

in the Material Name dropdown menu and one of the materials have to be

selected corresponding to a frame section. The user may define up to 100

frame sections to be used in the model. Once a few sections are defined and

added (++ button) the form may look like as shown in Figure 1.9

Figure 1.9

STEP 10: Click the Close button if all the required frame section properties

are filled in. Now we go back to the structural information box where we see

the previous red box next to Set Frame Sections button has now turned to




332

Figure 1.10

STEP 11: Click on the Set Sheathing Sections button and a form window

appears as shown in Figure 1.11

Figure 1.11



333

STEP 12: Fill in the Sheathing Section form. The form gives the option of

having only Shell, Plate and Membrane section type. The material property

name defined in the steps 5 to 7 may be seen in the Material Name

dropdown menu and one of the materials have to be selected corresponding

to a sheathing section.. The user is also required to fill in the section

thicknesses. The user may define up to 100 sheathing sections to be used in

the model. Once a few sheathing sections are defined and added (++ button)

the form may look like as shown in Figure 1.12

Figure 1.12

STEP 13: Click the Close button if all the required sheathing section

properties are filled in. Now we go back to the structural information box

where we see the previous red box next to Set Sheathing Sections button has

now turned to green as shown in Figure 1.13



334

Figure 1.13

STEP 14: Click on the Set Nail Properties button and a form window

appears as shown in Figure 1.14



335

Figure 1.14

STEP 15: Fill in the Nail Property form. The form gives the option of

defining nail mass, nail weight, and nail spring pair as oriented or non-

oriented. The form also gives an option of modeling nail springs using

linear, bilinear, or modified Stewart properties. The user may define up to

100 nail properties to be used in the model. Once a few nail properties are

defined and added (++ button) the form may look like as shown in Figure

1.15



336

Figure 1.15

STEP 16: Click the Close button if all the required nail properties are filled

in. Now we go back to the structural information box where we see the

previous red box next to Set Nail Properties button has now turned to green




337

Figure 1.16

STEP 17: Click on the Set Diaphragms button and a form window appears




338

Figure 1.17

STEP 18: Fill in the Diaphragms form. The form shown in Figure 1.17

corresponds to the input for a single diaphragm in the model. Once the

properties are defined for one diaphragm it is added to the container shown

on bottom right and another diaphragm (if any) needs to be defined and

added using the same form. This is continued till all the diaphragms (shear

walls and roofs) are added to the container. A diaphragm input consists of

diaphragm plane, diaphragm co-ordinates, framing data, sheathing data, nail



339

data and the diaphragms unique name. The only permitted diaphragm planes

are XY, XZ or YZ. As already mentioned that an individual diaphragm may

only be square or rectangular, hence four corner global co-ordinates are

required to define the diaphragm boundaries. These co-ordinates have to be

entered in a continuous fashion which may be clockwise or anticlockwise.

The framing data consists of (a) number of vertical studs which is parallel to

vector 23 (co-ordinates 2 and 3), (b) number of horizontal blockings which

is parallel to vector 12 (co-ordinates 1 and 2), (c) plate section, (d) sill

section, (e) side studs section, (f) interior studs section and (g) blockings

section. The sheathing data consist of all the sheathings in a diaphragm and

an individual sheathing in a diaphragm has (a) a unique sheathing number,

(b) sheathing section name, (c) width, (d) height and (e) co-ordinates of its

center with respect to global origin. The nail data consist of (a) nail spacing

on the perimeter of the diaphragm, (b) nail spacing in the field of the

diaphragm, (c) perimeter nail property, and (d) field nail property. The user

may define up to 100 diaphragms and each diaphragm may have up to 100

panels to be used in the model. Once a few diaphragms are defined and

added (++ button) to the container the form may look like as shown in

Figure 1.18



340

Figure 1.18

STEP 19: Click the Close button if all the required diaphragms are added to

the container in. Now we go back to the structural information box where we

see the previous red box next to Set Diaphragms button has now turned to




341

Figure 1.19

STEP 20: Click on the Set Connectors button if roof and shear wall

diaphragms needs to be connected using the nails defined in Nail Properties

else one may proceed to Step 23. If the user clicks on the Set Connectors

button then a form window appears as shown in Figure 1.20

Figure 1.20



342

STEP 21: Fill in the Diaphragm Connectors form. The forms show three

dropdown menus which are Diaphragm 1, Diaphragm 2 and Connector. The

diaphragm selected in Diaphragm 1 is connected to diaphragm selected in

Diaphragm 2 using the connector element selected in the Connector list.

