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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.