INTRODUCTION
Structures with inappropriate distributions of strength and stiffness have performed poorly in recent earthquakes and most of the observed collapses have been related to some extent to problematic configuration or a wrong conceptual design. A soft story has been observed in many collapsed structures because of having nonsuitable distribution of structural stiffness. Different types of strength and stiffness distributions are responsible for a deficient structural behavior. Concentrated drift and ductility in some stories are the worst conditions and the consequent results can be catastrophic.
Most buildings are preliminary designed on the basis of the equivalent static
forces under the governing code. It seems that the heightwise distribution
of these static forces (and therefore, stiffness and strength) is factually
based on the elastic vibration modes. However, structures do not remain elastic
during severe earthquakes and they usually undergo large nonlinear deformations.
Karami (2001) studied the effect of the conventional lateral loading pattern
(i.e., equivalent static method) specified by the different seismic codes (UBC,
1997; NEHRP, 1994; Anonymous, 1996) on height wise distribution of ductility
demand and drift in a number of steel shearbuilding frames. It was concluded
that the strength distribution patterns suggested by these seismic codes do
not lead to a uniform distribution of ductility and deformation in steel shearbuilding
frames subjected to catastrophic earthquakes. Therefore, the application of
such conventional heightwise distribution of seismic forces may not actually
cause the best seismic performance of a structure.
Chopra (1995) evaluated the ductility demands of several shear building elastoplastic models subjected to 1940 El Centro earthquake. The relative story yield strength of these models pertained to the heightwise distribution pattern of the earthquake forces which Uniform Building Code (UBC) clearly specified in 1994. It is perfectly realized that this distribution pattern does not make equal ductility demand in all stories possible and that the first story has the most ductility demand among all other stories. Moghaddam and Esmailzadeh Hakimi (1999) proportioned the relative story yield strength of a number of shear building models in accordance with some arbitrarily chosen distribution patterns as well as the distribution pattern suggested by the UBC. The ductility and displacement demands of these models were calculated. It was concluded that a uniform distribution of ductility will not be achieved by the suggested pattern as was offered by the UBC and other patterns can yield a uniform ductility distribution with a relatively smaller maximum ductility demand.
Ganjavi et al. (2005) considering a number of reinforced concrete buildings based on equivalent static loading patterns (Iranian Code of Practice for Seismic Resistance Design of Building, 1999) studied the heightwise hysteretic energy, drift and damage distribution subjected to four earthquakes. It has been concluded that this can lead to a relatively intense concentration of drift and damage in one or two stories of a building. In this study four reinforced concrete frames were considered. The seismic loading of these frames were applied according to two conventional patterns, namely equivalent static and spectral dynamic methods in accordance with the Iranian Code of Practice for Seismic Resistance Design of Building (2005). In the design of these samples a basic assumption has been considered, that is, a constant strength ratio (the ratio of the existing strength to the ultimate strength) has been applied in all stories. Although having a uniform distribution of strength ratio in the stories requires minor variations in beam and column crosssection and bar dimensions, great effort was made to achieve optimum conditions for arriving at a consistent value of 0.9 for this ratio in both methods.
The aim of this study is to investigate whether or not, reaching optimum condition mentioned above based on different lateral loading patterns specified by the governing seismic codes will result in reduction and optimum damage distribution subjected to severe earthquakes.
LATERAL LOADING PATTERNS
Equivalent static method: In most seismic building codes (UBC, 1997;
NEHRP, 1994; Anonymous, 1978; Anonymous, 1996; Iranian Code of Practice for
Seismic Resistance Design of Building, 2005), the height wise distribution of
lateral forces is determined from Eq. 1:
Where:
w_{i} and h_{i} 
= 
The weight and height of the ith floor above base level,
respectively 
N 
= 
The number of stories 
V 
= 
Total base chear 
k 
= 
The power that differs from one seismic code to another 
In some provisions such as NEHRP94 and ANSIASCE 795, k increases from 1
to 2 as the period varies from 0.5 to 2.5 s. In some codes such as UBC97, the
force at the top floor (or roof) computed from Eq. 1 is increased
by adding an additional force F_{t} = 0.07 TV for a fundamental period
T greater then 0.7 s. In such a case, the base shear V in Eq.
