Fragility Functions for Steel Plate Shear Walls
Nicole M. Baldvins,1) Jeffery W. Berman,1) M.EERI, Laura N. Lowes, 1)
M.EERI, Todd M. Janes1), Natalie A. Low1)
Fragility functions are developed to predict the method of repair required for
steel plate shear walls damaged due to earthquake loading. The results of previous
experimental studies are used to develop empirical relationships between damage
states and story drift. Damage states are proposed and linked deterministically
with commonly employed methods of repair; these damage states are
characterized by parameters such as yielding and tearing of the steel plate and
yielding, buckling and fracture of frame members. Lognormal probability
distributions are fit to the empirical data and evaluated using standard statistical
methods. The results of this effort are families of fragility functions that predict
the required method of repair for a damaged wall.
INTRODUCTION
The widespread adoption of performance-based seismic design (PBSD) requires reliable
performance-prediction models, often referred to as fragility functions, for a wide range of
structural systems. Such models enable engineers to select the structural system that is most
economical for their design space. The Applied Technology Council’s Project 58 (2009) is
currently developing fragility functions for a number of commonly employed structural
systems, including steel moment resisting frames, steel braced frames, concrete moment
frames, and concrete walls. That project is also developing an overall framework and
software tools to facilitate PBSD. The goal of the research presented here is to develop
fragility functions for steel plate shear walls, so that they may be added to the design
engineer’s list of structural systems that may readily be designed using PBSD procedures.
Steel plate shear walls (SPSWs) are stiff, ductile lateral load-resisting systems that are
well suited for use in buildings to resist seismic loads. The systems are typically composed of
steel plates that are either welded or bolted to beam and column boundary elements; these
boundary elements may serve as part of the gravity load-carrying system. Fig. 1 shows a
1)
Department of Civil and Env. Engineering, University of Washington, Box 352700, Seattle, WA 98195-2700.
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typical SPSW configuration and the accepted nomenclature that appears in the American
Institute of Steel Construction’s Seismic Design Provisions (AISC 2005), referred to herein
as The Provisions, where the infill plate is denoted the web plate and the beams and columns
are denoted horizontal and vertical boundary elements (HBEs and VBEs), respectively. A
survey of existing buildings with SPSWs and a review of literature reveals two distinct
design philosophies: (i) SPSWs that are stiffened such that the web plates can develop their
full shear yield strength, and (ii) SPSWs that are allowed to buckle in shear at low load levels
and rely on the development of web plate tension field action for strength, stiffness and
ductility. The latter type is more prevalent in new construction and is the only one for which
seismic design specifications are available in the United States. Therefore, SPSWs that rely
on tension field action are the focus of this paper.
The development of post-buckling tension field action is illustrated in Fig. 2, where a test
specimen is shown at a story drift of 1.2% (Berman and Bruneau 2003). As shown, shear
buckling waves develop that are orthogonal to the tension field. The tension field orientation
angle, , is dependent on the stiffness of the boundary frame elements, the thickness of the
web plate and the aspect ratio of the web plate, and can calculated using the approach in The
Provisions, which is based on a derivation by Timler et al. (1983). With increasing
displacement, the web plate yields along the inclined tension field. Upon load reversal the
web plate has little strength and stiffness until it completes shear buckling in the other
loading direction and again develops a tension field.
The force-deformation behavior of SPSWs depends largely on the HBE-to-VBE
connections. When simple or partially restrained connections are used, the cyclic response
can be quite pinched as shown in Fig. 3a. When fully restrained moment resisting
connections are used, there is a large contribution to the strength and energy dissipation from
frame action, as shown in Fig. 3b (note that the two experimental hysteresis shown are from
experiments with different scales and plate thickness, accounting for the difference in
ultimate strength). Currently, SPSWs with fully restrained moment resisting HBE-to-VBE
connections are designed with a response modification factor, R, of 7. SPSWs with partially
restrained connections must be designed with R equal to 3. In either system, the damage to
SPSWs subjected to earthquake loading is expected to be in the web plates, HBE-to-VBE
connections, and, in some cases, the HBEs and VBEs themselves. To ensure that SPSWs
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have the maximum possible energy dissipation and ductility, it is necessary that the HBEs
and VBEs and the HBE-to-VBE connections have the strength and stiffness to fully develop
tension field yielding of the web plates. Recommendations for achieving capacity design of
the boundary elements and connections are given in The Provisions, Sabelli and Bruneau
(2007), and Berman and Bruneau (2008).
To enable PBSD of SPSWs it is necessary to develop statistical models, i.e., fragility
functions, relating SPSW damage and a seismic demand parameter, such as story drift, that is
readily computed as part of the design process. Such fragilities enable engineers to design
SPSWs to achieve a specific likelihood of damage given the seismicity of the building site.
To achieve the goal of developing these essential fragilities for SPSWs, the study described
here sought to: (i) perform a thorough review of the literature to gather and synthesize
experimental data on the performance of SPSWs subjected to inelastic lateral loading, (ii) use
the literature to formulate an exhaustive list of damage observed during SPSW testing and
compile story drifts associated with the occurrence of the damage, (iii) group the observed
damage into logical damage states that consider the evolution, magnitude and location of the
damage, (iv) refine the damage states into repair states that consider the extent and
complexity of repairs required for various damage states, (v) study the impact of various
SPSW parameters on the damage and repair state data, and (vi) produce fragilities from
statistical analysis of the compiled repair state versus story drift data.
EXPERIMENTAL DATA
The objective of this study was to develop fragility functions for SPSWs with thin plates
for which response is determined by elastic plate buckling and plate yielding. Published
results of previous experimental tests of this type of SPSW were reviewed. The following
describes these test programs:
 Timler and Kulak (1983) tested a single, near full-scale, two-web plate specimen to
evaluate the post-buckling model proposed by Thorburn et al. (1983) as well as to
generate data characterizing SPSW system behavior under service-level and severe
loading. The specimen was essentially two single story specimens placed top-to-top that
shared the same upper HBE. It was loaded in a universal testing machine at the shared
HBE in a manner similar to a beam with a single point load. Timler and Kulak found
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minor inaccuracies in the principle stress angle calculations proposed by Thorburn et al.
(1983) and proposed a revised formula for the angle of inclination.
