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International Journal of Civil Engineering (2018) 16:1241–1259

https://doi.org/10.1007/s40999-018-0285-0

RESEARCH PAPER

Nonlinear Static and Dynamic Analyses of RC Buildings

Mehmet Inel1 · Bayram Tanik Cayci1 · Emrah Meral2

Received: 21 August 2017 / Revised: 22 December 2017 / Accepted: 15 January 2018 / Published online: 27 January 2018
© Iran University of Science and Technology 2018

Abstract
This study aims to compare pushover and nonlinear time history analyses for existing low- and mid-rise RC buildings to
better understand the applicability limits, advantages and disadvantages of nonlinear static and dynamic analyses. The 4-
and 7-story buildings designed according to the pre-modern and modern Turkish Earthquake Codes represent the existing
low- and mid-rise RC buildings based on inventory results of more than 475 real residential buildings located in Turkey.
Eight different ground motion records were used during nonlinear time history analyses selected from destructive earth-
quakes over past several decades. The outcomes show that the displacement demands obtained from time history analyses
differ over a wide range of values, emphasizing the importance of ground motion record selection. The pushover analysis
may cause underestimation of the maximum interstory drift ratio for the mid-rise buildings. Besides, it definitely misses
the beam damages at the first story. In conclusion, the pushover analysis seems to reflect the nonlinear time history analysis
confidently at moderate level earthquakes. However, the results start to deviate as the ground motions get stronger. It is hard
to specify a single value for the safe use of pushover analysis considering all parameters in the study. The outcomes of the
current study indicate that the pushover analysis provides reasonably well estimates up to 1 and 0.75% roof drift ratios which
approximately correspond to 1.5 and 1% interstory drift ratios for low- and mid-rise buildings, respectively. Beyond these
limits, the pushover analysis may give misleading demand estimates.

Keywords Building damage · Earthquake · Existing buildings · Nonlinear time history analysis · Nonlinear static
procedure · Pushover analysis · Reinforced concrete buildings

1 Introduction Seismic displacement estimates can be obtained using

either linear or nonlinear analysis methods. While linear
Significant reinforced concrete (RC) building damages in elastic analysis is used for seismic design of buildings, ine-
recent earthquakes emphasize nonlinear behavior during lastic behavior is intended in most structures subjected to
strong seismic events. It is widely accepted that both struc- infrequent earthquake loading. Therefore, the use of nonlin-
tural and nonstructural damages sustained during earth- ear analyses is essential to capture actual behavior of build-
quakes are mainly due to lateral displacements. Therefore, ings under seismic loads [1]. Nonlinear analysis is becoming
seismic displacement estimates are an important issue in more popular tool for seismic performance evaluation of
performance-based seismic design and assessment. existing and new structures. Although it is well known that
nonlinear time history analysis is the most accurate method
for seismic demand estimates and performance assessment
* Mehmet Inel of structures, nonlinear static analysis has been widely used
minel@pau.edu.tr
in structural engineering profession due to its simplicity.
Bayram Tanik Cayci Pushover and nonlinear time history analyses have their
bcayci@pau.edu.tr
own limitations, difficulties or uncertainties. It is widely
Emrah Meral accepted that pushover analysis provides an insight into
emrahmeral@osmaniye.edu.tr
structural aspects and useful information on the strength and
1
Department of Civil Engineering, Pamukkale University, displacement demands which cannot be obtained by elastic
Denizli, Turkey analysis [2, 3]. However, it is inaccurate for the structures
2
Department of Civil Engineering, Korkut Ata University, with higher mode effects or torsional irregularities [4–8].
Osmaniye, Turkey

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1242 International Journal of Civil Engineering (2018) 16:1241–1259

