nonlinear response analysis of soil layers under severe earthquake

Procedia Environmental Sciences 12 ( 2012 ) 940 – 948

1878-0296 © 2011 Published by Elsevier B.V. Selection and/or peer-review under responsibility of National University of Singapore.doi: 10.1016/j.proenv.2012.01.369

2011 International Conference on Environmental Science and Engineering

(ICESE 2011)

Nonlinear Response Analysis of Soil Layers Under Severe Earthquake *

Tao Lu1,a,Jingyan Huo1,a,Mianshui Rong2,b

1Department of Earthquake Engineering,Institute of Disaster Prevention,Yanjiao, Sanhe 065201, China 2Branch of composite Disaster Prevention,Institute of Crustal Dynamics, China Earthquake Administration,Changping, Beijing

100085, China a [email protected],[email protected]

* This work is partially supported by Teachers’ Scientific Research Fund of China Earthquake Administration Grant #20090111 to Tao Lu

Abstract

Equivalent Linear Method is a common way used in earthquake engineering to analyze nonlinear seismic response of soil layers, but the response under severe earthquake is underestimated by the way. For analyzing nonlinear response more veritably, in the study, a direct nonlinear method was proposed and used in a case for 1D seismic response analysis of soil layers under severe earthquake. The results obviously showed that, comparing with direct nonlinear method, the Equivalent Linear Method underestimated in the case not only the seismic response value but also response duration in natural period range of common civil engineering structures. The direct nonlinear method adopted in the study is more fitful for nonlinear response of soil layers under severe earthquake.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name organizer]

Keywords: Nonlinear earthquake response. Numerical analysis. Time domain. Severe earthquake. Response duration spectrum.

1.Introduction

Equivalent Linear Method is a common way to analyze nonlinear seismic response of soil layers in earthquake engineering, but nowadays studies and strong-motion observation data shows the real seismic response will be underestimated by equivalent linear method, especially in the cases under severe

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© 2011 Published by Elsevier B.V. Selection and/or peer-review under responsibility of National University of Singapore.

941 Tao Lu et al. / Procedia Environmental Sciences 12 ( 2012 ) 940 – 948

earthquake or of soft soil layers. The misestimate is due to the exaggerated resonance effect caused by equivalent linear method in the cases [1-4].

To avoid the misestimate, a direct nonlinear method is adopted and coded in a FORTRAN program, which is used to estimate nonlinear seismic response analysis of a soil layers under severe earthquake, and the nonlinear results are compared with the results by equivalent linear method.

The case study shows obviously that the equivalent linear method underestimates not only the seismic response value but also response duration in natural period range of common civil engineering structure, and the direct nonlinear method adopted in this study is more fitful for the cases under severe earthquake.

2.Fundamental Equations

2.1.Numerical Model

Figure 1 is proposed model for 1D nonlinear response analysis of a horizontal layered soil layer under lateral earthquake load, the problem is simplified to an initial boundary value problem; an integration scheme of space-time overlap based on central differential equation is used to solve the problem in this study [5].

In the model, soil layers is divided N sub-layer for numerical analysis in time domain, every sub-layer’s thickness is nh ( Nn ,,2,1 ), and time step is t , and they meet the requirements of stability and accuracy in numerical analysis.

Fig. 1 proposed model for 1D nonlinear response analysis of horizontal layered soil layers under lateral earthquake load

When tpt , the velocity on the upper surface of nth sub-layers

( , ) 1 , 2 , , pn nz p t n N 1


In Eq.1, the vertical coordinate of nth sub-layers upper surface 0 1 1n nz h h h00h When ( 1 2)t p t , the shear stress and strain at the midpoint of nth sub-layer could be

marked

1 , ( )

2 2 1 , 2 , , 1

1 , ( )

2 2

p nn n

p nn n

hz p t

n Nh

z p t2

According to free boundary condition and central differential equation, the velocity on the upper surface of nth sub-layers

1

1 1 1

1

1

1( ) 2 , 3 , ,

p p p

p p p p

n n n n

n

t

m

tn N

m

3

in (3)

1 1 1

n n n n-1 n-1

1

2

1( h h ) 2 , 3 , ,

2

m h

m n N

4

According to deformation compatibility condition, get the central differential equation

1 1 11

p p p pn n n n

nh t5

So

1 1 1

1( ) 1 , 2 , , p p p p

n n n n

n

tn N

h6

When ( 1 2)t p t , nonlinear stress-strain relationship at the midpoint of nth sub-layer is

11 pn

pn 7

(1)-(7) is numerical type of the proposed integration scheme of space-time overlap in this study.

2.2.Proposed Dynamics Stress-Strain Relationship of Soil

Dynamics stress-strain relationship of soil is first thing in the nonlinear seismic response analysis of soil layer, in this study; we adopted a direct nonlinear model in time domain for simulating 1D shear stress-strain relationship of soil under earthquake load.

The proposed stress-strain relationship is a dynamic skeleton curve including damp ratio degeneration coefficient based on hyperbolic line type, the hysteretic rules of this relationship could fit soil dynamic


parameter curve, which is used by equivalent linear method. The relationship is showed by equation (9), more details could be referenced [ 5-6] .

0

'

0

( )( )

( )12

( ) ( )

1

C M CC C

M CC

r

M CC M

M C

r

GK

G M

8

In (8), K is damp ratio degeneration coefficient, the equation is

00

' 2

0 ' '

2 ( )( )( )

2 4 ln 1 ( )( )2 2

M C M C

M C M Cr M C M C

r r

K

G9

2.3.Dynamic boundary condition

The dynamic boundary condition is another important thing, in this study, the transmitting boundary is used to treating incident wave input and wave scattering [7].

