viscous damping for the domain finite element analysis – nghiem manh hien & nien-yin chang

7/31/2019 Viscous damping for the domain finite element analysis Nghiem Manh Hien & Nien-Yin Chang

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VISCOUS DDAMPING FOR TIME DOMAIN FINITE ELEMENT ANALYSIS

Hien Manh Nghiem, PHD, Hanoi Architectural University, Km 10, Nguyen Trai St., Hanoi, Vietnam,

[email protected].

Nien-Yin Chang, Professor and Chair, Department of Civil Engineering, University of Colorado

Denver, Campus Box 113, P.O. Box 173364, 1220 Larimer St., Denver CO 80217-3364, USA,

[email protected], tel.: 303-556-2810, fax: 303-556-2368

ABSTRACT

Earthquake waves propagate mainly in rock mass from hypocenter to the bedrock directly

underneath a monitoring station. Then, it propagates as shear waves from the bedrock to a geophone,

where the surface motion is measured. For a deposit with uniform soil layers of horizontal interfaces,

one-dimensional finite element analysis can be performed to analyze the dynamic responses of a

horizontal soil deposit. In an ideal dynamic soil-structure interaction analysis, seismic waves are

propagated from the bedrock through soils and foundations, and then to structure. Thus, it is necessary

to obtain the bedrock motion from a measured surface motion registered in geophone. Conventionally

the process is called de-convolution. The de-convolution is treated as wave propagation is a

frequency domain involving damping factor, which is independent of motion velocity.

The time-domain analysis is usually used in assessing the effects of soil-structure interaction.

The time domain analysis requires the use of viscous damping proportional to motion velocity. Thus,

it is necessary to device a method for the evaluation of viscous damping that, when used in the time

domain analysis for the upward wave propagation from the bedrock back to ground surface, produces

a surface motion in close agreement to the measured surface motion. This paper presents a procedure

for evaluation of viscous damping from a given damping factors. This viscous damping successfully

produces a surface motion in close agreement with the measured surface motion in a time domain

analysis of upward wave propagation.

Introduction

Frequency domain analysis is widely used in site response analysis with 1-D model of soil

deposit. For soil-structure interaction with full soil-structure model, time-domain analysis must be

used. In time domain analysis, viscous damping is required instead of soil damping ratio which is a

given soil parameter and used in frequency domain analysis.

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mailto:[email protected]:[email protected]:[email protected]:[email protected]


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This paper presents a simple method to determine viscous damping based on matching

transfer function in frequency domain analysis using soil damping ratio.

1-D Wave Propagation Theory

Evaluation of Transfer Functions

The stress-strain relationship can be represented in following expression for a Kelvin-Voigtmodel, shown in Fig. 1:

Gt

= +

(1)

where is shear stress, is shear strain, and is viscosity.

Under harmonic motion, the shear strain can be written by:

0 sin t = (2)

The damping ratio, , for Kelvin-Voigt system related to the viscosity of the material can be

determined by:

2G

= (3)

The equation for a one-dimensional model of wave propagation for vertically propagating SH-

waves is:

2

2

u

t z

=

(4)

Substituting Eq. (1) to Eq. (4) with u z= , the wave equation leads to:2 2 3

2 2 2

u u uG

t z z t

= +

(5)

The one dimension system using thin element of a Kelvin-Voigt model is shown in Fig. 1.

2

G

dz

du

z

x


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( ) ( )0, i tm m mu t A B e= + (11)

The transfer function related to the displacement amplitude at the surface of layer m to that at

the surface of layer n is given by:

( )( )

( )

( ) ( )( )

( ) ( )( )

( ) ( )

( ) ( )

1

1

i tm mm m m mm

mn i tn n n n nn n

A a bA B e a buF

u A B e a bB a b

++ += = = =

+ ++(12)

The amplification factor in Eq. (12) is only dependent on soil properties and frequency of

motion. If the motion at any point in the soil profile is known, the motion at any other point can be

determined from Eq. (12).

Equation (12) also describes the amplification of velocities and accelerations from the surface

of layer m to the surface of layer n because2u u u = =&& & for harmonic excitation.

