The Pushover Analysis from basics - Rahul Leslie

1
The Pushover Analysis
– from basics
Presented by .
Rahul Leslie
Assistant Director,
Buildings Design,
DRIQ, Kerala PWD,
Trivandrum, India

2
Introduction
• Performance Based Design --- an emerging field
– To provide engineers with a capability to design buildings that have
predictable and reliable performance in earthquakes
– It employs concept of ‘performance objectives’, which is the
specification of an acceptable level of damage on experiencing a
earthquake of a given severity.
(FEMA 349)
• Seismic design for the future
– Presently a linear elastic analysis alone is sufficient for both its elastic
and ductile design
– In course of time, for large critical structures, a specially dedicated non-
linear procedure will have to be done, which finally influences the
seismic design as a whole.
The Pushover Analysis – from basics Presented by Rahul Leslie

3
Introduction
• Linear approach (IS:1893-2002)
is based on the Response
Reduction factor R.
– For example, R = 5, means that
1/5th of the seismic force is
taken by the Limit State capacity
of the structure.
– Further deflection is taken by the
ductile capacity of the structure.
– Reinforced Concrete (RC)
members are detailed (as per
IS:13920) to confirm its ductile
capacity.
– We never analyse for the ductile
part, but only follow the
reinforcement detailing
guidelines for the same.

4
Introduction
• The drawback is that the response
beyond the limit state is neither a
simple extrapolation, …
• … nor a perfectly ductile behaviour
with pre-determinable deformation
capacity, due to various reasons:
– Change in stiffness of members due
to cracking and yielding,
– P-delta effects,
– Change in the final seismic force
estimated (due to Change in
• time period ‘T’ and
• effective damping ratio ‘ζ’ (also
represented by ‘β’)
– etc.

5
Introduction
• Although elastic analysis gives a good indication of elastic capacity
of structures and shows where yielding will first occur,
– It cannot predict the redistribution of forces during the progressive
yielding that follows and predict its failure mechanisms.
• A non-linear static analysis can predict these more accurately.
– It can help identify members likely to reach critical states during an
earthquake for which attention should be given during design and
detailing.

6
Introduction
The Pushover Analysis (PA):
• PA is a non-linear analysis procedure to estimate the strength
capacity of a structure beyond its Limit State up to its ultimate
strength.
• It can help demonstrate how progressive failure in buildings most
probably occurs, and identify the mode of final failure.
• The method also predicts potential weak areas in the structure, by
keeping track of the sequence of damages of each and every
member in the structure.

7
PA can be useful under two situations:
 When an existing structure has deficiencies in seismic resisting
capacity,
 due to either omission of seismic design when built, or
 the structure becoming seismically inadequate due to a later
upgradation of the seismic codes,
is to be retrofitted to meet the (present) seismic demands,
PA can show where the retrofitting is required and how much.
 For a building in its design phase, PA results help scrutinise and
fine tune the seismic design based on SA.
Introduction

8
• For a new building, PA is meant to be a second stage analysis (The
first stage being a conventional Seismic analysis - SA).
• This is because the details of reinforcement provided are required to
calculate exact hinge properties (to be covered later)
• But one has to design the structure based on SA in order to obtain
the reinforcement details.
• This means that PA is meant to be a second stage analysis (The first
stage being a conventional SA).
• Thus the emerging methodology to an accurate seismic design is:
1. First a conventional linear seismic analysis based on which a primary
structural design is done;
2. Insertion of hinges determined based on the design/detail and then
3. A pushover analysis is done, followed by
4. Modification of the design and detailing, wherever necessary, based
on the latter analysis.
5. The above steps may have to be iterated, if required.
Introduction

9
Features of a Typical Pushover Approach
• The model, which is a Multi-degree of freedom (MDoF) model, is
used for the analysis
There are certain features common to all PA approaches:
1. An analysis model of the building, is generated using a common
analysis-design software package (having facility for PA), like
– STAAD.Pro,
– SAP2000, ETABS,
– MIDAS/Gen, etc.

