The system considering only the first three

The nonlinear behavior and response
of structures to earthquake loads have resulted in inadequacies over the
reliance on linear analysis for seismic design. Non-linear dynamic analysis
which is the ideal analysis for seismic analysis is complex and characterized
by lengthy repetition of structural response to each group of ground
acceleration. Engineers relied on pushover analysis which is a method of nonlinear
static analysis as it offers a simplistic approach of estimating inelastic seismic
response to structures. However, pushover analysis does not provide the exact
behavior of structures under lateral loads due to its inabilities in factoring
higher modes effects. More studies are being conducted to ascertain ways to
either perfect nonlinear static or to simplify nonlinear dynamics. This is a
review of 10 papers in recent years on pushover analysis using moment resisting
steel frames.

REVIEW
AND DISCUSSIONS

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Mahdi et al (2015) compared
structural response of 3-story (low rise) and 9-story (medium rise) steel
resisting frames using Modal Incremental Dynamic Analysis (MIDA) proposed by
Mofid et al in 2005 and Incremental Modified Pushover (IMP) proposed by AZIMI
et al in 2009 with Nonlinear Time-History Analysis (NTHA) which gives exact
structural response. In the MIDA approach, each multi-degree of freedom (MDOF)
frame was modelled as several equivalent single degree of freedom (SDOF)
systems and analyzed using modal analysis up to the nth mode effects
(ie 3 for 3-story and 9 for 9-story). The IMP method modelled the MDOF as one
equivalent SDOF system considering only the first three mode effects. The
resulting nonlinear Incremental Dynamic Analysis (IDA) curves were plotted and
compared with that of NTHA. Both methods (MIDA and IMP) were found to be
accurate for low rise frames with accuracy levels reducing for medium rise
frames. The MIDA method produced more accurate results for both low rise and
medium rise structures. The results for both methods were on the conservative
side with the IMP being more conservative and can be used in design due to its
simplicity and safety margins.

G Tarta et al (2012)
compared interstory drift obtained from standard and advanced pushover analysis
with exact results from nonlinear time history analysis on two moment resisting
steel frames of 8 and 12 story. This frames were loaded in accordance with Eurocodes
and seismic response spectra for three ground motion. Two standard pushover
analysis (PA) namely: PA with uniform distribution of vertical loads and PA
with a vertical distribution after first mode vibration were considered. Force
based adaptive pushover analysis, interstory drift based scaling pushover
analysis and method of modal combination were the advanced pushover analysis
considered. Interstory drift errors which is a measure of the difference
between the exact method and particular method under consideration were
computed. The least error was obtained for adaptive pushover analysis methods with
interstory drift based scaling adaptive pushover analysis being the most accurate.
The errors obtained from method of modal combination rose significantly for 12
story frames recording the highest error making this method unreliable for high
buildings.

Hariri et al (2012) studied structural
response obtained from Endurance Time Analysis (ETA), Time History Analysis
(THA) and Incremental Dynamic Analysis (IDA) for four steel moment resisting
frames (9, 11, 13 and 15 stories). ETA, a dynamic pushover analysis capable of
estimating both linear and nonlinear utilizes endurance time acceleration
functions (ETAFs) to evaluate structural response. Incremental Dynamic Analysis
(IDA) which utilizes nonlinear time history analysis at different levels of
intensity of a specified ground motion. The four frames were analyzed using three
set of second generation ETAFs for the ETA method and seven scaled ground
motions for THA and IDA methods. Results indicated that ETA method produced
results with acceptable accuracy compared with THA and IDA.

The inconsistencies associated with
nonlinear static analysis was the basis for Rofooei et al (2012) to propose two
separate pushover analyses namely modal spectra combination (MSC) and first
mode load pattern of ASCE41-06 as a combination rule using three 2-D (one 8
story and two 15 story) moment resisting as a case study. The MSC load pattern
combines mode shapes with their weighting factors which are based on modal
frequencies and spectral values. Sway, plastic moment capacity and inelastic
response envelope were developed from both procedure and compared to method of
modal combination (MMC), modal pushover analysis (MPA) and nonlinear dynamic
time history analysis to ascertain the efficiency of the proposed methods. Sway
results obtained for the MSC and combination rule were acceptable compared with
nonlinear time history analysis. They also concluded that the results for MPA
proposed by Chopra, et al in 2003 were also acceptable.

Structural response (ductility
based reduction factor and response modification factors) were computed by Izadinia
et al (2011) for 3, 9 and 20 moment resisting steel frames analyzed by Adaptive
Pushover Analysis (APA) using eight different constant and adaptive lateral
load pattern to estimate proportion earthquake loads imposed on structures. ADA
introduced by Antoniou et al (2004) utilizes multi-mode effects and considers
inelastic range of plastic hinges unlike the Conventional Pushover Analysis
(CPA) which normally deals with the first elastic mode. ADA gave higher
structural response recording 16% for response modification factor and 17% for
ductility ratio above that obtained from CPA (Izadinia et al 2011).