 Tromposch and Kulak (1987) tested a single, near full-scale, two-web plate SPSW
specimen under cyclic loading in a configuration similar to Timler and Kulak (1983).
Tromposch and Kulak concluded that the Thorburn et al. (1983) model as modified by
Timler and Kulak (1983) could be used with confidence to determine ultimate capacity.
They concluded also that if the model assumes pinned HBE-VBE connections it provides
a conservative estimate of strength.
 Chen (1991) tested ten small-scale three-story SPSW specimens under cyclic and
monotonic loading to study the impact of HBE-VBE connection type on response. Chen
concluded that SPSWs with web plates that are designed to buckle and develop tension
field action are a viable option for seismic load resisting systems and that for walls with
thicker plates the boundary frame members will limit the system’s strength and ductility.
Two tests (M14-3, M14-5) failed prematurely due to out-of-plane VBE buckling resulting
from insufficient stiffness in the lateral bracing system and fracture of the weld between
the web plate and boundary members due to poor weld quality, respectively. The data
from these tests was excluded.
 Driver et al. (1997) tested the first large-scale multi-story (four story) unstiffened steel
plate shear wall. The results of this test confirmed the researchers’ theory that the ductile
behavior of the thin, unstiffened plates is favorable for dissipating energy under extreme
cyclic loading. The load distribution used in this test applied essentially equal loads at all
stories. Coupled with the fact that the first two stories had the same web plate thickness,
this resulted in the first story having most of the inelastic behavior while the upper stories
remained essentially undamaged. Thus, the data extracted for this specimen comes from
the response of the first story only.
 Lubell (1997) tested three SPSW specimens, two one-story frames and one four-story
frame. Study objectives were to determine the load-deformation response of the systems,
evaluate design guidelines, and verify the use of steel plate shear walls for high seismic
areas. The study concluded that boundary frame members must have adequate stiffness to
develop the strength of the web plate and ensure a ductile response. Similar to the fourstory test by Driver et al. (1997), the four-story test here had concentrated damage on the
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first story due to the load distribution and plate thickness. Therefore, the data from this
specimen used here comes from the response of the first story.
 Behbahanifard (2003) tested a three-story SPSW, which consisted of the upper three
undamaged stories of the original specimen tested by Driver et al. (1997). Loading was
applied in a similar manner as the test by Driver et al. (1997). It was found that the lower
web plates absorbed far more energy than the top-story, with energy dissipation for the
first to third stories being 65%, 30%, and 5% of the total. It was concluded that SPSWs
have great potential for high-energy dissipation, as was demonstrated through the shear
wall’s high shear capacity and ductility. Because of their different web plate thicknesses
and the difference in loading and damage distribution, each story of this specimen is
treated as a different specimen here.
 Berman and Bruneau (2003) tested three one-story SPSW specimens to investigate the
behavior of SPSWs with light-gauge steel plates and develop simple retrofit solutions.
Two specimens (F1 and C1) had web plates connected to VBEs and HBEs using a
combination of bolts and epoxy while the third used a combination of bolts and welds
(F2). Results of the tests showed that the small-gauge web plates were effective in
reducing strength demands on the boundary elements. Only the test specimen (F2), which
had a welded connection of the web plate to fish plates that were bolted to the boundary
frame members was included in the current study (F2); the test with a corrugated web
plate and those with epoxy connections were excluded.
 Vian and Bruneau (2005) tested four one-story SPSW systems. The goal of the study was
to investigate new designs for SPSWs including top and bottom HBEs with reduced beam
sections (RBSs), a perforated web plate, and a reinforced corner cutout. Testing showed
that the HBEs with RBSs were effective in controlling boundary frame yielding.
Additionally, both the perforated and reinforced corner cutout web plates were found to
be practical alternatives to solid infill SPSWs; both of these configurations allow for
utility access. Three of the specimens are considered here: Specimen S2 with a solid web
plate, Specimen P with a perforated web plate, and Specimen CR with reinforced corner
cutouts. Specimen S1 was a trial specimen that had a number of fabrication and test setup
deficiencies and is not included in the data set here.
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 Zhao (2006) tested four specimens: two steel systems with different story height-to-span
ratios and two composite systems. Only data from the steel systems were used in the
current study.
 Park et al. (2007) tested five three-story SPSW specimens, three of which had
seismically compact VBEs and two of which did not. The objective of this study was to
determine the impact on seismic performance of web plate thickness, column strength,
and column compactness.
 Qu et al. (2008) tested a full-scale two-story SPSW. The testing was conducted in two
phases: Phase I consisted of five individual pseudodynamic tests using various amplitude
ground motions. After Phase I testing was completed, the web plates were replaced.
Phase II testing consisted of quasi-static cyclic loading under displacement control to
failure. Only the damage data at failure was used for the current study. Load was applied
at both stories in this testing and was proportioned based on demand in the
pseudodynamic testing and a triangular distribution in the quasi-static testing. Thus,
damage from all stories of this specimen is considered in this research.
 Choi and Park (2009) tested five three-story SPSWs with different infill web plate-toboundary element connections, infill web plate-to-boundary element connection lengths,
and wall openings. One specimen (FSPW5) was a coupled-wall; data from this test were
not used in this study.
Ultimately, experimental data were collected for more than 30 specimens from 12 test
programs (Table 1). Specimens are identified in Table 1 using the last name of the first
author for cited research reference, which is abbreviated when the name is long, and an
identification number that is similar to that used in the original research.
The specimens listed in Table 1 have designs representative of modern SPSW, i.e. they
rely on tension field action to resist applied lateral loads, and were subjected to simulated
earthquake loads in the laboratory. However, there are differences in specimen design and
load characteristics that could be expected to result in variability in the observed damage
patterns. Selected characteristics are listed in Table 1 and described briefly below.
 Scale: The specimen scale is computed as the laboratory specimen story height divided
by an assumed full-scale story height of 3.05 m.
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 VBE base fixity: The fixity of the VBE bases varied between experimental programs
depending on the research objectives and laboratory constraints. Some SPSW test
programs utilized fixed-base VBEs while others used large pins to allow free rotation of
the VBE bases. This factor impacts the development of plastic hinging at the VBE base,
which would be expected in SPSWs with fixed VBE bases but not in those with pinned
VBE bases.
