Seismic Capacity Design and Retrofit of Reinforced Concrete Staggered Wall Structures
International Journal of Concrete Structures and Materials
Seismic Capacity Design and Retrofit of Reinforced Concrete Staggered Wall Structures
This study investigates the seismic performance of a staggered wall structure designed with conventional strength based design, and compares it with the performance of the structure designed by capacity design procedure which ensures strong column-weak beam concept. Then the seismic reinforcement schemes such as addition of interior columns or insertion of rotational friction dampers at the ends of connecting beams are validated by comparing their seismic performances with those of the standard model structure. Fragility analysis shows that the probability to reach the dynamic instability is highest in the strength designed structure and is lowest in the structure with friction dampers. It is also observed that, at least for the specific model structures considered in this study, R factor of 5.0 can be used in the seismic design of staggered wall structures with proposed retrofit schemes, while R factor of 3.0 may be reasonable for standard staggered wall structures.
staggered wall structures; seismic performance; capacity design; friction dampers
Reinforced concrete (RC) buildings having vertical shear
walls both as partition walls and as load resisting systems
have advantage in economic use of structural materials and
ease of construction using slip forms. The seismic
performance of RC shear wall structures have been widely
investigated by many researchers (Wallace 2012; Kim 2016).
The shear walls are also effective in preventing spread of fire
(Kang et al. 2016). However the buildings with shear
partition walls are not preferred these days mainly because the
plan layouts fixed by the shear walls fail to meet the demand
of people who prefer spatial variability. A staggered wall
structure has story-high walls placed at alternate levels,
which makes the system easier to remodel and consequently
more sustainable while the economy and constructability of
shear wall structures still maintained. Fintel (1968) proposed
a staggered system for RC buildings in which staggered
walls with attached slabs resist the gravity as well as the
lateral loads as H-shaped story-high deep beams, and
observed that the staggered wall systems would be more
economical. Mee et al. (1975) investigated the structural
performance of staggered wall systems by carrying out
shaking table tests of 1/15 scaled models. Lee and Kim
(2013) investigated the seismic performance of staggered
wall structures with middle corridor; Kim and Baek (2013)
conducted seismic risk assessment of staggered wall system
structures; and Kim and Lee (2014) proposed a formula for
fundamental natural period of staggered wall structures.
Recently seismic behavior factors of the system were
investigated based on the procedure recommended in the
FEMA P 695 (2009) (Lee and Kim 2013, 2015), and ATC
19 (1995) (Kim et al. 2016). The seismic performance of a
similar structure system in steel, the staggered truss system,
has already been investigated (Kim et al. 2015; Kim and
Kim 2017), and the system has been applied in many real
The staggered wall systems, however, have not been
widely applied in practice due mainly to the lack of
knowledge in the structural performance of the system. This
study investigates the seismic performance of a staggered
wall structure designed with conventional strength based
design, and compares it with the performance of the
structure design by capacity procedure which intends to ensure
strong column-weak beam behavior. Then the seismic
reinforcement schemes such as addition of interior columns or
insertion of rotational friction dampers at the ends of
connecting beams are implemented using the capacity design
procedure. Their effects on enhancing seismic load-resisting
capacity are validated by comparing their seismic
performances with those of the standard model structure.
2. Application of Energy Dissipation Devices
in Shear Wall Structures
Even though there is no known example of a staggered
wall structure with energy dissipation devices, many
researchers have been investigating the possibility of
mitigating seismic response of structures with shear walls using
dampers. Madsen et al. (2003) investigated the seismic
performance of viscoelastic damping systems placed
between shear walls at the coupling beam locations. Finite
element methods were used to analyze the effects of dampers
in these structural systems under different earthquakes
records. The results of the analysis of the 20-storey structure
with dampers in all levels illustrated that dampers could be
used to improve the mitigation of seismic forces. Chung
et al. (2009) proposed a friction damper that was applied
between coupled shear walls in order to reduce the
deformation of the structure induced by earthquake loads. It was
found that the control performance of the proposed friction
damper was superior to that of a coupled wall with a rigid
beam. Mao et al. (2012) proposed a shape memory alloy
(SMA) damper to be located in the middle of a coupling
beam in a coupled shear wall building. In this study it was
intended that, after earthquakes, deformation of the dampers
can recover automatically because of the pseudoelasticity of
austenite SMA material. Nonlinear time history analysis was
conducted for an 18-story frame-shear wall structure with
such SMA dampers to verify seismic response control effect
of this damper. MacKay-Lyons (2013) developed the
viscoelastic coupling damper (VCD) for RC coupled wall
highrise buildings. These dampers were introduced in place of
coupling beams to provide distributed supplemental
damping in all lateral modes of vibration. A parametric study has
been conducted to determine the optimal number and
placement of the dampers to achieve enhanced seismic
performance. Results highlight the improved performance of
VCDs over RC coupling beams at all levels of seismic
hazard. Pant et al. (2015) developed viscoelastic coupling
dampers to be located at coupling beams between two shear
walls and at outrigger beams. They applied the system to a
40-story RC structure and found that the viscoelastic
coupling dampers can be effective in reducing both structural
and nonstructural damage under MCE level seismic events.
