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INFLUENCE OF VARYING STRENGTH, FROM STORY TO STORY, ON
MODELED SEISMIC RESPONSE OF WOOD-FRAME SHEAR WALL
STRUCTURES
Logan A. Perry1, Philip Line2, Finley A. Charney3
ABSTRACT: This paper presents a numerical study of the influence of varying story strength on the seismic
performance of multi-story wood-frame shear wall buildings. In the prior FEMA P695 studies of these buildings, the
non-simulated collapse limit-state was exceeded primarily in the first story [6]. This observation raised interest in
quantifying the influence of varying strength from story to story on seismic response. In this study, four different
distributions of strength are used as bounding cases. The Parabolic strength distribution (1) is based upon the ELF
method in ASCE 7 and assigns lateral forces to each level based on weight and story height. The Triangular strength
distribution (2) is based upon the simplified procedure in ASCE 7 and distributes lateral forces based on the seismic
weight at each level. The Constant strength distribution (3) assumes the same shear wall design was used on all levels.
The Baseline strength distribution (4) is from actual designs provided in the FEMA P695 wood-frame example and
represents the practical implementation of the ELF method for designed shear walls. The FEMA P695 methodology,
which quantifies seismic performance via adjusted collapse margin ratios, is employed in this study. The analytical
models include P-Delta effects and utilize the 10-parameter CASHEW hysteresis model. Based on the analysis of a
subset of index models from the FEMA P695 wood-frame example, it is observed that the Parabolic strength
distribution, which facilitates dissipation of energy along the entire height of the building, has larger adjusted collapse
margin ratios (lower collapse risk) than other strength distributions studied and reduces occurrence of concentrated
inelastic deformations in a single story from the onset of an applied lateral force.
KEYWORDS: Wood-frame Shear Walls, Seismic Analysis, Hysteresis Models
1 INTRODUCTION 123
Wood-frame shear wall buildings are one of the most
commonly used structural systems. Four and five story
wood-frame shear wall buildings are becoming
increasingly popular for multi-family residential
structures, especially in areas along the western coast of
the United States [1]. As wood shear wall structures
continue to grow in popularity and height, improvements
in understanding of their behaviour in seismic conditions
is increasingly important.
The objective of the research described in this paper was
to investigate the influence on collapse performance of
vertical variations in strength in wood-frame shear wall
1 Graduate Research Assistant, Dept. of Civil and
Environmental Engineering, Virginia Tech, 200 Patton Hall,
Blacksburg, VA 24061. Email: laperry@vt.edu
2Senior Director, Structural Engineering, American Wood
Council, 222 Catoctin Circle SE Suite 201, Leesburg, VA,
20174. Email: PLine@awc.org
3 Professor, Dept. of Civil and Environmental Engineering,
Virginia Tech, 200 Patton Hall, Blacksburg, VA, 24061.
Email: fcharney@vt.edu
buildings. More specifically, it focused on the resistance
provided by shear walls in individual building stories
when subjected to lateral loads induced by a seismic
event. The objective was achieved by systematically
varying the strength (with proportional change in
stiffness) of the stories of the building, and evaluating
the influence on the computed behaviour. In total, four
different vertical strength distributions were considered.
The first is specified in ASCE 7 Section 12.8.3 and
represents the distribution produced by the Equivalent
Lateral Force (ELF) method [2]. The force at each level
is determined based on the weight and height of the level
above grade, as shown in Equations 1 and 2,
(1)
(2)
where Fx is the force at level x, Cvx is the vertical
distribution factor, k is an exponent related to the period
of the structure (k=1.0 is used in this study), V is the base
shear, w is the portion of the total seismic weight at level
i or x, and h is the above grade height of level i or x. The
strength distribution along the height of the building
resulting from this force distribution resembles half a
parabola (if all story weights and heights are equal), and
is referred to as the “Parabolic Strength Distribution” in
this paper.
The second distribution, deemed the “Triangular
Strength Distribution” in this paper, is based on ASCE 7
Section 12.14.8.2 where the force at each level is
determined by the portion of the seismic weight at each
level, as shown in Equation 3,
(3)
where Fx is the force at level x, W is the total weight of
the structure, and wx is the portion of the total weight at
level x. It is important to note that the ASCE 7 criteria
for Equation 3, which limits applicability to 3 stories and
imposes scaling factors, were not used in this study.
The third distribution, referred to herein as the “Constant
Strength Distribution” results from an assumed vertical
force distribution that assigns a concentrated lateral force
at the uppermost level based on the total seismic weight.
For this distribution all stories have identical strength.
This scenario is plausible in cases where the same shear
wall design is used on every story.
The top part of Figure 1 presents the vertical distribution
of lateral forces, and the bottom of the figure shows the
vertical distribution of story shears for each of the three
strength distributions identified above. The name given
to each distribution is chosen based on the assumption
that mass at each level and the height of each story are
equal. In the buildings analysed in this study the story
heights are equal, but the mass at the top level (roof
level) is smaller than that in the lower levels (floor
levels).