 Web plate thickness: The web plate thickness of each test specimen is given. For the
multistory specimen, as noted in the discussion above, some specimen had significantly
different damage at each story due to the combination of applied load distribution and
web plate thickness. In these cases each story is treated as a separate test specimen, i.e.,
the specimen from Behbahanifard (2003), Driver et al. (1997) and Lubell (1997), where
in the latter two cases only the first story response is considered as there was little
damage elsewhere. Thus web plate and specimen properties are given for only the first
story in those cases. In the test by Qu et al. (2008) the damage was well-distributed and
both stories are treated as one specimen, with story drift corresponding to the maximum
of the first or second story drifts.
 Web plate steel yield strength: The web plate yield strength is the average yield strength
value from reported coupon tests. When coupon tests were not reported, the nominal
yield strength is listed.
 Aspect ratio (Lp/hp): The specimen aspect ratio is the length of the web plate, Lp, divided
by the height of the web plate, hp; the centerline dimensions of the frame are not
considered here.
 The Lp/tw ratio: The width-to-thickness ratio of the web plate length, Lp, divided by the
web plate thickness, tw.
 HBE-to-VBE connection: Some specimens had fully restrained HBE-to-VBE
connections, thereby ensuring complete participation of the boundary frame, while others
had partially restrained connections such as shear tab or web angle connections. In
general, those specimens with fully restrained connections are more likely to have
boundary frame damage as flexural demands from frame action can be significant.
 Meets certain seismic criteria: This Table 1 entry indicates whether the specimen
conformed to certain aspects of the AISC Seismic Provisions (AISC 2005). The specific
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requirements considered are the compactness requirements for VBEs and HBEs and the
VBE stiffness per the requirement:
twh 4
I c  0.00307
L
(1)
where Ic is the moment of inertia of the VBE, h is the centerline height of the VBE and
L is the centerline length of the HBE. Also considered was the HBE-to-VBE connection

types as only those systems
with fully restrained connections are identified as meeting
the criteria. However, not all fully restrained connections conform to the requirements
of the AISC Seismic Provisions, which requires connections used in SPSWs to be
prequalified for use in special or intermediate moment resisting frames. Because of
specimen scale and inadequate documentation it is difficult to determine whether the
connection details in older tests meet prequalification requirements, thus the only
distinction that could be made is whether the connections were fully or partially
restrained. Further, specimens denoted as meeting certain seismic criteria may have had
VBE base connections that did not conform to the AISC Seismic Provisions as those
details were again difficult to verify from the literature. Thus, the specimens that are
said to meet the certain seismic design criteria may not be completely representative of
new construction designed for large seismic demands.
DAMAGE STATES AND REPAIR METHODS
DAMAGE STATES
Twelve damage states were used to describe observed SPSW damage and identify the
damage that may be easily linked to repair options. These damage states group together
damage that: (i) occurred at comparable drift levels and (ii) resulted in similar severity of
damage to the system. The resulting twelve damage states, denoted DS 1 through DS 12, are
described below.
DS 1: Elastic Web Plate Buckling
Web plates undergo elastic shear buckling at low load and drift levels due to their high
slenderness ratio. The tension field immediately develops in the web plate as buckling
occurs. Typically, there is no strength degradation associated with web plate buckling; some
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stiffness degradation may occur if the web plate has a very high slenderness ratio. In
experiments, web plate buckling is identified visually (Fig. 4a) and is often accompanied by
audible popping sounds (Driver et al. 1997). Since this damage state occurs prior to web plate
yielding there is no residual web plate buckling when the SPSW returns to its initial position.
DS 2: Web Plate Yielding
The onset of web plate yielding may be difficult to detect in the laboratory. Researchers
often identify web plate yielding by “flaking of whitewash” or the formation of visible lines
in the material running across the plate (Fig. 4b). Here web plate yielding is taken as the first
mention of yielding in the experimental observations. Therefore, the data are highly variable
as some researchers note yielding that is more localized than that noted by others.
DS 3: Residual Web Plate Buckling
This damage state is defined as the first occurrence of residual buckles that can be
visually identified in the specimen after it has returned to zero applied lateral load and, in
many cases, is associated with residual drift. This permanent damage indicates that the web
plate has yielded and accumulated significant inelastic deformation. As with web plate
yielding, identification of the onset of residual web plate buckling is somewhat subjective,
with different researchers identifying the onset of this damage state at slightly different
buckling magnitudes. Fig. 4c shows residual web plate buckling after several post-yield
cycles.
DS 4: HBE and/or VBE Yielding
This damage state is defined by the first occurrence of yielding anywhere along the length
of any HBE or VBE. Here, the drift corresponding to the first mention of any yielding in any
of the boundary elements is taken as the drift associated with this damage state. Again
identification of onset of this damage state is somewhat subjective, and data are highly
variable. In the literature, research use both flaking of whitewash and strain gauge readings to
identify boundary elements yielding. Fig. 4d shows flaking of white wash on a VBE that is
taken to indicate VBE yielding.
DS 5: Initial VBE Local Buckling, and DS 6: Initial HBE Local Buckling
For the experiments considered here, local buckling of boundary elements is identified
visually. These damage states are reached when local buckling occurs anywhere along any
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VBE or HBE and on any part of the element’s cross-section. An example of the magnitude of
local buckling typically associated with these damage states is shown in Fig. 4e. Typically,
local buckling initiates in regions of plastic hinging; these regions are typically located at the
ends of the members in properly proportioned SPSWs. However, there are several specimens
in the data set in which plastic hinging of boundary elements occurred somewhere other than
the member ends.
DS 7: VBE Local Buckling Requiring Repair, and DS 8: HBE Local Buckling
Requiring Repair
These damage states are defined by significant local buckling of VBEs or HBEs requiring
repair by heat straightening but not requiring section replacement. Onset of these damage
states is determined from photographs and written descriptions of damage provided by
researchers. In general, this damage state is assigned when the local buckling is confined to
either the web or the flanges of the boundary element, does not result in large distortion of
the section, but is significant enough to require repair. While most researchers report the
onset of local buckling, most do not provide comprehensive documentation of the
progression of local buckling. Thus, identification of these damage states is difficult, and the
data set includes relatively few data for these damage states.