Based on (1)-(9) and transmitting boundary, a direct nonlinear method in time domain could be formed. In this study, a FORTRAN program- DynaSoil1D is coded by the method, and used in the nonlinear response analysis of a soil layer under severe earthquake followed, the validity of the method and the program could be referenced [5] .

3.Case Study

In the study, a soil layer’s nonlinear seismic response under severe earthquake is analyzed by DynaSoil1D, and same analysis is analysis by equivalent linear method to contrast.

TABLE I Parameters of Soil Layers

No. Soil Type Layer

Thickness ( m)

Shear Wave Velocity

(m/s) 1 miscellaneous fill 0.80 271.0 2 Sand 2.20 156.0 3 Fine sand 3.80 300.0 4 Silt sand 2.20 218.0 5 Clay 4.00 237.0 6 Silt 2.20 282.0 7 Fine sand 3.00 288.0 8 Silt Clay 12.30 319.0 9 Weathered rock 16.50 829.0


The physics soil layer parameter is showed in Table I, the G/G0- Curve and - Curve of every soil is showed in Fig.2 and Fig. 3.

Shear Strain

Fig. 2 G/G0- Curve

Shear Strain

Fig. 3 - Curve

TCU088, an accelerogram recorded in Chi-Chi earthquake, is used as an incident wave to calculate, time history and Fourier amplitude spectrum is showed in below.

Time(Second)

Fig.4 Time History of the incident accelerogram (TCU088)

Dam

ping

Rat

ion

Acc

eler

atio

n(M

/s2 )

G/G

0


Frequency (Hz)

Fig.5 Fourier Amplitude Spectrum of the incident accelerogram (TCU088)

3.1.Surface Response Time History and Normalized Acceleration Spectrum

Nonlinear seismic response of the soil layer above mentioned is analyzed by two method, equivalent linear method and direct nonlinear method. Figure 6 shows surface acceleration response time history by two methods, some obvious differences could be found

(1) Response time history curve from equivalent linear method is very smooth, and from direct nonlinear method is coarse;

(2) Relative duration of response time history from direct nonlinear method id longer than from equivalent linear method.

The differences above mentioned is the just exhibit of the exaggerated resonance effect caused by equivalent linear method, under severe earthquake, in the analysis process, equivalent linear method makes soil layers show exaggerated nonlinear character, too small stiffness and too large damping ratio, which chiefly filtering the relative high frequency components (such as >1.0Hz).

Time (Second)

Fig6. Acceleration history of surface response analysis results by different methods

In term of earthquake resistant, major aim of nonlinear seismic response of the soil layer is providing seismic ground motion parameters for civil engineering design. Normalized acceleration spectrum is one

Four

ier

Am

plitu

de

(M/s

)A

ccel

erat

ion

(cm

/s2 )

Acc

eler

atio

n(c

m/s

2 )


of the parameters; Normalized acceleration spectrum of surface response result by two methods is showed in Fig. 7.

Obviously, in period range from 0.6 seconds to 0.5 seconds, the spectrum value from equivalent linear method is smaller than from direct nonlinear method, and lots of civil engineering structures’ natural period is within the range, underestimating of the spectrum value in the period range is dangerous under severe earthquake.

Period (Second)

Fig7. Normalized acceleration spectrum of surface response result by two methods

3.2.Response Duration Spectrum

Normalized acceleration spectrum value above shows magnitude of earthquake loads of difference natural period SDOF suffering, but it is not only parameter leading structure to destroy, earthquake duration is another import one.

For exploring the influence of two methods result to earthquake resistant, a new spectrum- response duration spectrum is discussed.

Nor

mal

ized

Acc

eler

atio

n


Period (Second)

Fig. 8 Response duration spectrum of surface response by different methods

As an acceleration response spectrum is simply a plot of the peak or steady-state acceleration response of a series of oscillators of varying natural period, that are forced into motion by the same base vibration or shock, in the study, response duration spectrum is simply a plot of acceleration response time history ’s 90% energy duration of a series of oscillators of varying natural period.

According to definition on response duration spectrum above, response duration spectra of surface response result by two method are calculated, and deviation of equivalent linear method result’s spectrum and direct nonlinear method result’s are calculated, they are showed in Fig.8, in this case some tendency could be found

(1) Response duration spectra shape from two methods is quite similar, larger natural periods is leading to longer energy duration;

(2)In period range from 2 seconds to 10 seconds, response duration spectra value from two methods is same;

(3) In period range less than 2 seconds, duration spectra value from equivalent linear method result is less than value from direct nonlinear method, the maximum absolute value of deviation is about 17 seconds, and in most part of the period range, the normalized acceleration spectrum value from equivalent linear method is smaller than from direct nonlinear method.

According to discuss above, in the case, not only normalized acceleration spectrum value but also Response duration spectrum value is underestimating by equivalent linear method in natural period range of common civil engineering structures, it is dangerous.

4.Result and Discussions

In this study, a direct nonlinear method is proposed for nonlinear response analysis of soil layers under severe earthquake, and used to a case study. The case study shows obviously that the direct nonlinear method adopted in this study is more fitful for the cases under severe earthquake, in this case, not only the seismic response value but also response duration is underestimated by underestimating by equivalent linear method in natural period range of common civil engineering structures, it must be paid enough attention.

Res

pons

e D

urat

ion

(Sec

ond)

D

urat

ion

devi

atio

n (S

econ

d)


In the study, only one case is mentioned, some case data is being treated; the related conclusion will be mentioned in the papers followed.

5.Acknowledgment

Thanks my supervisors for support and discussion in the study.

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nonlinear response analysis of soil layers under severe earthquake

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