INTRODUCTION
Non uniform distributions of mass, stiffness and strength in an asymmetric
building cause the building to experience torsional moments and rotational deformations
around vertical axes. Rotational deformation causes non uniform distribution
demand in Lateral Force Resisting Elements (LFREs). Also, it leads to increased
damages in an asymmetric building. The experiences of past earthquakes such
as Mexico City 1986 (Chandler et al., 1996),
confirm these effects. The vulnerability of asymmetric buildings has been addressed
by building seismic design codes in the form of special torsional provisions.
In these provisions, the design eccentricity is defined as a combination of
the stiffness eccentricity and the accidental eccentricity. This eccentricity
is used to calculate the design torsional moments. Most of these torsional provisions
have been primarily developed based on studies of a single story building in
the linear range. Although the stiffness eccentricity is a suitable indicator
of torsional responses when the building behaves in the linear range, majority
of buildings are designed to behave in the nonlinear range during moderate and
large earthquakes. In such a situation, stiffness eccentricity is not a good
indicator of building torsional responses. Sadek and Tso
(1989) introduced the concept of strength eccentricity. The strength eccentricity
is defined as the offset of the center of strength to the center of mass. The
center of strength is the center of yield strength of the LFREs. The strength
eccentricity is an appropriate indicator for the torsional response of an asymmetric
building in the nonlinear range. During a ground motion, a building responds
in various ranges of behaviors, from full elastic to elastoplastic and full
plastic. During these excitations some LFREs behave in the linear to nonlinear
range while the other LFREs remain in the linear range. Therefore, it can be
expected that the torsional responses of a building during moderate and high
earthquake intensities depend on both strength and stiffness eccentricities.
In earlier researches, it has been mainly assumed that for every LFRE of a
building, its stiffness can be calculated based on the dimensions and before
assignment of strength to it. Therefore, the element stiffness is assumed to
be independent of its strength. These LFRE are called Ktype elements. For this
type of modeling, the location of center of rigidity can be determined before
the design procedure and as such, the design parameter is the location of center
of strength. Current design procedures and torsional provisions of seismic codes
have been developed based on the cited assumption. Recent researches (Aschheim,
2002) revealed that for many LFREs such as shear walls and moment resisting
frames, stiffness is depended on their strength and will be modified during
strength assignment. These LFRE are called Dtype elements. For buildings composed
of these elements, torsional provisions that assume location of center of resistance
can be determined before strength distributions are deficient.
Many researches in the past two decades studied inelastic behavior of asymmetric
buildings, mostly focused on single story buildings. Single story models can
appropriately show torsional behavior of equiheight asymmetric buildings. In
studies on single story buildings before 1997 only Ktype elements have been
considered. In these researches, assuming that the location of center of strength
remains unchanged during design procedure, the proper location of center of
strength was examined. For these types of buildings Tso and
Ying (1992) suggested that the strength eccentricity should be zero or near
to zero in order to reduce the ductility demand on a flexible edge element for
buildings that have nonuniform stiffness distribution. Rutenberg,
(1992) and De Stefano et al. (1993) tried
to find optimum location of center of strength relative to the center of mass
and the center of rigidity. They concluded that the best location of center
of strength is at the middle of centers of mass and stiffness. Based on the
plastic mechanism analysis, Paulay (1997, 2001)
considered the behavior of a single story asymmetric structure with Dtype elements.
He concluded that the current lateral strength distribution of seismic codes
is inappropriate. He suggested that an arbitrary strength distribution strategy
can be more effective for superior performance of an asymmetric structure in
the ultimate limit state. He proposed that an appropriate location of center
of strength is some place near the center of mass. Myslimaj
and Tso (2002, 2005) studied the effect of different
configurations of centers of mass, stiffness and strength on responses of a
single story asymmetric structure. They demonstrated that the response of asymmetric
structures during the earthquake excitation depends on the location of both
the centers of strength and stiffness. They proposed that the best configuration
of centers of mass, strength and stiffness is a configuration in which the center
of mass is between centers of strength and stiffness. This configuration was
termed as the balance configuration. According to their study, the balance configuration
will improve the interstory drift and diaphragm rotational responses of a building.
