Viscous Damping Formulation Publication

Viscous damping formulation and high frequency motion propagation

in non-linear site response analysis

Youssef M.A. Hashash*, Duhee Park

Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 N. Mathews Avenue, Urbana, IL 61801, USA

Accepted 1 January 2002

Abstract

Non-linear time domain site response analysis is widely used in evaluating local soil effects on propagated ground motion. This approach

has generally provided good estimates of field behavior at longer periods but has shortcomings at relatively shorter periods. Viscous damping

is commonly employed in the equation of motion to capture damping at very small strains and employs an approximation of Rayleigh

damping using the first natural mode only. This paper introduces a new formulation for the viscous damping using the full Rayleigh damping.

The new formulation represents more accurately wave propagation for soil columns greater than 50 m thick and improves non-linear site

response analysis at shorter periods. The proposed formulation allows the use of frequency dependent viscous damping. Several examples,

including a field case history at Treasure Island, California, demonstrate the significant improvement in computed surface response using the

new formulation. q 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Site response; Viscous damping; Deep deposits; Non-linear analysis; Amplification

1. Introduction

One-dimensional site response analysis is used to solve the

problem of vertical propagation of horizontal shear waves (SH

waves) through a horizontally layered soil deposit. Horizontal

soil layer behavior is approximated as a KelvinVoigt solid

whereby elastic shear moduli and viscous damping charac-

terize soil properties. Solution of wave propagation equations

is performed in the frequency or time domain (TD).

Seed, Idriss and co-workers introduced the equivalent

linear approximation method to capture non-linear cyclic

response of soil. For a given ground motion time series (T.S.

also referred to as time history) and an initial estimate of

modulus and damping values, an effective shear strain (equal

to about 65% of peak strain) is computed for a given soil layer.

Modulus degradation and damping curves are then used to

obtain revised values of shear modulus and damping. The

solution is performed in frequency domain (FD) and an

iterative scheme is required to arrive at a converged solution

(e.g. SHAKE, Ref. [1]). This approach provides results that

compare well with field measurements and is widely used in

engineering practice. More recently, Sugito et al. [2] and

Assimaki et al. [3] extended the equivalent linear approach to

include frequency and pressure dependence of soil properties.

Assimaki et al. [3] suggest that it is appropriate to assume soil

damping to be frequency dependent to truly represent non-

linear soil response in a FD analysis.

The equivalent linear approach is computationally easy

to use and implement but remains an approximation of non-

linear cyclic response of soils. Non-linear site response

analysis is employed by integrating the equation of motion

in TD. A non-linear constitutive relation is used to represent

the hysteretic behavior of soil during cyclic loading. The

simplest constitutive relations use a model relating shear

stress to shear strain, whereby the backbone curve is

represented by a hyperbolic function. Strain dependent

modulus degradation curves are used to define the backbone

curve. The Masing criteria [4] and extended Masing criteria

[5,6] define unloadingreloading criteria and behavior

under general cyclic loading. Lee and Finn [7] developed

a one-dimensional seismic response analysis program using

the hyperbolic model. Matasovic [8] and Matasovic and

Vucetic [9] further extended the model with a modification

of the hyperbolic equation. Plasticity models have also been

used to represent cyclic soil behavior. For example, Borja et

al. [10] used a bounding surface plasticity model to

represent cyclic soil response at the Lotung Site in Taiwan.

Hashash and Park [11] introduced an extension of the

0267-7261/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S0 26 7 -7 26 1 (0 2) 00 0 42 -8

Soil Dynamics and Earthquake Engineering 22 (2002) 611624

www.elsevier.com/locate/soildyn

* Corresponding author. Tel.: 1-217-333-6986; fax: 1-217-265-8041.E-mail addresses: [email protected] (Y.M.A. Hashash), dpark1@uiuc.

edu (D. Park).

modified hyperbolic model to capture the dependence of

modulus degradation and damping curves on confining

pressure.

A problem commonly noted in non-linear site response

analysis is that while it provides good estimate at relatively

long periods, the computed ground response underestimates

measured response at shorter periods [12].