Figure 1: Force distributions (top) and resulting strength
distributions (bottom)
The actual distribution of strength after the shear wall is
designed will most likely not match the strength
distribution suggested by the assumed force distribution.
Reasons for this difference stem from practical design
and construction considerations, including use of
discrete values of shear wall sheathing edge nail
spacings, preference to limit changes in sheathing
thicknesses, and available wall lengths for shear walls.
Strictly speaking, these practical construction and
materials considerations will cause the relative strength
from story to story to be different than the relative
strength from story to story in accordance with the
ASCE 7 ELF vertical force distribution procedure. To
account for this, additional Baseline cases resulting from
actual designs are analysed for each structure; in this
paper, the strength distribution associated with these
cases is referred to as the “Baseline Strength
Distribution.” Further information about these Baseline
models is provided in the Model Archetypes section in
this paper.
In total, these four vertical strength distributions,
including the Parabolic, Triangular, Constant, and
Baseline, are analysed and compared via static and
dynamic analyses. The four distributions are summarized
below:
1) Parabolic Strength Distribution – Represents
distribution from ELF method (k=1.0).
2) Triangular Strength Distribution – Vertical
distribution of strength resulting from vertical
distribution of forces that are proportional to the
seismic weight at each story.
3) Constant Strength Distribution – Vertical
distribution of strength resulting from a
concentrated lateral force at the uppermost story
based on the total seismic weight (i.e. story
strength at each story is equal to the strength at
the first story).
4) Baseline Strength Distribution – Actual
distribution of vertical strength in modeled
structures. This distribution represents the
practical implementation of the ELF method for
designed shear walls.
The effects of each distribution are compared using
adjusted collapse margin ratios (ACMRs) that are
discussed in the Methodology section.
2 METHODOLOGY
2.1 MODEL ARCHETYPES
This study uses a subset of index models (IMs) adapted
from the FEMA P695 wood-frame example [1]. The four
index models selected for this study contain three, four,
or five stories and are in Seismic Design Category D.
Structures for both residential and commercial
occupancy are considered, along with shear walls with
low and high aspect ratios. A summary of the index
models is presented in Table 1.
Table 1: Index model descriptions
Index
Model No. of
Stories
Shear Wall
Aspect
Ratio Occupancy
IM9
3
Low
Commercial
IM10
3
High
Multi-Family
IM13 4 High Multi-Family
IM15 5 High Multi-Family
Construction details for these index models, as specified
in FEMA P695, are provided in Table 2. These index
model designs, using 3.04 m story heights, were utilized
for the Baseline strength distribution cases in this study.
Table 2: Index model construction details
Index
Model
Story
No. of
Shear
Walls
Shear
Wall
Width
[m (ft.)]
Nominal
Sheathing
Thickness
Nailing at Panel
Edges
[mm (in.)]
IM9
3 3 3.05 (10) 7/16" OSB 8d at 152.4 (6)
2 3 3.05 (10) 7/16" OSB 8d at 50.8 (2)
1 3 3.05 (10) 19/32" PLWD
10d at 50.8 (2)
IM10
3 5 0.91 (3) 7/16" OSB 8d at 152.4 (6)
2 5 0.91 (3) 7/16" OSB 8d at 50.8 (2)
1 5 0.91 (3) 19/32" PLWD
10d at 50.8 (2)
IM13
4 6 1.01 (3.3) 7/16" OSB 8d at 152.4 (6)
3 6 1.01 (3.3) 7/16" OSB 8d at 50.8 (2)
2 6 1.01 (3.3) 19/32" PLWD
10d at 50.8 (2)
1 6 1.01 (3.3) 19/32" PLWD
10d at 50.8 (2)
IM15
5 6 1.13 (3.7) 7/16" OSB 8d at 152.4 (6)
4 6 1.13 (3.7) 7/16" OSB 8d at 76.2 (3)
3 6 1.13 (3.7) 7/16" OSB 8d at 50.8 (2)
2 6 1.13 (3.7) 19/32" PLWD
10d at 50.8 (2)
1 6 1.13 (3.7) 19/32" PLWD
10d at 50.8 (2)
The building models used in this study consist of
nonlinear shear wall elements and rigid diaphragms.
Because no torsional irregularities are present in any of
the index models, three-dimensional models are not
necessary. Instead, each of the four index models were
modelled in two-dimensions using nonlinear spring
elements with the CASHEW load-deformation hysteretic
relationship, discussed later in this section. This
hysteretic relationship allows the entire shear wall
subassembly to be represented in one zero-length
nonlinear spring element at each level, connected via
truss elements. In addition, the plan symmetry of each
index model allows the analysis to performed for half the
building for the sake of simplicity and reduced analytical
runtime.