However, it can cause an increase of ductility demand on elements at the stiff
side of structure.
To develop a methodology for performancebased design of asymmetric buildings,
there is a need for recognition of the proper distribution of the strength
between LFREs. Identifying the proper configuration of building centers
for a required hazard level and Engineering Demand Parameters (EDPs) associated
with various limit states can be a designer guide to improve the behavior
of an asymmetric building. Consequently, the main purpose of this study
is to investigate performance of different configurations of centers for
codedesigned one directional asymmetric building models undergoing nonlinear
dynamic analysis.
To identify the proper configuration of centers more precisely and to
evaluate performance of various sets of centers, fivestory building models
with different configurations of centers of mass, rigidity and strength
are used. Edge displacements, floor rotation, maximum interstory drift,
edge ductility demand and plastic rotation are considered as EDPs. The
building models are assumed to consist of Dtype elements. These models
are subjected to two directional far field ground motions. Subsequently,
the maximum values of drift, plastic rotation and ductility responses
are evaluated by conducting nonlinear dynamic analysis and the results
are presented in the form of different limit states fragility curves.
The presented study by using these fragility curves in multistory model
tries to more precisely determine the configuration of centers as a useful
tool to improve performance of asymmetric multistory buildings.
MATERIALS AND METHODS
To study the effects of different configurations of centers of mass, stiffness
and strength at different levels of hazard, fivestory buildings with rigid
beams are used. The models consist of rigid diaphragms with dimensions of 20x30
m and three shear walls in each direction with similar properties in the stories.
Buildings are symmetric in the xdirection and asymmetric in the ydirection.
The asymmetry in the ydirection is produced by changing width and strength
of the two edge walls in the ydirection (Fig. 1a, b).
Thus, the total strength in all Torsionally Balanced (TB) and Torsinally Unbalanced
(TU) models are similar. A symmetric model is also used as a reference TB system.

Fig. 1: 
(a) Plan and (b) 3D view of a TU building model 
Table 1: 
Building models configuration 

The design gravity loads of the TB system have been determined based on the
Iranian standard 519 (2006). The design earthquake loads
are calculated based on the Iranian Standard 2800 (2005).
The design base shear is equal to 320 tones. The lateral strengths of all the
TU systems are the same and equal to the lateral strength of the TB system.
Lateral strengths of the models in the x or ydirections are the same. To generate
TU models, the length of the left and the right walls in the ydirection are
changed in a way that the distance between yield center (Center of LFREs yield
displacement) of models were equal to 14.29% of the plan width in each story.
In all TU systems the shape and geometry of the walls are similar; the length
of the right element is 180 cm, left element is 420 cm and the length of all
other elements is 300 cm (Table 1).
For producing strength eccentricity, the strength of the left side wall has
been increased while the strength of the right side wall has been decreased
in such a way that the total strength remains constant.
With these assumptions, the stiffness eccentricity can be calculated
from the following equation:
where, B is the plan width, V_{i} and l_{i} are
strength and length of element i, respectively.
In Eq. 1, it is assumed that stiffness of each shear wall
is proportional to its strength and inversely related to the length. This assumption
is based on the equation for yield displacement of a shear wall element proposed
by Priestley and Kowalsky (1998).
Myslimaj and Tso (2002) showed that using different
approaches for assigning strength to the two lateral load resisting elements
of a building, the distance between centers of strength and stiffness remains
almost equal to the distance between centers of mass and yield displacement.
To investigate the effects of different configurations of centers of stiffness
and strength, behavior of a TB and seven TU models are examined in this study.