DS 9: Web Plate Tearing/Cracking
This damage state is reached when initial fractures develop in either the web plate or in
the welds that connect the web plate to the boundary elements. Tearing often initiates at the
welds in the corners of the web plates and may be quite small for several cycles following
identification, as shown in Fig. 4f. Tearing can also occur away from the corners of the web
plate due to plastic folding along buckling lines.
DS 10: VBE Cracking, and DS 11: HBE and HBE-to-VBE Connection Cracking
The initiation of cracking in a boundary element connection includes the development of
cracks in HBE-to-VBE connection elements such as angles, shear tabs or welds as well as
fractures developing in local buckling regions of VBEs or HBEs. The onset of these damage
states is associated with the first occurrence of fracture in any of these elements, with the
connection elements lumped together with the HBEs. The initial fractures are quite small, as
shown in Fig. 4g, but tend to propagate relatively quickly with increasing drift and result in
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rapid degradation of SPSW strength and stiffness. Most often seen in the literature are
cracking of the VBE-to-base plate connection and cracking at the location of local buckling
of the VBE flange.
DS 12: Connection and/or Boundary Frame Failure
The onset of this damage state is associated with buckling or fracture of any boundary
elements or connections that is significant enough to require member replacement or that
may endanger the stability of the system. This includes section distortions from severe local
buckling and significant fracture of HBE-to-VBE connections, as shown in Fig. 4h. This
level of damage is typically associated with the end of a test and typically causes a substantial
loss of system strength and stiffness. Researchers regularly associated this level of damage
with system failure.
METHODS OF REPAIR
The twelve damage states above were used as a basis for identifying a series of repair
states that describe repair activities required to approximately restore a SPSW to preearthquake strength and stiffness. The five proposed repair states, denoted RS 1 through RS
5, are described below and are related to the above damage states as shown in Table 2.
RS 1: Cosmetic Repair
The first repair state includes all damage states that do not require structural repair. Initial
yielding of web plates and boundary elements results in minimal permanent deformation and
should not necessitate repair. Thus, this repair state is associated with DSs 1, 2 and 4. DSs 5
and 6 (initial local buckling of boundary frame members) would also be represented in RS 1,
however, in all tests considered here DS 2 or 4 occurred prior to DS 5 or 6.
RS 2: Replace Web Plate
Web plate replacement is required when residual buckling or web plate cracking becomes
significant. Even though SPSW strength and drift capacity are not affected by residual
buckling, significant residual buckling can result in reduced SPSW stiffness that may lead to
unacceptable story drifts under wind loading. Web plate cracking can cause loss of strength
and stiffness. Thus, RS 2 is achieved when the first of DS 3 or 9 occurs and in all cases
considered here DS 3 occurred at lower drift than DS 9. Web plate replacement is typically
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accomplished without disturbing the fish plate, or other connection detail, used to connect the
web plate to the boundary elements and is a relatively simple repair.
RS 3: VBE Repair
FEMA 352 (FEMA 2000) recommends that frame members be repaired when local
buckling exceeds rolling tolerances. Heat or flame straightening is the preferred method of
repair and can be effective for repairing even large buckles. Stiffeners may also be added in
the case of web buckling. If heat straightening is not an option, or no workers skilled in the
method are available, the damaged portion of the VBE flange can be removed and replaced
with a new plate. If VBEs exhibit limited cracking, fracture or tearing (DS 10), repair is also
required. For these cases FEMA 352 (FEMA 2000) provides conceptual details for a number
of repair options, depending on the extent of the fracture, that range from replacing small
sections of flange to replacing longer section of both flange and web.
Repair of VBE local buckling or fracture defines a unique repair state as shoring costs
associated with VBE repair generally make VBE repair significantly more expensive than
HBE or connection repair. This repair state is reached with the onset of either DS 7 (local
buckling of VBE) or DS 10 (VBE cracking).
RS 4: HBE and Connection Repair
As with VBEs, local buckling or limited cracking or fracture of HBEs requires repair to
restore stiffness and strength. For HBEs or connections, the repair activities are the same as
for VBEs; however, shoring of the member to transfer gravity load around the damaged
region and to the foundation is not required. Thus, repair activities for HBEs and connections
are significantly less effort and expensive than for VBEs and a unique repair state is proposed
for HBE and connection repair. FEMA 352 (FEMA 2000) provides details and procedures
for repair of beams and beam-to-column connections in moment resisting frames that are
applicable to SPSWs.
RS 5: Replace Boundary Elements or Frame
This repair state consists of replacing VBEs, large sections of HBEs or the entire SPSW.
Typical damage associated with this upper-bound repair includes extreme local buckling of a
VBE, global buckling of a VBE, and full connection or member fracture. The damage states
associated with this repair state must be severe enough to cause substantial loss of capacity
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and thus require extensive repairs. For the current study, DS 12, connection and/or boundary
frame failure is assumed to trigger this repair state.
GENERATION OF FRAGILITY FUNCTIONS
In the current study, a fragility function is a cumulative distribution function (CDF) that
represents the likelihood a system or component will reach or exceed a specific damage state
and, as a result, require a specific method of repair given a specific level of earthquake
demand defined by an engineering demand parameter (EDP). The fragility functions
presented here define the likelihood that the boundary elements or web plate in a story of a
SPSW system will require repair or replacement given the maximum story drift experienced
under the simulated earthquake loading. Story damage and maximum story drift are
employed to enable use of the fragility functions for prediction of damage for any story in a
structure. The experimental data described above are used to determine appropriate
lognormal cumulative distribution function parameters, and a standard, statistical goodnessof-fit test is employed to verify the lognormal distribution and computed distribution
parameters.
OUTLIERS
Pierce’s criterion as described in ATC-58 (ATC 2009) is recommended for identifying
outliers in collected experimental data to be used in the development fragility functions for
performance-based seismic design. Outliers can occur randomly, but they can also be
indicative of an error in the testing procedure. In the latter case, it is important to remove the
outlier from the data set so that it does not affect the empirical model. Peirce’s criterion was
applied separately to the damage state data and repair state data using the full data set in both
cases. It was determined that the damage state data set included three outliers, one each in DS
5 (data point from Zhao (2006) Specimen 2), DS 9 (data point from Park et al. (2007)
Specimen F2) and DS 12 (data point from Park et al. (2007) F4). These outliers were
removed from the damage state statistics described below. No outliers were found in the
repair state data.
