Pushover Analysis of Structures Considering Strain Rate Effects

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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.

7. P. Pankaj, E. Lin:Engineering Structures, 2005, 27, (7), 1014-1023.

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Pushover Analysis of Structures Considering Strain Rate Effects