the effects of diaphragm flexibility on the seismic performance of light frame wood structures

SEISMIC PERFORMANCE OF LIGHT FRAME WOOD
STRUCTURES
Polytechnic Institute and State University in partial fulfillment of the requirements for the
degree of
Dr. Daniel P. Hindman
Dr. Elisa D. Sotelino
Dr. Raymond H. Plaut
Dr. W. Samuel Easterling
Analysis, Light Frame Wood Structure, Finite Element
@Copyright 2008, Rakesh Pathak
SEISMIC PERFORMANCE OF LIGHT FRAME WOOD
STRUCTURES
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
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
MOTIVATION 1
WOOD SHEAR WALLS, DIAPHRAGMS AND HOUSES: OVERVIEW
7
THE SEISMIC PERFORMANCE OF LIGHT FRAME WOOD
STRUCTURES, PART I: MODEL FORMULATION
27
FINITE ELEMENT MODELING METHODOLOGY 32
FINITE ELEMENTS 33
FE DIAPHRAGM MODEL 35
FE HOUSE MODEL 36
RECTANGULAR TYPE 1 37
NONLINEAR FINITE ELEMENT ANALYSIS PROGRAM
60
PROGRAM FEATURES 67
ELEMENT LIBRARY 68
EXAMPLES 74
ANALYTICAL MODEL
VERIFICATION WITH SAP2000 3D HOUSE MODEL 76
SUMMARY 77
REFERENCES 79
THE SEISMIC PERFORMANCE OF LIGHT FRAME WOOD
STRUCTURES II: PARAMETRIC STUDY
MASS MATRIX 110
DAMPING MATRIX 110
STIFFNESS MATRIX 111
VERIFICATION WITH DOLAN (1989) EXPERIMENTS
112
RESULTS COMPARISON 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
INVESTIGATION (1) 124
FUTURE WORK 166
INPUT FILE FORMAT
APPENDIX D: WHFEMG PROGRAM USERS MANUAL 321
APPENDIX E: ANALYSIS RESULTS 346
ix
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
non-checkerboard staggered manner (no blockings
present), thick line in the figure represents panel
boundary
46
3-6 Load distribution 48
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
in SAP2000)
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
WoodFrameSolver (dense system of equations, linear
static analysis)
WoodFrameSolver (sparse system of equations, linear
static analysis)
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
xi
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
modes 1 and 3
4-32 Finite element model of garage 98
xii
4-40 X direction base shear response history 102
5-1 Lateral force distribution in a shear wall under rigid
and flexible diaphragm assumption
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
Imperial Valley earthquake
Northridge earthquake
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
2-2 Finite element models 26
3-1 Number of finite elements and degree of freedoms in
parent house models
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
verification analysis
5-4 Direction aspect ratios and vibration periods of all the
models
158
5-6 Analysis cases 160
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
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
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 ABAQUS 1 , 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.
23
REFERENCES:
1. Bathe, K. J., Wilson, E. L., and Iding, R. H. (1974). "NONSAP - A Structural
Analysis Program for Static and Dynamic Response of Nonlinear Systems." National
Information Service Earthquake Engineering Computer Program Applications.
2. Cheung, C. K., and Itani, R. Y. (1983). "Analysis of Sheathed Wood-Stud Walls." 8th
Conference on Electronic Computation, New York, 683-696.
3. Collins, M., Kasal, B., Paevere, P., and Foliente, G. C. (2005a). "Three Dimensional
Model of Light-Frame Wood Buildings I: Model Description." Journal of Structural
Engineering, 131(4), 676-683.
4. Collins, M., Kasal, B., Paevere, P., and Foliente, G. C. (2005b). "Three Dimensional
Model of Light Frame Wood Buildings II: Experimental Investigation and Validation
of Numerical Model." Journal of Structural Engineering, 131(4), 684-693.
5. Dinehart, D. W., and Shenton III, H. W. (2000). "Model for Dynamic Analysis of
Wood Frame Shear Walls." Journal of Engineering Mechanics, 126(9), 899-908.
6. Dolan, J. D. (1989). "The Dynamic Response of Timber Shear Walls," PhD
Dissertation, University of British Columbia, Vancouver, B.C., Canada.
7. Dolan, J. D., and Filiatrault, A. (1990). "A Mathematical Model to Predict the Steady-
State Response of Timber Shear Walls." Proceedings of International Timber
Engineering Conference, Tokyo, Japan.
8. Dolan, J. D., and Foschi, R. O. (1991). "Structural Analysis Model for Static Loads
on Timber Shear Walls." Journal of Structural Engineering, 117(3), 851-861.
9. Easley, J. T., Foomani, M., and Dodds, R. H. (1982). "Formulas for Wood Shear
Walls." Journal of Structural Division, 108(ST11), 2460-2478.
10. Falk, R. H., and Itani, R. Y. (1989). "Finite Element Modeling of Wood
Diaphragms." Journal of Structural Engineering, 115(3), 543-559.
11. Folz, B., and Filiatrault, A. (2001). "Cyclic Analysis of Wood Shear Walls." Journal