The research results presented above confirm the
effectiveness of energy dissipation devices in the design of shear wall
3. Seismic Performance of Staggered Wall Structures
3.1 Configuration and Design of Staggered Wall Structures
In a typical staggered wall structure, the story-high RC
walls that span the width of the building are located along
the short direction in a staggered pattern. Figures 1 and 2
show the overall configuration and the structural plan of the
8-story example model structure, respectively, with 6 m long
staggered walls along the transverse direction and moment
frames located along the longitudinal direction. The two
staggered walls at both sides of the middle corridor are
connected with a 600 mm deep connecting beam. The
thickness of the staggered walls is 200 mm throughout the
stories. The staggered walls act like deep beams with the
depth of a story height, and are reinforced with vertical and
horizontal re-bars with diameter of 13 at 400 mm interval.
The horizontal shear force delivered from staggered walls
flows to the columns and staggered walls located below
through the 210 mm thick floor diaphragm.
In this section the seismic performance of the model
structure designed following the strength-based design procedure
currently specified in design codes is evaluated. In this
approach the structural members of the model structure are
designed in such a way that the ratio of the member force
demand to the design strength is maintained to be 0.8–0.9 for
combined gravity and seismic loads. The dead and live loads
are 7 and 2 kN/m2, respectively, and the design seismic load is
obtained using the seismic coefficients for short (SDS) and
1.0 s period (SD1) of 0.5 and 0.2, respectively, in the ASCE
7-13 (2013) format. The site class is assumed to be C and the
response modification factor of 3.0 is used. The fundamental
natural period of the model structure along the transverse
direction is computed to be 0.40 s. Table 1 shows the size and
rebar placement for the columns and the connection beams in
the first story of the analysis model structure. The X shaped
shear reinforcement composed of eight D13 rebars is provided
in the connecting beams as shown in Fig. 3 to prevent shear
failure prior to bending failure.
3.2 Analysis Modeling of the Structure
The staggered walls are modeled as deep beams using the
General Wall fiber elements provided in the PERFORM 3D
(2006) as shown in Fig. 4. The stress–strain material model of
Paulay and Priestley (1992) is used for concrete as shown in
Fig. 5a, in which the ultimate and yield strengths of concrete
are 24 and 14 MPa, respectively, and the residual strength is
defined as 20% of the ultimate strength. The strain at the
ultimate strength is 0.002, and the ultimate strain is defined as
0.004. The reinforcing steel is modeled with bi-linear force–
deformation relationship with the ultimate strength of
400 MPa as shown in Fig. 5b. The shear stress–strain
relationship of the staggered wall is modelled by bi-linear lines
with yield and ultimate strains of 0.004 and 0.012 respectively.
Overstrength factors of 1.5 and 1.25 are used for concrete and
reinforcing steel, respectively, in the nonlinear static and
Fig. 1 Schematic view of a staggered wall structure with
Fig. 2 Structural plan and elevation of the analysis model structure. a Structural plan and b elevation.
Fig. 4 Auto-sized fiber section for wall elements.
Nonlinear behavior of the RC columns located along the
perimeter in the longitudinal direction is modeled using the
‘FEMA Column, Concrete Type’ element in Perform 3D
developed based on an interpretation of the ASCE/SEI 41-13
(2013) Table 10-8. The nonlinear force–deformation
relationships are shown in Figs. 6 and 7. To define a column
plastic hinge, a moment-axial capacity interaction curve is
Fig. 3 X-shape rebars in the connecting beams.
Fig. 5 Nonlinear models for reinforced concrete. a Concrete and b reinforcing steel.
Fig. 6 Nonlinear model for columns.
calculated using the expected material properties. The limit
states defined in the ASCE/SEI 41-13 such as immediate
occupancy (IO), life safety (LS), and collapse prevention
(CP) are indicated in the curve. The back bone curve is
trilinear and My is assumed to be 80% of Mu. The unloading
stiffness was modeled to be equal to the initial elastic
stiffness. The energy degradation factor was set to be zero,
which results in the same unloading and reloading lines. The
analysis model for connecting beams located between the
two staggered walls are composed of two end rotation type
moment hinges and a middle shear hinge as shown in Fig. 8.
Fig. 8 Modeling of connecting beams.
Figure 9 shows the nonlinear model for connecting beams
defined in ASCE/SEI 41-13.