Besides, nonlinear static methods are inherently insuffi- study, including 475 real residential RC buildings, 40,351
cient to reflect behavior of structures under specific ground column and 3123 beam elements, was conducted prior to
motion records like near-fault [9]. Another important point the establishment of building models [22]. In this study,
is that damage pattern observed from nonlinear static analy- values of more than 30 key parameters like plan dimen-
sis may significantly differ compared to that of nonlinear sions, story height, total column area per unit area, total
dynamic analysis in certain cases [10]. Different nonlinear load carrying infill-wall area per unit area, overhang area
static analysis procedures were developed to exceed these over floor area ratio, floor story height over regular story
[11–13]. height for building level and section dimensions and rein-
There are many studies related to the nonlinear static and forcement detailing for member level were examined.
dynamic analyses. These studies evaluated the adequacy and Results of this inventory were taken into account for deter-
inadequacy of pushover analysis, complexity of nonlinear mination of building model features.
dynamic or nonlinear time history analysis [14–17]. The Two sets of 3-D RC buildings with 4 and 7 stories
researchers studied assessment of nonlinear static proce- were selected to represent low- and mid-rise residential
dures and they concluded that pushover analysis provides buildings located in the high-seismicity region of Turkey
good estimates of displacement demands for the first mode in the current study. All of the considered buildings are
dominant structures. Besides, carefully performed pushover typical beam–column RC frame buildings with no shear
analysis provides insight into structural aspects related to walls. Plan view of models can be seen in Fig. 1. Typical
structural performance [18]. Recent studies mention good- floor-to-floor height is 2.8 m. Column dimensions change
ness of nonlinear time history analysis for seismic demand along the building height; the changes in the pre-modern
estimates, emphasizing its complexities, difficulties in selec- buildings are more pronounced compared to the changes in
tion of an appropriate ground motion record set, required the modern code buildings. The column and beam dimen-
computational effort and required time for employment of sions are provided in Table 1. Beam dimensions do not
analysis and extracting the useful results [19]. However, change throughout the building height as in the practice.
most of these studies preferred two- or three-dimensional The amount of longitudinal reinforcement ranges from 1 to
fictive frames or buildings. They usually had equal bay 1.23% similar to the ratios in the existing buildings. Note
width, equal column dimensions and equal longitudinal rein- that the seismic weights consist of dead loads and 30% of
forcement ratio for columns and beams. Therefore, the mod- live loads at the time of the earthquake as given codes. The
els did not reflect the characteristics of existing buildings. soil structure is not taken into account and the buildings
Although a few studies used a particular existing building are assumed to have fixed-base at the ground level. No
model, it is not possible to reflect characteristics of exist- special effort was given to avoid plan irregularities in mod-
ing structures with a particular model. Since nonlinear time els since the average characteristics based on the inven-
history analysis has own complexities and additional com- tory were reflected in the building models. Nevertheless,
putational efforts, it is almost impossible to use numerous placement of structural members is almost symmetrical
real building models. for the selected models and torsional irregularity was not
This study aims to compare pushover and nonlinear time observed according to averages of inventory result. But
history analyses for existing low- and mid-rise RC build- it is known that the irregularities on plan and along the
ings to better understand the applicability limits, advantages height may increase damage vulnerability of structures as
and disadvantages of nonlinear static and dynamic analy- mentioned in the literature [23–25]. The irregularities are
ses using 3D existing building models. The 4- and 7-story out of scope of this study.
buildings designed according to the pre-modern and modern The selected reference buildings were designed per pre-
Turkish Earthquake Codes [20, 21] represent the existing modern (TEC-1975) and modern Turkish Earthquake Code
low- and mid-rise RC buildings. The building characteris- (TEC-1997) considering both gravity and seismic loads. A
tics are based on an inventory results of more than 475 real design ground acceleration of 0.4 g and soil class Z3 that is
residential buildings located in the highest earthquake zone similar to class C soil of FEMA-356 was assumed [26]. The
of Turkey. expected concrete strength values are 16 and 25 MPa for
pre-modern and modern code buildings, respectively, while
the expected yield strength of longitudinal and transverse
2 Description of Structures and Modeling steel values are 220 and 420 MPa for pre-modern and mod-
Approach ern code buildings, respectively. Characteristics of existing
buildings were reflected on the reference buildings. Detailed
Proper modeling of the existing building stock is impor- information about models is given in Table 2. Model ID
tant for consistence of results of any study with the actual includes number of story and design code; 4–75 represents
behavior of the buildings. For this reason, an inventory 4-story building designed according to pre-modern code.

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International Journal of Civil Engineering (2018) 16:1241–1259 1243

Fig. 1  Structural floor plan view

of 4- and 7-story buildings

The pushover analysis consists of the application of grav- a rectangular shape, may be used for the lateral load pat-
ity loads and a representative lateral load pattern. The frames tern. P–Delta effects and gravity loads were both taken into
were subjected to gravity analyses and simultaneous lateral account during pushover and time history analyses.
loading. Gravity loads were in place during lateral loading. Although both pre-modern and modern codes impose
In all cases, lateral forces were applied monotonically in ductile design requirements, most of pre-modern code
a step-by-step nonlinear static analysis. The applied lateral and few percent of modern code buildings do not comply
forces were proportional to the product of mass and the first with these requirements. Such buildings have consider-
mode shape amplitude at each story level under considera- ably lower deformation capacity than the buildings with
tion. Although the first mode shape is used in this study, a ductile design requirements. Since the aim of this study
non-modal shape vector, such as an inverted triangular or is the comparison of pushover and nonlinear time history

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Table 1  Typical column and beam dimensions of 4- and 7-story buildings
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Element type 4-story pre-modern code 4-story modern code

1st, 2nd stories 3th, 4th stories 1st, 2nd 3th, 4th

S1 250 × 500 250 × 400 300 × 600 250 × 500

S2 500 × 250 400 × 250 550 × 250 550 × 250
S3 250 × 600 250 × 500 250 × 600 250 × 550
S4 300 × 600 250 × 500 300 × 550 250 × 500
S5 600 × 250 500 × 250 600 × 300 250 × 500
S6 250 × 550 250 × 500 600 × 250 500 × 250
S7 – – 550 × 300 550 × 250
S8 – – 300 × 650 250 × 550
Beam 250 × 500 250 × 500 250 × 600 250 × 600
Element type 7-story pre-modern code 7-story modern code
1st, 2nd stories 3th, 4th, 5th stories 6th, 7th stories 1st, 2nd stories 3th, 4th, 5th stories 6th, 7th stories

S1 300 × 650 250 × 600 250 × 500 300 × 800 300 × 750 300 × 700
S2 700 × 300 600 × 300 500 × 250 750 × 350 700 × 350 700 × 300
S3 300 × 700 300 × 600 250 × 500 300 × 700 300 × 600 300 × 600
S4 750 × 300 700 × 300 600 × 300 700 × 300 600 × 300 600 × 300
S5 300 × 750 300 × 700 300 × 600 350 × 750 350 × 700 300 × 700
Beam 250 × 600 250 × 600 250 × 500 300 × 700 300 × 700 300 × 600
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Table 2  Properties of building models

ID: Weight (kN) H (m) T (s) Predominant Predomi- Trans. rein- Concrete Analysis
mode mass nant mode forcement strength direction
participation participation space (mm) (MPa)
factor factor