1 is replaced by VF_{t}. In this study, the value of k in Eq.
1 based on the Iranian Code of Practice for Seismic Resistance Design of
Building (2005) is taken as 1 (triangular loading pattern).
Spectral dynamic method: In this method, dynamic analysis is performed
assuming linear elastic behavior using maximum response from all vibration modes
which have considerable effect on response of the entire building. Maximum response
of each mode is obtained using its period from the standard design spectrum.
The heightwise distribution of lateral forced in spectral dynamic method is
determined from Eq. 2:
Where:
φ_{im} 
= 
The mth vibration component in the ith floor above the base 
V_{m} 
= 
The Shear force of the mth mode 
F_{im} 
= 
The horizontal force acting on the ith floor from the mth mode 
The maximum story and base shear forces in each mode are combined using one of the common statistical methods, namely: Complete Quadratic Combination (CQC), or Square Root of Sum of Squares (SRSS). In this study the Iranian Standard Design Spectrum (Iranian Code of Practice for Seismic Resistance Design of Building, 2005) is used for both, equivalent static and spectral dynamic methods.
DAMAGE ANALYSIS
In a nonlinear analysis, the correct choice of a hysteretic model is crucial
in forecasting the exact dynamic response of the structure. The model should
be able to describe a response similar to the actual hysteretic response of
the structure and parameters such as stiffness degradation, strength deterioration
and pinching behavior under cyclic loading are to be considered. In this study
IDARC 2D software (Valles et al., 1996) has been used to calculate structural
damage index and hysteretic energy and to conduct linear and nonlinear static
and dynamic analyses on reinforced concrete structures under seismic loading.
The software is also capable of conducting comprehensive damage analysis in
local and global scale and is able to arrive at a calibrated damage index. This
ability is an important step in quantitative evaluation of damage sustainability
of reinforced concrete buildings (Valles et al., 1996).
The current release of IDARC incorporates three models for damage index: a modified Park and Ang model (Park et al., 1984; Kunnath et al., 1992), introduced in the previous releases of the program, a new fatigue based damage model introduced by Reinhorm and Valles (1995) and an overall damage qualification based on the variation of the fundamental period of structure.
The Park and Ang and the fatigue based damage models can be used to calculate different damage indices: element, story (subassembly) and overall building damages.
Park and ang damage model: The Park and Ang damage model (Park et
al., 1984) was incorporated in IDARC since the original release of the program.
Furthermore, the Park and Ang damage model is also an integral part of the three
parameter hysteretic model since the rate of strength degradation is directly
related to the parameter β described below (Park et al., 1987b).
The Park and Ang damage index for a structural element is defined as:
Where:
δ_{m} 
= 
Maximum experienced deformation 
δ_{u} 
= 
Ultimate deformation of element 
P_{y} 
= 
Yield strength of element 

= 
Hysteretic energy absorbed by the element during response history 
β 
= 
A model constant parameter 
A value of 0.1 for the parameter β has been suggested for nominal strength deterioration (Park et al., 1987b). The Park and Ang damage model accounts for damage due to maximum inelastic excursions, as well as damage due to the history of deformations. Both components of damage are linearly combined.
Equation 3 is the basis for damage index computation, although
some considerations need to be taken into account.
Direct application of the damage model to a structural element, a story, or
to the overall building requires the determination of the corresponding overall
element, story, or building ultimate deformations. Since the inelastic behavior
is confined to plastic zones near the ends of some members, the relation between
element, story or top story deformations, with the local plastic rotations is
difficult to establish.
Table 1: 
Interpretation of overall damage index (Park et al.,
1987a) 

For the element end section damages, the following modifications
to the original model were introduced in Version 3.0 (Kunnath et al.,
1992):
Where:
θ_{m} 
= 
Maximum rotation attained during loading history 
θ_{u} 
= 
Ultimate rotation capacity of section 
θ_{r} 
= 
Recoverable rotation when unloading 
M_{y} 
= 
Yield moment 
E_{h} 
= 
Dissipated energy in section 
The element damage is then selected as the biggest damage index of end sections.