2. Numerical implementation of non-linear one-

dimensional wave propagation analysis

In non-linear analysis, the following dynamic equation of

motion is solved:

M{u} C{_u} K{u} 2M{I}ug 1where [M], mass matrix; [C], viscous damping matrix; [K],

stiffness matrix; {u}; vector of nodal relative acceleration;{u}; vector of nodal relative velocities; and {u}, vector ofnodal relative displacements. ug is the acceleration at the

base of the soil column and {I} is the unit vector. The [M],

[C] and [K] matrices are assembled using the incremental

properties of the soil layers. The properties are obtained

from a constitutive model that describes the cyclic behavior

of soil. The dynamic equilibrium equation, Eq. (1), is solved

numerically at each time step using the Newmark [13] bmethod. The geologic column is discretized into individual

layers using a multi-degree-of freedom lumped parameter

model shown in Fig. 1.

Each individual layer i is represented by a corresponding

mass, non-linear spring, and a dashpot for viscous damping.

Lumping half the mass of each of two consecutive layers at

their common boundary forms the mass matrix. The

stiffness matrix is updated at each time increment to

incorporate non-linearity of the soil.

The geologic material (soil or rock) is represented either

as a linear elastic material with constant value of damping or

using a non-linear constitutive model such as the pressure

dependent modified hyperbolic model described by Hashash

and Park [11]. The base of the soil column can be modeled

as either an infinitely stiff or a visco-elastic half space.

3. Limitation of current viscous damping formulation

In a non-linear soil model, soil damping is captured through

hysteretic loadingunloading cycles in the soil model. The use

of the damping matrix [C] may become unnecessary but is

commonly used as a mathematical convenience or to include

damping at very small strains where response of many

constitutive models is nearly linear elastic.

Hysteretic damping of the soil model defined by Hashash

and Park [11], as well other models (e.g. Ref. [8]), can

capture damping at strains larger than 1024 1022%,

depending on the values of material properties. However,

the hyperbolic model is nearly linear at small strains (less

than 10241022%) with practically no damping, which can

cause unrealistic resonance during wave propagation. These

models incorporate additional damping to the dynamic

equation in the form of the [C] matrix, as shown in Eq. (1).

Similarly, the model by Borja et al. [10] uses the viscous

damping matrix. The [C] matrix is derived from a

combination of the mass matrix and the stiffness matrix

[14]:

C aRM bRK 2The damping matrix is assumed in current formulations to

be only stiffness proportional since the value of aRM issmall compared to bRK: Small strain viscous dampingeffects are assumed proportional only to the stiffness of the

soil layers. This is further simplified to:

C bRKwhere bR 2j/v1 and v1 is the frequency of the firstnatural mode of the soil column.

The viscous damping matrix for a multi-layered soil is

expressed as [11]:

C 2vjiKi 2v

j1K1 2j1K1

2j1K1 j1K1 j2K2 2j2K22j2K2

2664

37753

where v is natural circular frequency of the first naturalmode and ji is the equivalent damping ratio for layer i atsmall strains. The viscous damping matrix is dependent on

the first natural mode of the soil column and the soil column

stiffness, which are derived from the shear wave velocity

profile of the soil column. [C] is commonly taken as

independent of strain level and the effect of hysteretic

damping induced by non-linear soil behavior can be

separated from (but added to) viscous damping.

The value of the equivalent damping ratio j is obtainedfrom the damping ratio curves at small strains. A constant

small strain viscous damping is used in some non-linear

models with a recommended upper bound value of 1.54%

for most soils, independent of confining pressure [8,15].

Hashash and Park [11] propose a pressure dependent

equation for the viscous damping ratio j.In order to assess the accuracy of the viscous damping

formulation approximation, a series of linear site response

analyses are conducted using four idealized soil columns 50,

100 and 500 m thick with constant stiffness and viscous

damping ratio profiles (1) shown in Fig. 2. The thick soil

columns with variable shear wave velocity and viscous

damping are representative of conditions in the Mississippi

Embayment in the Central US (New Madrid Seismic Zone).

The analyses compare linear TD wave propagation analysis

with linear FD wave propagation analysis. The FD analysis

represents the correct analysis as the solution of the wave

equations can be derived in closed form (e.g. Ref. [16]). Fig.

3 shows the computed surface response for a harmonic input

Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) 611624612

motion with soil thickness up to 500 m. TD analyses provide

identical results to FD analysis for the analyses using zero

viscous damping.

TD analysis with viscous damping ratio of 1% gives

results similar to FD analysis for the 50 m soil column.

However, for 100 and 500 m soil columns the TD analysis

gives a response lower than FD analysis. The quality of the

computed response deteriorates with increasing soil column

thickness. The approximation of the viscous damping

matrix in TD analysis may be acceptable for soil columns

less than about 50 m thick and when the contribution of the

viscous damping is very small.

