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2. The theory of pushover analysis considering strain rate effects
2.1. The duration of a 1/4 cycle
The structures reciprocate in earthquake. The time-history curve of top
displacement(or storey drift) has many zero-crossings. The process between two
zero-crossings is named semicycle. The duration of a semicycle correspond with
relatively large or maximum top displacement(or storey dtift) is approximately
equal to half of the fundamental period(T1) of the structure whose responses are
dominated by the fundamental mode on condition that there is no big difference
between the predominant period of the ground motion(Tg) and T1. A semicyclecan be divided into two 1/4 cycle, the duration of a 1/4 cycle is approximately
equal to 1/4 of T1.
2.2. The variation law of strain in a 1/4 cycle
The variation law of strain in a semicycle is studyed by nonlinear time-history analysis of a reinforced concrete structure by ABAQUS(no elaboration
here). It is discovered that the shape of strain curve is most like sine curve in
a semicycle.
2.3. How to consider strain rate effects
By pushover analysis, we can obtain the roof displacement-base shearcurve or storey drift-base shear curve. Before applying lateral load, the structure
is subjected to gravity load. As the applied lateral load becomes larger, the roof
displacement or storey drift becomes larger. When the target displacement is
reached, the process is over.The process can be regarded as a 1/4 cycle. In a 1/4
cycle, the time-history curve of strain is most like sine curve. Because the
duration of a 1/4 cycle and variation law of strain are approximate, it is
impossible to consider strain rate effects precisely in pushover analysis. Using
average strain rate is the most convenient and rational method. It is convenient
to adjust the moment-curvature properties of the members according to average
strain rate.
2.4. Rationality of using average strain rate
Before applying lateral load, the structure is subjected to gravity load.
Every point of the structure has initial strain. Before the target displacement is
reached, the strain changes all the time. When the target displacement is reached,
the variation of strain reaches the maximum. In most cases, the initial strain is
micro. Therefore, it is neglected in the following.
We will illustrate the rationality of using average strain rate accoring to the
constitutive relations of rebar.The strain of rebar during pushover analysis can
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be expressed as follows:
tT
t1
max
2sin)(
= 1
Where max is the peak value of strain in a semicycle; 1T is the
fundamental period,4
0 1T
t
The strain rate of rebar during pushover analysis can be expressed as
follows:
tTT
t11
max
2cos
2)(
=
2
The average strain rate can be expressed as follows:
1
max4
T
=
3
Based on the equations above, we can obtain some conclusions: when the
strain is zero, the strain rate arrive at the peak value; when the strain arrive at
the peak value, the strain rate is zero; the maximun strain rate is 1.57 times as
large as average strain rate.
As the strain rate increases, the variation of rebars elastic modulus is
micro; the yield strength increases and can be expressed as follows[6]
:
ysfyd fcf )lg1(
0
+=
4
ysffc 410289.31709.0 = 5
Where
is the current strain rate;
0 is the quasi-static strain rate,
s/105.2 40
= ; ysf is the yield strength at quasi-static strain rate; ydf is
the dynamic yield strength at the current strain rate.
Elastic-perfectly plastic model for the rebar is shown in Fig.4.1yd
f is the
dynamic yield strength at the average strain rate in a 1/4 cycle, and 2ydf is the
dynamic yield strength at the maximum strain rate in a 1/4 cycle; ys is the
yield strain at the quasi-static strain rate; max is the maximun strain in the 1/4
cycle; 1yd is the yield strain at the average strain rate; hs is the ultimate
strain.
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If max is known, we can obtain the relations of stress and strain in the
1/4 cycle using (1) to (5). Before the strain exceed ys , the Constitutive
relations at different strain rate is identical(line AB in Fig.1);when the strain
is equal to max , a point F can be obtained according to the constitutive
relations at the quasi-static strain rate; there is another point E that can be
determined using (1) to (5) in line BD; E and F can be connected with astraight line approximatly. Curve ABEF is the relations of stress and strain
in the 1/4 cycle.When1max
297.1yd
= (there is no big difference between
1yd and ys in earthquake)E coincides with Cwhen 1max 297.1 yd ,
E can be determined between C and D. Generally, it is rational to useaverage strain rate.
Fig.1. Constitutive relations of rebar
3. The approach of pushover analysis considering strain rate effects
According to the theory above, strain rate effects can be considered in
pushover analysis. The initial state of structure and duration of a 1/4 cycle is
known, but the end-state is unknown. However, the end-state can be abtained
approximately by pushover analysis without considering strain rate effects,
which is called pseudo end-state. The step-by-step approach of this method is as
follows:
1. Determine the initial state of the structure and model.
2. Obtain the pseudo end-state by pushover analysis without considering strain
rate effects.
3. The average strain rate of every point in the 1/4 cycle can be obtained
according to the initial state, the pseudo end-state and the duration of the
process, then the model can be modified according to the average strain rate of
every point. In the end, perform pushover analysis for the second time. Plastic-
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hinge modle is used widely in pushover analysis. Therefore, it is necessary to
modify the settings of plastic-hinge only.
4. Numerical cimulation
According to the procedure above, a four-story reinforced concrete frame
structure is analysed by SAP2000(no elaboration here). We considered strain
rate effects by adjusting the moment-curvature properties of the members. The
pushover curve is shown in Fig.2.
By comparing the two curves, some conclusions can be obtained. First, the
base shear and story drift increase when the structure shows nonlinearcharacteristics after considering strain rate effects. Second, the structure can
endure larger base shear after considering strain rate effects.
Fig.2. Pushover Curve of the Structure
References
1. P. Fajfar, P. Gaspersic:Earthquake Engineering and Structural Dynamics,
1996, 25, (1), 31 46.
2. J. M. Bracci, S. K. Kunnath, A. M. Reinhorn: Journal of Structural
Engineering, 1997, 123, (1), 3 10.
3. B. Gupta, S. K. Kunnath:Earthquake Spectra, 2000, 16, (2), 367 391.
4. A. K. Chopra, R. K. Goal: Earthquake Engineering and Structural
Dynamics, 2002, 31, (3), 561582.
5. P. H. Bischoff, S. H. Perry:Materials and Structures, 1991, 24, (6), 425-440.
6. M. Li, H. N. Li: China Civil Engineering Journal, 2008, 43, (4), 70-75.
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