4–75 6216 11.20 0.566 0.83 1.31 100 16 X

4–97 6473 11.20 0.459 0.85 1.30 100 25 X
7–75 18,622 19.60 0.829 0.76 1.38 100 16 X
7–97 20,066 19.60 0.639 0.80 1.34 100 25 X

analyses, the buildings with ductile design requirements

are within the scope of the study. The ductile design of the
code is reflected using transverse reinforcement spacing
as 100 mm.
SAP2000, a general-purpose structural analysis program,
was used for nonlinear time history and static pushover
analyses [27]. Beam and column elements were modeled as
nonlinear frame elements with lumped plasticity by defining
plastic hinges at both ends of beams and columns. SAP2000
provides default or the user-defined hinge properties options
to model nonlinear behavior of components. The default Fig. 2  Force–deformation relationship for a typical plastic hinge
hinge properties of SAP2000 are implemented from FEMA-
356 or ATC-40 [28]. Inel and Ozmen studied possible dif-
ferences on the results of pushover analysis due to default is in the order of 800 and 1800 for the 4- and 7-story build-
and user-defined nonlinear component properties [29]. They ings, respectively.
observed that although the model with default hinge proper- Newmark average acceleration method was used during
ties seemed to provide reasonable displacement capacity for nonlinear time history analyses as time integration approach
the well-confined case, the displacement capacity was quite for solution of Eq. 1. Rayleigh (mass and stiffness propor-
high compared to that of the poorly confined case. Thus, this tional) damping is considered. Mass and stiffness propor-
study implemented user-defined hinge properties. tional coefficients are calculated for 5% damping.
The definition of user-defined hinge properties requires
moment–curvature analysis of each element. The Mander (1)
{ }
[M]{̈u} + [c]{u}
̇ + [K]{u} = − [M] ü g
model [30] for unconfined, confined concrete and typical
steel stress–strain model with strain hardening for steel is where [M], [C] and [K] denote mass, damping {and}stiff-
implemented in moment–curvature analyses. Plastic hinge ness matrices, respectively. {u}, ̇ {u} and ü g are
̈ {u},
length was assumed to be equal to half of the section depth acceleration, velocity, displacement and ground accelera-
as recommended in 2007 Turkish Earthquake Code [31] and tion vectors, respectively.
other documents (such as ATC-40, FEMA-356 etc.). Also,
effective stiffness values were obtained per the code; 0.4EI
for beams and values between 0.4 and 0.8 EI depending on 3 Ground Motion Records
axial load level for columns. Beside plastic hinges, shear
hinges were defined at the middle of columns to reflect brit- Eight different ground motion records were used during
tle behavior of members. Shear hinges were not effective nonlinear time history analyses selected from destructive
on results in the scope of this study since none of column earthquakes over past several decades. Table 3 lists major
members reached the shear capacity. As shown in Fig. 2a, attributes of records considered in this study. Figure 3 illus-
five points labeled A, B, C, D and E define force–deforma- trates spectrum of the selected ground motion records for
tion behavior of a typical plastic hinge. The points B and 5% damping. The figure also plots average spectrum of
C are related to yield and ultimate curvatures. The values eight ground motions and demand spectrum provided in
assigned to each of these points vary depending on type 2007 Turkish Earthquake Code for design earthquake with
of element, material properties, longitudinal and transverse 10% probability of exceedance in 50 years for soil class Z3.
steel content, and axial load level on the element. Note that As seen in the figure, no special effort has been given to fit
number of plastic hinges to be generated for each building the average of selected records to the code spectrum that

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1246 International Journal of Civil Engineering (2018) 16:1241–1259

Table 3  Ground motion records selected from past earthquakes

Record ID Record Date Station Component PGA (g) PGV (cm/s) Vs30 (m/s) Dist. to
source
(km)

Italy-Stu270 Italy 23.11.1980 Sturno 270 0.358 52.7 1000.0 10.84

Impvall-E05140 Imperial V 15.10.1979 El C.Array #5 140 0.519 46.9 205.6 12.11
Chichi-Tcu45w Chi–Chi 20.09.1999 TCU45 W 0.474 36.7 704.6 73.50
Gazli-Gaz000 Gazli 17.05.1976 Karakyr 000 0.608 65.4 659.6 5.46
Dzc-Bol090 Duzce 12.11.1999 Bolu 090 0.822 62.1 326.0 12.04
Morgan-Cyc285 Morgan Hill 24.04.1984 C. Lake Dam 285 1.298 80.8 597.1 1.50
North-Nwh360 Northridge 17.01.1994 Newhall 360 0.590 96.9 269.1 13.90
Norh-Syl090 Northridge 17.01.1994 Sylmar O. 090 0.604 78.2 440.5 15.00

Fig. 3  Elastic spectrum of 2.5 Italy-Stu270

selected ground motions for 5% Impvall-H-E05140
damping 2 Chichi-Tcu045W
Gazli-Gaz000
Acceleration (g)

Dzc-Bol090
1.5 Morgan-Cyc285
Northr-Syl090
1 Northr-Nwh360
Average
0.5 TEC-2007 Design Spectrum

0
0 0.5 1 1.5 2 2.5 3
Period (s)

is provided to visualize the demand of selected records. to capture the displacement demand estimates of nonlinear
The figure clearly shows that the average spectrum of the time history analyses. Figure 4 illustrates typical capac-
selected ground motions is slightly higher than the code ity curves that represent the relationship between the base
demand spectrum within the period range of the low- and shear force and the displacement of the roof. The base shear
mid-rise RC buildings selected in this study. force is normalized by seismic weight of the building. The
increase in lateral load and deformation capacities is obvious
for the modern code buildings.
4 Results The nonlinear time history analyses of the selected
building models were subjected to ground motions listed
Static pushover and time history analyses are compared in Table 3 without any scaling. Several additional analyses
for displacement demands, displacement and interstory were conducted using scaled ground motions to visualize
drift profiles and plastic hinge patterns. The displacement the damage patterns for larger displacement demand levels.
demands obtained from nonlinear time history analysis is
used for proper comparison of the displacement profiles and
damage states. The average spectrum of the selected ground 5 Seismic Displacement Demands
motions is taken into account for the displacement demands
of pushover analyses. The 4- and 7-story buildings per the Seismic displacement demands of the considered buildings
pre-modern and modern earthquake codes were subjected to were determined for nonlinear static and nonlinear dynamic
both pushover and nonlinear time history analyses. analyses. Nonlinear time history analysis subjected to the
The pushover analysis consists of the application of selected ground motion set was used to obtain displacement
gravity loads and a representative lateral load pattern, pro- demand estimates for nonlinear dynamic analysis. Displace-
portional to the product of mass and the first mode shape ment demands for nonlinear static analyses were estimated
amplitude. Gravity loads were in place during lateral load- using capacity curves obtained from pushover analysis and
ing. P-Delta effects were taken into account. Target displace- the average spectrum of the selected ground motions. Dis-
ment of each building model was considered as large enough placement demands were also estimated using the design