The simplified damping formulation in Eq. (3) introduces

excessive damping in the TD analysis that increases with

increasing column thickness. The contribution of higher

modes is small for relatively short soil columns but may

become important for deeper soil columns and when

propagating high frequency motion. The simplified damp-

ing formulation depends only on the first mode of the

deposit and is proportional to the stiffness matrix. If only

stiffness proportional damping is used [17,18], then the

effective damping ratio being used for higher modes is:

jn bRvn2

j vnv1

4

This implies that the effective damping ratio is increasing at

higher natural modes. This would explain the underestimate

of surface ground motion for TD analysis shown in Fig. 3.

Fig. 1. Multi-degree-of freedom lumped parameter model representation of horizontally layered soil deposit shaken at the base by a vertically propagating

horizontal shear wave. The model is used in the solution of the dynamic equation of motion in TD.

Fig. 2. Shear wave velocity and viscous damping profiles used in analyses. The variable profile properties are representative of conditions encountered in the

Mississippi Embayment, Central US. Bedrock shear wave velocity is 2700 m/s.

Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) 611624 613

4. Proposed extension of viscous damping formulation

In the original damping formulation proposed by

Rayleigh and Lindsay [14], also in Clough and Penzien

[17], Chopra [18], aR and bR coefficients of Eq. (2) can becomputed using two significant natural modes m and n:

1

2

1/vm vm

1/vn vn

" #aR

bR

( )

jm

jn

" #5

This matrix can be solved for aR and bR:

aR 2vmvnvmjn 2 vnjm

v2m 2 v2n

!

bR 2vmjm 2 vnjn

v2n 2 v2n

! 6

If the damping ratio j is frequency independent then:

aR 2jvmvn

vm vn

!bR 2j 1vm vn

!7

Eq. (3) is obtained from Eqs. (5)(7) by assuming m is the

first natural mode and vn 0; implying that the secondrelevant mode occurs at zero circular frequency. This is

acceptable for short soil columns where only the first mode

dominates. For thicker columns, such an assumption will

filter out high frequency components due to the resulting

large value of viscous damping matrix. Therefore, vn 0should be included to represent the contribution of higher

modes.

When choosing higher modes, the mass matrix com-

ponent will counter-balance part of the contribution of the

stiffness matrix component. As higher modes are used, aRincreases and bR decreases. Contribution of the mass matrixcannot be ignored as using only the stiffness proportional

damping will result in different damping ratio for the

corresponding modes as shown in Eq. (4). The Rayleigh

damping formulation using two significant modes has been

incorporated for example in the two-dimensional finite

element program QUAD4M by Hudson et al. [19]. For a

multi-layered soil with frequency independent damping

ratio, Eq. (2) can be expanded as follows:

C 2 vmvnvm vn

! j1M1j2M2

2664

3775

2 1vm vn

! j1K1 2j1K12j1K1 j1K1 j2K2 2j2K2

2j2K2

2664

37758

with frequency dependent damping ratio [3], Eq. (2) can be

Fig. 3. Computed surface ground motion, linear frequency and TD site response, Vs profile (1). Linear TD analysis uses first natural mode approximation only

of viscous damping formulation. Harmonic input motion, amplitude 0.3 g, period 0.2 s, duration 1 s.


expanded as follows:

The viscous damping matrix is therefore dependent on

stiffness, mass and the natural modes of the soil column. In

most current applications, [C] is taken as strain independent

and has a constant value throughout an analysis. The natural

modes and the soil column stiffness are derived from the

shear wave velocity profile of the soil column. In a non-

linear analysis, the mass matrix remains constant but the

stiffness matrix is related to the strain level in the soil

column. The natural periods or frequencies vary with the

stiffness variation of the soil column. In the Netwon-b

method, Eq. (1) is integrated or linearized over a given time

increment. All matrix quantities including the [C] matrix

should correspond to soil properties at that given time

increment and strain level. The [C] matrix has to be updated

to accurately represent the viscous damping ratio (j)

property of the soil layers. Therefore, the [C] matrix is

strain dependent and its strain dependent components

including frequencies of natural modes and stiffness matrix

are updated at each time increment.

This formulation of the damping matrix is implemented

in the non-linear site response analysis program DEEPSOIL,

developed by Hashash and Park [11]. The proposed viscous

damping formulation is necessary to improve the accuracy

of the solution of wave equations in TD. In addition, the

formulation allows for the use of frequency dependent

viscous damping ratio.