of Structural Engineering, 127(4), 433-441.
12. Folz, B., and Filiatrault, A. (2004a). "Seismic Analysis of Woodframe Structures I:
Model Formulation." Journal of Structural Engineering, 130(9), 1353-1360.
24
13. Folz, B., and Filiatrault, A. (2004b). "Seismic Analysis of Woodframe Structures II:
Model Implementation and Verification." Journal of Structural Engineering, 130(9),
1361-1370.
14. Foschi, R. O. (1977). "Analysis of Wood Diaphragms and Trusses. Part I:
Diaphragms." Canadian Journal of Civil Engineering, 4(3), 345-352.
15. Gupta, A. K., and Kuo, G. P. (1987). "Wood-Framed Shear Walls with Uplifting."
Journal of Structural Engineering, 113(2), 241-259.
16. Gupta, A. K., and Kuo, P. H. (1985). "Behavior of Wood-Framed Shear Walls."
Journal of Structural Engineering, 111(8), 1722-1733.
17. Gutkowski, R. M., and Castillo, A. L. (1988). "Single and Double-Sheathed Wood
Shear Wall Study." Journal of Structural Engineering, 114(6), 1268-1284.
18. He, M., Lam, F., and Foschi, R. O. (2001). "Modeling Three-Dimensional Timber
Light-Frame Buildings." Journal of Structural Engineering, 127(8), 901-913.
19. Itani, R. Y., and Cheung, C. K. (1984). "Nonlinear Analysis of Sheathed Wood
Diaphragms." Journal of Structural Engineering, 110(9), 2137-2147.
20. Itani, R. Y., Tuomi, R. L., and McCutcheon, W. J. (1982). "Methodology to Evaluate
Racking Resistance of Nailed Walls." Forest Products Journal, 32(1), 30-36.
21. Judd, J. P., and Fonseca, F. S. (2005). "Analytical Model for Sheathing-to-Framing
Connections in Wood Shear Walls and Diaphragms." Journal of Structural
Engineering, 131(2), 345-352.
22. Kamiya, F., and Itani, R. Y. (1998). "Design of Wood Diaphragms with Openings."
Journal of Structural Engineering, 124(7), 839-848
23. 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.
24. Kasal, B., and Leichti, R. J. (1992a). "Nonlinear Finite-Element Model for Light-
Frame Stud Walls." Journal of Structural Engineering, 118(11), 3122-3135.
25. Kasal, B., and Leichti, R. J. (1992b). "Incorporating Load Sharing in Shear Wall
Design of Light-Frame Structures." Journal of Structural Engineering, 118(12),
25
3350-3361.
26. 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.
27. McCutcheon, W. J. (1985). "Racking Deformations in Wood Shear Walls." Journal
of Structural Engineering, 111(2), 257-269.
28. Pang, W. C., Rosowsky, D. V., Pei, S., and Lindt, J. W. v. d. (2007). “Evolutionary
Parameter Hysteretic Model for Wood Shear Walls.” Journal of Structural
Engineering, 133(8), 1118-1129.
29. Patton-Mallory, M., and McCutcheon, W. J. (1987). "Predicting Racking
Performance of Walls Sheathed on both Sides." Forest Products Journal, 37(9), 27-
32.
30. Pei, S., and Lindt, J. W. v. d. (2006). "SAP WOOD: A Seismic Analysis Package for
Wood Frame Structures.”
31. Phillips, T. L. (1990). "Load Sharing Characteristics of Three-Dimensional Wood
Diaphragms," MS Thesis, Washington State University, Pullman, Washington.
32. Schmidt, R. J., and Moody, R. C. (1989). "Modeling Laterally Loaded Light-Frame
Buildings." Journal of Structural Engineering, 115(1), 201-216.
33. Stewart, W. G. (1987). "The Seismic Design of Plywood Sheathed Shear Walls," PhD
Dissertation, University of Canterbury, Christchurch, New Zealand.
34. Tarabia, A. M., and Itani, R. Y. (1997a). "Static and Dynamic Modeling of Light-
Frame Wood Buildings." Computers and Structures, 63(2), 319-334.
35. Tarabia, A. M., and Itani, R. Y. (1997b). "Seismic Response of Light-Frame Wood
Buildings." Journal of Structural Engineering, 123(11), 1470-1477.
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.
YEAR SHEAR FLOOR & HOUSE STATIC DYNAMIC
WALL ROOF
McCutcheon 1985 √
Gupta and Kuo 1987 √
Patton-Mallory and McCutcheon 1987 √
Schmidt and Moody 1989 √ RIGID √
Dolan and Filiatrault 1990 √
Kamiya and Itani 1998 √ √
Dinehart and Shenton 2000 √ √ √
Folz and Filiatrault 2001 √ √ √
Folz and Filiatrault 2004a √ RIGID √ √ √
Pei and Lindt 2006 √ RIGID √ √ √
ANALYSISASSEMBLAGE
AUTHORS
YEAR SHEAR FLOOR & HOUSE STATIC DYNAMIC
WALL ROOF
Gutokowski and Castillo 1988 √ √
Falk and Itani 1989 √ √ √
Dolan and Foschi 1991 √ √
Kasal and Leichti 1992a √ √
Kasal and Leichti 1992b √ √ √ √
Kasal et al. 1994 √ √ √ √
White and Dolan 1995 √ √ √
Tarabia and Itani 1997a √ √ √ √ √
Collins et al. 2005a √ √ √ √ √
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.
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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,
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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

the effects of diaphragm flexibility on the seismic performance of light frame wood structures

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