The two additional indices: story and overall damage indices are computed using
weighting factors based on dissipated hysteretic energy at component and story
levels, respectively:
Where:
λ_{i} 
= 
Energy weighting factors 
E_{i} 
= 
Total absorbed energy by the component or the ith story 
The Park and Ang damage model has been calibrated with observed structural
damage of nine reinforced concrete buildings (Park et al., 1987a). Table
1 shows the calibrated damage index with the degree of observed damage in
the structure.
STRUCTURAL SYSTEMS AND GROUND MOTIONS
Structural systems: Reinforced concrete frames of 3, 5, 10 and 15story
structures with identical bays and story heights have been used in present study.
Table 2: 
Some characteristics of reinforced concrete frames 

Table 3: 
Earthquakes records used in this study 

The total height to the total building dimension ratio in these samples varies
from 0.96 to 4.8 for 3and 15story frames, respectively. These models have
been chosen to represent three common building behaviors (shear, flexural and
shearflexural behavior). A sample of 5story frame is shown in Fig.
1. In order to correctly compare the effects of two lateral loading patterns
(equivalent static and spectral dynamic methods) on heightwise distribution
of hysteretic energy, drift and damage, analysis and design processes have been
completely similar for both patterns. Some characteristics of the selected frames
are given in Table 2. Other details of analysis and design
are as follow: The vertical and lateral loadings of the structures were applied
according to Minimum Design Loads for Buildings (Iranian National Building Code
for Structural Loadings, 2004) and Iranian Code of Practice for Seismic Resistant
Design of Buildings (2005), respectively.
Soil type II (gravel and compacted sand, very stiff clay) was used in the analyses and it was also assumed that the structures are located in a region with relatively high seismic risk and relative design base acceleration of A = 0.35 g. The frames are moment resisting with medium ductility. ETABS (2001) software (Computers and Structures, 2001) was used for linear dynamic elastic analysis and design and IDARC 2D version 6.0 software (Valles and Reinhorn, 2004) was used for nonlinear dynamic analysis. All the analyses were performed with damping model corresponding to stiffness and damping ratio of 5%. Trilinear hysteretic model of Takada was used in nonlinear analyses (Valles et al., 1996).
Ground motions: For input ground motions, 7 observed ground motions
are used. Emphasis is placed on those recorded at a low to moderate distance
from epicenter (less than 45 km), with rather high local magnitudes (i.e., M>6).
The recorded ground motions cover a broad variety of conditions in terms of
frequency content, peak ground acceleration and velocity, duration and intensity.
Real characteristics of earthquake records used in this study are shown in Table
3. In order to eliminate the influence of peak ground acceleration, all
of them are scaled to a ground acceleration of 0.35 g based on Iranian Code
of Practice for Seismic Resistance Design of Building (2005).
RESULTS AND DISCUSSION
Heightwise distribution of hysteretic energy, drift and damage index in
samples: In order to study the heightwise distribution of hysteretic energy
(Eh%) and story damage index (DI story) in the frames, the beams and columns
were chosen as the consisting elements of each story. According to UBC (1997),
if seven or more timehistory analyses are performed, then the average value
of the response parameter of interest may be used for design. Therefore, in
this regard, the average values of heightwise distribution of Eh%, drift and
DI story, subjected to 7 strong ground motions in two lateral loading patterns
known as Equivalent Static (ES) and Spectral Dynamic (SD) methods, were calculated
and then compared (Fig. 2, 3). It should
be noted that the hysteretic energy of each story is shown as the percentage
ratio of hysteretic energy in each story to total hysteretic energy in each
frame (Eh%).

Fig. 2: 
Comparison of the average values of heightwise distribution
of hysteretic energy, drift and damage index in 3and 5story frames from
ES and SD methods 
3and 5story frames: In the 3story frame, the amount and the form
of heightwise distribution for Eh% are completely identical in both ES and
SD methods. The qualitative distribution of drift and DI story in this frame
is identical. However, ES method has a larger drift and consequently, a greater
amount of damage is caused in the first and second stories as compared to SD
method (Fig. 2).