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Fig. 4  Capacity curves for 4- 0.45 0.45

and 7-story building models
0.40 0.40

0.35 0.35

0.30 0.30

0.25 0.25

V/W

V/W
0.20 0.20

0.15 0.15

0.10 0.10
4-story pre-modern code 7-story pre-modern code
0.05 0.05
4-story modern code 7-story modern code
0.00 0.00
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
Roof drift (%) Roof drift (%)

earthquake spectrum with 10% probability of exceedance Maximum roof displacement demand of time history
in 50 years to have an idea for the demands of the average analyses for the selected ground motion set and inelastic
spectrum of the selected ground motions. spectral displacement demand of the average spectrum
The inelastic spectral displacement demand of each build- determined using the code approach and inelastic spec-
ing was obtained for the average spectrum using displace- tral displacement demand for design earthquake with 10%
ment coefficient method given in TEC-2007, similar to the probability of exceedance in 50 years for soil type Z3 (TEC
procedure of FEMA-356. In this method, the elastic spectral demand) are given in Table 4. TEC demands are given to
displacement values are increased by multiplying a factor visualize the consistency of displacement demands for the
(CR1) as given in Eqs. (2a) and (2b). According to the TEC- average spectrum. Table 4 shows that the displacement
2007, if the building period is greater than the characteristic demands for the average spectrum are slightly higher than
period of the soil type the equal displacement rule is valid displacement demands for the code spectrum consistent with
and the inelastic displacement demand is taken equal to the their spectrum.
elastic one (CR1 is equal to 1). The characteristic period of Table 4 shows that the displacement demands for the
the average spectrum is similar to the characteristic period nonlinear time history analyses differ over a wide range of
of design spectrum for soil class Z3 given 2007 Turkish values for the selected records; the ratios of maximum and
Earthquake Code. Therefore, the characteristic period of the minimum demands are 7.5, 5.6, 3.0 and 5.2 for pre-modern
average spectrum is taken as 0.6 s. If the building period is code 4-story, modern code 4-story, pre-modern code 7-story,
smaller than the soil characteristic period (TB), the elastic modern code 7-story, respectively. The table also indicates
displacement demand is increased by multiplying a factor the separation of displacement demands into two groups: the
(CR1) which is given as demands are significantly smaller for four records while they
are significantly higher for the other four records. There-
1 + (Ry1 − 1) ⋅ TB ∕T fore, the earthquakes are grouped as set 1 and set 2 records
CR1 = (2a) for the seismic performance evaluation purposes to reflect
Ry1
moderate and strong earthquakes, respectively. The average
demand of eight records is also provided in Table 4. The
Sae1 difference between nonlinear dynamic and static analyses
Ry1 =
ay1 (2b) is significant for the pre-modern 4-story building displace-
ment demand. The average roof displacement demand of
In Eqs. 2a and 2b, CR1 is the ratio between inelastic and nonlinear dynamic analysis is 130.5 mm while the inelas-
elastic displacements, Ry1 is the strength reduction factor, T tic displacement demand of the average spectrum of eight
is the building period, Sae1 is the spectral acceleration, and records is 208.4 mm. The difference is reasonable for the
ay1 is the acceleration at the yield point of the building, in pre-modern 7-story building. The differences for the both
other words lateral strength at yielding point divided by seis- modern code buildings are negligible.
mic weight of building. The inelastic displacement demand Figure 5 illustrates the roof drift demands of 4- and
determined using the code approach is called as nonlinear 7-story buildings for the ground motion set considered
static displacement demand in the paper. This value is com- in this study. The average displacement demands of eight
pared with nonlinear time history analyses demands. ground motion records obtained by nonlinear time history

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1248 International Journal of Civil Engineering (2018) 16:1241–1259

Table 4  Roof displacement demands of the model buildings

Sets Record Roof displacement (mm)
Pre-modern code 4-story Modern code 4-story Pre-modern code 7-story Modern code 7-story

SET 1 (moderate) Chichi-Tcu45w

58.6 56.4 100.6 59.4
Impvall-E05140
76.6 39.6 92.2 73.9
Italy-Stu270
35.6 29.2 147.9 47.3
Gazli-Gaz000
78.8 55.1 140.1 93.3
Set 1 average 62.4 45.0 120.2 68.5
SET2 (Strong) Dzc-Bol090 217.0 140.2 176.3 207.6
North-Syl090 162.8 105.4 230.5 175.4
Morgan-Cyc285 147.5 144.1 232.6 148.4
North-Nwh360 267.1 164.6 280.0 246.9
Set 2 average 198.6 138.6 229.9 194.6
Average of set 1 and Set 2* 130.5 (1.17%) 91.8 (0.82%) 175.0 (0.89%) 131.5 (0.67%)
Demand for the average spectrum* 208.4 (1.86%) 89.5 (0.81%) 206.8 (1.05%) 144.5 (0.74%)
Demand for TEC spectrum* 184.8 (1.65%) 80.9 (0.72%) 182.9 (0.93%) 129.3 (0.66%)

* The values in parenthesis are roof drift ratios (roof displacement demands normalized by building height)