3.3 Seismic Performance of the StrengthDesigned Structure
To investigate the seismic performance and the collapse
mode of the model structure, pushover analysis is carried out
with the lateral load gradually increased proportional to the
fundamental vibration mode shape vector. Figure 10 depicts
the pushover curve which presents the base shear–roof
displacement relationship of the model structure. The points of
major plastic hinge formations and the maximum inter-story
drift of 2.5% are marked on the pushover curve. It is noticed
that plastic hinges occur first at the 5th story connecting
beams. After reaching the maximum strength, the strength
drops abruptly due to the formation of plastic hinges at the
5th story columns. Further strength drop occurs due to
formation of plastic hinges at the 4th story columns. It can be
observed that the maximum roof drift of the model structure
at the major strength drop is significantly smaller than that of
the structure at the maximum allowable inter-story drift of
2.5%. This implies that the strength-designed structure may
not have enough ductility to satisfy the required seismic
performance. Figure 11 shows the plastic hinge formation in
an exterior frame and the adjacent interior frame at the
maximum inter-story drift of 2% of the story height. The
magnitude of the plastic rotation is also indicated as a
percentage of the rotation corresponding to the CP state. It can
be observed that many plastic hinges exceeding CP state
form at the upper story columns and the plastic deformations
in the connecting beams are relatively small. Based on the
observation, it can be concluded that a typical staggered wall
structure designed by current design code behaves as a
Fig. 9 Nonlinear model for connecting beams. a Moment–rotation relationship and b shear force–deformation relationship.
Fig. 10 Pushover curve of the strength-designed structure.
Fig. 11 Plastic hinge formation in the strength-designed
structure at the maximum inter-story drift ratio of 2%.
4. Capacity Design Procedure
It is observed in the previous section that structural
damage is concentrated in the exterior columns rather than in the
connecting beams in the staggered wall structure designed
following the code-based approach. In this section a capacity
design procedure is applied to achieve the strong
columnweak beam design of the model structure so that the damage
in columns and the brittle failure mode observed in the
conventional design are prevented. To this end the model
structure is designed in such a way that the plastic hinges are
concentrated at the connecting beams while the other
members remain elastic. Similar approach has been
successfully applied to the design of special truss moment
frames by Chao and Goel (2006), who designed the special
segment in the truss girders using the plastic design
procedure. The design process was adopted to the AISC Seismic
The capacity design of the staggered wall system starts
from the design of the connecting beams based on their
required flexural strength at the ends. The target deformation
is set to be 2% of the story height in each story. The target
deformed shape and the desired plastic hinge formation are
depicted in Fig. 12. The required total bending capacity of
plastic hinges formed at the end of the connecting beams,
i¼1 Mpi; can be calculated from the following equilibrium
Fig. 12 Desired deformed configuration of the
where Fi is the equivalent seismic load obtained using the
vertical seismic force distribution method in the ASCE 7-13,
di ¼ hphi; hi is the height from the ground to the ith story, Mpi is
the required plastic moment of the connecting beams in Level i,
Mpc is the required plastic moment of columns in the first story, L
is the length of the transverse side of the structure, Lb is the length
of the connecting beam, hp is the given target drift, and h0p is the
rotation of the connecting beam which is LLb hp: In the above
equation the lateral seismic force and the geometric information
of the model structure are given values and the moment
capacities of the columns and the connecting beams are to be
determined. The moment capacity of the first story columns,
Mpc, can be obtained from the equivalence of the external and
internal works assuming that plastic hinges form at the base and
the top of the first story columns. In this study the total moment
capacity of the connecting beams obtained above is distributed
to each story proportional to the seismic story shear as follows:
Mpi ¼ Pn Vi
where Vi is the story shear in level i. The connecting beams
in each story can be designed using the required moment
capacity determined above using a resistance factor specified
in the design code. It is expected in this design process that
most connecting beams yield when a given lateral drift
occurs in the structure.
Fig. 13 Free body diagram of the model structure.