The Rayleigh viscous damping formulation represents an

approximate solution and has some important features and

limitations [17,18]. The use of the Rayleigh damping

formulation results in an effective frequency dependent

damping as shown in Fig. 4 even when using Eq. (8). In the

figure, the first mode is assumed to occur at 1.0 s and the

higher mode at 0.1 s with corresponding damping ratio of

1%. Fig. 4 shows that for 0.1 , T , 1.0 s the resultingdamping ratio is less than 1% while at T . 1 s or T , 0.1 sthe resulting damping ratio increases significantly. There-

fore, the user has to carefully choose the relevant modes to

capture ground motion response in the desired period/

frequency range. The choice is significant for the frequen-

cies higher than the higher mode used in the Rayleigh

damping. At frequencies higher than the frequency of the

higher mode, ground motion content can be filtered out.

However, this may not be significant from an engineering

point of view if the natural modes in the Rayleigh damping

cover the range of frequencies of interest. The following

sections illustrate the significance of the new viscous

damping formulation.

5. Validation of new viscous damping formulation

A series of analyses are presented to validate the new

viscous damping formulation and evaluate its impact on

non-linear site response analysis. The analyses use a range

of idealized as well as representative input motions and soil

profiles. Four typical soil columns, 50, 100, 500 and 1000 m

deep, are used in the analyses as shown in Fig. 2. Two

typical shear wave velocity and viscous damping profiles

are used. The non-linear material properties used are those

published in Ref. [11].

C 2vmvnv2m 2 v2n

! vmj1n 2 vnj1mM1vmj2n 2 vnj2mM2

2664

3775 2 v2m 2 v2n

!

vmj1m 2 vnj1nK1 2vmj1m 2 vnj1nK12vmj1m 2 vnj1nK1 kvmj1m 2 vnj1nK1 vmj2m 2 vnj2nK2l 2vmj2m 2 vnj2nK2

2vmj2m 2 vnj2nK2

2664

3775 9

Fig. 4. Variation of viscous damping as a function of period and frequency

using Rayleigh damping formulation (after Clough and Penzien [17] and

Chopra [18]).


Harmonic input motion, constant shear wave velocity

and viscous damping profiles (1), linear analysis. In this set

of analyses the results of linear TD wave propagation

analysis (DEEPSOIL) using a harmonic input motion with a

duration t 1 s, period T 0.2 s, and amplitude 0.3 gare compared to those obtained from linear FD wave

propagation analysis. Fig. 5(a) shows that for a 50 m profile

the TD analysis using the first natural mode only

(conventional viscous damping formulation) matches the

result of the FD analysis. For soil profiles greater than 50 m

in thickness the use of TD analysis with first mode

approximation significantly underestimates the surface

response. For a 100 m profile, Fig. 5(b), TD first mode

analysis gives lower response than FD analysis. However,

TD analysis using first and second modes in the proposed

viscous damping formulation, Eq. (9), matches that of the

FD analysis. For the 500 m profile, Fig. 5(c), TD analysis

using first mode only as well as first and second modes

underestimates the surface response compared to FD

analysis. A good match is obtained when the first natural

mode and the second mode at T 0.2 s (corresponding tothe period of the ground motion) are used in the TD

analysis.

Recorded input ground motion, shear wave velocity and

viscous damping profiles (2) representative of the Mis-

sissippi Embayment, linear analysis. In this series of

analyses, the shear wave velocity and viscous damping

profiles representative of the conditions in the Mississippi

Embayment, profiles (2) in Fig. 2, are used. A transient

ground motion recording from Hector Mine earthquake

(1999) in California with a peak ground acceleration

PGA 0.0073 g is used as the input motion. Computed5% damped surface response spectra obtained from linear

TD and FD analyses are plotted in Fig. 6. Computed surface

ground motions for t 1015 s are plotted in Fig. 7. Theanalyses show that with increasing soil column thickness the

use of TD analysis with first natural mode (conventional

formulation) results in an increasing underestimate of

surface motion.

The computed surface response spectra from FD and TD

Fig. 5. Computed surface ground motion, linear frequency and TD site response, Vs and viscous damping profiles (1). Linear TD analysis uses first natural mode

approximation as well as proposed full Rayleigh viscous formulation. Harmonic input motion, amplitude 0.3 g, period 0.2 s, duration 1 s.


first mode analyses compare well for the 50 m column, Fig.