Chichi-Tcu45w significantly lower compared to that of set 2 records. Fig-

3
Impvall-E05140 ure 5 also illustrates the average displacement demands
Italy- Stu270
Gazli-Gaz000 of eight ground motions determined by nonlinear time
2.5
Dzc-Bol090 history analyses and inelastic spectral displacement esti-
North-Syl090
Morgan-Cyc285 mates of nonlinear static analyses. The demands for non-
North-Nwh360 linear static analyses are based on the average spectrum of
NTH-Avg. eight ground motion records. The difference between two
2 NSA-Avg. Spectrum
demands is more pronounced for pre-modern code build-
Roof Drift (%)

ings, especially pre-modern code 4-story building. The

1.5 differences between displacement demands of nonlinear
dynamic and nonlinear static analyses are negligible for
the modern code buildings. Figure 5 shows that the roof
1 drift ratios tend to decrease for modern code buildings.
This is related to increase in stiffness and strength capac-
ity of TEC-1997 buildings compared to that of TEC-1975
0.5 buildings. Another observation is that the 4-story building
roof drift ratios are higher than those of 7-story build-
ings. Unlike the general trend, the displacement demand
0 of TEC-1997 7-story building seems to be higher than that
Pre-modern Modern code Pre-modern Modern code of TEC-1975 7-story building for Dzc-Bol090 record. This
code 4-story 4-story code 7-story 7-story is mainly due to interaction between system and charac-
teristic of ground motion record. The increase in stiffness
Fig. 5  Comparisons of roof drift demands of 4- and 7-story buildings and strength capacity changes the building characteristic
such as period that may cause a shift in spectrum. Ground
(NTH) analyses and inelastic spectral displacement esti- motion records may cause considerably higher demands
mates using pushover capacity curves and the average for buildings with certain periods, especially close to res-
spectrum (called as nonlinear static analysis, NSA) are onance region. Another remarkable result was observed
also plotted in the figure. The displacement demands for Italy-Stu270 earthquake record. Although this record
under individual records changes over a wide range, espe- seems to be non-destructive for the considered buildings,
cially for pre-modern code 4-story buildings. The varia- the maximum roof displacement value for 1975 TEC
tion reduces for the 7-story buildings. It is obvious that 7-story building is considerably high, more than three
the displacement demands of set 1 records (4 records) are times larger than that of TEC-1997 7-story building due
to the dynamic amplification.

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6 Displacement Profile they start to deviate as the displacement demands (set 2)

increase independent of the number of story and code.
Displacement profile along the building height is an Figure 7 compares mode shape (modal) and normalized
indicator for interstory drift ratios. It also shows sudden displacement profiles obtained as story displacement divided
changes of story displacement for irregular structures. In by roof displacement values for pushover and time history
general, pushover analysis concentrates on roof displace- analyses. Since the pushover analysis generally uses the roof
ments assuming a mode shape compliant displacement displacement and assumes mode shape compliant displace-
profile along the building height. The displacement pro- ment along the building height, this comparison is important
files of the selected buildings are compared for the pusho- to validate this assumption. Modal and pushover displace-
ver and nonlinear time history analyses (NTHA) at the ment patterns of the pre-modern code buildings are very
same roof displacement as well as the predominant mode similar for the ground motions with smaller displacement
shapes. demands (set 1) while the time history displacement pattern
Displacement profiles of NTHA and pushover analysis slightly deviates indicating higher story displacements. As
are compared at the peak displacement demand obtained the earthquakes get stronger (set 2), the pushover displace-
from NTHA. Displacement profiles of NTHA for each ment pattern approaches to that of time history.
ground motion are used to obtain average profiles for Modal, pushover and time history displacement patterns
moderate and strong ground motion sets (set 1 and set 2). of the modern code buildings do not deviate much for the
Figure 6 plots displacement profiles of the 4- and ground motions with smaller displacement demands as it is
7-story buildings for pushover and time history analyses. seen in Fig. 7. For the stronger earthquakes, the deviation
Time history displacement demands are the average of set among the displacement patterns increases and the high-
1 or set 2 ground motions to compare pushover displace- est deviation is in the time history displacement pattern.
ment profiles. The figure clearly indicates that the dis- Although the differences in displacement patterns of push-
placement profiles of pushover and time history analyses over and time history are tolerable for small displacement
are similar at smaller displacement demands (set 1) while demands, the differences may not be reasonably small for the
larger displacement demands. It should be kept in mind that
the displacement demands of the modern code buildings are
smaller than that of the pre-modern code buildings. Figure 8
11.2 11.2
4-story
pre-modern code
4-story
modern code plots the displacement pattern for the 4-story modern code
building subjected to Syl090 record scaled to obtain 2.5%
8.4 8.4
roof drift to see displacement pattern of modern code build-
ings for larger displacement demands. The general trend in
Height (m)

Height (m)

5.6 5.6
displacement pattern observed from Figs. 6, 7 and 8 is that
as the earthquakes get stronger, the pushover displacement
2.8 2.8
pattern approaches to that of time history while they both
0.0
deviate from the predominant mode shape. The observations
0.0
0 50 100 150 200 0 50 100 150 from the comparisons illustrate that mode shape-compliant
Displacement (mm) Displacement (mm)
19.6 19.6
displacement profile along the building height is not always
a valid assumption in pushover analysis.
16.8 16.8

14.0 14.0
7 Interstory Drift Ratio and Profiles
Height (m)
Height (m)