10
Non-linear Building model & Non-linear Hinges
Pushover analysis uses a non-linear computer model
for the analysis:
– This is done by incorporated in the form of non-linear hinges
inserted into an otherwise linear elastic model which one
generates using a common analysis-design software package
(STAAD.Pro, SAP2000, ETABS, MIDAS/Gen, etc.)
– Hinges are points on a structure where one expects cracking
and yielding to occur in relatively higher intensity so that they
show higher flexural/shear displacement, under a cyclic loading

11
- These are locations where one expects to see cross
diagonal cracks in an actual building structure after a
seismic mayhem
– they would be at either ends of beams and columns, the ‘cross’
being at a small distance from the joint
– this is where one inserts hinges in the corresponding computer
model.

12
• Basically a hinge represents localised force-displacement relation of
a member through its elastic and inelastic phases under seismic
loads.
• A flexural hinge represents the moment-rotation relation of a beam.
• Hinges are of various types – namely,
– (1) flexural hinges,
– (2) shear hinges
– (3) axial hinges.

13
• The flexural and shear hinges
are inserted into the ends of
beams and columns.
• Since the presence of masonry
infills have significant influence
on the seismic behaviour of
the structure, modelling them
using equivalent diagonal
struts (of ‘truss’ elements) is
common in PA
• The axial hinges are inserted
at either ends of the diagonal
struts

14
Typical Moment Hinge property:
• AB represents the linear range
from unloaded state (A) to its
effective yield (B),
• Followed by an inelastic but
linear response of reduced
(ductile) stiffness from B to C.
• CD shows a sudden reduction
in load resistance, followed by
a reduced resistance from D to
E, and
• finally a total loss of resistance
from E to F.
Flexural Hinge

15
• These hinges have non-linear
states defined within its ductile
range as
– ‘Immediate Occupancy’ (IO),
– ‘Life Safety’ (LS) and
– ‘Collapse Prevention’ (CP)
• This is usually done by dividing
B-C into four parts and
denoting IO, LS and CP, which
are states of each individual
hinges
Flexural Hinge

16
2. The model is pushed monotonically with an invariable distribution of lateral load with some predefined
distribution pattern such as:
– Proportional to 1st
mode (or SRSS combination of modes)
– Inverted triangle / Uniform distribution
– Power distribution (for example, parabolic)
∑
= n
i
k
jj
k
ii
bi
hW
hW
VQ

17
2. (Continuation …)
• Unlike conventional SA, in Pushover analysis, analysis for Gravity
loads is done first, continued by an analysis for Lateral loads.
• Since PA is done to simulate the behaviour under actual loads, the
Gravity loads applied are not factored, but in accordance with
Cl.7.3.3 and Table 8 of IS:1893-2002 :
[DL + 0.25 LL≤3kN/sq.m + 0.5 LL>3kN/sq.m]

18
3. A pushover curve is obtained, which is a Base shear (Vb) vs. Roof
top displacement (Δrt) curve
– Base shear is sum of all horizontal support reactions in that
direction
– Roof top displacement is the displacement at the centre of
mass of the general roof

19
4. A single-degree of freedom (SDoF) model, corresponding to the MDoF model, and
the rules to convert the parameters of the MDoF model (Vb & Δrt) to those of the
SDoF model (Sa & Sd) are defined

20
4. (Continuation…) A single-degree of freedom (SDoF) model,
corresponding to the MDoF model, and the rules to convert the
parameters of the MDoF model (Vb & Δrt) to those of the SDoF model
(Sa & Sd) are defined
– In ATC-40 and FEMA440, the conversion is
(where ), and
– In EC 8 (where Sa and Sd are denoted by F* and d* respectively)
and
α
WV
Sa b /
= M
Mk 1=
=α
rtkkP
rt
Sd
@,11 == ⋅
∆
=
φ
1=
=
k
b
P
V
Sa
1=
∆
=
kP
rt
Sd

21
5. The Sa-Sd curve has to be converted to an equivalent bi-linear
curve (equal energy) by a suitable method
– Different codes follow different methods
ATC-40 and FEMA440
– ATC-40 and FEMA440 follows the
method of keeping the 1st
line as initial
tangent stiffness and adjusts the 2nd
line (to the point under consideration)
such that to get the ‘equal area’.

22
EC 8
– EC8 (EuroCode 8) follows the method of
keeping the 2nd
line (to the point under
consideration) as ‘perfectly plastic’, ie.,
horizontal and adjusts the 1st line such
that to get the ‘equal area’.