the combination of factored gravity loads and the maximum
vertical shear force developed at the connecting beams, Vp,
which is obtained as follows:
where Ry is the overstrength factor of 1.25 recommended in
the ACI 318-14 (2011). The required balancing lateral forces
applied on the left and right free bodies can be obtained as
follows, respectively, using the moment equilibrium:
In this study the total lateral forces obtained above are
vertically distributed using the vertical distribution factor
specified in the ASCE 7-13. The seismic story forces acting
on the left- and right-hand side free bodies used to design the
members outside the vierendeel panel are obtained as
Once the beams are designed using the plastic moment
obtained in Eq. (2), the next step is to design the exterior
columns and staggered walls in such a way that they remain
elastic when the connecting beams yield. The staggered
walls, which are deep beams with depth of the story height,
have significantly large stiffness and bending and shear
capacities, and therefore remain elastic at application of
design seismic load. Figure 13 shows the free body diagram
of a frame of staggered wall structure when all connecting
beams yield. To concentrate the plastic hinges at the
connecting beams when subjected to seismic load, the elements
other than the connecting beams should be designed to resist
The vertical distribution factor at level x, Cvx, specified in
the ASCE 7-13 is given by
where wx is the effective seismic weight of the structure at
level x, hx is the height from the base to level x, and k is an
exponent related to the structure period. The structural
elements other than the connecting beams are designed to
respond elastically for the gravity loads and the lateral load
computed above. The size and rebars of the first story
columns and the second story beams of the structure
designed by the capacity design procedure (model CD) are
presented in Table 1. It can be observed that the column
sizes of the model CD are slightly increased and the
longitudinal rebars in the connecting beams are slightly
reduced compared with those of the strength designed
structure (model SD). According to the eigenvalue analysis
results, the fundamental natural period along the transverse
direction is 0.41 s, which is almost the same with that of
the model SD.
Figure 14 shows the pushover curve of the model CD, in
which it can be observed that both the yield and the
maximum strengths of the structure are smaller than those
of the model SD. As in the model SD, the first yield of the
model CD occurs at the 5th story beam. However the
major drop of strength occurs as a result of yield of the 4th
story beam, and the residual strength is larger than that of
the model SD. Figure 15 depicts the maximum inter-story
drifts of the model structures at the point of the maximum
strength and at the maximum inter-story drift of 2.5%
obtained from pushover analysis. It can be noticed that
large inter-story drifts occur at mid-height, and the drift
patterns of the strength and the capacity-designed structures
are similar to each other. Figure 16 depicts the plastic
hinge formation in the model structures at the maximum
inter-story drift of 2.5% of the story height. It can be
observed that, even though the drift patterns are similar to
each other, the plastic hinge formations which cause the
drift are quite different. In the model SD, most columns are
subjected to CP (collapse prevention) level plastic
deformation, whereas significant plastic deformations are
concentrated in the connecting beams in the model CD. Even
though the plastic hinge formation and the failure mode
correspond well with those assumed in the design stage of
the model CD, significant increase in ductility cannot be
achieved due mainly to the insufficient plastic rotation
capacity of the beams. In the following section the
validities of two different seismic retrofit schemes are
investigated for enhancing seismic-load resisting capacity
of the staggered wall structure.
5. Seismic Retrofit Schemes for Staggered
5.1 Addition of Interior Columns
In a typical staggered wall structure, columns exist along
the perimeters in the longitudinal direction and the
connecting beams are located in alternate floors between two
staggered walls. As a means of enhancing seismic load
resisting capacity of the system, internal columns with
200 9 600 mm in cross section reinforced with 4-D19
rebars are added along the corridor. In the stories where
staggered walls exist, the internal end of the wall is
reinforced as a column. The internal columns are continuous
from the second to the top stories, and are designed to resist
only seismic load. In the structure with interior columns,
connecting beams can be placed in every story between the
internal columns. Figure 17 shows the structural plan and
elevation of the analysis model structure with interior
columns (model CD_IC). The same capacity design procedure
applied previously to design the model CD is applied to the
design of the retrofitted structure with internal columns. The
energy equilibrium equation applied in this model is similar
to that of the original model, except that the deformation of
each connecting beam located every story contributes to the
internal work as follows:
Also the required balancing lateral forces, obtained
similarly to Eqs. (4) and (5) considering the plastic moments of
the added connecting beams, are applied to the design of the
interior as well as the exterior columns. The member sizes of
the model structure with interior columns are presented in
Table 1, where it can be observed that both the column size
and the beam rebars decrease as a result of the addition of
interior columns. The fundamental natural period along the
transverse direction is reduced to 0.31 s due to the increased
5.2 Addition of Rotational Friction Dampers
In the second retrofit scheme, rotational friction dampers
are installed at the ends of connecting beams as shown in
Fig. 18a. Figure 18b depicts the bending moment–rotation
relationship of a typical rotational friction damper. The
insertion of rotational friction dampers significantly
Fig. 17 Analysis model structure with interior columns.
a Structural plan and b structural elevation.
increases the rotational capacity of the connecting beams.
Rotational friction dampers can be manufactured to have
larger deformation and energy dissipation capacity than
those of typical plastic hinges formed at beam ends. They
also can be reused after experiencing small to medium
earthquakes. The effectiveness of the friction dampers has
been verified by many researchers. Morgen and Kurama
(2008) carried out a seismic response evaluation tests of
Fig. 18 Rotational friction damper considered in this study.
a Installed configuration between staggered wall and
connecting beam and b moment–rotation
Table 2 Member size and rebars of the second story
coupling beams in the frame A.
unbonded posttensioned precast concrete moment frames
with friction dampers at selected beam ends. Mualla et al.