11.2 11.2
Interstory drift ratio is defined as the relative drift between
8.4
7-story
8.4
7-story
two consecutive stories normalized by story height. It is an
pre-modern code modern code
important demand parameter and indicator of structural per-
5.6 5.6
Set1 TH
formance, especially for structures with nonlinear response.
2.8 2.8
Set1 PO It also indicates the damage concentrated story. The varia-
Set2 TH tion is examined for different ground motion records. Then,
Set2 PO
0.0 0.0 the average values of interstory drift ratio profiles obtained
0 50 100 150 200 250 0 50 100 150 200 250
Displacement (mm) Displacement (mm) from nonlinear time history analyses are compared to those
obtained using the pushover analyses.
Fig. 6  Displacement profiles of the 4- and 7-story buildings for push- The interstory drift ratios obtained in nonlinear time
over and time history analyses history analyses along the building height are illustrated

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1250 International Journal of Civil Engineering (2018) 16:1241–1259

11.2 4-story 11.2 4-story 11.2 4-story 11.2 4-story

pre-modern code pre-modern code modern code modern code

8.4 8.4 8.4 8.4

Height (m)

Height (m)

Height (m)
Height (m)
5.6 5.6 5.6 5.6

2.8 2.8 2.8 2.8

Modal Modal Modal Modal
Set1 TH Set1 TH
Set2 TH Set2 TH
Set1 PO
0.0 Set2 PO Set1 PO Set2 PO
0.0 0.0 0.0
0 0.25 0.5 0.75 1
Norm. Displacement
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1
Norm. Displacement Norm. Displacement Norm. Displacement
19.6 19.6 19.6 19.6
7-story 7-story 7-story 7-story
pre-modern code pre-modern code modern code modern code
16.8 16.8 16.8 16.8

14.0 14.0 14.0 14.0

Height (m)

Height (m)
Height (m)

Height (m)

11.2 11.2 11.2 11.2

8.4 8.4 8.4 8.4

5.6 5.6 5.6 5.6

Modal Modal Modal Modal

2.8 2.8 2.8 2.8
Set1 TH Set2 TH Set1 TH Set2 TH

Set1 PO Set2 PO Set1 PO Set2 PO

0.0 0.0 0.0 0.0
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1
Norm. Displacement Norm. Displacement Norm. Displacement Norm. Displacement

Fig. 7  Comparison of mode shapes and normalized displacement profiles of the 4- and 7-story buildings for pushover and time history analyses

in Fig. 9 for the buildings considered in this study. Due to The interstory drift ratios vary in a very large band. Both
significant displacement demand differences, the interstory the building model and ground motion record are important
drift ratios are plotted separately for two ground motion sets. parameters in the variation. The figure obviously shows that
set 2 ground motions have significant story displacement
demands compared to that of set 1 ground motions. Besides,
4-story the pre-modern 4-story buildings are the most critical build-
11.2
modern code ing set having an interstory drift ratio up to 3.8%. The high-
est drift ratios are experienced on the first or second stories
for the 4-story buildings while the second or third stories
8.4
are critical for the 7-story buildings. Size reduction of col-
umn members along the building height causes the changes
Height (m)

on stiffness capacity of inter-stories. Thus, IDR values may

5.6 have concentrated at upper stories for 7-story models.
Figure 10 compares the interstory drift ratios of nonlin-
ear time history and pushover analyses for the set 1 and set 2
2.8 ground motions. The interstory drift ratio profiles obviously
Modal
differ as the number of stories and earthquake set changes.
North. Syl. TH
The pushover and time history analyses of both pre-modern
North. Syl PO
0.0
and modern code 4-story buildings have very similar drift pro-
0 0. 25 0.5 0. 75 1 files throughout the building height for set 1 ground motions.
Norm. Displacement The estimates of 4-story buildings per pre-modern and mod-
ern codes are reasonably good under strong earthquake set,
Fig. 8  Displacement pattern of the 4-story modern code building sub- except higher estimates of the pushover analysis at the third
jected to Syl090 record scaled for 2.5% roof drift floor. The general trend for the 7-story buildings is that the

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International Journal of Civil Engineering (2018) 16:1241–1259 1251

11.2 Chi-Chi 11.2 Duzce 11.2 Chi-Chi 11.2 Duzce

Imp. Val. North. Syl. Imp. Val. North. Syl.
Italy Mor. Hill Italy Mor. Hill
8.4 Gazli 8.4 North. Nwh. 8.4 Gazli 8.4 North. Nwh.
Avg. Avg. Avg.
Avg.
Height (m)

Height (m)

Height (m)
Height (m)
5.6 5.6 5.6 5.6

2.8 2.8 2.8 2.8

0 0 0 0
0 0.5 1 1.5 0 1 2 3 4 5 0 0.25 0.5 0.75 1 0 1 2 3
IDR (%) IDR (%) IDR (%) IDR (%)
4-story pre-modern code 4-story pre-modern code 4-story modern code 4-story modern code
19.6 Chi-Chi 19.6 Duzce 19.6 Chi-Chi 19.6 Duzce
Imp. Val North. Syl. Imp. Va North. Syl.
Italy Mor. Hill Italy Mor. Hill
Gazli North. Nwh Gazli North. Nwh
16.8 Avg. 16.8 Avg. 16.8 Avg. 16.8
Avg.