6(a). The match between the two analyses can be improved

by using TD analysis with first and second modes. For the

100 m profile, Fig. 6(b), a good match is obtained using first

and second modes in TD analysis. For the 500 m column,

Fig. 6(c), a good match is obtained when using first and fifth

modes. For the 1000 m column, Fig. 6(d), match is obtained

when using first and eighth modes.

Fig. 7 shows that the use of the full Rayleigh viscous

damping formulation improves the results from the TD

analysis, but does not provide an exact match of the results

of the FD analysis.

Fig. 6. Computed 5% damped surface response spectra, linear frequency and TD site response, Vs and viscous damping profiles (2). Linear TD analysis uses

first natural mode approximation as well as proposed full Rayleigh viscous damping formulation. Input motion: Hector mine earthquake, PGA ,duration 18 s.

Fig. 7. Computed surface ground motion, linear frequency and TD site response, Vs and viscous damping profiles (2). Linear TD analysis uses first natural mode

approximation as well as proposed full Rayleigh viscous damping formulation. Input motion: Hector mine earthquake, PGA 0.007 g, duration 18 s.


Synthetic input ground motion, shear wave velocity and

viscous damping profiles (2) representative of the Mis-

sissippi Embayment, non-linear analysis. The analyses

provided so far focus on the performance of TD analysis

whereby soil response is assumed linear, i.e. soil at a given

depth has a constant stiffness (obtained from shear wave

velocity) and a constant viscous damping value. In non-

linear (TD) site response analysis, damping is primarily a

result of hysteretic soil response, and the contribution of the

viscous damping term may become relatively small.

In the model proposed by Hashash and Park [11] the

cyclic response of the soil is dependent on the in situ

confining pressure. Fig. 8 shows the modulus degradation

and total damping curves at a range of confining pressures.

The model was designed to fit the data from Laird and

Stokoe [20]. The total damping curves are the sum of

hysteretic damping from the non-linear soil model and the

viscous damping profile (2) shown in Fig. 2.

Fig. 912 show the surface ground motion and computed

5% damped surface response spectra for non-linear TD

analyses using the program DEEPSOIL. Ground motion input

used in the analyses are synthetic ground motions generated

using the program SMSIM [21] for M 5 R 20 km(PGA 0.063 g) and M 7, R 20 km (PGA 0.59 g)with input parameters appropriate for the New Madrid

Seismic Zone. The analyses for a given soil profile are

conducted in two steps, (a) a linear TD analysis, with

viscous damping only, is performed to select the relevant

modes (always consisting of the first mode and a higher

mode) to match a similar FD analysis within the range of

frequencies of interest, and (b) a non-linear analysis is then

performed with viscous damping using the selected modes

from the previous step. Step one analyses show that for a

given soil profile (column thickness, shear wave velocity

and viscous damping profile) the choice of relevant modes

that provides the correct linear response is not very sensitive

to the input ground motion within the range of frequencies

of engineering interest.

Figs. 9 and 10 show comparisons of the computed non-

linear response for the four soil columns using the

Fig. 8. Influence of confining pressure on modulus degradation and damping ratio curves in DEEPSOIL non-linear model used for modeling of site response in

the Mississippi Embayment. Data from Laird and Stokoe [20] shown for comparison.


conventional viscous damping formulation (first mode only)

and the proposed viscous damping (first and a higher mode

as described in the earlier paragraph). For the 100 m profile,

the contribution of the higher modes in non-linear analysis

is small. For the 500 and 1000 m profile the contribution is

significant for periods less than 1 s. Use of the conventional

viscous damping formulation will filter out a significant

portion of the high frequency content of the ground motion.

The analysis results in Figs. 11 and 12 correspond to

higher input motion (PGA 0.59 g) whereby the contri-bution of the non-linear model and hysteretic damping is

also higher relative to the viscous damping component. The

Fig. 9. Computed surface ground motion, non-linear TD site response, Vs and viscous damping profiles (2), non-linear soil properties. Non-linear TD analysis

uses first natural mode approximation as well as proposed full Rayleigh viscous damping formulation. Synthetic input motion, M 5, R 20, New MadridSeismic Zone parameters, PGA 0.063 g, duration 4 s.