In attempt to reduce
computational demand of nonlinear plastic design, Liu et al (2009) proposed a
pushover analysis which utilizes a single element per member for seismic design
considering initial imperfection, P-? and P-? effects.
Two buildings consisting of 3 and 9 story steel moment resisting frames were
analyzed in accordance with FEMA 440 and Nonlinear Integrated Design and
Analysis (NIDA). Base shear/weight were plotted against roof
displacement/height for FEMA 440, NIDA without initial imperfection and NIDA
with initial imperfection for both frames. NIDA without initial imperfection
gave more accurate results (very close to FEMA 440) than NIDA with initial
imperfection. NIDA procedure were less time consuming as compared to FEMA 440 (Liu
et al. 2009).

Azim et al (2009) developed the
Incremental Modified Pushover (IMP) which considers high mode vibration by
modelling each 4, 8, 12 and 16 story moment resistant frames as SDOF system and
one pushover analysis performed per ground motion from seventeen different
scaled earthquake as against Incremental Dynamic Analysis (IDA) which utilizes
nonlinear time history analysis at different levels of intensity of a specified
ground motion. The resulting responses (maximum roof displacement, inter-story
drifts and plastic hinge rotation for each ground motion) were compared with
that obtained from IDA. The results indicated that IMP method gave higher structural
response than the IDA.

Progressive collapse occurs when
failure/failures of any element of a structure results in partial or total
collapse of the structure either by serviceability or ultimate limit state.
Jinkoo et al (2007) studied progressive collapse capacity of steel resisting
frames in accordance with General Services Administration (GSA) and Department
of Defense guidelines using both linear static and nonlinear dynamic analysis
and concluded that the nonlinear dynamic procedure may be used as more precise
procedure whiles the linear procedure gave conservative decision for
progressive collapse potential of modeled structures.

Rofooei et al (2006) analyzed structural
response of dynamic nonlinear time history analysis on five steel resisting
moment frames (2, 5, 10, 15 and 20 stories) modelled as a single degree of
freedom (SDOF) system using five different load patterns for each of the five
earthquake records considered. Pushover curves were constructed from nonlinear
static analysis using (FEMA 360) as well as nonlinear dynamic SDOF analysis
using both bilinear and trilinear approximation to assess maximum roof
displacements (target displacement). Percentage error which measures the ratio
of the differences between studied procedures analyzed as SDOF and THA on MDOF
system, increases from low rise to high rise structures. Nonlinear static in
the first mode shape gave the least error for low rise buildings (2 and 5 stories)
with trilinear approximation having least error for high rise buildings (15 and
20 stories). First mode shape analysis is therefore the preferred nonlinear
static procedure for low rise structures compared with the complicated static
procedure in codes as both results are structurally close (Rofooei et al. 2006).

R. Hasan et al (2002), simplified
performance based design pushover analysis of buildings subjected to increasing
lateral loads using both linear and nonlinear analysis. Two buildings consisting
of three story and nine story steel moment frames representing low and high
rise buildings respectively were used for the study. Loading was done by using
constant gravity loads and lateral loads applied increasingly based on the different
spectral ground acceleration. The buildings were analyzed with the nodal
displacement, member displacement and forces measured. This analysis was
repeated with incremental lateral loads until the horizontal displacement at
the roof reached the collapse prevention displacement. They concluded that the
above described procedure and method offer adequate structural response (both
elastic and inelastic), information and results for performance based of
seismic design for new moment resisting frame structures as well as
rehabilitation of existing steel frame structures.

CONCLUSIONS

The studied methods provided varied
limits for structure height classifications. The upper limit for high rise
structures of most of the reviewed papers was 12 story which is not a good
representation of high rise building although Izadinia et al (2011) and Rofooei
et al (2006) pegged it at 20 story. Structural response of buildings to lateral
loads are more severe for taller buildings, therefore there is a probability
that the method being proposed may not be application to buildings with stories
above 20.

All the methods proposed and
reviewed in the above mentioned publications produced similar structural
response with errors increasing from low rise buildings to high rise buildings
when compared with the exact time history nonlinear dynamic analysis. Most of
the errors for low rise buildings were on the conservative side and hence can
be acceptable in design due to high factor of safety, however relying of these
methods will results in higher cross-sectional elements and uneconomical
designs. Since high errors were recorded for medium and high rise building, it
is not acceptable to consider the use of the above methods for design although
most of the above mentioned methods were on conservative side. Also, there is
the risk of the proposed methods not being on conservative side for structures
above 20 story which are yet to be reviewed using the above mentioned methods. More
studies need to be done for structural response to the proposed methods of
medium to high rise building (including buildings above 20 story).

Almost all the reviewed papers
except for Liu et al (2009) were silent on the actual time involved in using
the proposed methods to analyze the various structures compared with the
nonlinear dynamic analysis. Whiles argument were made that nonlinear dynamic
analysis is time consuming and therefore not practical for everyday design,
little was said about the associated time used in the proposed methods thus
making it difficult to ascertain if the proposed methods are less time
consuming.