The models neglect axial-flexural interaction, as well as
in-plane rigid body rotations, reducing the number of
degrees of freedom to one per story: in-plane shear
deformations of the shear wall. All deformation at each
level is concentrated in the nonlinear spring elements. To
consider the effects of gravity, P-Delta effects are
included via leaning columns for all analyses. When
vertical gravity loads are applied (before any lateral
loads are introduced), the influence of the P-Delta effects
is to reduce stiffness and strength of the system. Figure 2
illustrates a two-story example of a wood-frame shear
wall building model that incorporates nonlinear spring
elements and leaning columns (shown at the right of the
figure).
Figure 2: Two-story wood-frame shear wall analytical model
Dashpots are shown in Figure 2 to represent the damping
set at each level. These dashpots represent the physical
basis of Rayleigh damping, which is assumed to be
linear viscous, and proportional to mass and stiffness
The mass-proportional and stiffness-proportional
constants, α and β, respectively, were determined to
produce an effective viscous damping ratio of 1%
critical.
Though the modelling method used in this study captures
the primary behaviours and limit states found in wood-
frame shear wall buildings, it must be noted that not all
of the observed characteristics of an actual full-scale
wood shear wall system are explicitly represented. In a
multi-story building, the shear walls are typically
connected through tension rods that run from the bottom
of the structure and terminate in the top wall. Excessive
yielding or rupture of these tension walls can occur
anywhere along the height of the structure, a limit state
that is not explicitly modelled in this study. The model
also does not explicitly consider overturning since
rotations are not considered. In realistic loading
scenarios, overturning causes tension on one side of the
wall (resisted by the tension rods) and compression on
the other. Compression of the wall can lead to
compression failures in the studs, causing the wall to
buckle out of plane. The analytical model does not
explicitly consider these limit states.
The model also does not consider limit states associated
with shear transfer through the floor platform that is
situated between adjacent upper and lower story shear
walls. In multi-story structures, large shear forces must
be transferred through the rim board or other framing
detail at the edge of the floor deck. Shear failures in this
area are not explicitly modelled. Through-thickness
crushing in this area, due to the large compression
forces, is also not considered.
Though the limit states listed in this section should not
be forgotten, it is generally accepted that the primary
limit state in wood shear walls buildings is associated
with yielding of the nails, the crushing of wood fibers
adjacent to the nails, and in-plane shear deformations of
the shear wall. These response characteristics are
specifically captured in this study. Buildings properly
designed to modern building codes are generally less
susceptible to failures from the limit states noted in this
section Also, while the analytical model does not
explicitly address specific limit states previously noted,
the model is based on the tested response of shear walls
that incorporate effects associated with a designed load
path for overturning and shear (i.e. appropriately sized
hold-downs, shear anchorage, and framing members are
used in shear wall testing that forms the basis of the
analytical model).
2.2 MODIFICATION FACTORS
In engineering practice, the design strength of each story
would be based upon the design loads provided by
ASCE 7. As mentioned, however, the exact strength
distribution provided by the methods in ASCE 7 will not
be the same as the strength distribution in the final
design due to practical constraints in construction (actual
nail spacing, sheathing thickness, available wall lengths
for shear walls, etc.). For simplicity, and to most closely
Table 3: Parabolic and triangular strength distribution
modification factors
Index
Model
Story
10-Parameter Model
Modification Factors
Parabolic
Distribution
Triangular
Distribution
IM9
(3-story)
3
2
1
0.33
0.78
1.00
0.20
0.60
1.00
IM10
(3-story)
3
2
1
0.33
0.78
1.00
0.20
0.60
1.00
IM13
(4-story)
4
3
2
1
0.25
0.63
0.88
1.00
0.14
0.43
0.71
1.00
IM15
(5-story)
5
4
3
2
1
0.20
0.52
0.76
0.92
1.00
0.11
0.33
0.56
0.78
1.00
evaluate the distribution calculated in ASCE 7, exact
strength and stiffness modification factors representing
the ratio of the seismic base shear at each story are used
to establish the strength and stiffness of upper stories of
each model. More specifically, the modification factors
match the exact strength distribution calculated as
opposed to designing each story individually and
attempting to create shear wall designs close to the exact
ASCE 7 distribution. Modification factors for the
Parabolic and Triangular strength distributions can be
found in Table 3.
The Baseline index models are based on constructible
designs provided in FEMA P695. These designs were
based on the ELF force distribution (Parabolic strength
distribution) and their final strength distribution is
presented in Table 4. Table 4 also provides
modifications factors for the Constant strength
distribution.