The input motion is assigned at the surface of a soil layer with the form given in Eq. (11) and

rewritten as follows:

( ) ( )0, Re Im i tm mu t X i X e= + (13)

Where ReX and ImX real part and imaginary part of the input motion. The parameters for

the input motion on the ground surface are related to real and imaginary parts as:

( )( ) ( )

1 1

Re Imm

m m

X i XA B

a b

+= =

+(14)

If the input motion is located on the ground surface, Eq. (14) can be written as:

1 1

Re Im

2

X i XA B

+= = (15)

Since the complex functions A1 and B1 are known, complex functions at all other layers can be

determined by Eqs. (9) and (10).

Equivalent Linear Analysis

In frequency domain analysis, the input motion is represented by the Fourier series and uses

transfer functions for the solution of wave equations based on the principle of superposition so that

the nonlinear stress-strain behavior of soil is not allowed. The nonlinear behavior of soil can be

represented by the shear modulus reduction curve in which shear modulus depends on shear strain

level. The actual nonlinear, inelastic behavior of soil can be approximated by equivalent linear soil

properties (Kramer, 1996). In the linear approach, shear modulus and damping factor are the constants

in each soil layer and the equivalent values need to be determined, which are consistent with the level

of strain induced in each layer during earthquake. For the transient motion of an earthquake, it is

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common to characterize the strain level in terms of an effective shear strain, which is usually taken as

65 percent of the peak strain.

In equivalent linear analysis, the iterative solution is employed. In the first solution, the shear

strain and damping ration are calculated from zero shear strain. In the following iteration, effective

shear strain is taken as 65 percent of peak shear strain and the shear modulus and damping ratiocorresponding to effective shear strain are then use for the next iteration. This process is repeated until

the effective shear strain does not change much from one iteration to the next.

1-D Finite Element Method

Governing Equations

In the finite element method, the continuous displacement in Eq. (5) is approximated by u in

terms of nodal displacements, 1u and 2u , through simple shape functions as follows:

1 1 2 2u N u N u= +%

(16)

Equation (29) can be written in matrix form as:

[ ] [ ] { }1

1 2

2

uu N N N u

u

= =

% (17)

where 1N and 2N are linear functions of variable z as:

1 1z

Nh

= ; 2z

Nh

= (18)

In matrix form, the finite element equilibrium equation can be expressed as:

[ ] { } [ ] { } [ ] { } [ ] { }gm u c u k u m u+ + = && & && (19)

where [ ]k , [ ]c and [ ]m are stiffness, damping, and mass matrices of a finite element, respectively

are given as follows:

[ ]1 1

1 1

Gk

h

=

(20)

[ ]1 1

1 1c

h

=

(21)

[ ]2 1

1 26

hm

=

(22)

where h is element length.

Each soil layer with the same soil properties is modeled by a spring and dashpot system, as

shown in Fig. 3.

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Figure 3: Soil Layer and Finite Element Model Subjected to a Horizontal Seismic Motion at

Its Base

The element matrices need to be assembled into global matrices to make the equilibrium

equation for whole system which expressed as following equation:

[ ] { } [ ] { } [ ] { } [ ] { }gM U C U K U M u+ + = && & && (23)

where: [ ]M , [ ]C , and [ ]K are global mass, damping, and stiffness matrices, respectively; and gu&& is

bed-rock motion.

Consider the mass-proportional damping and stiffness-proportional damping:

c m k = + (24)

where the constants and have unit sec-1

and sec, respectively. In Eq. (21), damping matrix has

the same form as stiffness matrix. Compare Eqs. (21) and (24), the constants, and in Eq. (24)

will be:

0= and 2G = = (25)

The use of constant will be discussed in next Section.