All the TU models have similar geometric configurations, element sizes and lateral
strengths. The unbalanced conditions of the TU models are created by an asymmetric
distribution of strength between the left and right edge elements. Table
1 shows the characteristics of the TB model along with the seven TU models.
Model number one is TB, the reference model, with uniform geometry, mass, strength
and stiffness distributions in its plan. Model number 2 has its center of stiffness
close to its center of mass.
Table 2: 
Far field earthquake records 

Model 3 to 5 are the models introduced by Myslimaj
and Tso (2002) as having balanced configurations. In all theses models,
the center of mass is between centers of stiffness and strength. In model 3,
the center of mass is located in an approximate distance of 0.75 C_{v}C_{s} (distance between centers of strength and stiffness) from the center of strength.
This distance changes to 0.5 and 0.25 of the distance between C_{v} and C_{s} for model 4 and 5, respectively. Model 6 is a model with symmetric
strength distribution and unbalanced stiffness distribution. Model 7 and 8 are
optimized models, as suggested by De Stefano et al.
(1993) with the center of strength located between centers of mass (C_{m})
and stiffness (C_{s}). For model 7, the distance between C_{v} and C_{m} is almost equal to 0.25 of the distance between C_{s} and C_{m}, while for model 8 the center of strength is approximately
in the middle of the distance between centers of mass and stiffness.
To compare performance of models in a given limit state, a probabilistic approach
based on fragility curves has been used. Each point of the fragility curve for
an assigned earthquake peak ground acceleration is defined by a relationship
as below (De Stefano et al., 2004; Reinhorn
et al., 2001).
Fragility= P[EDP>AC  IM] 
(2) 
where, IM is the earthquake intensity measure that in this study it is
assumed as the peak ground acceleration and AC is corresponding acceptable
criteria for the assumed limit state.
A normal statistical distribution is assumed for every Engineering Demand
Parameter (EDP) in each specific ground motion intensity. To evaluate
the probability of accedence from a specific limit state, the average
and standard deviation of each EDP is calculated for the ensemble of 15
earthquake records. Then, using the cumulative distribution function of
normal distribution, the probability of exceeding of each EDP from the
given limit state is estimated.
Fifteen earthquake ground motion records have been selected for conducting
dynamic nonlinear analysis. All the records are scaled to ten different
peak ground accelerations from 0.05 to 0.6 g. The characteristics of ground
motion records are shown in Table 2.
Dynamic nonlinear analysis of models are done using OPENSEES software (OpenSees,
2005). Five percent damping ratio for the first mode proportional to the
mass is included in the analysis.
RESULTS AND DISCUSSION
Using the OPENSEES software, the nonlinear dynamic analysis are performed.
All models analyzed for 15 two directional (x, y) farfield ground motions
(Table 2). The forcedeformation relationship of concrete
shear walls assumed to be bilinear model with post yield stiffness equal
to 2% of the elastic stiffness As response parameters, diaphragm rotation,
the maximum interstory drift, edge ductility demand and plastic rotation
of shear walls are considered.
The interstory drift and plastic rotation of shear walls as obtained from nonlinear
dynamic analysis have been compared with limit states recommended by FEMA356
(2000) for each performance level. For the immediate occupancy level, the
maximum interstory drift suggested by FEMA 356 (2000)
is 0.5% and the plastic hinge rotation is 0.005 radian, while for the life safety
level these values are 1% and 0.01 radian and for the collapse prevention level
these quantities are 2% and 0.015 radian, respectively.
Rotational responses: Diaphragm rotation is a suitable measure
of torsional response of building models. The averages of maximum rotation
of diaphragms during earthquake excitation for 15 farfield records are
shown in Fig. 2. Models with LFRE in both principle
directions of building are subjected to two directional earthquake records.

Fig. 2: 
Average of maximum diaphragm rotation for TU models
for PGA 0.05 g till 0.6 g 
As model 1 is torsionally balanced, it does not have rotational responses
and is not represented in the figures.