Table 4: Baseline and constant strength distribution
modification factors
Index
Model
Story
10-Parameter Model
Modification Factors
Constant
Distribution
Baseline
Distribution
IM9
(3-story)
3
2
1
1.00
1.00
1.00
0.33
0.90
1.00
IM10
(3-story)
3
2
1
1.00
1.00
1.00
0.33
0.90
1.00
IM13
(4-story)
4
3
2
1
1.00
1.00
1.00
1.00
0.33
0.90
1.00
1.00
IM15
(5-story)
5
4
3
2
1
1.00
1.00
1.00
1.00
1.00
0.33
0.62
0.90
1.00
1.00
2.3 MATHEMATICAL MODEL
This study uses the load-deformation relationship
referred to as CASHEW (Cyclic Analysis of SHEar
Walls) developed as a part of the Consortium of
Universities for Research in Earthquake Engineering
(CUREE) Caltech Wood-Frame project. The nonlinear
model captures pinching as well as strength and stiffness
degradation, all characteristics of wood shear wall
behaviour. CASHEW uses 10 unique parameters to
characterize the load-deformation response of
experimental wood shear wall testing results. The
parameters are calibrated according to shear wall
characteristics including size, sheathing properties, and
nail properties [3]. Figure 3 illustrates the CASHEW
force-displacement model. The parameters for each
Baseline case index model are listed in Table 5. The first
story parameters are used in conjunction with the
modification factors presented in Tables 3 and 4 for the
Parabolic, Triangular, and Constant strength
distributions. Strength and stiffness are modified by
adjusting the F0, FI, and K0 parameters.
Figure 3: CASHEW force-deformation model [4]
Table 5: Baseline case – index model CASHEW parameters
F
0
F
I
δ
U
K
0
r
1
r
2
r
3
r
4
α Β
KN KN m KN/m - - --- -
376.45 10.90 0.07 7642.09 0.04 -0.04 1.01 0.02 0.72 1.14
2226.73 31.64 0.08 15017.87 0.04 -0.06 1.01 0.02 0.74 1.13
1223.83 36.65 0.08 18862.18 0.05 -0.06 1.01 0.02 0.59 1.15
347.70 6.68 0.09 4357.15 0.04 -0.06 1.01 0.02 0.73 1.16
2131.41 18.87 0.10 8702.03 0.04 -0.08 1.01 0.02 0.75 1.12
1131.88 21.81 0.10 10846.93 0.05 -0.08 1.01 0.02 0.60 1.14
4147.38 7.12 0.09 4905.51 0.04 -0.07 1.01 0.02 0.69 1.14
3147.38 20.92 0.10 9944.17 0.04 -0.07 1.01 0.02 0.71 1.12
2147.39 24.28 0.09 12442.52 0.05 -0.07 1.01 0.02 0.59 1.16
1147.39 24.28 0.09 12442.52 0.05 -0.07 1.01 0.02 0.59 1.16
557.36 8.10 0.08 5834.06 0.04 -0.05 1.01 0.02 0.72 1.15
4112.14 15.26 0.08 9316.97 0.04 -0.06 1.01 0.02 0.72 1.15
3167.81 23.41 0.09 11566.50 0.04 -0.08 1.01 0.02 0.75 1.14
2166.56 27.23 0.08 14532.47 0.05 -0.06 1.01 0.02 0.59 1.15
1166.56 27.23 0.08 14532.47 0.05 -0.06 1.01 0.02 0.59 1.15
IM13
IM15
Index
Model
Sto ry
IM9
IM10
2.4 PERFORMANCE EVALUATION METRICS
Primary response evaluation metrics used in this paper
are static pushover and incremental dynamic analysis.
All analyses are performed in OpenSEES (Open System
for Earthquake Engineering Simulation), an open-source,
non-linear analysis program developed by the University
of California, Berkeley [5].
2.4.1 Static Pushovers
Displacement-controlled static pushovers are used in this
study. To perform the pushover, lateral point loads are
applied at each level and are scaled up (or down) to
ensure a constant increase in lateral drift with each
increment of force applied. The load at each level is
proportional to the first mode shape of the model, as
shown in Equation 4.
(4)
The pushover proceeds in increments of 0.0001 inches
and continues until an ultimate roof displacement of 10%
of the total building height is reached. Figure 4 illustrates
an idealized pushover curve.
Figure 4: Idealized pushover curve
2.4.2 Incremental Dynamic Analyses
Incremental dynamic analyses (IDAs) are performed
using 44 unique ground motions as specified in FEMA
P695. These 44 ground motions are classified as “Far
Field” ground motions and consist of two orthogonal,
horizontal components of 22 different seismic events.
All ground motions have been normalized by their peak
velocities to eliminate extreme variability. During
analysis, they are collectively adjusted again so that the
median spectral acceleration of the ground motions is
equal to the spectral acceleration of the index model at
its fundamental period. This adjustment factor is referred
to as the “anchor factor.” To begin, each ground motion
is applied to the index model at a low scale factor (or
intensity). Once the analysis has finished, the maximum
drift at each story is recorded, along with the scale factor
at which the ground motion was ran. The scale factor is
then increased by 0.1, and the ground motion is applied
again. This process continues until there is a dynamic
instability or any story in the structure reaches a drift of
7%, at which point the analysis stops.
