Viscous damping formulation and high frequency motion propagation in non-linear site response analysis

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1 Soil Dynamics and Earthquake Engineering 22 (2002) 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 Kelvin Voigt solid whereby elastic shear moduli and viscous damping characterize 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 * Corresponding author. Tel.: þ ; fax: þ addresses: hashash@uicu.edu (Y.M.A. Hashash), dpark1@uiuc. edu (D. Park). 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 nonlinear soil response in a FD analysis. The equivalent linear approach is computationally easy to use and implement but remains an approximation of nonlinear 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 unloading reloading 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 /02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S (02)

2 612 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) 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 onedimensional wave propagation analysis In non-linear analysis, the following dynamic equation of motion is solved: ½MŠ{ u} þ½cš{_u} þ½kš{u} ¼ 2½MŠ{I} u g ð1þ where [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 of nodal relative displacements. u g 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] b method. 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 loading unloading 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 %, depending on the values of material properties. However, the hyperbolic model is nearly linear at small strains (less than %) 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]: ½CŠ ¼a R ½MŠþb R ½KŠ ð2þ The damping matrix is assumed in current formulations to be only stiffness proportional since the value of a R ½MŠ is small compared to b R ½KŠ: Small strain viscous damping effects are assumed proportional only to the stiffness of the soil layers. This is further simplified to: ½CŠ ¼b R ½KŠ where b R ¼ 2j/v 1 and v 1 is the frequency of the first natural mode of the soil column. The viscous damping matrix for a multi-layered soil is expressed as [11]: 2 3 j 1 K 1 2j 1 K 1 ½CŠ ¼ 2 v ½j ik i Š¼ 2 6 2j v 1 K 1 j 1 K 1 þ j 2 K 2 2j 2 K j 2 K 2 ð3þ where v is natural circular frequency of the first natural mode and j i is the equivalent damping ratio for layer i at small 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 obtained from 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 1.5 4% 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

3 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) 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. 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 damping 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: j n ¼ b Rv n 2 ¼ j v n v 1 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. ð4þ 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.

4 614 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) Fig. 3. Computed surface ground motion, linear frequency and TD site response, V s profile (1). Linear TD analysis uses first natural mode approximation only of viscous damping formulation. Harmonic input motion, amplitude ¼ 0.3 g, period ¼ 0.2 s, duration ¼ 1 s. 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], a R and b R coefficients of Eq. (2) can be computed using two significant natural modes m and n: " #( ) " 1 1/v ðmþ v ðmþ ar ¼ j # ðmþ ð5þ 2 1/v ðnþ v ðnþ b R j ðnþ This matrix can be solved for a R and b R :! v ðmþ j ðnþ 2 v ðnþ j ðmþ a R ¼ 2v ðmþ v ðnþ v 2 ðmþ 2 v2 ðnþ b R ¼ 2 v! ð6þ ðmþj ðmþ 2 v ðnþ j ðnþ v 2 ðnþ 2 v2 ðnþ If the damping ratio j is frequency independent then:!! v ðmþ v ðnþ 1 a R ¼ 2j b v ðmþ þ v R ¼ 2j ð7þ ðnþ v ðmþ þ v ðnþ Eq. (3) is obtained from Eqs. (5) (7) by assuming m is the first natural mode and v ðnþ ¼ 0; implying that the second relevant 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, v ðnþ 0 should be included to represent the contribution of higher modes. When choosing higher modes, the mass matrix component will counter-balance part of the contribution of the stiffness matrix component. As higher modes are used, a R increases and b R decreases. Contribution of the mass matrix cannot 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: 2 3! j 1 M 1 v ðmþ v ðnþ ½CŠ ¼2 6 j v ðmþ þ v 4 2 M ðnþ 2 3! j 1 K 1 2j 1 K 1 1 þ2 6 2j v ðmþ þ v 1 K 1 j 1 K 1 þ j 2 K 2 2j 2 K ðnþ 2j 2 K 2 ð8þ with frequency dependent damping ratio [3], Eq. (2) can be

