A Visco-Elastic Model with Loading History Dependent Modulus and Damping for Seismic Response Analyses of Soils. Zhiliang Wang 1 and Fenggang Ma 2.

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2 A Visco-Elastic Model with Loading History Dependent Modulus and Damping for Seismic Response Analyses of Soils Zhiliang Wang 1 and Fenggang Ma 2. 1 Senior Associate, AMEC Environment & Infrastructure, Inc., Oakland, CA 94612; zhi-liang.wang@amec.com 2 Environmental Engineer, Washington State Dept. of Ecology, Dam Safety Office, Spokane, WA 9925 ABSTRACT: It is well established that soils exhibit non-linear behavior even at small strain levels. Yet, most evaluations of the seismic response of soil deposits utilize an equivalent linear methodology, i.e. elastic solutions incorporating constant damping. In such solutions the modulus and damping constants are adjusted by means of an iterative approach to correlate to the maximum strain. In doing so, the smaller amplitude, high-frequency component motions are forced to use the same modulus degradation and damping as that of the lower frequency motions. As a consequence, the computed motions at the surface of a deposit often exhibit unrealistic low amplitudes at high frequencies, when the strong input motions are applied. This article presents a modified Kelvin model, in which the modulus and damping are treated as loading history dependent coefficients for each loading-unloading branch. This model works in the program FLAC in a time domain integration procedure. Based on the peak strain level experienced in the previous half cycle, the modulus degradation and viscosity are updated for the current branch. Numerical analyses for a soft soil site are presented and compared with those obtained using the equivalent linear method implemented in the program SHAKE for a recorded motion at Treasure Island. The site response analysis of another soft clay site for a higher design input motion shows that the computed PGA and response spectral could be higher than those obtained from a typical equivalent linear analysis. INTRODUCTION It is well known that soil response exhibits strong non-linearity during monotonic and cyclic loading. From soil cyclic loading tests, two key parameters are determined: the secant shear modulus and the damping ratio. The secant modulus corresponds to the slope of the diagonal of a cyclic stress-strain loop and the damping ratio from the area of the loop (or energy loss) as shown in Figure 1. The non-linearity of soil is reflected in the degradation of the shear modulus and an increase in the damping ratio as shear strain amplitude increases, i.e. they are both functions of the strain amplitude. Typical shear modulus reduction and associated damping ratio curves for clays as a function of shear strain are shown in Figure 2 (Vucetic, M. and Dobry, R.,1991). Page 1

3 Shear Stress τ (ksf) Kelvin Model: τ=τ e + τ v = Gγ+ηγ A' B' τ v = ηγ τ e = Gγ Shear Strain γ FIG 1. Stress strain loops based on Kelvin model FIG 2. Typical modulus reduction and damping ratio curves of clay (Vucetic and Dobry, 1991) O γ A B Program SHAKE is a visco-elastic model with constant modulus and damping ratio (Schnabel, et al, 1972). An initial estimate of the shear modulus and damping ratio is made and the problem solved. From this solution an effective strain is defined as a fraction of the calculated maximum shear strain, typically between.5 and.7. That effective strain is used to guide a more refined estimate of the dynamic soil properties and the problem rerun. This solution process is repeated until the modulus and damping used is within an acceptable tolerance of strain compatible modulus and damping calculated. The simplicity of the approach and its ability to yield site response predictions that reasonably agree with recorded motions and spectral content have made it the program of choice for most site response analyses. However, there has been a progressive increase in the intensity of strong ground shaking. These greater accelerations are testing the limits of SHAKE and in particular the appropriateness of the effective shear strain scheme to use a constant shear modulus and damping ratio for the duration of shaking. In particular SHAKE results appear to underestimate the response when high input motions induce very high strains in the soft soil layers. Recently the geotechnical profession has begun investigating other schemes to more realistically model the shear modulus and damping response of soils in the equivalent linear visco-elastic approach. Kausel and Assimaki (2) proposed replacing the constant modulus and damping ratio for the entire duration of shaking in an equivalent linear approach with frequency-dependent moduli and damping ratios. At about the same time, another new procedure was proposed based on the time domain integration of the motion equations using the concept of cycle-wise equivalent linear analysis (CELA). In this Page 2