IDAs are used to determine the median collapse
intensity, collapse margin ratio (CMR) and adjusted
collapse margin ratio (ACMR) for each index model for
the specified strength and stiffness. The collapse margin
ratio is defined as the ratio of the median spectral
acceleration of the collapse level ground motions, ŜCT,
and the spectral acceleration of the maximum considered
earthquake (MCE) ground motion, ŜMT, at the
fundamental period of the index model. Equation 5
illustrates this relationship.
(5)
Once calculated, the CMR is adjusted by the Spectral
Shape Factor (SSF), a parameter that accounts for the
fact that some ground motions have a spectral shape that
is not accurately represented by the design spectrum
provided in ASCE 7. Once calculated, the SSF is applied
to the CMR to calculate ACMR as shown in Equation 6.
(6)
ACMR is calculated for each strength distribution
variation in all index models. It is a critical measure of
seismic performance and is used as the primary
evaluation metric in this paper.
3 RESULTS AND DISCUSSION
3.1 ACMR – ADJUSTED COLLAPSE MARGIN
RATIO
The results of this study provide insight into the
performance of four different distributions of vertical
strength for four wood-frame index models from FEMA
P695. Table 6 summarizes the ACMR (with P-Delta)
based on 44 ground motions for each vertical strength
distribution and each index model.
Table 6: ACMR results
Index
Model
ACMR
Parabolic
Distribution
Triangular
Distribution
Constant
Distribution
Baseline
Distribution
IM9 1.68 1.23 1.43 1.53
IM10 2.37 1.63 2.04 2.18
IM13 2.21 1.38 1.92 2.02
IM15 1.94 1.14 1.68 1.79
Table 7 summarizes the average ACMR values for each
distribution and index model to facilitate comparison of
average effect of a given strength distribution on ACMR
for all index models studied and average effect on
ACMR of a given index model subject to all strength
distributions studied.
Table 7: ACMR averages
ACMR - Averages
Average for given
Index Model (over all
strength distributions)
Index Model
IM9 1.47
IM10 2.06
IM13 1.88
IM15 1.64
Average for given
Strength Distribution
(over all models)
Strength Distribution
Parabolic 2.05
Triangular 1.35
Constant 1.77
Baseline 1.88
3.2 STRENGTH DISTRIBUTIONS
3.2.1 Parabolic Strength Distribution
The results indicate that the Parabolic strength
distribution has higher ACMR values, and thus lower
collapse risk than the three other strength distributions
considered. Much of this superior performance is
attributed to its ability to evenly dissipate energy over
the height of the structure. Figure 5 presents a plot
illustrating the displacement at each level during a
pushover analysis. The displaced shape is depicted at
five evenly spaced increments during the analysis.
Figure 5: Pushover displacement progression for the IM13
parabolic strength distribution
For the Parabolic case, the drift at each floor is roughly
equal until about 2.5% roof drift is achieved. After this
point, the drift becomes concentrated on the first level.
Other distributions do not see similar drifts initially
along the height of the structure, and typically begin to
have a drift concentration in a single story at much lower
levels of roof drift.
Also highlighting the Parabolic strength distribution’s
ability to efficiently dissipate energy along the height of
the structure is the percentage of the peak strength
reached at each story during pushover analyses: for
IM13, when the first story reaches 100% of its peak
strength, the second story reaches 97% of its peak
strength, the third story reaches 96% of its peak strength,
and the fourth story reaches 97% of its peak strength.
For comparison, the Constant strength distribution only
reaches 81%, 53%, and 19% peak strength on the
second, third, and fourth stories, respectively.
3.2.2 Triangular Strength Distribution
The Triangular strength distribution has the lowest
ACMR of the distributions considered with its average
value just over 65% of the average for the Parabolic
strength distribution. Figure 6 plots the pushover curves
for all four distributions for IM 13. The Triangular
strength distribution develops a much lower base shear
strength than the three other distributions. Relative poor
performance in this distribution is attributed to the low
levels of strength in the upper stories of the models.
When compared to the Parabolic strength distribution,
the Triangular strength distribution has significantly less
strength at each story, with the strength in the fourth
story being almost half of the strength at the fourth story
of the Parabolic strength distribution model. This
reduction in strength subjects the model to a collapse in
the upper story at very low levels of strength.
Figure 6: IM13 pushover curves
Figure 7 plots the pushover progression of total
displacement for the Triangular strength distribution of
IM13. As expected, the drift is concentrated in the top
story, where the strength is only 14% of the strength of
the first story.