5 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) expanded as follows: 2 3 ½CŠ ¼ 2v! ðv ðmþ j 1ðnÞ 2 v ðnþ j 1ðmÞ ÞM 1! ðmþv ðnþ v 2 ðmþ ðv ðmþj 2ðnÞ 2 v ðnþj 2ðmÞÞM þ 2 v2 ðnþ v 2 ðmþ 2 v2 ðnþ 2 3 ðv ðmþ j 1ðmÞ 2 v ðnþ j 1ðnÞ ÞK 1 2ðv ðmþ j 1ðmÞ 2 v ðnþ j 1ðnÞ ÞK ðv ðmþ j 1ðmÞ 2 v ðnþ j 1ðnÞ ÞK 1 kðv ðmþ j 1ðmÞ 2 v ðnþ j 1ðnÞ ÞK 1 þðv ðmþ j 2ðmÞ 2 v ðnþ j 2ðnÞ ÞK 2 l 2ðv ðmþ j 2ðmÞ 2 v ðnþ j 2ðnÞ ÞK ð9þ 2ðv ðmþ j 2ðmÞ 2 v ðnþ j 2ðnÞ ÞK 2 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 nonlinear 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 resulting damping ratio is less than 1% while at T. 1sorT, 0.1 s the resulting damping ratio increases significantly. Therefore, 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 frequencies 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 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]). 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].

6 616 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) Fig. 5. Computed surface ground motion, linear frequency and TD site response, V s 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 ¼ 0.3 g, period ¼ 0.2 s, duration ¼ 1 s. 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 ¼ 1 s, period T ¼ 0.2 s, and amplitude ¼ 0.3 g are 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 ¼ 0.2 s (corresponding to the 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 Mississippi 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 ¼ g is used as the input motion. Computed 5% damped surface response spectra obtained from linear TD and FD analyses are plotted in Fig. 6. Computed surface ground motions for t ¼ s are plotted in Fig. 7. The analyses 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

7 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) Fig. 6. Computed 5% damped surface response spectra, linear frequency and TD site response, V s 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 ¼, duration ¼ 18 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. 7. Computed surface ground motion, linear frequency and TD site response, V s 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 ¼ g, duration ¼ 18 s.

8 618 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) 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. Synthetic input ground motion, shear wave velocity and viscous damping profiles (2) representative of the Mississippi 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 nonlinear (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 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 ¼ 5 R ¼ 20 km (PGA ¼ g) and M ¼ 7, R ¼ 20 km (PGA ¼ 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 nonlinear response for the four soil columns using the

9 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) Fig. 9. Computed surface ground motion, non-linear TD site response, V s 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 ¼ 5, R ¼ 20, New Madrid Seismic Zone parameters, PGA ¼ g, duration ¼ 4 s. 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 ¼ 0.59 g) whereby the contribution of the non-linear model and hysteretic damping is also higher relative to the viscous damping component. The Fig. 10. Computed 5% damped surface response spectra, non-linear TD site response, V s and viscous damping profiles (2), non-linear soil properties. Nonlinear TD analysis uses first natural mode approximation as well as proposed full Rayleigh viscous damping formulation. Synthetic input motion, M ¼ 5, R ¼ 20, New Madrid Seismic Zone parameters, PGA ¼ g, duration ¼ 4 s.

10 620 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) Fig. 11. Computed surface ground motion, non-linear TD site response, V s 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 ¼ 7, R ¼ 20, New Madrid Seismic Zone parameters, PGA ¼ 0.59 g, duration ¼ 17 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 frequencies/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. 12. Computed 5% damped surface response spectra, non-linear TD site response, V s and viscous damping profiles (2), non-linear soil properties. Nonlinear TD analysis uses first natural mode approximation as well as proposed full Rayleigh viscous damping formulation. Synthetic input motion, M ¼ 7, R ¼ 20, New Madrid Seismic Zone parameters, PGA ¼ 0.59 g, duration ¼ 17 s.

11 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) Fig. 13. Computed surface Fourier spectra, non-linear TD site response, V s 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 ¼ 7, R ¼ 20, New Madrid Seismic Zone parameters, PGA ¼ 0.59 g, duration ¼ 17 s. 6. Use of constant versus variable [C] matrix in nonlinear 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 ¼ 7 analyses 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 E W and N S components. 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 E W 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)

12 622 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) Fig. 14. Soil profile and soil properties used in the non-linear analysis of the Treasure Island case history. 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 N S 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 formulation 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

13 Y.M.A. Hashash, D. Park / Soil Dynamics and Earthquake Engineering 22 (2002) 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. (b) first natural mode of the column. The other relevant mode corresponds to the second or a higher natural mode. 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 recordings 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 ; 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; [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): [3] Assimaki D, Kausel E, Whittle AJ. Model for dynamic shear modulus and damping for granular soils. J Geotech Geoenviron Engng 2000; 126(10): [4] Masing G. Eignespannungen und Verfestigung beim Messing. Second International Congress on Applied Mechanics, Zurich, Switzerland; [5] Pyke RM. Nonlinear soil models for irregular cyclic loadings. J Geotech Engng Div 1979;105(GT6): [6] Vucetic M. Normalized behavior of clay under irregular cyclic loading. Can Geotech J 1990;27: [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,

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