14 14 14 14

Height (m)
Height (m)

Height (m)
Height (m)

11.2 11.2 11.2 11.2

8.4 8.4 8.4 8.4

5.6 5.6 5.6 5.6

2.8 2.8 2.8 2.8

0 0 0 0
0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 0 0.25 0.5 0.75 1 0 0.5 1 1.5 2 2.5
IDR (%) IDR (%) IDR (%) IDR (%)
7-story pre-modern code 7-story pre-modern code 7-story modern code 7-story modern code

Fig. 9  The interstory drift ratios along the building height for 4- and 7-story buildings subjected the selected ground motion records

pushover analysis tends to underestimate the interstory drift 8 Comparisons of Plastic Hinge Pattern
ratios at the lower stories while the interstory drift ratios are
overestimated at the upper stories except 7-story modern code Plastic hinge pattern shows damage distribution of struc-
building subjected to moderate earthquake set, probably due tural elements in each story. This study compares plastic
to significantly small displacement demand levels at which the hinge patterns for different ground motion records as well
building is at the beginning of nonlinear behavior. as for different peak roof drifts. It provides to observe dif-
Although there are obvious differences among interstory ferences on demand distribution on structural members.
drift profiles of time history analyses subjected to the consid- Ground motion records with similar displacement demands
ered ground motion records, the average profiles of moderate were chosen to compare plastic hinge patterns as shown in
and strong earthquake sets reasonably fit the pushover profiles. Figures from 11 to 18. For example, Impvall-E05140 and
The general trend in Fig. 10 is that the pushover analyses are Gazli-Gaz000 records have displacement demands of 76.6
successful in estimating interstory drift ratios and location of and 78.8 mm, respectively. The plastic hinge patterns at
peak values for the 4-story buildings while they are partially displacement of smaller one (76.6 mm) has been used for
successful for the 7-story buildings.

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1252 International Journal of Civil Engineering (2018) 16:1241–1259

11.2 11.2 11.2 11.2

4-story 4-story 4-story 4-story
pre-modern code pre-modern code modern code modern code
8.4 8.4 8.4 8.4
Height (m)

Height (m)

Height (m)

Height (m)
5.6 5.6 5.6 5.6
Set1 TH Set2 TH Set1 TH Set2 TH
Set1 PO Set2 PO Set1 PO Set2 PO
2.8 2.8 2.8 2.8

0 0 0.0 0.0
0 0.25 0.5 0.75 1 0 1 2 3 0 0.25 0.5 0.75 0 0.5 1 1.5 2
IDR (%) IDR (%) IDR (%) IDR (%)
19.6 7-story 19.6 7-story 19.6 7-story 19.6 7-story
pre-modern code pre-modern code modern code modern code

16.8 16.8 16.8 16.8

14 14 14 14

11.2 11.2 11.2 11.2

Height (m)

Height (m)

Height (m)

Height (m)
8.4 8.4 8.4 8.4
Set1 TH Set2 TH Set1 TH Set2 TH
Set1 PO Set2 PO Set1 PO Set2 PO
5.6 5.6 5.6 5.6

2.8 2.8 2.8 2.8

0 0 0 0
0 0.5 1 0 0.5 1 1.5 2 0 0.25 0.5 0.75 0 0.5 1 1.5 2
IDR (%) IDR (%) IDR (%) IDR (%)

Fig. 10  The comparison of the interstory drift ratios of nonlinear time history and pushover analyses for the set 1 and set 2 ground motions

comparison. Since the roof displacement drift demands motions to obtain around 2% roof displacement drifts. A
of the selected buildings except pre-modern code 4-story typical interior frame in the analysis direction is used to
are not large enough to visualize significant damage, the reveal the diversity of nonlinear static and time history
modern code 4-story and both pre-modern and modern analyses as well as differences for the selected records.
codes 7-story buildings were subjected to scaled ground

Fig. 11  Plastic hinge pattern of the pre-modern code 4-story building frame at 0.67% roof drift demand

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International Journal of Civil Engineering (2018) 16:1241–1259 1253

Figure 11 plots plastic hinge pattern of the pre-mod- Figure 14 plots plastic hinge patterns of the modern code
ern code 4-story building frame for Gazli-Gaz090 and 7-story building frame for Dzc-Bol090 and North-Nwh360
Impvall-E05140 records and pushover analysis at similar records and pushover analysis at similar displacement
small displacement demands (0.67% roof drift) where the demands of about 1.05% roof drift ratio. Although both
building is at the beginning of nonlinear behavior. The records have similar hinge pattern, there are differences in
plastic hinge patterns of two records are very similar beam damages. The differences in damage patterns of two
except one beam at the first story and small differences records as well as pushover analysis are obvious. Moreover,
of yielded elements at the upper stories. The comparison the pushover analysis differs in damage levels and column
of time history and pushover analyses obviously shows hinging at lower stories; pushover analysis has no yielding
that both analyses have similar plastic hinge formation column at the top of first story and both ends of second story.
at the lower stories while the pushover analysis does not Plastic hinge patterns of the pre-modern code 4-story
correctly estimate plastic hinge formation of time history building frame are illustrated at roof displacement drift
analysis at the upper stories. ratio of 2% in Fig. 15 for North-Nwh360 and Dzc-Bol090
Figure 12 plots plastic hinge pattern of the modern code records and pushover analysis. The plastic hinge formation
4-story building frame for Dzc-Bol090 and Morgan-Cyc285 and its distribution are similar for all three cases. It is obvi-
records and pushover analysis at roof displacement drift ously observed that damages concentrate at the base of the
demand of 1.25%. Except the first story beams, the plastic first story columns and the first story beams while there are
hinge patterns are similar. Despite the same displacement limited damages on the second and third stories and no dam-
demands, two time history cases have differences in plastic ages at the top story. However, there are some differences
hinging for the first story beams. The pushover analysis defi- related to damage stages at the lower stories depending on
nitely misses the beam damages at the first story. the records. The pushover indicates similar column damages
Figure 13 illustrates plastic hinge patterns of the pre- at the base while it gives the heaviest beam damage on the
modern code 7-story building frame for North-Syl090 and second story.
Morgan-Cyc285 records and pushover analysis at similar Figures 16, 17 and 18 show plastic hinge patterns of
displacement demands of 1.17% roof drift ratio. The figure modern code 4-story, pre-modern code 7-story and modern
points out that there are significant differences in damage code 7-story buildings subjected to scaled Gazli-Gaz090 and
patterns under two records. Although both records demand North-Syl090 records at 2% roof drift ratio. The pushover
very similar displacements, Morgan-Cyc285 record causes plastic hinge pattern of each building at 2% roof drift ratio is
more damages at beams and more yielding at columns of also provided in the same figures for comparison. The dam-
lower stories while column yielding is more at the upper sto- ages of modern code 4-story building distributed throughout
ries for North-Syl090 record. Except several differences in the building height for Gazli-Gaz090 record while damages
column and beam yielding at the upper stories the hinge pat- are at the first three stories for North-Syl090 record. The
tern of pushover analysis is similar to that of North-Syl090 damage pattern of pushover analysis is somehow different
record. and damages concentrate at columns of the first and third