Figure 7: Pushover displacement progression for the IM13
Triangular strength distribution
3.2.3 Constant Strength Distribution
Though higher than the Triangular strength distribution,
the Constant strength distribution also has low ACMR
values relative to the Parabolic strength distribution. Its
poorer performance is generally a result of the
strengthened upper stories of the structure, which cause
all drifts to be concentrated in the first story at very low
levels of roof drift. Figure 8 illustrates the drift of each
story throughout the pushover analysis. From the onset
of loading, the first story experiences higher drifts than
any other story, a phenomenon that continues throughout
the entire analysis. The Constant strength distribution
case for IM13 only reaches 81%, 53%, and 19% peak
strength on the second, third, and fourth stories,
respectively, during the pushover analysis.
Figure 8: Interstory drifts vs. roof drift for the IM13 constant
strength distribution
3.2.4 Baseline Strength Distribution
As expected, the Baseline strength distribution performs
similarly to the Parabolic strength distribution, with its
average ACMR reaching about 92% of the average
ACMR for the Parabolic strength distribution. This
similarity is attributed to the Baseline strength
distribution being designed using the ELF method, on
which the Parabolic strength distribution is based. Still,
however, the Parabolic strength distribution’s average
ACMR is 8% higher, suggesting that the actual strength
distribution resulting from the design of the lateral
system reduced collapse performance compared to the
idealized distribution produced by the Parabolic case.
For convenience, the modification factors for the
Parabolic and Baseline cases for IM13 have been
reproduced in Table 8.
Table 8: Modification factors for the Parabolic and Baseline
strength distributions of IM13
Index
Model Story
Parabolic
Strength
Distribution
Baseline
Strength
Distribution
IM13
(4-story)
4
3
2
1
0.25
0.63
0.88
1.00
0.33
0.90
1.00
1.00
As shown in Table 8, the actual strength distribution
between the two cases is quite different, even though the
Baseline case design was based on the Parabolic (ELF)
strength distribution in ASCE 7. In the Baseline case the
designer chose to use the same shear wall design on the
second story as the first story. In terms of
constructability, this is logical – the contractor can
purchase the same materials for both stories and the
construction of the shear walls is repetitive. Though the
Parabolic case had almost the same drift at each story
until about 2.5% roof drift, the first story of the Baseline
case is already experiencing a drift concentration in the
first story well before 1% roof drift is achieved.
Figure 9: Pushover displacement progression for the IM13
Baseline strength distribution
The Baseline strength distribution case demonstrates the
sensitivity of the building models to minor changes in
the strength distribution. Even small, practical design
decisions (such as repeating a shear wall design for
multiple stories) impacts on the modelled performance
of the structure.
3.3 OTHER OBSERVATIONS
Also important to note is the performance of the index
models themselves. Indicating a higher risk of collapse
than the other models, IM9 has the lowest ACMR values
for all four strength distributions. This higher collapse
risk is most likely due to its classification as a low-aspect
ratio shear wall. Unlike high aspect-ratio shear walls,
low-aspect shear walls are not subjected to a strength
reduction factor in design. The application of the
strength reduction factor for high aspect ratio shear
walls, to account for reduced stiffness of high aspect
ratio shear walls, leads to greater overstrength and better
overall ACMR values. A prior sensitivity study [6]
conducted on FEMA P695 wood-frame index models
indicates a strong dependence between overstrength and
ACMR. In this study, for a given strength distribution,
increased ACMRs are associated with increased levels of
overstrength provided in the index model. The level of
overstrength of index models in order from low to high
is IM9, IM15, IM13 and IM10.
3.4 STORY IN WHICH COLLAPSE OCCURS
In addition to ACMR, another metric in evaluating each
distribution is the story in which collapse occurs. This
section discusses the story that led to collapse for both
pushover and IDA analyses.
3.4.1 Pushover Collapse Story Trends
Table 9 presents a summary of the collapse floor for
each distribution during static pushover analyses, both
with and without P-Delta effects.
Table 9: Story of collapse for pushover analyses
IM9
(3-Story)
IM10
(3-Story)
IM13
(4-Story)
IM15
(5-Story)
With P-Delta 1 1 1 1
No P-Delta 3311
With P-Delta 3345
No P-Delta 3345
With P-Delta 1111
No P-Delta 1111
With P-Delta 1111
No P-Delta 1111
Baseline
Collapse Story
Strength Distribution
Parabolic
Triangular
Constant
As discussed in the previous section, the Triangular
strength distribution always collapses in the top story
due to very low levels of strength in the upper stories. As
expected, however, a majority of the index models
collapsed on the first floor for the Parabolic, Constant,
and Baseline strength distributions. Two cases – the
Parabolic distribution for IM9 and IM10 – did not follow
this trend when P-Delta effects were not included;
instead, they collapsed in the top story. Reasons for this
difference stem from the fact that all stories reach almost
100% of their peak strength when the Parabolic strength
distribution is used. The extra shear produced by the P-
Delta leaning column is not considered without P-Delta
effects, causing the first story to see less shear than when
P-Delta effects were included (since the first story is
most affected by the accumulating P-Delta forces). This
change caused the top story to reach its peak strength
before the first story, inducing a collapse at the top of the
structure. This same behaviour is not observed in other
distributions where the upper stories do not come close
to reaching their peak strength.
