October 2-7, 28, Beijing, China COMPARISON OF FREQUENCY DEPENDENT EQUIVALENT LINEAR ANALYSIS METHODS Dong-Yeo Kwak Chang-Gyun Jeong 2 Duhee Park 3 and Sisam Park 4 Graduate student, Det. of Civil Engineering, Hanyang University, Seoul, Korea 2 Graduate student, Det. of Civil Engineering, Hanyang University, Seoul, Korea 3 Assistant Professor, Det. of Civil Engineering, Hanyang University, Seoul, Korea Email: dark@hanyang.ac.kr 4 Manager, Technical Division, GS Engineering & Construction Cor., Seoul, Korea ABSTRACT: Equivalent linear one dimensional site resonse analysis, which aroximates the nonlinear soil behavior within the linear analysis framework, is widely used in ractice due to its simlicity and ease of use. However, the equivalent linear method has been known to be unreliable in case of roagating strong ground motion or in soft soil columns. Various frequency deendent algorithms have been roosed to imrove the accuracy of the aroximate solution and better simulate the wide range of shear strains within the duration of the seismic loading. However, the effectiveness of the modified schemes has not yet been fully verified. In this study, the results of the modified equivalent linear analyses are comared to evaluate the degree of imrovement and the alicability of the modified algorithms. Results show that there is a significant difference between the develoed rocedures and they do not always rovide imroved estimates. KEYWORDS: Equivalent linear, Frequency deendent, Ground resonse analysis. INTRODUCTION One-dimensional (D) seismic site resonse analysis is routinely erformed to characterize the site amlification effects under an earthquake ground motion. D analysis, which simulates the vertical roagation of horizontal shear waves through horizontally layered soil rofile, has been known to give reasonable estimates of ground vibration under a seismic event (Idriss, 99). D site resonse analysis is either erformed in frequency or time domain. In a nonlinear analysis, which is erformed in time domain, the dynamic equation of motion is integrated at each time ste and the nonlinear soil behavior is accurately modeled. However, the non-linear site resonse analysis is not widely used due to difficulty in erforming the analysis and high comutational cost. Instead, the equivalent linear analysis, which
October 2-7, 28, Beijing, China aroximates the nonlinear soil behavior within the linear framework, is widely used in engineering ractice (Schnabel et al., 972). The main limitation of the rocedure is that a constant linear shear modulus and daming at a reresentative level of strain is used throughout the analysis. Frequency-deendent equivalent linear algorithms have been roosed to overcome the limitation and better simulate the nonlinear hysteretic soil resonse under the seismic loading (e.g. Sugito et al., 994; Yoshida et al., 22; Kausel and Assimaki, 22). It has been shown that the frequency-deendent equivalent linear methods result in imroved estimates of the ground motion roagation. However, it is still now clearly understood on how the distinctively different schemes all result in imroved solution, and whether the accuracy of the solution always increases universally for all cases. This study comares the comuted resonses using the schemes by Yoshida et al. (22) and Kausel and Assimaki (22) with those from equivalent linear and nonlinear analyses to evaluate whether the degree of imrovement of the frequency-deendent rocedures over a standard equivalent linear scheme. 2. FREQUENCY-DEPENDENT EQUIVALENT LINEAR ALGORITHM Figure shows a schematic lot of hysteretic loos under large (A) and small (B) shear strain amlitudes, resectively. Assuming that the strain increment at each time ste is similar for both loos, the frequency of vibration of a large amlitude hysteretic loo will be lower than for a small amlitude loo. This indicates that the frequency of vibration is associated with the amlitude of shear strain. Figure 2, which shows the Fourier sectra of two shear strain time histories, demonstrates the strong deendence of the shear strain amlitude on the frequency. The amlitude of the shear strain decays quickly with increase in frequency. Figure and Figure 2 show that the frequency-deendence of shear strain is a distinct henomenon that needs to be incororated in a site resonse analysis. Standard equivalent linear rocedure ignores such deendence and alies constant shear modulus and daming throughout entire frequency range. τ A B O γ Figure Comarison of hysteretic loos of large(a) and small(b) strain amlitudes (modified after Yoshida et al., 22)
October 2-7, 28, Beijing, China. Strain (%)... Lotung Treasure Island. Figure 2 Frequency-deendence of shear strain Fourier sectra The frequency-deendent equivalent linear rocedures roose use of the shear strain Fourier sectrum, such as shown in Figure 2, in defying the frequency deendence of the shear strain in a site resonse analysis in characterizing. However, the direct use of the Fourier sectrum in a site resonse analysis can lead to numerical instability (Kausel and Assimaki, 22) and hence, the Fourier sectrum is always smoothed. The difference in the frequency-deendent algorithms lie in how the Fourier sectrum is smoothed. Normalized Strain Fourier Amlitude.5 (a) Yoshida et al.. (b) Kausel and Assimaki. Figure 3 Smoothed Curves Yoshida et al. (22) roosed the following equation to characterize the smoothed frequency-deendent shear strain: γeff = γmax f > f m log f log f γ γ eff = f < fe eff = γmax f f f e log fe log f (2.)