4 procedure (Shiomi et al, 2, 28), the equivalent linear method is applied cyclewise instead of the entire duration of the earthquake. The modulus and damping change for each cycle, based on the peak strain level experienced in the previous cycle. The authors claim this cycle-wise equivalent linear analysis (CELA) should theoretically have better accuracy. In the present article, a modified Kelvin model (visco-elastic model) is described. The strain time history is modeled as sets of loading-unloading branches (or, half cycles). The response of the model to each loading-unloading branch is assessed with a time domain integration scheme. The modulus and damping are defined in each loading unloading branch based on the load induced and shear strain time history experienced prior to the current branch. The model has been written for the program FLAC in both the FISH language and as a compiled dynamic linked library, Kelvin-Wang.FIS and Kelvin-Wang.dll, respectively. To demonstrate the capabilities of the site response analysis, a soft clay site was assessed with both the proposed model and SHAKE. A comparison of the results shows important similarities and notable differences. CLASSICAL VISCO-ELASTIC MODEL As a notation convention, we use symbols s ij and e ij to denote deviatoric stress and strain components, i.e.,, (1) (2) Then, the classical Kelvin visco-elastic model is 2 2 (3) In which the first term on the right side of the equation is the elastic term representing the resistance of soil to the deformation (e ij ). G is the shear modulus of soils. The second term is the soil resistance to the strain rate de ij /dt. Based on the Kelvin model, the total resistance is the summation of the elastic term and viscous term. The volumetric strain vs. mean stress relation is elastic (4) For simplicity, consider a simple shear condition, then, the Kelvin model is (5) If we apply a sinusoidal strain loading wave sin, the viscous resistance is (6) Using the relationship 1, we can demonstrate the relation between τ v and γ is an ellipse: 1 (7) The area of the ellipse (see Figure 1) is the energy loss, dw=π( ωγ ο )γ ο. The elastic part is completely recoverable and does not contribute to energy losses. In soil dynamics (Das, 1993), the shear modulus of a soil in cyclic simple shear tests can be determined as,, The damping ratio at a given shear strain amplitude can be obtained from the hysteretic stress-strain properties. Referring to Figure 1 the damping ratio can be (8) Page 3

5 expressed as Now, we have Then, the viscosity can be determined by damping ratio D and period T (frequency f =1/T) by MODIFIED VISCO-ELASTIC MODEL The original Kelvin model has constant modulus and viscosity. In the traditional equivalent linear approach, such modulus and damping parameters were determined to correlate to the loading cycle that produces the highest strain amplitude. This is conducted through an iterative procedure as in the program SHAKE. This approach may generate a better simulation of the loading cycle at effective strain level (e.g., 65% of the maximum strain), but could be unrealistic for cycles smaller or larger than such cycle. The alternative approach presented in this paper is as follows: a) First treat the strain loading time history as a sequence of many loading-unloading branches in the time domain. b) Each loading-unloading branch is defined as the load path from zero shear strain to a peak strain and the unloading path from the peak strain back to zero strain. c) A complete branch may include a partial loading that starts from a nonzero strain and a partial unloading that ends with a nonzero strain. d) The modified Kelvin model assumes that modulus and damping parameters can be different for each branch. e) A postulation about soil s memory is made that the modulus and damping parameters for the current branch are dependent on the loading history prior to this branch. Specifically, the parameters are treated as a function of peak shear strain in the immediate prior loading branch. Note that when strain peak is reached (or unloading starts) in a branch, the strain rate is zero. The peak strain γ mi for the current branch is recorded, and the modulus G=G(γ mi ), damping D(γ mi ) as well as the period T=4*(t mi -t i ) can be determined to modify the Kelvin model in the following manner. f) Modulus G will be changed at the beginning of the next branch, i.e. when zero strain appears. Doing it at this time avoids causing a discontinuity in the computed stress. No modulus reduction is assumed for the first branch, i.e. the modulus is the maximum modulus assigned the layer (user input value) when starting the computations. g) Viscosity will be changed based on Eq. (11), i.e.,. In the above equation, damping, modulus and period will be changed at the reverse point indicated by zero shear strain rate, which means it is changed at the peak strain level. It is easy to verify that changing viscosity at zero strain rate level will ensure (9) (1) (11) Page 4