Once a few diaphragms are connected and added (++ button) to the

container the form may look like as shown in Figure 1.21

Figure 1.21

STEP 22: Click the Close button if all the required diaphragms are defined

to be connected and added to the container. Now we go back to the structural

information box where we see the previous red box next to Set Connectors

button has now turned to green as shown in Figure 1.22



343

Figure 1.22

STEP 23: Assuming that all the information needed for the house model is

complete we close the Model Input form and go back to main form shown in

Figure 1.1. We now click on Create menu and click on House Model. This

brings a progress bar as shown in Figure 1.23 on the main form which

moves as the diaphragm objects are being meshed and the model is being

created inside. Once the progress bar is complete it means the input file is

created.



344

Figure 1.23

The input file is generated in the same folder in which the program is

executed. The example input file created in Steps 1 to 23 is presented in the

example folder of the program. This file may be viewed in SAP2000

provided it does not have (1) bilinear spring properties, (2) modified Stewart

hysteresis properties, and (3) oriented nails defined. This part of the input

file is not readable by SAP2000 and is only applicable to WoodFrameSolver

models. One may remove these properties in order to just view the

generated model in SAP2000. We do this for our example input file and

rename it as LFWS1_SAPVIEW.S2K. This file when viewed in SAP2000

version 7.4 looks like as shown in Figure 1.24. Also, to use the model for

any analysis one may still need to perform various additions to the generated

file. This typically would involve adding static and dynamic load cases. On

how to add static and dynamic load cases to the input files one should refer

to WoodFrameSolver input file format document.



345

Figure 1.24

346

APPENDIX E ANALYSIS RESULTS

347

Table E-1: Type 1 house, X direction peak base shear (dpbs) using Imperial Valley

earthquake (kips)

TYP1M1 TYP1M2 TYP1M3 TYP1M4 TYP1M1 TYP1M2 TYP1M3 TYP1M4

WALL1 1.84 2.11 2.66 3.26 2.13 2.29 2.73 3.19

WALL2 1.84 2.11 x x 2.13 2.29 x x

WALL3 1.84 2.11 1.85 2.57 2.13 2.29 2.14 2.75

WALL4 1.84 2.10 1.86 2.57 2.13 2.29 2.14 2.75

WALL5 2.15 x 2.29 x 2.04 x 2.20 x

WALL6 0.05 0.07 0.08 0.12 0.05 0.06 0.07 0.10

WALL7 0.05 0.07 0.05 0.09 0.05 0.06 0.05 0.08

WALL8 0.05 0.07 0.08 0.12 0.05 0.06 0.07 0.10

WALL9 0.05 0.07 0.05 0.09 0.05 0.06 0.05 0.08

FLEXIBLE RIGID

* x indicates wall is not present in the model

Table E-2: Type 1 house, X dpbs using Northridge earthquake (kips)


WALL1 2.02 2.45 3.13 3.73 2.27 2.66 2.96 3.57

WALL2 2.02 2.45 x x 2.27 2.66 x x

WALL3 2.02 2.43 1.97 2.81 2.27 2.66 2.33 3.03

WALL4 2.02 2.44 1.97 2.81 2.27 2.66 2.34 3.03

WALL5 2.16 x 2.58 x 2.17 x 2.50 x

WALL6 0.08 0.10 0.11 0.15 0.08 0.09 0.09 0.12

WALL7 0.08 0.10 0.08 0.12 0.08 0.09 0.08 0.10

WALL8 0.08 0.09 0.11 0.16 0.08 0.09 0.09 0.12

WALL9 0.08 0.10 0.09 0.12 0.08 0.09 0.08 0.10

FLEXIBLE RIGID


Table E-3: Type 1 house, Y dpbs using Imperial Valley earthquake (kips)


WALL1 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.02

WALL2 0.02 0.02 x x 0.01 0.01 x x

WALL3 0.02 0.02 0.01 0.02 0.01 0.01 0.01 0.02

WALL4 0.02 0.02 0.03 0.03 0.01 0.01 0.03 0.03

WALL5 0.02 x 0.02 x 0.01 x 0.02 x

WALL6 0.83 0.87 0.65 0.87 0.87 0.89 0.72 1.03

WALL7 0.83 0.87 0.65 0.87 0.87 0.89 0.72 1.03

WALL8 0.83 0.87 1.19 1.33 0.87 0.89 1.52 1.66

WALL9 0.83 0.87 1.19 1.33 0.87 0.89 1.52 1.66

FLEXIBLE RIGID


348

Table E-4: Type 1 house, Y dpbs using Northridge earthquake (kips)