DAMAGE & METHOD OF REPAIR DATA
Damage and demand (maximum story drift) data were collected for each specimen in the
data set, and damage was related to the method of repair required to restore the system to pre13
earthquake condition. Table 3 lists the median story drift at the onset of each damage state,
the coefficient of variation in the drift, and the number of data points. Table 4 provides
similar statistics for the repair states, and Fig. 5a shows repair state versus drift. Statistics are
provided for the entire dataset and for three groups of specimens: SPSWs designed to meet
certain seismic design criteria as described above, non-seismically designed SPSWs, and
SPSWs with RBS beam-to-column connections. Specimens were organized into these three
groups as these design parameters were found to impact damage progression and required
repair. Additional discussion regarding the characteristics of the SPSWs included in these
groups is provided below. Note that in using damage state-drift data to generate repair statedrift data sets, if for a single test specimen multiple damage state-drift data points were
associated with a particular repair state, only the data point with the lowest drift was included
in the repair state dataset.
All SPSWs in the total data set with RBS connections also satisfy the seismic design
requirements described above and no SPSWs with RBS connections are included in the
“meets certain seismic design criteria” dataset. Further, all SPSWs that meet the specified
seismic design criteria and do not have RBS connections had fixed VBE bases and thus
would be expected to exhibit proper VBE damage states due to the development of plastic
hinges at the bases. Therefore, the statistics for the group “meets certain seismic design
criteria” should be applicable to most modern SPSW designs that do not have RBS
connections. The dataset of SPSWs with RBS connections contains data from four tests with
different characteristics: three single story SPSW tests from Vian and Bruneau (2005) that, as
described above, had different web plates (standard, perforated, and with reinforced corner
cutouts), and one two-story SPSW with a fixed base from Qu et al. (2008) that was tested
with pseudo-dynamic loading prior to cyclic loading to failure. Since these tests have such
different characteristics, their damage state and repair state statistics are reported for the
group as a whole but fragility functions will not be developed due to the lack of data for
similar configurations. In the dataset for non-seismically designed SPSWs there are three
specimens with pinned VBE bases, and all of these also had large boundary elements which
remained essentially elastic during the tests. Damage and repair state data for those
specimens are included in the statistics and fragility functions for the non-seismic design
group. Although the pinned base specimens did not suffer boundary frame damage and do
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not appear in the statistics for those damage states, they did have web plate damage, and in
one case had HBE-to-VBE connection damage, therefore contributing to the statistics for
those damage states.
A preliminary review of the data in Tables 3 and 4 and Fig. 5a shows the following. First,
the median drift at onset increases with increasing damage and repair state number, indicating
that the chosen ordering of the states is appropriate and is consistent with progressive
damage. Second, the data in Table 4 and Fig. 5 show, for some repair states, a relatively
small drift differential between the onset of one repair state and the next; for some
applications it may be appropriate to combine the damage data into fewer repair states so that
the drift differential between repair states is larger. Third, the data in Table 4 and Fig. 5a
show that for most repair states there are a sufficient number of data points to enable
calibration of fragility functions, with the exception of the SPSW with RBS connection group
which was discussed above.
IMPACT OF SPSW PARAMETERS ON REPAIR STATE DATA
The SPSW test specimens in the assembled data set span a wide range of design
parameters. The impact of various parameters on damage progression was investigated to
determine if different suites of fragility functions should be developed for SPSWs with
different design parameters. Specimen parameters investigated and discussed here include the
whether the SPSW met certain seismic design criteria as described above (which includes
HBE and VBE compactness requirements), VBE base fixity, HBE-to-VBE connection type,
web plate aspect ratio, width-to-thickness ratio of web plates, and test specimen scale.
Fig. 5b shows repair state versus story drift with data sorted on the basis of whether the
specimens met the specific seismic design criteria described above or not with SPSWs with
RBS HBE-to-VBE connections separated; Table 4 shows the median and COV values for the
sorted data. These data show that RS 2 (Replace Web Plate) occurs at lower drifts in
specimens that met certain seismic design criteria and that RS 3 occurs at larger drifts. The
lower drifts associated with RS 2 are likely due to the larger stiffness of the boundary frame
members in SPSWs that meet the seismic design requirements, such frames would not be
expected to deform significantly under the pull-in action of the tension field in the web plate
and, as a result, could be expected to develop larger web-plate tension strains and exhibit
15
earlier web plate yielding at lower story drifts. The increase in drift for RS 3 (VBE Repair)
for specimens that met the specified seismic design criteria could also be expected since local
buckling is delayed with the more stringent seismic compactness requirements. However,
there are only 5 data points defining the median drifts for RS 3, so this observation cannot be
considered statistically significant.
Fig. 5b and Table 4 provide repair state data for test specimens that have RBS HBE-toVBE connections. These data show that only specimens with RBS connections exhibit HBE
damage requiring repair (RS 4). For specimens without RBS connections, accumulation of
VBE damage typically resulted in system failure. The purpose of the RBS connections is to
protect the VBEs, and failure of SPSW test specimens with RBS connections resulted from
fracture at the RBS rather than buckling or fracture of the VBEs. Thus, the data show that the
introduction of RBS connections was successful in protecting the VBEs.
The impact on damage and repair of the VBE base connection was also examined. While
collecting drift-damage data from the literature, it was observed that failure of the VBE baseplate connection and/or weld was common. This failure mode was observed in multiple
specimens tested by Choi and Park (2009), Park et al. (2007), Zhao (2006), and Chen (1991).
Failure of the VBE base-plate connection was typically observed at story drifts of 2-2.5%
when it occurred. Fig. 5c compares the occurrence of the five repair states for specimens with
and without fixed base VBE conditions. As shown, base fixidity appears to affect RS 2, RS 4
and RS 5. However, there are only six non-fixed base specimens in the data set (one each
from Timler and Kulak (1983), Tromposch and Kulak (1987), and Berman and Bruneau
(2003), and three from Vian and Bruneau (2005)). Of those, the specimens from Timler and
Kulak (1983) and Tromposch and Kulak (1987) were not tested to drifts large enough to
reach either RS 4 or RS 5. The specimen from Berman and Bruneau (2003) had partially
restrained HBE-to-VBE connections and the specimens from Vian and Bruneau (2005) had
RBS VBE-to-HBE connections. In the latter four tests there was no VBE damage as they
were protected by the weaker connections and ultimate failure was due to the connections.