According to Fig. 2, for PGA equal to 0.05 g where
models dominantly respond in the linear range, the rotational response
of building model 2 and 3 have the least value in comparison to the other
TU models. These two are the models with the least stiffness eccentricity
among the TU models. As the earthquake intensity increases, the rotational
responses of all models increase. For a PGA equal to 0.1 g where buildings
start to behave in the nonlinear range, the difference between three balance
models (model 35) is clearly significant. For PGAs equal to 0.1 and 0.15
g, models 3 and 4 perform better but with higher earthquake intensities
and more nonlinear behavior of models. The model with a smaller strength
eccentricity performs better. In general, model 5 behaves better than
other TU models. Model 8, in which the center of strength is located approximately
in the middle of the center of rigidity and the center of mass, experiences
the maximum number of rotational responses. This can be due to the fact
that this model has the highest strength and stiffness eccentricity among
TU models. Therefore, during earthquake excitations the largest torsional
moments in the linear and nonlinear ranges were developed in this model.
The interstory drift: Although, the building diaphragm rotation
is a good indicator of the torsional response but it is not a suitable
index for building damages during earthquake. For nonstructural and structural
damages, the interstory drift is a better indicator of damages. The maximum
interstory drift of the eight selected building models are calculated
and used to establish the fragility curves for three performance levels,
the Immediate Occupancy (IO), Life Safety (LS) and Collapse Prevention
(CP). In Fig. 3, the drift fragility curves of seven
TU models are compared with each other and with the TB model (model 1).
The drift fragility curves of Fig. 3 shows the CP, LS
and IO levels based on the values proposed by FEMA 356 guide line.
As shown in Fig. 3ac, the probability of accedence
for the symmetric model is lower than the TU models. Interstory drift
of TU models possesses similar trends to those of the building rotation
except that for models with balance configuration, in particular, interstory
drift fragility curves are similar. This can be contributed to the fact
that these models are translational dominant models. Model 5 (thick curve
on Fig. 3) has the least probability of accedence of
the assumed limit states. After this model, model 4 for the lower limit
state (IO) and model 6 for the higher limit states (CP) perform better.
Plastic rotation: Plastic hinge rotation is the damage measure used
in different guidelines such as FEMA 356/357 (2000) and
ATC40 (1996) as a main parameter for evaluation of structural
members in nonlinear methods. In Fig. 4ac, plastic rotation
fragility curves of building models have been presented for far field ground
motion records and IO, LS and CP limit states. Figure 4 shows
that model 5 (presented with the thick line) outperforms the other models. Subsequently,
model 6 with symmetric strength has the minimum probability of accedence of
the assumed limit state among other asymmetric building models.
Between different models, model 2, which is a stiffness symmetric model
and model 8 have the maximum probability of accedence of the assumed limit
state. The difference between fragility curves for models with balance
configuration in the case of plastic rotation increases in comparison
to the case of interstory drift. This is due to a higher dependency of
plastic rotation to the yield displacement distribution. Consequently,
balance configuration that yields a good performance with regard to the
displacement and interstory drift for plastic rotation, does not perform
well especially for configurations with higher strength eccentricities.
The balance model with a higher strength in the left side element (element
with less yield displacement) and thus with less strength eccentricity
(i.e., model 5), displays the best performance among TU models.
Ductility demand ratio: Ductility demand ratio is a damage parameter
that has been widely applied to asymmetric buildings in previous studies.
This parameter is related to both displacement and yield displacement
of elements. Thus, it can be expected that the corresponding results significantly
differ from those of other EDPs.

Fig. 3: 
Drift fragility curves for (a) IO, (b) LS and (c) CP
limit states for TB and TU models 

Fig. 4: 
Plastic rotation fragility curves for (a) IO, (b) LS
and (c) CP limit states for TB and TU models 

Fig. 5: 
Ductility demand fragility curves for ductility ratio
limit state equal to 7 for TB and TU models, (a) ductility ratio 4.5
and (b) ductility ratio 6 
In Fig. 5, ductility demand fragility curves of selected
models are shown. In Fig. 5, fragility curves are derived
based on models responses to an ensemble of 2D farfield earthquake excitations.