6 the continuity of computed stresses. Of course, for the loading part before reaching the current peak, the viscosity is that used in the previous branch. Shear Stress τ (ksf) τ v = η i+1 γ τ e = G i -1 γ (i+1) th Branch i th Branch τ e = G i γ τ v =η i γ Shear Strain γ FIG 3. Loading history dependent modulus and damping from the modified Kelvin model γ mi The foregoing is illustrated in Figure 3. The modulus of the i-th loading branch is determined by the peak strain in the (i- 1)th loading branch, so that the elastic resistance follows τ e =G i-1 γ. When unloading starts, the peak strain for this i- th branch is determined, so are the damping ratio D i, modulus G i and Period T i. Note that the viscosity η is refreshed when un-loading starts from where the strain rate is zero. It will be kept unchanged to the loading part of next (i+1)-th branch while the modulus will be changed in the (i+1)-th branch based on the peak strain at i-th branch G i. NUMERICAL IMPLEMENTATION OF MODIFIED KELVIN MODEL The numerical implementation of the Kelvin model Eq. (3) is described below. Kelvin-Wang.FIS is the subroutine written in FISH (FLAC s programming language) that implements the modulus and damping rules proposed in the previous section. Although the FLAC manual cites a number of different algorithms to carry out the integration of the model subject to the earthquake time histories, the authors found that the subroutine worked best with the Euler forward method. The elastic and viscosity part of Eq. (3) are as follows: 2 (12) (13) And (14) In which dt is constant, and de ij is given for each new time step. The peak strain is measured by (15) The strain crossing zero is judged by (16) Within the branch, sume is positive. If sume<=, then, the strain is defined as having crossed the zero point. Then, the modulus will be changed, and a new loading branch begins. The moment of crossing is the starting time of this branch. The strain rate crossing zero is judged by (17) If sumv<=, an unloading is defined using an index Load(i) =-1. Expressing this index for the previous time step as Load(i-1) which is 1 (loading) or -1 (unloading). The reverse time is detected by Load(i)*Load(i-1) <. At that time, the strain rate is Page 5

7 crossing zero, and the viscosity will be changed for the i-th branch based on Eq.(11). SIMULATION FOR A SOIL ELEMENT UNDER CYCLIC LOADING Shear strain FIG 4. Time history of shear strain as input for one element simple shear Shear stress (ksf) Time in second Shear strain FIG 5. Computed stress strain loops using the modified Kelvin model for one element simple shear The modified Kelvin model is applied to simulate one zone s response to a sinusoidal shear strain loading in FLAC. The input shear strain time history is presented in Figure 4. The computed stress-strain loops are shown in Figure 5. It can be seen that when the shear strain amplitude increases for each cycle, the secant modulus reduces and the damping ratio increases. When the peak strain cycle is passed, the shear strain amplitude decreases for each cycle, so that the secant modulus increases and the damping ratio decrease. The modulus and damping ratio time histories are shown in Figure 6. It is demonstrated that the modulus and damping ratio vary for each of the loading branches or half-cycles. Computed shear modulus ratio (G/Gmax) and damping ratio G/Gmax Damping ratio Time in second FIG 6. Time histories of computed modulus and damping ratio for one element simple shear Page 6

8 COMPARISON OF COMPUTED RESPONSE WITH RECORDINGS AT TREASURE ISLAND Spectral Acceleration in g Recorded and Computed Sa Recorded at Treasure Island from SHAKE analysis from FLAC with Kelvin-Wang model Period in second FIG. 7. Comparison of computed (dashed line) and recorded response spectra at Treasure Island Using the model described above via the program FLAC, a site response analysis was performed. The site conditions are consistent with those at Treasure Island (CSMIP STA 58117). The input motion (9 components) was recorded at Yerba Buena Island (CSMIP STA 58163) which is 2 km to the north and is classified as a rock site. The input data for the SHAKE analysis is reported in Long Sen (212) and used here too for the new analysis using the proposed model. Figure 7 presents the spectral accelerations computed using the modified Kelvin model and those from the recordings. In the same figure, equivalent linear (SHAKE) results are also included for comparison. It is clear that the analytical results reasonably match the values from recordings. The new analysis is expected to be close to the SHAKE results, because the input motion is at a low level (PGA=.6g). Both analyses are reasonably close in this Treasure Island case study as shown in Figures 7. COMPARISON WITH EQUIVALENT LINEAR METHOD IN A SITE SEISMIC RESPONSE ANALYSIS Pseudo acceleration in g from Kelvin-Wang model from SHAKE Period in second FIG. 8. Computed spectral acceleration at ground surface compared with results from SHAKE analysis A site response analysis was conducted using the equivalent linear method. The profile and all parameters for SHAKE are presented in Tables 1 and 2. These parameters are all applied for this modified Kelvin model and a new response analysis was completed using this model coded for FLAC. The input acceleration has apeak value about.6g. The computed ground surface response spectra are presented in Figure 8 together with those from SHAKE. It can be seen that the PGA and higher frequency spectral values obtained using the modified Kelvin model are higher than those obtained from the SHAKE analysis. Page 7