(2010) developed a rotational friction damper which can
produce maximum friction force as high as 5000 kN
For model structures with rotational friction dampers at the
ends of connecting beams, the same capacity design
procedure is applied to lead the formation of plastic hinges in the
connecting beams where the friction dampers are installed.
The slip forces of the dampers are determined in such a way
that their moment capacities are equal to the maximum
moment of the connecting beams. To ensure yielding of
dampers prior to other structural elements when the model
structure is subjected to design seismic load, the structures
are designed in such a way that the plastic hinges are
concentrated at the connecting beams and the other members
remain elastic. The failure point of the friction dampers at
which the friction force is lost is conservatively assumed to
be 0.3 rad based on the experimental results of rotational
friction dampers (Chung et al. 2009).
5.3 Seismic Performance of the Retrofitted Structures
Figure 19 shows the pushover curves of the model
structure retrofitted with the two schemes described above. The
pushover curve of the performance or capacity-designed
structure (model CD) is also presented for comparison. The
pushover curve of the structure with interior columns (model
CD_IC) is presented in Fig. 19a, where it is observed that
the maximum strength is significantly increased due to the
addition of the interior columns. The roof displacement at
the maximum inter-story drift of 2.5% is larger than that of
the model CD, which is more desirable in the sense that
plastic damage is not concentrated in a few stories but is
more uniformly distributed. Figure 19b depicts the pushover
curve of the model with friction dampers at beam ends
(CD_FD). It can be observed that, as the yield moments of
the friction dampers are equal to those of the beam ends of
the model CD, the yield strengths of the two models are
similar to each other. However, as the rotational capacity of
the friction dampers is much larger than that of the
connection beams, ductility is significantly increased compared
with the model CD. The increase in the roof displacement at
Fig. 19 Pushover curves of the retrofitted staggered wall structures. a Model CD_IC and b Model CD_FD.
Fig. 20 Plastic hinge formation in the structure with friction dampers. a Model CD_IC and b Model CD_FD.
the maximum inter-story drift of 2.5% is also significant
compared with that of the model CD. Figure 20 shows the
plastic hinge formation in the retrofitted structures at the
maximum inter-story drift of 2% of the story height. It can
be observed that most connecting beams yield as assumed in
the design process. However in the model CD_IC many
plastic hinges also form in the exterior and interior columns,
which results in sharp drop of strength compared with the
behavior of the model with friction dampers.
Figure 21 compares the hysteretic energy dissipated by
each structural element obtained by nonlinear dynamic
analysis of the models CD_IC and CD_FD using the
Northridge earthquake ground motion. It can be noticed that
the hysteretic energy dissipated in the beams of the model
CD_IC is only 93% of the total hysteretic energy, whereas
about 99% of the hysteretic energy is dissipated in the
friction dampers in the model CD_FD. This implies that
most structural elements are free from any damage during
the earthquake. It also can be observed that the total
hysteretic energy is significantly smaller in the model CD_IC
due to the increased redundancy and decreased plastic
6. Seismic Safety of the Model Structures
6.1 Collapse Margin of the Model Structures
In this section the validity of the capacity design approach
and the retrofit schemes for staggered wall structures is
verified by statistical seismic performance evaluation
procedure proposed in the FEMA P695 (2009). In this approach
nonlinear incremental dynamic analyses are conducted to
establish the median collapse capacity and collapse margin
ratio (CMR) for the analysis models. The adjusted collapse
margin ratio (ACMR) is obtained by multiplying the
collapse margin ratio (CMR), which is the ratio of the median
collapse intensity (SdCT ) and the MCE (maximum considered
earthquake) intensity (SMT), and the spectral shape factor.
Acceptable values of adjusted collapse margin ratio are
based on total system collapse uncertainty, bTOT, and
established values of acceptable probabilities of collapse.
To evaluate the seismic performance of the model structures
following the FEMA P695 process, incremental dynamic
analyses of the model structures are carried out using the 22
pairs of scaled far-field records provided by the PEER NGA
Database (2006). The general procedure for incremental
Fig. 21 Hysteretic energy dissipated by each structural element. a CD_IC and b CD_FD.