Fig. 12  Plastic hinge pattern of the modern code 4-story building frame at 1.25% roof drift demand

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1254 International Journal of Civil Engineering (2018) 16:1241–1259

Fig. 13  Plastic hinge pattern

of the pre-modern code 7-story
building frame at 1.17% roof
drift demand

stories and beams of the first and second stories. The pusho- of the pre-modern 7-story building except damage states at
ver analysis indicates the heaviest beam damages among few locations. Both Gazli-Gaz090 and North-Syl090 records
three plastic hinge patterns as shown in Fig. 16. Similar plas- have almost identical damage distributions. The pushover
tic hinge formations and damage patterns of the pre-modern analysis pattern has significant differences for both loca-
7-story building are observed for the scaled Gazli-Gaz090 tion and damage state of plastic hinges. Although the hinge
and North-Syl090 records except few differences in column formation distribution is better than that of the pre-modern
and beam damage states in Fig. 17. The heaviest damage is code 7-story building, the plastic hinge distributions of time
under the scaled Gazli-Gaz090 record. On the other hand, history and pushover analyses differ in columns of the first
the plastic hinge formation is at the base of the first story three stories.
columns, at the top of the sixth story columns and at the both
sides of beams at the first six stories. As opposite to the time
history analysis, the damages of pushover analysis lump at 9 Summary and Conclusions
the beams. The pushover analysis obviously indicates the
plastic hinge pattern of an ideal frame behavior while this is This study compares nonlinear static known as pushover
not the case for the time history under two selected records. and nonlinear time history analyses for existing low- and
Figure 18 plots the plastic hinge pattern of the modern code mid-rise RC buildings to better understand the applicability
7-story building for the scaled Gazli-Gaz090 and North- limits, advantages and disadvantages of these approaches.
Syl090 records and pushover analysis at 2% roof drift ratio. It is also aimed to check the validity of common knowledge
The overall behavior of time history patterns are similar that about time history and pushover analyses that are mentioned

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International Journal of Civil Engineering (2018) 16:1241–1259 1255

Fig. 14  Plastic hinge pattern of

the modern code 7-story build-
ing frame at 1.05% roof drift
demand

Fig. 15  Plastic hinge pattern of the pre-modern code 4-story building frame at 2% roof drift demand

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1256 International Journal of Civil Engineering (2018) 16:1241–1259

Fig. 16  Plastic hinge pattern of the modern code 4-story building frame at 2% roof drift demand

Fig. 17  Plastic hinge pattern

of the pre-modern code 7-story
building frame at 2% roof drift
demand

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International Journal of Civil Engineering (2018) 16:1241–1259 1257

Fig. 18  Plastic hinge pattern

of the modern code 7-story
building frame at 2% roof drift
demand

on previous studies using existing building models. Dis- tions from the comparisons pointed out that the mode
placement demands, displacement profiles throughout the shape compliant displacement profile along the build-
building height, interstory drift ratios and profiles and plastic ing height is not always a valid assumption in pushover
hinge patterns for different damage levels are evaluated for analysis.
existing low- and mid-rise RC buildings. • The variation in the interstory drift ratios is obvious
The important findings of this study are summarized as from the outcomes. Both the building model and ground
follows: motion record are important parameter in the variation.
The results obviously illustrated that the strong (set 2)
• The displacement demands under individual records ground motions have significant story displacement
change over a wide range. Dynamic interaction between demands compared to that of moderate (set 1) ground
structure and ground motion record may cause unex- motions.
pected demand estimates for particular cases whereas • The general trend for the 7-story buildings is that the
nonlinear static methods are inadequate reflecting these pushover analysis tends to underestimate the interstory
effects as expected. drift ratios at the lower stories while they are overesti-
• Displacement profiles of pushover and time history anal- mated at the upper stories. This may cause underestima-
yses are similar at smaller displacement demands. As tion of the maximum interstory drift ratio for the mid-rise
the earthquakes get stronger, the pushover displacement buildings.
pattern approaches to that of time history while they both • Although there are obvious differences among interstory
deviate from the predominant mode shape. The observa- drift profiles of time history analyses subjected to the

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1258 International Journal of Civil Engineering (2018) 16:1241–1259

considered ground motion records, the average profiles 6. Kalkan E, Kunnath SK (2007) Assessment of current nonlinear
of moderate and strong earthquake sets reasonably fit the static procedures for seismic evaluation of buildings. Eng Struct
29(3):305–316
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• Plastic hinge patterns of time history and pushover analy- history analysis evaluation of a RC building collapsed during
ses differ in both location and number of plastic hinges. the Van (Turkey) earthquake on October 23, 2011. Nat Hazards
The differences become more apparent with the increas- 70(1):657–673
8. Goel RK, Chopra AK (2005) Role of higher-“mode” pusho-
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• While time history and pushover analyses have similar 21(4):1027–1041
plastic hinge formation at the lower stories, the pushover 9. Liao W, Loh C-H, Wan S (2001) Earthquake responses of RC
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• The pushover analysis indicates the plastic hinge pattern mance and retrofit evaluation of reinforced concrete structures.
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the nonlinear time history analysis confidently at moderate sis for earthquake applications. Struct Eng Mech 12(1):51–69
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