23
ATC-40 and FEMA440 EC8

24
PA procedures can generally be classified to two:
1. DCM (Displacement Coeff. Method): These procedures
estimates a Target displacement prior to the analysis, to which
the model has to be pushed, and on analysis, checked for the
intended (good) performance at that displacement. The method
is nevertheless, iterative. Ref:-
- FEMA356,
- FEMA440 (Ch.5),
- EC 8
2. CSM (Capacity Spectrum Method): The analysis is done, and
each pt. on the pushover curve (known as Capacity curve) is
consecutively checked to see whether the Sa-Sd at that pt.
meets (or intersects) the Response Spectrum curve (known as
Demand curve), reduced by a factor. (continued…)
Different Pushover Approaches

25
PA procedures can generally be classified to two:
2. CSM : … For each point on the Capacity curve, the Demand
curve to be checked with, for intersection, is a Response
Spectrum curve reduced by a reduction factor calculated
corresponding to that point under consideration on the Capacity
curve. When the curves intersects (or meet), that meeting point
is known as the Performance Pt. Ref:-
- ATC-40,
- FEMA440 (Ch.6)
- EC8 (Optional method)
Different Pushover Approaches

26
The steps for the CSM method are:
1. First, the Response Spectrum (RS) curve has to be modified: from its ordinates of Sa vs. Time period ‘T’, to its
‘Acceleration Displacement Response Spectrum’ (ADRS) form, which is an Sa vs. Sd curve.
• This to facilitate the super-imposing the pushover curve over the RS (which is in its ADRS form)
RS ADRS
Steps for CSM method of Pushover Analysis

27
The steps for the CSM method are:
1. First, the Response Spectrum (RS) curve has to be modified:
from its ordinates of Sa vs. Time period, to its ‘Acceleration
Displacement Response Spectrum’ (ADRS) form, which is an
Sa vs. Sd curve.
• This is done by using the relation
RS ADRS
Sa
T
Sd 2
2
4π
=

28
2. Super-impose the converted Pushover curve on the ADRS curve:

29
3. With the Capacity curve (Pushover curve) superimposed on the Demand
curve (ADRS), each point on the former is consecutively checked to :
i. Get the yield point ordinates (Say & Sdy)
ii. Calculate the ductility μ and the 2nd
tangent stiffness coeff. α
ATC-40, FEMA440 EC8

30
iii. Determine the reduced ADRS for the above parameters corresponding to that pt. on the Capacity curve
as:
 ATC-40/FEMA440 : Calculate damping β from ductility μ and 2nd
tangent stiffness coefficient α. Reduce ADRS
corresponding to β
 EC 8 : Reduce ADRS corresponding to ductility μ
ATC-40, FEMA440 EC8

31
• For example, in ATC-40, for the reduction of the Demand (ADRS)
curve, the ‘effective’ damping ratio β is determined from the
formula :
y
p
d
d
=µ
( )( )
( )ααµπµ
αµ
κβ
−+
−−
+=
1
112
05.0eff
init
nd
K
K2
=α
y
y
init
d
a
K =
yp
yp
nd
dd
aa
K
−
−
=2

32
• …where the Damping Modification Factor κ is determined from
the building type
Table : Structural behaviour types Table : Values for Damping
Modification Factor κ
Shaking
Duration
Essentially
New
Building
Average
Existing
Building
Poor
Existing
Building
Short Type A Type B Type C
Long Type B Type C Type C
Structure
behaviou
r type
βeq(%)
κ
Type A
≤16.25 1.0
>16.25
Type B
≤25 0.67
>25
Type C
Any
value
0.33
( ) 02/51.013.1 βπ−
( ) 02/446.0845.0 βπ−
( )( )
( )ααµπµ
αµ
κβ
−+
−−
+=
1
112
05.0eff

33
( )
12.2
681.021.3 (%)effeLog
SRa
β−
=
( )
65.1
41.031.2 (%)effeLog
SRv
β−
=
• From the effective damping ratio β, the factors for reducing the
ADRS curve are determined from the formula :

34
4. Include the reduced ADRS Demand curve in the super-imposed
graph:

35
Step by step through each method
1. The conventional SA procedure is explained to highlight the
difference in approaches between SA & PA
2. Trace the progress of a PA from beginning to end,
• both demonstrates plots of Vb vs Δroof top and RS curve in its
– separate and uncombined form and
– also their transformed and super-positioned ADRS plot.