Equivalent Viscous Damping

Dynamic testing on soil indicated that the soil damping ratio associated with the area bounded

by hysteretic stress-strain loops is essentially independent of cyclic frequency. In time domain

analysis, which is used in soil-structure interaction analysis, only velocity-proportional viscous

damping can be used in the form of dashpot embedded within the material elements (Kwok et al.,

2007). Currently, the use of viscous damping in time domain analysis is not a convenient

approximation in comparison to frequency domain analysis. Kwok et al. (2007) and Stewart and

Kwok (2008) recommended the procedures for the specification of Rayleigh damping in time domain

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z

u(z,t)dz

dudz

(z,t)=0 Ground Surface

Base (bed rock)

u(z,t)g

Column of uniform

cross-sectional

area

k c1 12m

m 32k 2c

mN+1

3k 3c

kN cN

m1

mN

h1

h2

h3

hN

Layer 1

Layer 2

Layer 3

Layer N


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analysis using mode superposition method. There two sources of viscous damping including modal

damping ratio and viscous damping. Modal damping ratio is used in mode superposition, and viscous

damping is used in direct method of solving the dynamic equilibrium equation. The selection of

modal damping ratio presented detail by Kwok et al. (2007).

Currently, there is no guideline for using viscous damping in direct method. In the following,recommended procedures for the specification of viscous damping will issues for current practice.

Effects of soil damping ratio on transfer function is shown in Eq. (12). In the frequency

domain, damping ratio, , is constant represents the soil dynamic property, so varies with circular

frequency (Eq. 3). In the time domain analysis, is constant and related to Rayleigh damping, by

Eq. (25). The wave propagation analysis in frequency domain also can be made for varied damping

ratio by keeping constant, and in general, it gives different result from constant damping ratio

analysis. To match results from these two analyses, the transfer functions need to be matched.The transfer functions from varied and constant damping ratios can be matched by choosing

the appropriate predominant frequency, . Consider one soil layer in soil profile, it can be seen that,

upward transfer function from underlain layer to this layer has some peak values at certain

frequencies. From some analyses, it can be concluded that the signal can be match if the frequencies

used to determine the viscosity, are picked at first or second peak location of upward transfer

function.

Method Verification

The site is located at latitude 32.7920 and longitude 115.5640 with a soil profile represented by

weathered bed-rock underlying 19.8 m of surficial clayey silts and sands. The dynamic properties-

shear modulus, shear wave velocity, and damping-are summarized in Table 1. The properties are

calculated from curves given in Fig. 4.

Deconvolutions are performed for this site with soil properties given in Table 1. The

acceleration time histories used in the analyses are shown in Fig. 5 in both the horizontal directions X

and Z. The equivalent shear modulus and damping ratio at strain level of 65 percent of peak strain

from equivalent linear analyses are given in Table 2. To determine the viscous and Rayleigh damping

in each soil layer, the frequency is selected based on the location of the peak value in transfer

function. The transfer functions are shown in Fig. 6. The appropriate frequencies, viscous damping

and Rayleigh damping are also given in Table 2.

The deconvoluted motions at the bed-rock level are shown in Fig. 7 and used in time domain

analyses to determine far-field motions. In time domain analysis, soil damping is constant in each soil

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layer and Rayleigh damping determined from deconvolution procedure is used. The computed far-

field motions are compared to the measured far-field motions. Figure 8 show the response spectral

accelerations with 5 percent damping of free-field motions in both the X and Z directions,

respectively. The predicted result is lower than the measured result in periods less than 0.1 s because

the high frequencies are removed manually in deconvolution. The response spectral accelerations inboth directions are almost the same for high period structures, and only minor differences exist at

period from 0.1 s to 1 s.

8

PI>80

PI=40-80

PI=20-40PI=10-20PI=0-10

1E-4 1E-3 1E-2 1E-1 1E-0 1E+1

G

/G

sec

0

Shear Strain (%)

0.0

0.2

0.4

0.6

0.8

1.0


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Figure 4: Modulus degradation and damping curves for clay soil (Sun et al., 1988 and Vucetic and

Dobry, 1991)

9

1E-31E-4 1E-2 1E-1 1E-0 1E+1

Shear Strain (%)

0

5

Damping(%)

15

10

20

25

PI=10-20

PI=0-10

PI=20-40

PI=40-80

PI>80

-6

-4

-2

0

2

4

6

0 10 20 30 40 50 60

Time (sec)

Acceleration(m/s2)


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(a)

(b)

Figure 5: Time History Function (a) in the X Direction and (b) in the Z Direction

Table 1: Soil properties

Depth

(m)

sV

(m/s)

uS

(kN/m2)

(kN/m3)

G

(kN/m2)

0-3 152.4 72 19.0 45057

10

-3

-2

-1

0

1

2

3

0 10 20 30 40 50 60

Time (sec)