3.4.2 IDA Collapse Story Trends
Tables 10 and 11 present a summary of the IDA collapse
data for all distributions and index models. The number
of collapses for each story is included, each of which
designates the number of ground motions that caused the
collapse to occur in that story. The average drifts prior to
collapse and maximum drifts are also summarized for
each distribution and index model. These values are
averaged over the 44 ground motions considered.
Table 10: IDA collapse data for the Parabolic and Triangular
strength distributions
Number of
Collapses
Average
Drifts Prior
to Collapse
Average
Maximum
Drifts
Number of
Collapses
Average
Drifts Prior
to Collapse
Average
Maximum
Drifts
133 4.85% 4.89% 2 2.06% 2.06%
23 2.89% 3.11% 4 3.09% 3.17%
38 3.34% 3.73% 38 6.29% 6.35%
126 5.51% 5.55% 1 2.07% 2.07%
25 3.71% 4.03% 2 3.15% 3.20%
313 4.25% 4.67% 41 7.08% 7.13%
124 5.22% 5.28% 3 1.68% 1.69%
22 2.86% 3.12% 0 1.98% 2.00%
33 3.07% 3.47% 1 2.72% 2.77%
415 4.56% 4.92% 40 6.62% 6.73%
123 5.16% 5.20% 3 1.59% 1.59%
20 2.51% 2.68% 0 1.45% 1.48%
30 2.46% 2.59% 0 1.83% 1.86%
41 2.87% 3.22% 2 2.86% 2.91%
520 4.99% 5.34% 39 6.65% 6.78%
IM13
Parabolic
Triangular
Sto ry
Index
Model
IM15
IM9
IM10
Table 11: IDA collapse data for the Constant and Baseline
strength distributions
Number of
Collapses
Average
Drifts Prior
to Collapse
Average
Maximum
Drifts
Number of
Collapses
Average
Drifts Prior
to Collapse
Average
Maximum
Drifts
144 4.96% 4.99% 35 4.73% 4.76%
20 1.45% 1.58% 0 1.65% 1.85%
30 0.43% 0.47% 9 3.15% 3.46%
144 6.49% 6.55% 33 6.19% 6.21%
20 1.89% 2.15% 0 2.06% 2.35%
30 0.49% 0.56% 11 4.00% 4.36%
141 6.01% 6.04% 40 6.07% 6.12%
23 2.37% 2.57% 2 2.39% 2.66%
30 1.25% 1.39% 0 1.44% 1.61%
40 0.40% 0.45% 2 2.26% 2.48%
142 6.15% 6.18% 38 5.61% 6.00%
22 2.63% 2.89% 4 2.30% 2.67%
30 1.47% 1.59% 0 1.75% 2.10%
40 0.94% 1.07% 2 2.18% 2.57%
50 0.34% 0.38% 0 1.24% 1.61%
Index
Model
Sto ry
Constant
Baseline
IM9
IM10
IM13
IM15
Important to note are the assumptions made for this data,
and how these assumptions may affect the results.
Instabilities are often observed during IDA analyses at
the collapse level intensity. In other words, very large
story drifts often occur during the final analysis (or
intensity level) of each ground motion. To address this
issue, the largest drift at the previous intensity factor is
considered indicative of the collapse story.
Considering Tables 10 and 11, several trends can be
identified regarding the story of collapse:
1) The Parabolic strength distribution tends to have
most of its collapses on the first or top story. This is
because the Parabolic strength distribution tends to
spread drifts out over the height of the structure,
meaning almost all stories come close to reaching
their peak strength. Variations in the ground
motions themselves then account for fluctuation in
the collapse story, depending on which exceeds its
peak strength first.
2) The Triangular strength distribution has almost all
its collapses on the top story. This is expected due to
the extremely weak nature of the top story in all
models.
3) The Constant strength distribution has almost all its
collapses on the first story. This is also expected
because of the strong upper stories, which tend to
cause drift concentrations at the lowest story.
4) The Baseline strength distribution tends to be a
“hybrid” between the Parabolic and Constant
strength distributions. Though all Baseline index
models are based on the ELF distribution in ASCE 7
(as is the case with the Parabolic strength
distribution), shorter three-story index models (IM9
and IM10) behave more similarly to the Parabolic
strength distribution, whereas taller four- and five-
story index models (IM13 and IM15) behave more
similarly to the Constant strength distribution. This
is because the taller index models use the same
design on the second floor as the first floor - a
distribution of strength more similar to the Constant
strength distribution.