36
• In SA, the maximum DBE force acting on the structure is Z/2.(Sa/g),
(assuming I = 1) with Sa/g corresponding to the estimated time
period.
• Its envelop is the RS curve marked q
• The RS curve for the Limit State design is plotted in terms of Z/2R.
(Sa/g), and is marked as curve p.
-- SA Method

37
• Fig. shows the Vb vs Δroof top displacement.
– The point P represents the Vb and Δroof top for the design lateral load
(ie., of 1/R times full load)
– The point Q represents the same for the full load, had the building
been fully elastic
– Point Q' for a perfectly-elastic perfectly-ductile structure.
– The slope of the line OP represents
the stiffness of the structure in a
global sense. Since the analysis is
linear, the stiffness remains same
throughout the analysis, with Q
being an extension of OP.
-- SA Method

38
-- SA Method
• The same is represented in Fig.(left) where, for the time period Tp of
the structure,
– the full load is represented by Q (Saq), and
– the design load by P (Sap).

39
-- SA Method
• The ADRS representation of SA is as in Fig.(left).
– the full load is represented by Q (Saq),
– the design load by P (Sap).

40
-- PA Method
Now we shall see how differently the PA approaches the same
scenario :-
• The segment OA in Fig.(left) is equivalent to OP in Fig.(right), with the
slope representing the global stiffness in its elastic range.

41
-- PA Method
The RS curve : Segment OA has time period Ta, curve ‘a’ representing
the RS curve and Saa is the lateral load demand, in its elastic range.

42
-- PA Method
• ADRS representation:

43
-- PA Method
• As the analysis progresses, the lateral load is monotonically
increased beyond its elastic limit of A, and the first hinges are formed.
This decreases the overall stiffness of the structure. This is
represented by the segment AB.
• The decrease in slope of OB from that of OA shows the change in
secant stiffness.

44
-- PA Method
• The first hinges are formed, decreasing the overall stiffness of the
structure, which in turn increases T and β, represented by point B in
the plots.

45
-- PA Method
• The change in the x-axis value of point B from that of point A shows
the shift of time period from Ta to Tb.
• The increase in β of the structure calls for a corresponding decrease
in the RS curve, reduced by a factor calculated from β, which has thus
come down from curve a to b.

46
-- PA Method
• ADRS representation: Note the angular shift from Ta to Tb .
• The increase in β of the structure calls for a corresponding decrease
in the RS curve, reduced by a factor calculated from β, which has thus
come down from curve a to b.

47
-- PA Method
• As the lateral load is further increased monotonically, more hinges are
formed and the existing hinges have further yielded in its non-linear
phase represented by point C
• This has further reduced the stiffness (the slope of OC),

48
-- PA Method
• (Here are the two graphs overlapped – a possibility

49
-- PA Method
• This has further reduced the stiffness, and increased T (from Tb to
Tc).

50
-- PA Method
• More hinges are formed and the existing hinges have further yielded
in its non-linear phase, represented by point C
• Note the angular shift from Tb to Tc.

51
-- PA Method
• Here the point C is where the capacity curve OABC extending
upwards meets the demand curve in, which was simultaneously
descending down to curve c.
• Thus C is the point where the total lateral force expected Sac is same
as the lateral force applied ~Vbc
• This point is known as the performance point.

52
-- PA Method
• It is also defined as the point where the ‘locus of the performance
point’, the line connecting Saa, Sab and Sac, intersects the capacity
curve

53
PA Method – Reviewing results
• Once the performance point is found, the overall performance of the
structure can be checked to see whether it matches the required
performance level, based on inter-storey drift limits specified in ATC-
40, which are
– 0.01h for IO,
– 0.02h for LS, and
– 0.33(Vb/W)∙h for CP, (h = height of the building).
• The performance level is based on the importance and function of the
building. For example, hospitals and emergency services buildings
are expected to meet a performance level of IO.