Fig. 3: 
Comparison of the average values of heightwise distribution
of hysteretic energy, drift and damage index in 10and 15story frames from
ES and SD methods 
It is seen that with an increase in the height
to dimension ratio (h/d = 1.6) in 5story frame, the distribution pattern of
the mentioned parameters in this frame is completely different from those of
3story frame. The heightwise distribution patterns of these parameters are similar in both
SD and ES methods and the maximum drift and damage occurs in the second story.

Fig. 4: 
Effect of ground motion on heightwise distribution of hysteretic
energy, drift and damage index in 5and 15story frames 
However, considering an increase in drift and damage values of stories of 3,
4 and 5 from ES method comparing to those of SD method, it can be concluded
that the frame loaded by SD method has a better performance in this case.
10and 15story frames: As indicated in Fig. 3, distribution
patterns of drift, Eh% and Di_{story} in 10 and 15story frames are
completely different from those of 3and 5story frames, in a way that with
an increase in h/d concentration of the mentioned parameters occurs in one or
two stories especially in ES patterns. In other words, although ES frames are
made of larger beam and column crosssections compared to those of SD frames,
the difference between maximum and minimum of the mentioned parameters in height
is much higher in ES frames than to SD frames for both 10and 15story frames.
An intense concentration of drift, Eh% and Di_{story} occurs in the
8th and 13th story of 10and 15story frames, respectively. Thus it can be said
that although frames with dynamic spectrum loading patterns do not lead to uniform
distribution of drift and DI_{story} in height, they generally show
better performance compared to frames with equivalent linear loading pattern.
Second, roof floors of all models (3, 5, 10 and 15story frames) show the least
damage compared to other floors from both SD and ES patterns. Also, the amount
of absorbed hysteretic energy (Eh%) for the roof is negligible and approximately
zero in value, so it can be stated that most of the elements of this story remain
in elastic state. The minor damage caused in the story is only due to the drift.
Thus, applying Ft in the equivalent static method (Eq. 1)
which describes, in someway, the effect of higher modes seems to be prone to
discuss. This story, on the other hand, undergoes the least damages compared
to other stories.
Effect of ground motion on heightwise distribution of hysteretic energy,
drift and damage index: The average values of drift, Eh% and DI_{story}
obtained due to seven earthquakes were used in order to prevent the scattering
of the results from various ground motions. None of two earthquakes, even those
occurring in the same region, have completely similar characteristics. Thus,
considering the fact that the earthquakes chosen in this study cover a broad
variety of conditions in terms of intensity, duration, frequency content and
peak ground acceleration, the effect of ground motion on heightwise distribution
of drift, Eh% and DI_{story} in 5and 15story frames is investigated
as shown in Fig. 4. This figure indicates that the qualitative
distribution of Eh% is similar in different earthquakes and as shown in this
figure, an average value of these parameters from seven earthquakes may be considered.
It can be noted from the distribution pattern of drift and DI_{story}
that in severe earthquakes such as Northridge, Manjil and ChiChi, the concentration
of drift and damage index are observed in one or two stories while other stories
have a relatively uniform distribution. The fact that most of the elements reach
inelastic deformations in such earthquakes leads to a nonuniform damage distribution.
In addition, earthquakes with lower intensity (i.e., Naghan and El Centro) compared
to those mentioned previously have a relatively uniform distribution of the
mentioned parameters in a way that they follow a uniform heightwise distribution
of strength ratio in an elastic state. These findings are confirmed by the results
reported elsewhere (Moghaddam et al., 2003; Moghaddam and Hajirasouliha,
2004). They studied the effect of the conventional lateral loading pattern (i.e.,
equivalent static method) specified by the different seismic codes (UBC, 1997;
NEHRP Recommended Provisions, 1994; Anonymous, 1996) on height wise distribution
of ductility demand and drift in a number of steel shearbuilding and concentric
bracedsteel frames. It was concluded that the strength distribution patterns
suggested by these seismic codes do not lead to a uniform distribution of ductility
and deformation in steel shearbuilding and concentric bracedsteel frames subjected
to severe earthquakes.
Comparison of overall structural damage index from spectral dynamic and
equivalent static methods: We have already discussed the distribution patterns
of damage index in stories, based on beam and column damage indices from each
story. Park et al. (1984) computed an overall structural damage index
(DI_{overall}) using story damage indices (DI_{story}) and weighting
factors based on dissipated hysteretic energy at component and story levels.