Table 3 Estimation of collapse margin of the model structures.
dynamic analysis is well documented in Vamvatsikos and
Cornell (2002). Figure 22 depicts the incremental dynamic
analysis results of the model structures using the 44
earthquake records. The collapse margin ratios (CMR) of the model
structures are obtained from the spectral accelerations at which
dynamic instability of the structures occurred for more than 22
earthquake records. The spectral accelerations corresponding
to the 16, 50, and 84% of failure probability are indicated in the
IDA curves. It can be observed that the median collapse
intensity (SdCT ), which is the 50% probability of failure,
increases in the retrofitted structures. Especially the
capacitydesigned structure with rotational friction dampers (model
CD_FD) shows the largest median collapse intensity. Similar
trend can be observed in the other failure probabilities. In this
study the total system collapse uncertainty is evaluated as 0.7
in accordance with Table 7-2 of FEMA P695. Table 3 shows
all parameter values used in the computation of the adjusted
collapse margin ratios of all model structures obtained from
the incremental dynamic analysis results. It can be observed
that the adjusted collapse margin ratios (ACMR) of all model
structures are larger than the acceptable values of ACMR 20%
provided in the FEMA P695 and the parameters used in the
seismic design of the model structures are valid. It also can be
noticed that the collapse margin of the model structure
designed with friction dampers, model CD_FD, is
significantly larger than those of the other model structures.
Fig. 23 Fragility curves of the model structures for reaching
the state of dynamic instability.
6.2 Fragility Analysis
Fragility analysis is carried out to investigate the failure
probability of each model structure for a given seismic
intensity. The seismic fragility is described by the
conditional probability that the structural capacity, C, fails to resist
the structural demand, D, given the seismic intensity hazard
and is modeled by a lognormal cumulative distribution
function as follows (Cornell et al. 2002):
C ¼ Uðln½D=C^ =bcÞ
where U½ = Standard normal probability integral,
C^ = median structural capacity associated with the limit
state, and bC = uncertainty in C. In this study the median
structural capacity is obtained from the incremental dynamic
analysis results of the model structures, and the fragility for a
given spectral acceleration is computed from Eq. (10) using
the total system collapse uncertainty of 0.7. The state of
dynamic instability is considered as the failure state, at
which the stiffness decreases lower than 20% of the initial
stiffness in the incremental dynamic analysis.
Figure 23 depicts the probability of reaching the failure
state, where it can be observed that, for the same spectral
acceleration, the collapse probability is highest in the
strength designed structure (model SD) and is lowest in the
structure with friction dampers (model CD_FD). The
spectral acceleration at the 50% probability of failure
(dynamic instability) is 0.7 and 1.3 g, respectively, for the
model SD and CD_FD. FEMA P695 requires that the
probability of failure of a structure corresponding to the
MCE level earthquake, which is 3/2 of the design level
spectral acceleration, be smaller than 0.1 so that the seismic
design variables used for the model structures are valid. This
condition is satisfied in all model structures and therefore it
can be concluded that the response modification factor of 3.0
is valid in the seismic design of the staggered wall structure.
Fragility analyses are also carried out for the four damage
states defined in the HAZUS (2010), which are Slight,
Moderate, Extensive, and Complete damage. The Complete
damage state is defined as the maximum inter-story
displacement at which the strength decreases to 80% of the
maximum strength in the pushover curve. The states of the
Slight damage and the Moderate damage were defined as the
spectral displacements corresponding to the 70 and the 100%
of the yield point, respectively. The Extensive Damage was
defined as the quarter point from the Moderate to the Complete
damage. Figure 24 depicts the fragility curves of the analysis
model structures, where it can be observed that the structures
designed with friction dampers have significantly lower
probability of reaching the collapse state than the
strengthdesigned structure. It can be observed that the probability of
the model CD to reach the Moderate damage state is slightly
larger than that of the model SD. However those to reach the
Extensive and the Complete damage states are somewhat
decreased. This implies that the capacity design applied in this
study is only effective for large earthquakes which cause
severe damage to structures. In the model CD_IC the
Table 4 Member size and rebars of the first story exterior
columns designed using R = 5.
probabilities of reaching the Slight and the Moderate damage
states are somewhat lower than those of the other models, and
those of reaching the Extensive and the Complete damage
states are between the models SD/CD and the CD_FD. The
probabilities of the model CD_FD to reach the Slight and the
Moderate damage states are similar to those of the model SD.
However decrease in the probabilities of reaching the
Extensive and the Complete damage states are most significant in
7. Response of Structures Designed
with Higher R Factor
It is observed in the previous section that both the standard
and the retrofitted model structures designed using the
response modification (R) factor of 3 satisfy the FEMA P695
requirements. In this section all model structures are
redesigned with increased R factor of 5 (i.e. with reduced seismic
load) and their seismic performances are compared with
those of the structures designed with R = 3. Tables 4 and 5
show the size and rebars of the selected members in the
structure designed using R=5. Figure 25 shows the pushover
curves of the model structures designed with reduced
seismic load, where it can be observed that the overall strengths
of all model structures are significantly reduced as a result of
using increased R factor. The strength of the model structure
Fig. 25 Pushover curves of the model structures designed
using R = 5.