54
• The next step is to review the hinge formations at performance point.
One can see the individual stage of each hinge, at its location.
• Tables are obtained showing the number of hinges in each state, at
each stage, based on which one decides which all beams and
columns to be redesigned.
• The decision depends whether the most severely yielded hinges are
formed in beams or in columns, whether they are concentrated in a
particular storey denoting soft story, and so on.
PA Method – Reviewing results
OA AB BC

55
Adaptation for the Indian Code
Adapting of Pushover Analysis (PA) for IS:1893-2002
• The PA has not been introduced in the Indian Standard code yet.
However the procedure described in ATC-40 can be adapted for the
seismic parameters of IS:1893-2002.
• The RS curve in ATC-40 is
described by parameters
– Ca and
– Cv,
where the curve just as in IS:1893,
is having a flat portion of intensity
2.5 Ca and a downward sloping
portion described by Cv/T.
Resp. Spec (ATC-40)

56
• The seismic force in IS:1893-
2000 is represented by
(ZI/2R).(Sa/g), where Sa/g is
obtained from the RS curve,
which in our code is
represented by
– 2.5 in the flat portion &
– the downward sloping
portion by
• 1/T for hard soil,
• 1.36/T for medium
soil and
• 1.67/T for soft soil.
Resp. Spec (IS:1893-2002)






=
g
Sa
R
ZI
Ah
2

57
• On comparison it can be inferred that
– Ca = Z/2 and
– Cv = Z/2 for hard,
1.36∙Z/2 for medium and
1.67∙Z/2 for soft soil
• Here ‘I’ is not considered, since in PA, the criteria of importance of
the structure is taken care of by the performance levels (IO, LS & CP)
• R is also not considered since PA is always done for the full lateral
load.
Resp. Spec (ATC-40) Resp. Spec (IS:1893-2002)






=





=
g
SaZ
g
Sa
R
ZI
Ah
22

58
• The ‘Limit State’ inter-storey drift limit specified in IS:1893-2002, being
0.004, when accounted for
– R = 5 for ductile design and
– I = 1.5 for important structures (IO performance level)
= 1.0 for ordinary structures (LS performance level)
gives 0.004∙R/I = 0.02 and 0.0133 for IO and LS respectively
• The drift limit can be compared with those specified in ATC-40 (0.01
and 0.02 for IO and LS respectively). The limit for IO in IS:1893-2002
is more relaxed than that in ATC-40.
• This 0.004∙R/I can be taken as the IS:1893-2002 limits for pushover
drift, where I takes the values corresponding to Important and
Ordinary structures for limits of IO and LS respectively.

59
• Presented in this section are the results of a pushover analysis done
on a 10 storey RCC building of a shopping complex using the
structural package of SAP2000.
Example of a building analysis

60
• In the model, beams and columns were modelled using frame
elements, into which the hinges were inserted.
• Diaphragm action was assigned to the floor slabs to ensure integral
lateral action of beams in each floor.
• Although analysis was done in both transverse and longitudinal
directions, only the results of the former are discussed here.
• The lateral load was applied in pattern of that of the 1st mode shape
in the transverse direction of the building, with an intensity for DBE as
per IS:1893-2002, corresponding to zone-III in hard soil.

61
• The ADRS plot shows the Sa and Sd at performance point as 0.085g
and 0.242m.
• The corresponding Vb and Δroof top are 1857.046 kN and 0.287m.
The value of effective T is 3.368s.
• The effective β at that level of the demand curve which met the
performance point is 26%.

62
• Table shows the hinge state details at each step of the analysis.
Step
Δroof top
(m)
Vb
(kN)
Hinge States
Total
Hinges
A to
B
B to
IO
IO to
LS
LS to
CP
CP
to C
C to
D
D to
E > E
0 0 0 1752 0 0 0 0 0 0 0 1752
1 0.058318 1084.354 1748 4 0 0 0 0 0 0 1752
2 0.074442 1348.412 1670 82 0 0 0 0 0 0 1752
3 0.089645 1451.4 1594 158 0 0 0 0 0 0 1752
4 0.26199 1827.137 1448 168 136 0 0 0 0 0 1752
5 0.41105 2008.48 1384 144 136 88 0 0 0 0 1752
6 0.411066 1972.693 1384 146 136 86 0 0 0 0 1752
7 0.411082 1576.04 1376 148 136 39 0 0 53 0 1752
8 0.411098 1568.132 1376 148 136 37 0 0 55 0 1752
9 0.411114 1544.037 1375 149 136 31 0 0 61 0 1752
10 0.40107 1470.133 1375 149 136 31 0 0 61 0 1752