Fig. 10. Computed 5% damped surface response spectra, non-linear TD site response, Vs and viscous damping profiles (2), non-linear soil properties. Non-

linear TD analysis uses first natural mode approximation as well as proposed full Rayleigh viscous damping formulation. Synthetic input motion, M 5,R 20, New Madrid Seismic Zone parameters, PGA 0.063 g, duration 4 s.


results still show that for deeper soil columns the use of the

conventional viscous damping formulation will filter out

important components of ground motion at high frequen-

cies/short periods compared to the proposed viscous

damping formulation. Fig. 12(a) includes an additional

analysis for the 50 m column using frequency dependent

damping whereby the damping ratio of the second modes is

taken as half the damping ratio of the first mode. The figure

shows that the computed response is higher for periods less

than 0.1 s.

Fig. 11. Computed surface ground motion, non-linear TD site response, Vs and viscous damping profiles (2), non-linear soil properties. Non-linear TD analysis

uses first natural mode approximation as well as proposed full Rayleigh viscous damping formulation. Synthetic input motion, M 7, R 20, New MadridSeismic Zone parameters, PGA 0.59 g, duration 17 s.

Fig. 12. Computed 5% damped surface response spectra, non-linear TD site response, Vs and viscous damping profiles (2), non-linear soil properties. Non-

linear TD analysis uses first natural mode approximation as well as proposed full Rayleigh viscous damping formulation. Synthetic input motion, M 7,R 20, New Madrid Seismic Zone parameters, PGA 0.59 g, duration 17 s.


6. Use of constant versus variable [C] matrix in non-linear analysis

In the proposed damping formulation, Eq. (9), the [C]

matrix is updated at every time step. This implementation is

in contrast to the common implementation where [C] is

constant and based on the initial soil properties. All the

examples presented above use the updated formulation of

the [C] matrix. The same analyses were also performed

using a constant form of [C] dependent on the initial soil

properties only. The analysis results using variable and

constant [C] were very similar for most cases. The variable

[C] analyses give slightly higher response than the constant

[C] analyses. The difference was more noticeable for the

analyses with the largest ground motion input (M 7analyses of Fig. 12). The main difference is in the high

frequency range of the ground motion and is best seen in

plots of the Fourier amplitude of computed surface ground

motion in Fig. 13. This difference is a result of the

significant non-linear effects and reduction in stiffness

experienced by the soil.

7. Application of new viscous damping formulation to

case history of Treasure Island and Yerba Buena Island

recordings

During the 1989 Loma Prieta Earthquake in Northern

California ground motion recordings were obtained on fill

material underlain by sediments at Treasure Island and on

rock at adjacent Yerba Buena Island. Several site response

studies using these recordings were made using the

equivalent linear analysis [22,23] and the non-linear

analysis [8,24] methods. In these studies, the computed

surface response spectra were in general agreement with

measured spectra. However, several of the analyses had

some difficulty in capturing the recorded response in the

short period/high frequency range.

A similar site response analysis using the Yerba Buena-

Treasure Islands recordings is made using the non-linear site

response analysis program DEEPSOIL. The analysis is

presented to further illustrate the significance of viscous

damping formulation on computed site response. The soil

shear wave velocity profile used, Fig. 14, is based on data

from Gibbs et al. [25] and Pass [26]. Shear wave velocity of

the rock is taken as 2700 m/s. Non-linear hyperbolic model

parameters were obtained from modulus degradation and

damping curves published in Ref. [27]. Fig. 14 shows the

modulus degradation and damping curves obtained from the

calibrated modified hyperbolic model. The recording at

Yerba Buena Island is used as the input motion at the base of

the column. The site response analysis was conducted by

first obtaining a best estimate of soil parameters using

available data without any attempt to match model results to

recorded motions. Fig. 15 shows plots of the computed and

recorded response spectra of the EW and NS com-

ponents. For both motion components the analysis results

using the first mode (conventional) viscous damping

formulation significantly underestimates ground motion

response at periods less than 0.5 s.

For the EW motion component using the first and

second mode (proposed) viscous damping formulation

significantly improves ground motion response at periods

below 0.5 s and captures the high frequency (low period)

Fig. 13. Computed surface Fourier spectra, non-linear TD site response, Vs and viscous damping profiles (2), non-linear soil properties. Non-linear TD analysis

uses proposed full Rayleigh viscous damping formulation with and without the update of the [C] matrix. Synthetic input motion, M 7, R 20, New MadridSeismic Zone parameters, PGA 0.59 g, duration 17 s.


peaks in the response spectra. The use of the new

formulation also improves the response at the highest

peak of the response spectrum. Use of higher modes

(first and eighth natural modes) does not result in

significant improvement of the results and is indicative of

the convergence of the solution at higher modes.