Fig. 27 Fragility curves of the structures designed using
R = 5 for reaching dynamic instability.
with interior columns is the highest among the all model
structures, and the ductility is highest in the structure with
friction dampers. As observed in the previous sections, the
major strength drop of the strength-designed model SD
occurs due to plastic hinges in the exterior columns whereas
the other structures designed by capacity design process fail
by concentration of plastic hinges in beams. These
observations can be verified by the plastic hinge formations at the
maximum inter-story drift of 2.0% presented in Fig. 26,
where it can be noticed that the plastic hinge formations of
the redesigned structures are quite similar to those of the
structures designed using R = 3 except the fact that plastic
hinges are more concentrated in the connecting beams in the
model CD_IC. The plastic hinge formation in the model
CD_FD is the same with that of the model CD.
Figure 27 depicts the fragility curves of the four model
structures obtained from incremental dynamic analyses
using the 44 earthquake records. Compared with the fragility
curves of the model structures designed with R = 3 shown
in Fig. 23, the collapse probabilities are significantly
increased in the structures designed using R = 5. It can be
noticed that the collapse probabilities of the models SD and
CD exceed 0.1 which is the limit state specified in the
FEMA P 695 to validate the seismic design parameters used,
while those of the retrofitted structures are still far below the
limitation. Therefore based on the analysis results it can be
concluded that the structures retrofitted with interior
columns or friction dampers may be designed using higher R
factor than 3.0.
This study investigated the seismic performance of a
staggered wall structure designed with conventional strength
based design, and compared it with the performance of the
structures designed by capacity design procedure. Then the
seismic reinforcement schemes such as addition of interior
columns or insertion of rotational friction dampers at the
ends of connecting beams were validated by comparing their
seismic performances with those of the standard model
According to pushover analysis, the strength-designed
structure failed due mainly to failure of exterior columns,
whereas in the capacity designed structures major strength
drop occurred due to plastic hinge formation in beams.
Fragility analysis showed that the probability to reach the
dynamic instability was highest in the strength designed
structure and was lowest in the structure with friction
dampers. The capacity design applied in this study turned out to
be most effective for large earthquakes which cause severe
damage to structures. In the model with interior columns, the
probabilities of reaching the Slight and the Moderate damage
states were quite significant, and the probability of reaching
the Extensive and the Complete damage states decreased
most substantially in the structure with friction dampers. The
collapse probabilities of all model structures designed with
the R factor of 3.0 were smaller than 0.1, which confirmed
that the seismic design variables used for the model
structures were valid. However in the structures designed with the
R factor of 5.0, the collapse probabilities turned out to be
less than 0.1 only in the structures retrofitted with interior
columns or friction dampers. Based on the analysis results of
the specific analysis model structures considered in this
study, it was concluded that R factor of 5.0 might be used in
the seismic design of staggered wall structures with
proposed retrofit schemes, while R factor of 3.0 might be
reasonable for standard staggered wall structures.
This paper was supported by Sungkyun Research Fund,
Sungkyunkwan University, 2016.
This article is distributed under the terms of the Creative
C o m m o n s A t t r i b u t i o n 4 . 0 I n t e r n a t i o n a l L i c e n s e
permits unrestricted use, distribution, and reproduction in any
medium, provided you give appropriate credit to the original
author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
ACI 318. ( 2014 ). Building code requirements for structural concrete (318-14) and commentary . Farmington Hills, MI: American Concrete Institute.
AISC. ( 2010 ). Seismic provisions for structural steel buildings , AISC 341-10. Chicago: American Institute of Steel Construction.
ASCE 41- 13 . ( 2013 ). Seismic Rehabilitation of Existing Buildings . Reston: American Society of Civil Engineers.
ASCE 7-13 . ( 2013 ). Minimum design loads for buildings and other structures . Reston: American Society of Civil Engineers.
ATC- 19 . ( 1995 ). Structural response modification factors . Redwood City, CA: Applied Technology Council.
Chao , S.-H. , & Goel , S. C. ( 2006 ). Performance-Based Plastic Design of Seismic Resistant Special Truss Moment Frames. Report No. UMCEE 06-03. Department of Civil and Environmental Engineering , University of Michigan, Ann Arbor, MI.
Chung , H. , Moon , B. , Lee , S. , Park, J. , & Min , K. ( 2009 ). Seismic performance of friction dampers using flexure of RC shear wall system . The Structural Design of Tall and Special Buildings , 18 , 807 - 822 .
Cornell , C. A. , Jalayer , F. , Hamburger , R. O. , & Foutch , D. A. ( 2002 ). The probabilistic basis for the 2000 SAC/FEMA steel moment frame guidelines . ASCE Journal of Structural Engineering , 128 ( 4 ), 526 - 533 .
FEMA P695 . ( 2009 ). Quantification of building seismic performance factors . Washington, DC: Federal Emergency Management Agency.