Also of interest are the drifts presented in Tables 10 and
11. These averages tend to validate the trends above,
since the highest drifts are in the story that tends to
collapse most frequently. The magnitude of these story
drifts are usually below the collapse limit of 7%
indicating occurrence of dynamic instability prior to the
occurrence of the non-simulated collapse drift limit.
4 SUMMARY AND CONCLUSIONS
4.1 SUMMARY
Wood-frame shear walls buildings continue to grow in
numbers and size and improved understanding of their
seismic behaviour is increasingly important. This study
focused on intentional strengthening and weakening of
building stories along the building’s height to quantify
the effects on collapse performance. Four unique
distributions of strength were used to address a range of
vertical distribution of strength. The first two, termed the
Parabolic and Triangular strength distributions, were
based on the lateral force distributions found in ASCE 7.
The Parabolic strength distribution assigned forces to
each story based on weight and story height. The
Triangular strength distribution distributed lateral forces
based simply on the seismic weight at each level. The
Constant case assumed the same shear wall design was
used on all levels, effectively strengthening the upper
stories of the structure. The Baseline case was based on
actual designs provided in FEMA P695 and reflected a
realistic vertical strength distribution resulting from the
ELF method presented in ASCE 7.
4.2 CONCLUSIONS
This study aimed to quantify the effects of four different
vertical strength distributions on collapse performance. It
focused on four specific index models from FEMA P695
and summarized their collapse risk through adjusted
collapse margin ratios. Strength distributions that
allowed for a dissipation of energy along the entire
height of the building had larger ACMRs (lower collapse
risk) than buildings with concentrated deformations in a
single story from the onset of an applied lateral force.
The Parabolic strength distribution, based on the ELF
method presented in ASCE 7, most effectively
distributed drifts among all stories and thus had the
largest ACMRs. The approximate 10% difference in
ACMR between the Parabolic strength distribution and
the Baseline strength distribution highlights the
sensitivity of ACMR to changes brought about by actual
designs.
The Triangular and Constant strength distributions, on
the other hand, saw drift concentrations from the onset of
analysis and had the lowest ACMR values. The
Triangular distribution’s relative poor performance is
attributed to weakened upper stories, which leads to
premature collapse (collapse prior to development of the
base shear strength). Conversely, the Constant strength
distributions lower ACMR values stem from
strengthened upper stories that produce concentrated
drifts in the bottom story. These bounding cases
illustrate that improved ACMR performance is not as
simple as adding or reducing strength, but doing so in a
calculated manner that leads to more even energy
distribution over the height of the structure.
Regardless of the strength distribution used, all models
eventually experienced drift concentrations in a single
story at large displacements. In general, the story in
which these displacements were concentrated was
simply the first to reach its peak strength. Finally, it
should be noted that in addition to the strength variations
in the designated shear wall system due to factors such
as practical design and construction considerations,
presence of other materials including exterior finish such
as stucco and siding, interior finish such as gypsum wall
board and interior wall panels, and presence of
oversheahting (i.e. wood structural panel sheathing on all
exterior wall surfaces) have an effect on actual story
strength. In prior seismic studies, the added strength of
finish materials has been found to be beneficial in terms
of increasing strength and stiffness but consideration
should also be given to possibility of such materials
reducing ACMR and creating a weak or soft story.
ACKNOWLEDGEMENTS
The Authors would like to thank the American Wood
Council (AWC) for the financial support and guidance
provided for the research described in this paper. The
valuable contributions of Dr. Jeena Jayamon are
especially appreciated.
REFERENCES
[1] FEMA P695: Quantification of Building Seismic
Performance Factors, Federal Emergency
Management Agency, Washington, D.C., 2009.
[2] ASCE/SEI 7-10: Minimum Design Loads for
Buildings and Other Structures, American Society
of Civil Engineers, Reston, VA., 2010.
[3] Folz, B., and Filiatrault, A. A Computer program for
Seismic Analysis of Woodframe Structures.
CUREE Publication No. W-21, Consortium of
Universities for Research in Earthquake
Engineering, Richmond, CA, 2001.
[4] Folz B., Filiatrault A.: Seismic Analysis of
Woodframe Structures I: Model Formulation,
Journal of Structural Engineering, 130:1353-1360,
2001.
[5] McKenna, F. and Fenves, G.L.: Open System for
Earthquake Engineering Simulation (OpenSEES),
Pacific Earthquake Engineering Research Center,
University of California, Berkeley, CA., 2017.
[6] Jayamon, J., Line, P., and Charney, F.: Sensitivity of
Wood-Frame Shear Wall Collapse Performance to
Variations in Hysteric Model Parameters, Journal of
Structural Engineering (in press), 2018.