For the NS motion component the proposed (first

and second natural modes) viscous damping formulation

results in a dramatic improvement of computed ground

motion response at periods below 0.5 s compared to

conventional viscous damping formulation. The results

nearly match computed ground response and capture the

peaks in this low period/high frequency range. At longer

periods, the computed results underestimate recorded

motion. This result was also reported in other studies and

is attributed by Finn et al. [24] to ground motion

incoherency between the Yerba Buena Island and

Treasure Island recordings.

Fig. 15(c) and (d) also shows the influence of

frequency dependent formulation on computed surface

response spectra. The frequency dependent formulation

produces slightly higher response in the short period

range. However, it does not significantly alter the overall

response.

The case history shows that the new damping formu-

lation significantly improves computed ground motion

response, especially at short periods, for a soil column

less than 100 m thick.

8. Summary and conclusions

The use of the full form of the Rayleigh damping to

represent viscous damping significantly improves the

performance of non-linear site response analysis in TD.

The proposed formulation addresses a long-standing

problem whereby non-linear site response performance

appeared to underestimate ground motion response at short

periods/high frequencies. The proposed viscous damping

formulation suggests that in addition to the first mode of the

soil column, higher modes have an important contribution to

the viscous damping component. The use of the new

formulation ensures that when the soil behavior is linear the

computed ground motion response in TD analysis is similar

to that computed in the FD. The new formulation allows the

option of using a frequency dependent viscous damping

component in non-linear analysis.

The use of the proposed viscous damping procedure in

non-linear site response analysis consists of two steps:

(a) Viscous damping modes identification. For the selected

soil column perform a linear TD analysis using the

column shear wave velocity profile and viscous

damping component profile only. The two relevant

natural modes are identified in the viscous damping

formulation such that the computed response matches a

similar linear analysis in FD over the frequency range

of interest. One of the modes always corresponds to the

Fig. 14. Soil profile and soil properties used in the non-linear analysis of the Treasure Island case history.


first natural mode of the column. The other relevant

mode corresponds to the second or a higher natural

mode.

(b) Non-linear site response analysis. A non-linear

analysis is then performed with viscous damping

formulation using the two selected modes from the

previous step.

Examples of site response calculations using the

proposed formulation show that for soil columns greater

than 50 m thick ground response in the low period/high

frequency range is higher than what would be obtained

using conventional viscous damping formulation. Analysis

of site response using Yerba-Buena-Treasure Island record-

ings demonstrate the significant improvement achieved in

computed surface response using the new viscous damping

formulation in non-linear analysis.

Acknowledgments

This work was supported primarily by the Earthquake

Engineering Research Centers Program of the National

Science Foundation under Award Number EEC-9701785;

the Mid-America Earthquake Center. The authors gratefully

acknowledge this support. All opinions expressed in this

paper are solely those of the authors.

References

[1] Schnabel PB, Lysmer JL, Seed HB. SHAKE: a computer program for

earthquake response analysis of horizontally layered sites. Berkeley,

CA: Engineering Research Center; 1972.

[2] Sugito M, Goda H, Masuda T. Frequency dependent equi-linearized

technique for seismic response analysis of multi-layered ground.

Doboku Gakkai Rombun-Hokokushu/Proc Japan Soc Civil Engng

1994;493(3-2):4958.

[3] Assimaki D, Kausel E, Whittle AJ. Model for dynamic shear modulus

and damping for granular soils. J Geotech Geoenviron Engng 2000;

126(10):85969.

[4] Masing G. Eignespannungen und Verfestigung beim Messing. Second

International Congress on Applied Mechanics, Zurich, Switzerland;

1926.

[5] Pyke RM. Nonlinear soil models for irregular cyclic loadings.

J Geotech Engng Div 1979;105(GT6):71526.

[6] Vucetic M. Normalized behavior of clay under irregular cyclic

loading. Can Geotech J 1990;27:2946.

[7] Lee MK, Finn WDL. DESRA-2, Dynamic effective stress response

analysis of soil deposits with energy transmitting boundary including

assessment of liquefaction potential. Soil mechanics series no. 36,

Fig. 15. Computed 5% damped surface response spectra for the Treasure Island case history. Non-linear TD analysis uses first natural mode approximation as

well as proposed full Rayleigh viscous damping formulation.


Vancouver, Canada: Department of Civil Engineering, University of

British Columbia; 1978.