Fintel , M. ( 1968 ). Staggered transverse wall beams for multistory concrete buildings . ACI Journal , 65 ( 5 ), 366 - 378 .
HAZUS-MH 2. 1 . ( 2010 ). Technical Manual Washington. Washington, DC: Federal Emergency Management Agency.
Kang , J. , Yoon , H. , Kim , W. , Kodur , V. , Shin , Y. , & Kim , H. ( 2016 ). Effect of wall thickness on thermal behaviors of RC walls under fire conditions . International Journal of Concrete Structures and Materials , 10 , 19 - 31 .
Kim , D. K. ( 2016 ). Seismic response analysis of reinforced concrete wall structure using macro model international journal of concrete . Structures and Materials , 10 ( 1 ), 99 - 112 .
Kim , J. , & Baek , D. ( 2013 ). Seismic risk assessment of staggered wall system structures . Earthquake and Structures , 5 , 607 - 624 .
Kim , J. , Jun , Y. , & Kang , H. ( 2016 ). Seismic behavior factors of RC staggered wall buildings . International Journal of Concrete Structures and Materials , 10 ( 3 ), 355 - 371 .
Kim , J. , & Kim , S. ( 2017 ). Performance-based seismic design of staggered truss frames with friction dampers . ThinWalled Structures , 111 , 197 - 209 .
Kim , J. , & Lee , M. ( 2014 ). Fundamental period formulae for RC staggered wall buildings . Magazine of Concrete Research , 66 ( 7 ), 325 - 338 .
Kim , J. , Lee , J. , & Kim , B. ( 2015 ). Seismic retrofit schemes for staggered truss structures . Engineering Structures , 102 ( 1 ), 93 - 107 .
Lee , J. , & Kim , J. ( 2013 ). Seismic performance evaluation of staggered wall structures using FEMA P695 procedure . Magazine of Concrete Research , 65 ( 17 ), 1023 - 1033 .
Lee , J. , & Kim , J. ( 2015 ). Seismic response modification factors of reinforced concrete staggered wall structures . Magazine of Concrete Research , 67 ( 20 ), 1070 - 1083 .
MacKay-Lyons , R. ( 2013 ). Performance-based design of RC coupled wall high-rise buildings with viscoelastic coupling dampers . Master's thesis , Department of Civil Engineering , University of Toronto.
Madsen , L. P. B. , Thambiratnam , D. P. , & Perera , N. J. ( 2003 ). Seismic response of building structures with dampers in shear walls . Computers & Structures , 81 ( 4 ), 239 - 253 .
Mao , C. X. , Wang , Z. Y. , Zhang , L. Q. , & Li , H. ( 2012 ). Seismic performance of RC frame-shear wall structure with novel shape memory alloy dampers in coupling beams . In 15th World congress of earthquake engineering (15 WCEE) , Lisbon, Portugal.
Mee , A. L. , Jordaan, I. J. , & Ward , M. A. ( 1975 ). Dynamic response of a staggered wall-beam structure . Earthquake Engineering and Structural Dynamics , 3 ( 4 ), 353 - 364 .
Morgen , B. G. , & Kurama , Y. C. ( 2008 ). Seismic response evaluation of posttensioned precast concrete frames with friction dampers . Journal of Structural Engineering , 134 ( 1 ), 132 - 145 .
Mualla , I. H. , Jakupsson , E. D. , & Nielsen , L. O. ( 2010 ). Structural behavior of 5000 kN damper . In European conference on earthquake engineering , ECEE, Ohrid, Macedonia.
PEER , NGA Database. ( 2006 ). Pacific Earthquake Engineering Research Center , University of California, Berkeley. http://peer.berkeley.edu/nga.
Pant , D. R. , Montgomery , M. , Berahman , F. , & Christopoulos , C. ( 2015 ). Resilient seismic design of tall coupled shear wall buildings using viscoelastic coupling dampers . In 11th Canadian conference on earthquake engineering (11CCEE) , Victoria, Canada.
Paulay , T. , & Priestley , M. J. N. ( 1992 ). Seismic design of reinforced concrete and masonry building . New York : Wiley.
PERFORM-3D . ( 2006 ). Nonlinear analysis and performance assessment for 3D structures-user guide . Berkeley, CA, USA: Computers and Structures.
Vamvatsikos , D. , & Cornell , C. A. ( 2002 ). Incremental dynamic analysis . Earthquake Engineering and Structural Dynamics , 31 ( 3 ), 491 - 514 .
Wallace , J. W. ( 2012 ). Behavior, design, and modeling of structural walls and coupling beams-Lessons from recent laboratory tests and earthquakes . International Journal of Concrete Structures and Materials , 6 ( 1 ), 3 - 18 .