9 Acceleration in g Velocity in ft/sec Displacement in ft) FLAC with Kelvin-Wang model SHAKE Time in second FIG. 9. Computed surface acceleration, velocity and displacement time histories compared with results from SHAKE analysis The computed surface acceleration, velocity and displacement time histories, compared with those from SHAKE analyses are presented in Figure 9. It is interesting that the peak acceleration values from Kelvin-Wang model are higher than those from SHAKE analyses in general. The time histories for modulus and damping are presented in Figures 1 and 11 while these values are constants in SHAKE. The period for each half cycle is closely dependent on the strain time history. 1 Kelvin-Wang model SHAKE.2 Shear modulus ratio (G/Gmax) Critical damping ratio Time in second FIG. 1. Computed modulus reduction time history in a layer compared with results from SHAKE analysis Kelvin-Wang model SHAKE Time in second FIG. 11. Computed damping ratio time history in a layer compared with results from SHAKE analysis Page 8

10 Table 1. Soil Profile and Input Parameters for a Site Response Analysis Layer No. Curve No. Sub layer No. Thickness, ft Initial damping Unit weight, kcf Vs, fps Table 2. Modulus Reduction and Damping Curves Soil Type Curve Number Curves used in equivalent Linear Analysis (SHAKE) Fill and YBM 3 YBM Site-Specific Curves Silty Fine Sand 1 Mid-range Sand, Seed & Idriss, 197 Dense Silty Sand 2 Upper G/Gmax, Seed & Idriss, 197; Damping, Idriss, 199 OBC 4 OBC Site-Specific Curves Fine Sand 2 Upper G/Gmax, Seed & Idriss, 197; Damping, Idriss, 199 Gravel 5 Upper G/Gmax, Seed & Idriss, 197; Damping, Idriss, 199 Page 9

11 CONCLUSIONS Conventional equivalent linear visco-elastic analysis for seismic response is computed using an iterative procedure to get the modulus and damping correlated to the maximum strain (Schnabel et al., 1972). Such determined parameters are applied to the entire duration, which means the modulus reduction and damping ratio could be overestimated for the cycles smaller than the cycle of effective strain level. The present article proposes a scheme to update the shear modulus and damping over the course of shaking as a function of time history of loading. These values for the current loading branch (half cycle) are determined by the peak strain level experienced by the soil during the past loading branches (half cycles). Such determined modulus from the current loading branch are delayed and applied to the immediate subsequent loading branch. The viscosity is updated when the strain rate is zero, or at the peak strain level in the current branch when unloading starts. In such a time domain integration algorithm, there is no need for iterations. Numerical examples are provided for explanations. The site response analysis of a soft clay site using the Kelvin-Wang model shows that the computed PGA and response spectral could be higher than those obtained from a typical equivalent linear SHAKE analysis. Of course, the validity of all computed results should be judged by comparisons with field recordings when they become available. REFERENCES Das, B.M., (1993) Principles of Soil Dynamics, PWS-KENT Publishing Company. Dickenson, S.E. (1994). The dynamic response of soft and deep cohesive soils during the Loma Prieta earthquake of October 17, 1989, Ph.D. Dissertation, Univ. of California, Berkeley. Kausel, E and Assimaki, D., (22). Seismic simulation of inelastic soils via frequency-dependent moduli and damping, J. Engrg. Mech., Vol. 128 (1): Schnabel, P B., Lysmer, J., and Seed, H B. (1972). "SHAKE: A computer program for aeolian soil." Geomaterial earthquake response analysis of horizontally layered sites," Report No. EERC 72-12, University of California, Berkeley, Calif., Earthquake Engineering Research Centre. Seed, H. B., and Idriss, I. M. (1969). Soil moduli and damping factors for dynamic response analyses. Report No. EERC 7-1, Earthquake Research Center, University of California, Berkeley, Calif. Sen, Long, (212., Seismic response analysis considering ground motion variability and site property variability, Dessertation, Hong Kong University of Sience and Technology. Shiomi,T., Chan, A., Nukui, Y., Hijikata, K., and Koyama, K., (2). Comparison of equivalent linear analysis and nonlinear analysis for a liquefaction problem. 12 th WCEE, Paper Shiomi, T., Nukui, Y., and Yahishita, F., (28). Practical effective stress analysis and determination of its dynamic property, 14 th WCEE. Vucetic, M. and Dobry, R. (1991). Effect of Soil Plasticity on Cyclic Response. J. Geotechnical & Geoenv. Engrg., Vol. 117 (1): Page 1