WALL1 0.04 0.04 0.03 0.05 0.03 0.03 0.03 0.03

WALL2 0.04 0.03 x x 0.03 0.03 x x

WALL3 0.04 0.04 0.03 0.03 0.03 0.03 0.02 0.03

WALL4 0.04 0.04 0.04 0.04 0.03 0.03 0.04 0.04

WALL5 0.04 x 0.05 x 0.03 x 0.03 x

WALL6 0.86 0.83 0.66 1.01 0.93 0.92 0.77 1.22

WALL7 0.86 0.83 0.66 1.01 0.93 0.92 0.77 1.22

WALL8 0.87 0.85 1.23 1.33 0.93 0.92 1.35 1.41

WALL9 0.86 0.85 1.23 1.33 0.93 0.92 1.35 1.41

FLEXIBLE RIGID


Table E-5: Type 1 house, % difference in flexible and rigid X dpbs using Imperial Valley

earthquake


WALL1 16.13 8.68 2.29 -2.15

WALL2 16.17 8.53 x x

WALL3 15.75 8.79 15.33 6.77

WALL4 15.73 8.89 15.20 6.77

WALL5 -4.94 x -4.01 x

Table E-6: Type 1 house, % difference in flexible and rigid X dpbs using Northridge

earthquake


WALL1 12.51 8.56 -5.52 -4.08

WALL2 12.45 8.56 x x

WALL3 12.28 9.77 18.32 7.87

WALL4 12.34 9.10 18.66 7.79

WALL5 0.46 x -2.83 x

Table E-7: Type 1 house, % difference in flexible and rigid Y dpbs using Imperial Valley

earthquake


WALL6 5.03 1.77 10.11 18.64

WALL7 5.03 1.76 10.11 18.84

WALL8 5.22 1.97 28.04 24.19

WALL9 5.22 1.98 28.07 24.17

349

Table E-8: Type 1 house, % difference in flexible and rigid Y dpbs using Northridge

earthquake


WALL6 8.48 11.48 17.49 20.69

WALL7 8.40 11.41 17.52 20.51

WALL8 7.82 8.27 9.35 5.46

WALL9 8.14 8.38 9.40 5.53

Table E-9: Type 2 house, X dpbs using Imperial Valley earthquake (kips)