Thus, the data are not statistically significant and do not provide conclusive evidence of the
impact of VBE base fixity on performance.
In addition to the parameters discussed above, the impact on damage and repair of
specimen scale, panel aspect ratio, and web plate width-to-thickness ratio was also
16
investigated. These parameters were found to have no discernable impact on the drift levels
at which the various repair states occur. It should be noted that the range of SPSW panel
aspect ratios in the data set is small, ranging from 1.0 to 2.5; thus, results may differ for panel
aspect ratios outside this range. Also, the web plate width-to-thickness ratios for the
specimens in the data set are all large, ranging from 223 to 3350, with the result that
specimens exhibited negligible shear buckling strength; thus, results may differ for SPSWs
that have smaller width-to-thickness ratios and, therefore, larger shear buckling strengths.
On the basis of the above observations about the impact on damage progression and
required repair of various SPSW design parameters, suites of fragility functions were
developed using the entire data set as well as data for specimens that i) satisfy the selected
seismic design requirements described above and do not have RBS connections, and ii) do
not satisfy the selected seismic design requirements. Because so few data points existed for
non-fixed-base specimens, a unique suite of fragilities was not developed for these specimens
and data for non-fixed-base specimens were merged with those for fixed-base specimens.
Similarly, fragilities were not developed for SPSWs with RBS connections at this time due to
insufficient data. However, the authors are developing an online database, to be uploaded to
NEESHub (http://nees.org/resources/databases), which will enable the development and
update of the fragility parameters as new data becomes available.
CALIBRATION OF FRAGILITY FUNCTIONS USING THE METHOD OF MAXIMUM
LIKELIHOOD
Cumulative distribution functions (CDF) were developed using the data sets for each RS.
To facilitate use in practice, drift data were assumed to be lognormally distributed with CDF
 ln  x  x  
Fx   

 x

where 
(2)
 is the standard normal distribution, x is the median drift, and x is the standard
deviation of the natural log of the drift, which is often referred to as the dispersion.
Distribution parameters were determined from the data using the Method of Maximum
Likelihood as implemented in Matlab (Mathworks 2008). The Method of Maximum
Likelihood employs the likelihood function, L:
17
n
L   f x ( xi , x ,  x )
(3)
i 1
where f x ( xi , x ,  x ) is the lognormal probability distribution function for the random
variable. The method determines the distribution parameters that “make the observed data the
most likely”. Using this approach, the computed distribution parameters do not include errors
associated with estimating the population mean and standard deviation from the sample data
set.
Table 5 lists distribution parameter, x and x, computed using experimental data and the
Method of Maximum Likelihood. These values are similar to those listed in Table 4. With the
exception of the fragility for RS1 and the entire data set, none of the lognormal fragilities
defined by the x and x values listed in Table 5 pass the Lilliefors goodness-of-fit test at the
5% significance level.
For use in practice, a second dispersion values, β, was determined following the
recommendations of ATC-58 (ATC 2009); here β = √β2x + β2u where βx is the dispersion
computed using the collected data and the Method of Maximum Likelihood and βu is taken
equal to 0.25 for all cases to account for additional uncertainty associated with representation
of in situ conditions and, as appropriate, the limited nature of the experimental data set. This
second dispersion value, , is also listed in Table 5.
RECOMMENDED FRAGILITY FUNCTIONS
Table 6 lists lognormal distribution parameters recommended for use in practice for
predicting the performance of SPSWs. These parameters were determined from the data
listed in Table 5 using engineering judgment. Recommend values are provided for SPSWs
that meet the certain aspects of the AISC Seismic Design Provisions (AISC 2005) described
above, without RBS connections. These fragilities are considered to be appropriate for use
for some new and some existing construction. Recommendations are provided also for
SPSWs that do not meet AISC Seismic Design Provisions, as described above; these are
considered to be appropriate for use for existing SPSWs that have non-moment resisting
HBE-to-VBE connections.
18
The parameters in Table 6 were developed from the empirical data in Table 5 using
engineering judgment and reflect the impact of key design characteristics on SPSW
performance. For RS 1 (Cosmetic Repair), the dispersion values in Table 5 were reduced to
define a uniformly high rate of uncertainty for all types of SPSWs. For RS 2 (Web Plate
Replacement), the data in Table 5 show that SPSWs designed to satisfy specified seismic
provisions require web plate replacement at lower story drifts than non-seismic frames. As
previously discussed, SPSW meeting the specified seismic requirements have compact and
stiffer HBEs and VBEs, resulting in smaller frame deformations, larger web plate strains and
onset of RS 2 at lower drifts. Note that the data for the SPSWs with RBS connections in
Table 4 would imply larger median drifts for RS 3 and 5 when such connections are used,
however, due to the lack of data, unique fragilities were not developed for them. Further,
HBE damage requiring repair was not observed in SPSW without RBS connections; thus, for
these systems, no fragility function was developed for RS 4. However, as shown in Table 4,
RS 4 was observed for SPSWs with RBS connections suggesting that when additional data
becomes available, fragility functions for RS 4 for such systems should be developed.
Figure 6 shows the recommended lognormal fragility functions for each RS and each of
the three data sets described above as well as the discrete fragility functions defined by the
raw experimental data. As previously discussed, for RS 3 and RS 4, fragilities were
developed and experimental data exist only for subsets of the complete data set. The data
show that the recommended functions provide a reasonable fit to the experimental data, with
additional dispersion introduced per the recommendations of ATC-58.
It is important to note that improved SPSW design for optimal seismic performance is
still an active research topic. There are a number of ongoing experimental and analytical
research projects that will likely improve the data set, and in particular, result in larger drifts
at failure when particular details are employed. The use of RBS HBE-to-VBE connections
represents one such innovation that has already resulted in improved performance, although
the specifics of the design process are still being developed to maximize the drifts at failure.