The limit state for ductility demand is assumed to be equal to 6 for collapse
prevention level 4.5 for the life safety level. Performance of building
models for this damage measure parameter differs significantly compared
to the interstory drift and plastic rotation of LFREs. In this case, fragility
curves of TU models show higher discrepancy and their probability of accedence
of assumed limit state in similar levels is larger than the interstory
drift and plastic rotation parameter. This is due to a high dependency
of this parameter to the yield displacement of LFREs. This dependency
reduces the randomness (mid part slope of fragility curves) of ductility
fragility curves in comparison to other two EDPs.
In all TU models, the left side element yield displacement is lower than the
right side element. Therefore, with an increase of strength in the left side
performance of models improves and model 8 performs the best between all TU
models. In this model, the center of strength is located in the middle of centers
of mass and stiffness. This is the model suggested by earlier researchers for
building with Ktype elements (Destefano et al., 1993,
Rutenberg, 1992). Among fragility curves of TU models, this model shows
minimum difference with the TB model ductility fragility curves.
CONCLUSIONS
In this study, using fragility representation, the performance of symmetric
and asymmetric five story shear models affected by far field ground motions
was studied. Diaphragm rotation, interstory drift, plastic rotation and
ductility demand were selected as damage measure parameters and fragility
curves for each model were generated for two component far field ground
motion excitations. Based on the results of this study the following conclusions
can be drawn:
• 
In torsionally stiff buildings, for each specific pattern of strength
distribution (each of the 8 models) similar trends of responses were
observed for LS and CP limit states. In lower limit states such as
IO, the effect of strength distribution on improvement of building
performance decreases (less difference in the results of different
models) and in higher limit states such as CP, the effect of strength
distribution increases 
• 
Performance of different strength distributions and configurations
of centers of strength, stiffness and mass differ significantly among
different response parameters. For interstory drift and plastic rotation
parameters, models with lower diaphragm rotation perform better. However,
this situation is not valid for the ductility demand parameter 
• 
Based on results drawn from translational dominant models of this
study, for plastic rotation and interstory drift responses, models
with balance configuration and strength eccentricity equal to the
one forth of C_{v}C_{s} perform better. For the ductility
demand, models with center of strength in the middle of centers of
mass and stiffness perform better 
• 
For limit states where a moderate to high amount of damage is expected
(e.g., LS and CP), the proper configuration of centers are similar
to the case of item 4. However, in limit states that a low amount
of damage is expected (e.g., IO), models with a low stiffness eccentricity
perform better 
More exact configurations of centers for each important common demand
parameters were identified and studied. These configurations can be used
as a proper way to reduce the adverse torsional effect in design or rehabilitation
of asymmetric buildings. These configurations can also be used as a new
reference point for identifying acceptable code design eccentricity. For
each level of eccentricity, using these configurations as the reference,
the designer is capable of predicting the effect of torsional responses
on performance of an asymmetric building.
NOTATIONS
AC 
= 
Acceptable criteria 
B 
= 
Plan width 
C_{m} 
= 
Center of mass 
C_{P} 
= 
Collapse prevention 
C_{s} 
= 
Center of stiffness 
C_{v} 
= 
Center of strength 
EDP 
= 
Engineering Demand Parameters 
IM 
= 
Intensity Measure 
IO 
= 
Immediate Occupancy 
l_{i} 
= 
Length of element (i) 
LFRE 
= 
Lateral force resisting elements 
LS 
= 
Life Safety 
PGA 
= 
Peak Ground Acceleration 
TB 
= 
Torsionally Balanced 
TU 
= 
Torsionally Unbalanced 
V_{i} 
= 
Strength of element (I) 