WALL1 2.19 2.31 2.62 3.40 2.22 2.47 2.72 3.35

WALL2 2.19 2.31 x x 2.22 2.47 x x

WALL3 2.20 2.31 2.12 2.86 2.22 2.47 2.17 2.89

WALL4 2.19 2.31 2.12 2.85 2.22 2.47 2.17 2.89

WALL5 2.21 x 2.29 x 2.07 x 2.27 x

WALL6 0.07 0.07 0.07 0.12 0.05 0.07 0.06 0.11

WALL7 0.05 0.07 0.07 0.12 0.05 0.07 0.06 0.11

FLEXIBLE RIGID




WALL1 2.26 2.62 3.00 3.66 2.31 2.66 2.93 3.56

WALL2 2.26 2.63 x x 2.31 2.66 x x

WALL3 2.26 2.62 2.30 3.00 2.31 2.66 2.42 3.09

WALL4 2.26 2.61 2.30 2.98 2.31 2.66 2.42 3.09

WALL5 2.25 x 2.57 x 2.25 x 2.45 x

WALL6 0.10 0.12 0.11 0.15 0.08 0.09 0.09 0.12

WALL7 0.10 0.11 0.10 0.14 0.08 0.09 0.09 0.12

FLEXIBLE RIGID




WALL1 0.04 0.04 0.04 0.04 0.03 0.03 0.03 0.02

WALL2 0.04 0.04 x x 0.03 0.03 x x

WALL3 0.04 0.04 0.04 0.04 0.03 0.03 0.03 0.02

WALL4 0.04 0.04 0.05 0.07 0.03 0.03 0.05 0.07

WALL5 0.04 x 0.05 x 0.02 x 0.03 x

WALL6 1.50 1.58 1.41 1.37 1.38 1.43 1.25 1.07

WALL7 1.49 1.58 2.08 2.48 1.38 1.43 2.17 2.65

FLEXIBLE RIGID


350



WALL1 0.06 0.07 0.05 0.05 0.04 0.04 0.04 0.04

WALL2 0.06 0.06 x x 0.04 0.04 x x

WALL3 0.06 0.06 0.06 0.05 0.04 0.03 0.03 0.04

WALL4 0.06 0.08 0.08 0.09 0.04 0.04 0.06 0.06

WALL5 0.06 x 0.06 x 0.03 x 0.03 x

WALL6 1.93 1.97 1.61 1.38 1.47 1.51 1.17 1.32

WALL7 1.92 1.96 2.62 2.78 1.47 1.51 2.43 2.55

FLEXIBLE RIGID


Table E-12: Type 2 house, % difference in flexible and rigid X dpbs using Imperial

Valley earthquake


WALL1 1.28 7.07 3.78 -1.47

WALL2 1.23 7.02 x x

WALL3 1.05 7.07 2.46 1.30

WALL4 1.09 7.07 2.50 1.33

WALL5 -6.41 x -0.79 x


earthquake


WALL1 2.53 1.41 -2.20 -2.68

WALL2 2.34 1.18 x x

WALL3 2.34 1.49 5.44 3.00

WALL4 2.57 1.65 5.53 3.69

WALL5 0.00 x -4.67 x

Table E-14: Type 2 house, % difference in flexible and rigid Y dpbs using Imperial