The fragility functions developed here represent the use of the data available to date and can
be updated with available statistical procedures when new data becomes available.
19
CONCLUSIONS
Data on the performance of SPSW has been collected and analyzed. Observations from
experiments and an understanding of the system’s behavior have led to the development of
damage states and repair states, the latter linking the observed damage to the difficulty of the
subsequent repair. Five repair states are proposed that cover the full range of SPSW repairs
that may be expected following a seismic event, from cosmetic repair to member or frame
replacement. A data set consisting of results from 33 individual tests was developed and
damage and repair state information were extracted where available.
Using the collected experimental data, two sets of fragility functions were developed for
the proposed repair states. First, the Method of Maximum Likelihood was used to directly
calculate the fragility parameters from the data. Then, using the statistical results, engineering
judgment regarding the behavior of SPSWs, and increased dispersion as recommended by
ATC-58, recommended fragility functions for use in performance-based designed were
proposed. The resulting functions may be used in performance-based design of SPSWs to
predict the performance state of a SPSW following earthquake loading and, in combination
with cost-estimating data, to predict the cost and time required to complete structural repair
of the earthquake damaged system. The resulting functions can be easily updated as new data
become available.
REFERENCES
AISC, 2005. Seismic Design Provisions for Steel Buildings, ANSI/AISC 341-05, American
Institute of Steel Construction, Chicago, IL.
ATC, 2009. Guidelines for Seismic Performance Assessment of Buildings, ATC 58, 50%
Draft, Applied Technology Council, Redwood, CA.
Behbahanifard, M.R., 2003. Cyclic Behaviour of Unstiffened Steel Plate Shear Walls. Ph.D.
Dissertation. University of Alberta.
Berman, J.W. and Bruneau, M. (2003). Experimental Investgation of Light-Gauge Steel Plate
Shear Walls for the Seismic Retrofit of Buildings. Report MCEER-03-0001, MCEER,
Buffalo, NY.
Berman, J.W. and Bruneau, M. (2008). Capacity Design of Vertical Boundary Elements in
Steel Plate Shear Walls. Engineering Journal, 45, 57-71.
20
Chen, R., 1991. Cyclic Behavior of Unstiffened Thin Steel Plate Shear Walls. Ph.D.
Dissertation. University of Maine.
Driver, R.G., Kulak, D.J, Kennedy, D.J.L. and Elwi, A.E., 1997. Seismic Behavior of Steel
Plate Shear Walls. Structural Engineering Report 215, University of Alberta.
FEMA, 2000. FEMA 352 Recommended Post-earthquake Evaluation and Repair Criteria for
Welded Steel Moment-Frame Buildings, Building Seismic Safety Council for the Federal
Emergency Management Agency, Washington, D.C.
Lubell, A.S., 1997. Performance of Unstiffened Steel Plate Shear Walls Under Cyclic QuasiStatic Loading. MS Thesis. University of British Columbia.
Mathworks, 2008. MatLab User’s Manual. Mathworks Inc., Natick, MA.
Park, H.G., Kwack, J.H., Jeon, S.W., Kim, W.K. and Choi, I.R., 2007. Framed Steel Plate
Shear Wall Behavior under Cyclic Lateral Loading. Journal of Structural Engineering 133,
378-388.
Choi, I.R. and Park, H.G., 2009. Steel Plate Walls with Various Infill Plate Designs. Journal
of Structural Engineering 135, 785-796.
Qu, B., Bruneau, M., and Tsai, K.C., 2008. Experimental Investigation of Full-Scale TwoStory Steel Plate Shear Walls with Reduced Beam Section Connections. Report MCEER-080010, MCEER, Buffalo, NY.
Sabelli, R. and Bruneau, M., 2007. AISC Design Guide 20: Steel Plate Shear Walls,
American Institute of Steel Construction, Chicago, Illinois, 2007.
Thorburn, L. J., Kulak, G. L., and Montgomery, C. J., 1983. Analysis of Steel Plate Shear
Walls. Structural Engineering Rep. No. 107, University of Alberta.
Timler, P. A. and Kulak, G. L., 1983. Experimental Study of Steel Plate Shear Walls.
Structural Engineering Rep. No. 114, University of Alberta.
Tromposch, E.W. and Kulak, G. L. 1987. Cyclic and Static Behavior of Thin Panel Steel
Plate Shear Walls. Structural Engineering Rep. No. 145, University of Alberta.
Vian, D. and Bruneau, M., 2005. Steel Plate Shear Walls for Seismic Design and Retrofit of
Building Structures. Report MCEER-05-0010, MCEER, Buffalo, NY.
Zhao, Q. Experimental and Analytical Studies of Cyclic Behavior of Steel and Composite
Shear Wall Systems. Ph.D. Dissertation. University of California, Berkley. 2006.