A comparison between the average values of DI_{overall} subjected to
seven earthquakes for ES and SD methods has been made as shown in Fig.
5. This comparison indicates that in all structures, despite having smaller
beam and column crosssections, DI_{overall} resulting from ES patterns
are slightly larger than those obtained from SD patterns. This may be due to
a somewhat uniform heightwise distribution of damage from SD method compared
to that of ES method.
In addition, from a comparison between total bar areas used in equivalent static
(A_{ES}) and spectral dynamic (A_{SD}) shown in Table
4 can be concluded that increasing h/d leads to a considerable lower A_{SD}
compared to A_{ES}. For example, A_{SD} is 15.3% lower than
A_{ES} in 15story frame.
Table 4: 
Comparison of the total bar area in ES and SD methods 


Fig. 5: 
Comparison of the average values of overall structural damage
indices in spectral dynamic and equivalent static methods 
Moreover, considering the average values of DI_{overall} from both
methods and the relation between DI_{overall} and the state of the building
from Table 1, it can be observed that Di_{overall}
is lower than 0.2, i.e., the structure does not undergo severe damage. However,
since DI_{overall} is only a description of general damages exerted
to the structure and does not explain the energy dissipation and drift and damage
distribution patterns in stories, therefore it is necessary to investigate the
drift and damage indices in stories. As shown in Fig. 4, although
the average values of DI_{story} are acceptable (less than 0.4), in
catastrophic earthquakes such as Manjil and ChiChi, having high intensity and
damage potential, values of drift ratio and DI_{story} in one story
of 15story frame exceed 4% and 0.7, respectively. This may lead to the formation
of a soft story and collapse in the story which in turn causes an overall collapse
of the structure. Thus, beside controlling overall structural damage index,
the maximum drift and stories damage indices must be checked.
CONCLUSIONS
In this study, the effects of two lateral loading patterns (equivalent static
and spectral dynamic) on heightwise distribution of hysteretic energy, drift
and damage subjected to severe earthquakes with different characteristics have
been studied. The results of the study can be summarized as follow:
• 
In severe earthquakes with high intensity, despite uniform
distribution of strength ratio in elastic loading, heightwise distribution
of Eh%, drift and damage are nonuniform and an intense concentration of
mentioned parameters occurs in one or two stories especially in frames with
equivalent static loading pattern. Furthermore, although SD frames have
smaller dimensions (crosssection and total bar area) compared to those
of ES frames, considering a lower overall structural damage index and rather
a uniform distribution compared to ES frames, a better performance by these
frames can be concluded. 
• 
Roof floor of all models shows the least damage compared to other floors
from both equivalent static and spectral dynamic patterns. Also, the amount
of absorbed hysteretic energy (Eh%) for roof is negligible and approximately
zero in value, so it can be stated that most of the elements of this story
remain in elastic state. The minor damage caused in the story is only due
to drift. Thus, applying Ft in the equivalent static method (Eq.
1) which describes, in some way, the effect of higher modes seems to
be prone to discuss. This story, on the other hand, undergoes the least
damages compared to other stories. 
• 
Although the average value of overall structural damage indices of 7 earthquakes
indicates that the structures do not undergo severe damages according to
Park and Ang's damage calibration, a study of drifts and damage indices
in stories especially in earthquakes with high intensity like Northridge,
Manjil and ChiChi shows that the structures undergo severe damages in one
or two stories, which it can in turn lead to complete collapse of the building.
Therefore, in addition to controlling overall structural damage indices,
drift and structural damage indices in stories must also be checked. In
strong ground motions, nonuniform distributions of drift and damage indicate
that considering a unique strength parameter in seismic loading patterns,
even in optimum conditions, is not capable of guaranteeing building safety.
Thus, simultaneous consideration of strength, energy and drift (deformation)
parameters should be considered in an optimum seismic design. 
ACKNOWLEDGMENTS
We would like to acknowledge Babol University of Technology for partial support of this study. The authors are grateful to Engineer Mehdi Omidvar for helpful discussions.
NOTATIONS