Valley earthquake


WALL6 -7.67 -9.79 -11.16 -22.30

WALL7 -7.39 -9.51 4.42 6.61


earthquake


WALL6 -23.81 -23.40 -27.11 -4.43

WALL7 -23.76 -23.00 -6.96 -8.06

351



WALL1 2.24 2.31 2.56 3.21 2.22 2.44 2.64 3.18

WALL2 2.24 2.31 x x 2.22 2.44 x x

WALL3 2.24 2.31 2.26 2.93 2.22 2.44 2.28 2.91

WALL4 2.24 2.31 2.25 2.93 2.22 2.44 2.28 2.91

WALL5 2.21 x 2.25 x 2.03 x 2.29 x

WALL6 0.05 0.59 0.06 0.10 0.05 0.06 0.06 0.09

WALL7 0.54 0.06 0.06 0.10 0.05 0.06 0.06 0.09

FLEXIBLE RIGID




WALL1 2.08 2.60 2.89 3.53 2.34 2.66 2.83 3.52

WALL2 2.09 2.60 x x 2.34 2.66 x x

WALL3 2.09 2.60 2.48 3.05 2.34 2.66 2.49 3.19

WALL4 2.09 2.60 2.48 3.05 2.34 2.66 2.49 3.19

WALL5 2.05 x 2.46 x 2.17 x 2.45 x

WALL6 0.08 0.10 0.09 0.12 0.04 0.05 0.05 0.08

WALL7 0.08 0.10 0.09 0.12 0.04 0.05 0.05 0.08

FLEXIBLE RIGID




WALL1 0.04 0.05 0.05 0.05 0.03 0.03 0.02 0.02

WALL2 0.04 0.05 x x 0.03 0.03 x x

WALL3 0.04 0.05 0.05 0.05 0.03 0.03 0.02 0.02

WALL4 0.04 0.05 0.05 0.06 0.03 0.03 0.05 0.05

WALL5 0.05 x 0.06 x 0.26 x 0.03 x

WALL6 1.52 1.67 1.64 1.53 1.32 1.37 1.09 1.05

WALL7 1.53 1.67 1.85 2.33 1.32 1.37 1.96 2.23

FLEXIBLE RIGID


352



WALL1 0.06 0.07 0.06 0.06 0.02 0.02 0.02 0.02

WALL2 0.07 0.07 x x 0.02 0.02 x x

WALL3 0.07 0.07 0.06 0.06 0.02 0.02 0.02 0.02

WALL4 0.06 0.07 0.09 0.09 0.02 0.02 0.04 0.05

WALL5 0.07 x 0.08 x 0.02 x 0.02 x

WALL6 1.87 1.94 1.61 1.60 1.40 1.46 1.10 1.31

WALL7 1.88 1.94 2.38 2.53 1.40 1.46 2.16 2.23

FLEXIBLE RIGID



Valley earthquake


WALL1 -1.25 5.72 2.81 -1.00

WALL2 -1.25 5.63 x x

WALL3 -1.16 5.53 1.06 -0.82

WALL4 -1.12 5.63 1.51 -0.75

WALL5 -8.07 x 1.87 x


earthquake


WALL1 12.48 2.27 -2.11 -0.31

WALL2 12.32 2.23 x x

WALL3 12.32 2.04 0.20 4.36

WALL4 12.37 2.19 0.24 4.49

WALL5 5.76 x -0.45 x


Valley earthquake


WALL6 -13.61 -18.36 -33.38 -31.54

WALL7 -13.75 -18.27 5.90 -4.42

Table E-23: Type 3, % difference in flexible and rigid Y dpbs using Northridge

earthquake


WALL6 -25.48 -24.53 -31.55 -18.13

WALL7 -25.56 -24.81 -9.48 -11.64

353



WALL1 2.29 2.32 2.49 3.18 2.22 2.44 2.56 3.18

WALL2 2.29 2.32 x x 2.22 2.44 x x

WALL3 2.29 2.32 2.34 3.09 2.22 2.44 2.38 3.03

WALL4 2.29 2.32 2.35 3.09 2.22 2.44 2.38 3.03

WALL5 2.20 x 2.26 x 2.03 x 2.32 x

WALL6 0.05 0.05 0.06 0.10 0.05 0.06 0.06 0.09

WALL7 0.05 0.05 0.06 0.10 0.05 0.06 0.06 0.09

FLEXIBLE RIGID




WALL1 2.12 2.54 2.76 3.37 2.29 2.57 2.79 3.39

WALL2 2.12 2.54 x x 2.29 2.57 x x

WALL3 2.12 2.52 2.55 3.19 2.29 2.57 2.58 3.18

WALL4 2.13 2.53 2.52 3.17 2.29 2.57 2.58 3.18

WALL5 1.98 x 2.43 x 2.13 x 2.44 x

WALL6 0.08 0.08 0.09 0.11 0.04 0.05 0.05 0.08

WALL7 0.08 0.10 0.08 0.11 0.04 0.05 0.05 0.08

FLEXIBLE RIGID




WALL1 0.05 0.06 0.06 0.06 0.03 0.03 0.02 0.02

WALL2 0.05 0.06 x x 0.03 0.03 x x

WALL3 0.05 0.06 0.05 0.05 0.03 0.03 0.02 0.02

WALL4 0.05 0.06 0.06 0.07 0.03 0.03 0.04 0.04

WALL5 0.08 x 0.08 x 0.02 x 0.03 x

WALL6 1.53 1.69 1.45 1.43 1.25 1.29 1.07 1.09

WALL7 1.53 1.69 1.79 2.06 1.25 1.29 1.85 1.91

FLEXIBLE RIGID


354



WALL1 0.06 0.07 0.07 0.07 0.02 0.02 0.02 0.02

WALL2 0.06 0.07 x x 0.02 0.02 x x

WALL3 0.06 0.07 0.06 0.07 0.02 0.02 0.02 0.02

WALL4 0.06 0.07 0.08 0.09 0.02 0.02 0.03 0.04

WALL5 0.08 x 0.08 x 0.02 x 0.02 x

WALL6 1.80 1.79 1.47 1.45 1.34 1.46 1.09 1.36

WALL7 1.81 1.78 2.00 2.03 1.34 1.46 1.91 1.93

FLEXIBLE RIGID



Valley earthquake


WALL1 -2.97 5.18 2.86 0.00

WALL2 -2.93 5.14 x x

WALL3 -3.01 4.87 1.80 -2.13

WALL4 -3.01 4.96 1.11 -2.10

WALL5 -7.50 x 2.65 x


earthquake


WALL1 7.72 1.26 0.94 0.59

WALL2 7.82 1.50 x x

WALL3 7.77 1.98 1.14 -0.25

WALL4 7.67 1.90 2.30 0.35

WALL5 7.41 x 0.58 x


Valley earthquake


WALL6 -18.54 -23.76 -26.02 -24.09

WALL7 -18.33 -23.95 3.16 -7.20


earthquake


WALL6 -25.21 -18.59 -26.13 -6.26

WALL7 -25.65 -18.18 -4.31 -4.82

the effects of diaphragm flexibility on the ......the effects of diaphragm flexibility on the...

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