[8] Matasovic N. Seismic response of composite horizontally-layered soil

deposits. PhD Thesis, University of California, Los Angeles; 1993.

p. xxix, 452 leaves.

[9] Matasovic N, Vucetic M. Seismic response of soil deposits composed

of fully-saturated clay and sand layers. First International Conference

on Earthquake Geotechnical Engineering, Tokyo, Japan; 1995.

[10] Borja RI, Chao HY, Montans FJ, Lin CH. Nonlinear ground response

at Lotung LSST site. J Geotech Geoenviron Engng 1999;125(3):187

97.

[11] Hashash YMA, Park D. Non-linear one-dimensional seismic ground

motion propagation in the Mississippi embayment. Engng Geol 2001;

62(13):185206.

[12] Idriss IM. Personal communications; 2000.

[13] Newmark NM. A method of computation for structural dynamics.

J Engng Mech Div 1959;85:6794.

[14] Rayleigh JWS, Lindsay RB. The theory of sound, 1st American ed.

New York: Dover Publications; 1945.

[15] Lanzo G, Vucetic M. Effect of soil plasticity on damping ratio at small

cyclic strains. Soils Foundations 1999;39(4):12141.

[16] Kramer SL. Geotechnical earthquake engineering. Prentice-Hall

international series in civil engineering and engineering mechanics,

Upper Saddle River, NJ: Prentice Hall; 1996.

[17] Clough RW, Penzien J. Dynamics of structures, 2nd ed. New York:

McGraw-Hill; 1993.

[18] Chopra AK. Dynamics of structures: theory and applications to

earthquake engineering. Prentice-Hall international series in civil

engineering and engineering mechanics, Englewood Cliffs, NJ:

Prentice Hall; 1995.

[19] Hudson M, Idriss IM, Beikae M. University of California Davis,

Center for Geotechnical Modeling, and National Science Foundation

(US). QUAD4M: a computer program to evaluate the seismic response

of soil structures using finite element procedures and incorporating a

compliant base. Center for Geotechnical Modeling, Department of

Civil and Environmental Engineering University of California Davis:

Davis California; 1994.

[20] Laird JP, Stokoe KH. Dynamic properties of remolded and

undisturbed soil samples test at high confining pressure. Electric

Power Research Institute; 1993.

[21] Boore DM. SMSIM Fortran programs for simulating ground motions

from earthquakes: Version 1.87. Users manual, US Geological

Survey; 2000. p. 73.

[22] Seed RB, Dickenson SE, Riemer MF, Bray JD, Sitar N, Mitchell JK,

Idriss IM, Kayen RE, Kropp A, Harder LF, Power MS. Preliminary

report on the principal geotechnical aspects of the October 17, 1989.

Loma Prieta Earthquake; 1990.

[23] Hryciw RD, Rollins KM, Homolka M, Shewbridge SE, McHood M.

Soil amplification at Treasure Island during the Loma Prieta

earthquake. Second International Conference on Recent Advances

in Geotechnical Earthquake Engineering and Soil Dynamics, St Louis,

MO; 1991.

[24] Finn WDL, Ventura CE, Wu G. Analysis of ground motions at

Treasure Island site during the 1989 Loma Prieta earthquake. Soil

Dynamics Earthquake Engng 1993;12:38390.

[25] Gibbs JF, Fumal TE, Boore DM, Joyner WB. Seismic velocities and

geologic logs from borehole measurements at seven strong-motion

stations that recorded the 1989 Loma Prieta earthquake. USGS: Menlo

Park 1992;139.

[26] Pass DG. Soil characterization of the deep accelerometer site at

Treasure Island, San Francisco, California. MS Thesis in Civil

Engineering, University of New Hampshire; 1994.

[27] Hwang SK, Stokoe KH. Dynamic properties of undisturbed soil

samples from Treasure Island, California. Geotechnical Engineering

Center, Civil Engineering Department, University of Texas at Austin;

1993.


Viscous damping formulation and high frequency motion propagation in non-linear site response analysisIntroductionNumerical implementation of non-linear one-dimensional wave propagation analysisLimitation of current viscous damping formulationProposed extension of viscous damping formulationValidation of new viscous damping formulationUse of constant versus variable [C] matrix in non-linear analysisApplication of new viscous damping formulation to case history of Treasure Island and Yerba Buena Island recordingsSummary and conclusionsAcknowledgmentsReferences

Viscous Damping Formulation Publication

Documents

Transcript of Viscous Damping Formulation Publication