Acceleration(m/s2)


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3-9.1 243.8 132 17.9 108456

9.1-19.8 365.8 223 18.6 253706

19.8-29.8 609.6 500 19.0 720902

Table 2: Equivalent linear soil parameters

Depth(m)

G

(kN/m2)

(%)

(kN.s/m2)

f

(Hz)

Rayleigh

damping,

(s)

0-3 38391 4.7 48.9 11.8 0.00127

3-9.1 91982 4.9 247.4 5.8 0.00269

9.1-19.8 221974 2.8 511.2 3.9 0.0023

19.8-29.8 641772 1.53 913.9 3.4 0.0014

11

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25

U1/U2


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Figure 6: Transfer Functions

12

0

2

4

6

8

10

12

0 5 10 15 20 25

U2/U3

0

2

4

6

8

10

12

14

0 5 10 15 20 25

U3/U4

0

5

10

15

20

25

30

35

0 5 10 15 20 25

Frequency (Hz)

U1/U4

Equivalent Damping Constant Damping

-2.000

-1.500

-1.000

-0.500

0.000

0.500

1.000

1.500

2.000

0 10 20 30 40 50 60

Time (s)

Acceleration(m/s2)


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(a)

(b)

Figure 7: Bed-rock Motion: (a) in the X Direction and (b) in the Z Direction

13

-1.500

-1.000

-0.500

0.000

0.500

1.000

1.500

0 10 20 30 40 50 60

Time (s)

Acceleration(m/s2)

0

2

4

6

8

10

12

14

16

18

20

0.00 0.01 0.10 1.00 10.00

Period (s)

ResponseSpectralAcceleration(m/s2)

Farfield Measurement Analysis


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(a)

(b)

Figure 8: Response Spectral Acceleration

(a) in the X Direction; (b) in the Z Direction

Conclusion

In the time domain analysis allowing full soil-pile-structure interaction, only equivalent

viscous damping can be used instead of soil damping ratio. The measured ground motion is de-

convoluted to any depth below pile tip in frequency domain analysis using the damping ratio of soil to

obtain the input motion for the full soil-pile-structure interaction analysis. The equivalent viscous

damping of soil in time domain analysis can be evaluated by matching the transfer functions, soil

14

0

1

2

3

4

5

6

7

8

9

0.00 0.01 0.10 1.00 10.00

Period (s)

ResponseSpectralAccelera

tion(m/s2)

Farfield Measurement Analysis


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damping ratio and soil damping value in frequency domains. The viscous damping is verified by

comparing the calculated far-field motion from time domain analysis using bed-rock motion from

frequency domain analysis and measued far-field motion. There no significant difference between two

motions. It indicates that this method is effective to determine viscous damping.

References

Kramer, S. L. (1996), Geotechnical Earthquake Engineering. Prentice Hall, New Jersey.

Kwok, A. O., Stewart, J. P., Hashash, Y. M. A., Matasovic, N., Pyke, R., Wang, Z. and Yang, Z.

(2007), Use of Exact Solution of Wave Propagation Problems to Guide Implementation of Nonlinear

Seismic Ground Response Analysis Procedures. J. Geotech. Geoenv. Eng., ASCE, Vol. 133, No. 11,

pp. 1385-1398.

Stewart, J. P. and Kwok, A. O. (2008),Nonlinear seismic ground response analysis: code usage

protocols and verification against vertical array data. Geotechnical Engineering and Soil Dynamics

IV, May 18-22, 2008, Sacramento, CA, ASCE Geotechnical Special Publication No. 181, Zeng D.,

Manzari M. T. and Hiltunen D. R. eds., 24 pages.

Sun, J.I., Golesorkhi and Seed, H.B. (1988),Dynamic Moduli and Damping Ratios for Cohesive Soils.

Report No. UCB/EERC-88/15, Earthquake Engineering Research Center, University of California,

Berkeley.

Vucetic, M. and Dobry, R. (1991),Effect of Soil Plasticity on Cyclic Response. J. Geotech. Eng.,

ASCE, Vol. 117(1), pp. 89-107.

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viscous damping for the domain finite element analysis – nghiem manh hien & nien-yin chang

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