63
• For the performance point, taken as step 5 (which actually lies
between steps 4 and 5),
– 95% of hinges are within LS and IO performance levels
– 88% within IO performance level.
Step
Δroof top
(m)
Vb
(kN)
Hinge States
Total
Hinges
A to
B
B to
IO
IO to
LS
LS to
CP
CP
to C
C to
D
D to
E > E
0 0 0 1752 0 0 0 0 0 0 0 1752
1 0.058318 1084.354 1748 4 0 0 0 0 0 0 1752
2 0.074442 1348.412 1670 82 0 0 0 0 0 0 1752
3 0.089645 1451.4 1594 158 0 0 0 0 0 0 1752
4 0.26199 1827.137 1448 168 136 0 0 0 0 0 1752
5 0.41105 2008.48 1384 144 136 88 0 0 0 0 1752
6 0.411066 1972.693 1384 146 136 86 0 0 0 0 1752
7 0.411082 1576.04 1376 148 136 39 0 0 53 0 1752
8 0.411098 1568.132 1376 148 136 37 0 0 55 0 1752
9 0.411114 1544.037 1375 149 136 31 0 0 61 0 1752
10 0.40107 1470.133 1375 149 136 31 0 0 61 0 1752

64
• Following figures shows the hinge states during various stages in
course of the analysis.
Fig: Hinge states in the structure model at (a) step 0 & (b) step 3

65
Fig: Hinge states in the structure model at (c) step 5 & (d) step 8

66
Fig: Hinge states in the structure model at (e) step 10

67
• Hinge properties
– Determining hinge properties (beams, columns, diagonal struts)
– Determining hinge properties for flat-slab and shear walls
• Seismic analysis  design/detailing  hinge property
calculation  insertion of hinges  Pushover Analysis
– Doing the above manually at a practically acceptable speed
– Non-availability of a semi-automatic method in standard Analysis
Packages (STAAD, ETABS, etc.) :
Facility to quickly define details of provided
reinforcement bars for beams & columns and have the package
to automatically insert appropriately calculated hinges not
available.
Issues

68
• Inclusion of building torsion (no standardized guidelines
available)
• Inclusion of higher modes in PA
– PA with vectors that represent the effects of multiple modes
(FEMA 356)
– Explicit consideration of Multiple Modes
• Modal Pushover Analysis (Chopra and Goel, (2001).
• Incremental Response Spectrum Analysis (Aydinoglu, 2003)
• Consecutive Modal Pushover (Poursha et al., 2009)
– Progressive changes in the load vector pattern applied to the
structure.
• Displacement Adaptive Pushover (Antoniou and Pinho, 2004)
• IS:1893-2002 is yet to include the method
Limitations

69
References:
• IS 1893 (Part 1)–2002, “Indian Standard Criteria for Earthquake Resistant
Design of Structures, Part 1: General Provision and Buildings”, Bureau of
Indian Standards, New Delhi.
• FEMA 356 (2000) “Prestandard and Commentary for the Seismic Rehabilitation
of Buildings”, Federal Emergency Management Agency, Washington, DC, USA.
• ATC-40 (1996) “Seismic Analysis and Retrofit of Concrete Buildings”, vol. I,
Applied Technology Council, Redwood City, CA, USA.
• FEMA-440 (2005) “Improvement of Nonlinear static seismic analysis
procedures”, Federal Emergency Management Agency, Washington, DC, U.S.A.
• prEN 1998-1 (2003) “Eurocode 8 Part 1: Design of structures for earthquake
resistance”, European Committee for Standardization, Brussels.
• Jisha S. V. (2008), Mini Project Report “Pushover Analysis”, Department of Civil
Engineering, T. K. M. College of Engineering, Kollam, Kerala.

70
A write up on this topic can be found at …
http://rahulleslie.blogspot.in/p/blog-page.html
… but covers only the ATC-40 method of pushover
analysis.
Note

71
An effort has been made to present the topic as simple as
possible…
…presume, at least to some extend, the aim has been fulfilled.
Conclusion
Thank you
rahul.leslie@gmail.com

The Pushover Analysis from basics - Rahul Leslie

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