21
Table 1. Design Details for Experimental Test Specimens
Specimen Scale
ID
Behb 1
Behb 2
Behb 3
Berman
Chen M22
Chen M14
Chen M12
Chen S22
Chen S14
Chen W
Chen O
Driver1
Lubell 1
Lubell 2
Lubell 41
Park F2
Park F4
Park B1
Park B2
Park S2
Park S4
Park S6
Park W4
Park W6
Timler 1
Tromp 1
Vian S2
Vian P
Vian CR
Zhao 1
Zhao 2
Qu2
VBE
Base
Fixity
Web Plate
Thickness
(mm)
60%
-
Fixed
60%
31%
31%
31%
31%
31%
31%
31%
60%
30%
30%
30%
38%
38%
38%
38%
40%
40%
40%
40%
40%
82%
72%
66%
66%
66%
50%
50%
130%
Pinned
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Fixed
Pinned
Pinned
Pinned
Pinned
Pinned
Fixed
Fixed
Fixed
4.8
3.4
3.4
1.0
0.8
1.8
2.5
0.8
1.8
0.8
0.8
4.8
1.5
1.5
1.5
4.0
4.0
4.0
4.0
2.0
4.0
6.0
3.0
6.0
5.0
3.4
2.5
2.5
2.5
6.4
9.1
1st:3.2
2nd: 2.3
Web Aspect
Lp/tw
HBE-VBE
Plate Ratio
Connection
Fy (Lp/hp)
(MPa)
359
1.8
571
Fully
269
812
269
812
241
2.5
3350
Partially
306
1.3
1510
Fully
291
1.3
603
Fully
295
1.3
430
Fully
306
1.3
1507
Partially
332
1.3
603
Partially
261
1.5
1770
Fully
276
1.3
1510
Fully
359
1.8
571
Fully
320
1.0
551
Fully
320
1.0
551
Fully
320
1.0
551
Fully
300
2.2
552
Fully
300
2.2
552
Fully
300
1.6
552
Fully
300
1.6
552
Fully
351
1.6
750
Fully
392
1.6
376
Fully
377
1.6
250
Fully
392
1.5
376
Fully
377
1.5
250
Fully
228
1.7
688
Partially
262
1.6
751
Partially
165
2.3
1360
RBS
165
2.3
1360
RBS
165
2.3
1360
RBS
248
2.0
324
Fully
248
2.0
223
Fully
248
1.0
1st:1160
RBS
2nd: 1610
Peak
Story
Drift
(%)
3.00
-
Meets
Certain
Seismic
Criteria
NO
-
3.65
1.86
2.02
1.72
1.82
1.20
2.02
2.02
3.08
6.70
5.44
1.64
5.40
5.30
3.60
3.60
3.40
2.60
2.60
1.70
1.30
1.12
0.80
3.00
3.00
4.00
3.20
3.20
5.20
NO
YES
YES
NO
NO
NO
YES
YES
NO
NO
NO
NO
YES
YES
YES
YES
YES
YES
YES
NO
NO
NO
NO
YES
YES
YES
YES
YES
YES
Notes: 1. Only data for the 1st story of the test specimen were used. 2. Web plate thickness was
different for 1st and 2nd stories; these thickness are provided. Data for both both stories were used
22
Table 2: Damage States vs. Repair States
1
Associated
Damage State
1, 2, 4, 5, 6
2
3, 9
3
7, 10
4
8, 11
5
12
Repair State
Table 3: Damage state data (in % drift).
Meets Certain
Seismic Criteria
All Data
DS
Median COV
Non-Seismic Design
RBS
# of
# of
# of
# of
Median COV
Median COV
Median COV
points
points
points
points
1
0.19
0.71
17
0.19
0.64
8
0.55
0.51
6
0.10
0.00
3
2
0.34
0.44
17
0.50
0.39
5
0.35
0.49
10
0.30
0.00
2
3
0.73
0.37
11
0.50
0.29
3
0.88
0.14
5
0.30
0.43
3
4
0.76
0.48
12
0.85
0.25
4
0.77
0.53
5
0.30
0.43
3
5
0.90
0.21
3
-
-
0
0.90
0.21
3
-
-
0
6
1.35
0.39
4
-
-
0
1.18
0.00
1
1.50
0.39
3
7
1.60
0.23
3
1.60
0.00
1
1.44
0.32
2
-
-
0
8
-
-
0
-
-
0
-
-
0
-
-
0
9
1.61
0.42
16
1.64
0.37
8
1.32
0.45
7
2.00
0.00
1
10
1.75
0.42
3
1.75
0.42
3
-
-
0
-
-
0
11
2.75
0.13
2
-
-
0
-
-
0
2.75
0.13
2
12
3.00
0.23
13
2.96
0.24
7
3.08
0.27
4
3.25
0.11
2
23
Table 4: Repair state data (in % drift).
Meets Certain
Seismic Criteria
All Data
RS Median COV
Non-Seismic Design
RBS
# of
# of
# of
# of
Median COV
Median COV
Median COV
points
points
points
points
1
0.35
0.48
19
0.50
0.39
5
0.37
0.51
11
0.30
0.00
3
2
0.73
0.37
9
0.50
0.29
3
0.88
0.14
5
0.60
0.00
1
3
1.60
0.16
5
1.60
0.17
3
1.44
0.32
2
-
-
0
4
2.75
0.13
2
-
-
0
-
-
0
2.75
0.13
2
5
3.00
0.28
12
3.00
0.30
6
3.08
0.28
4
3.25
0.11
2
Table 5: Repair state statistics (in % drift).
Meets Certain
Seismic Criteria
All Data
Non-Seismic Design
RS
x 
x

x 
x

x 
x

1
0.40
0.49
0.55
0.42
0.44
0.50
0.43
0.56
0.61
2
0.72
0.26
0.36
0.58
0.27
0.37
0.85
0.14
0.29
3
1.46
0.21
0.33
1.51
0.18
0.31
1.40
0.33
0.41
4
2.74
0.13
0.28
-
-
-
-
-
-
5
2.83
0.28
0.37
2.73
0.31
0.40
2.85
0.30
0.39
Table 6: Recommended values (in % drift).
Meets Certain
Seismic Criteria
Non-Seismic Design
RS
rec
rec
rec
rec
1
0.40
0.40
0.40
0.40
2
0.60
0.30
0.90
0.30
3
1.50
0.30
1.40
0.30
4
-
-
-
-
5
2.75
0.30
2.75
0.30
24
Figure Captions
Figure 1. SPSW configuration and nomenclature.
Figure 2. Shear buckling and diagonal field orientation during testing by Berman and Bruneau
(2005).
Figure 3. Base shear vs. drift for SPSWs with (a) partially restrained HBE-to-VBE connections from
Berman and Bruneau (2005) and (b) fully restrained HBE-to-VBE connections (Vian and Bruneau
2005).
Figure 4. (a) Elastic web plate buckling during testing (Berman and Bruneau 2005); (b) web plate
yielding (Vian 2005); white arrow shows a visible line in the web plate material; (c) residual plate
buckling (Berman and Bruneau 2005), (d) VBE yielding (Vian 2005), (e) HBE local buckling (Vian
2005), (f) web plate cracking (Vian 2005), (g) HBE cracking (Vian 2005), and (g) RBS connection
failure (Vian 2005).
Figure 5. Repair state data vs. story drift for: (a) all data, (b) all data sorted by whether the specimens
were designed to meet certain seismic design criteria and had RBS connections, and (c) all data sorted
by fixed and non-fixed base tests.
Figure 6. Recommended SPSW fragility functions and comparison with experimental data for: (a)
SPSWs meeting certain seismic design criteria and (b) non-seismic design.
25