October 2-7, 28, Beijing, China where f = frequency at γ max, f e = minimum frequency, m = constant. Yoshida et al., 22 used f e = 5 Hz and m = 2. Kausel and Assimaki (22) roosed the following equation: ( ) γ ω where ω and γ are defined as follows: γ ω ex α ω ω ω = β ω ω ω > ω (2.2) ω = ωr( ω) dω r( ω) dω r r( ) ω (2.3) = ω ω dω (2.4) α and β are constants that are determined by least square method. The smoothed curve is constant u to ω, and decreases at higher frequencies, as shown in Figure 3. Deth (m) Fill Sand Clayey Silt 2 Silty Sand Clay 3 Clayey Silt (a) Acceleration (g) Acceleration (g).2 -.2. -. 2 4 Vs (m/s) (b) Synthetic 2 3 Time (sec) (d) Parkfield 2 4 6 Time (sec) Fourier am. (g-sec) Fourier am. (g-sec). (c) Synthetic.5..5 (e) Parkfield.25. Figure 4 Shear wave velocity rofile, Inut motion acceleration time histories and Fourier sectrum Yoshida et al. (22) showed that the frequency-deendent algorithm results in lower eak ground acceleration (PGA) comared to the standard equivalent linear anlaysis, since maximum shear strain is alied u to f
October 2-7, 28, Beijing, China and hence, resulted in a softer resonse. Kausel and Assimaki (22) showed that the frequency-deendent algorithm resulted in imroved match with the nonlinear solution, since low energy dissiation at high frequencies is modeled. 3. VERIFICATION OF ACCURACY OF FREQUENCY-DEPENDENT ALGORITHMS This study erformed a series of standard equivalent linear (EQL), nonlinear (NL), and frequency-deendent equivalent-linear analysis (FDEQL) at a selected site, as shown in Figure 4. The rofile is 34 m in thickness, and is comosed of layers of sands, silts, and clays. The curves by Seed and Idriss (97) are used for sands, while the curves by Vucetic and Dobry (99) are used for silts and clays. The shear wave velocity of the bedrock is assumed to be 76 m/sec. Two inut motions were used: a synthetic motion and a recorded motion during Parkfield earthquake (M = 6., U.S.A.), also shown in Figure 4. The synthetic motion is rich in high frequency comonents, while the recorded motion dislays strong energy concentration between Hz and low high frequency content. All analyses are erformed by newly develoed D site resonse analysis rogram GeoSHAKE. (a) Synthetic (b) Parkfield Deth (m) 2 3.2.4 PGA (g). PGA (g).2 Figure 5 PGA rofiles Figure 5 to Figure 8 show the calculated resonses. The PGA rofiles calculated from all analyses show that the FDEQL resulted in higher estimates comared to the EQL and NL. The results contrast the findings of Yoshida et al. (22), which showed that the FDEQL results in lower resonse than the EQL. Figure 6 shows the comuted shear strain stress relationshis extracted from the surface layer using the synthetic motion. The grey lines reresent the backbone curves. EQL, which uses the effective shear strain reresentative of 65% of maximum shear strain, showed stiffer resonse at maximum shear strain comared to the backbone-curve. The
October 2-7, 28, Beijing, China mean secant stiffness of the FDEQL comares better with the backbone curve. However, the alication of lower shear strain amlitude for high frequencies resulted in highly irregular stress strain relationshis. This henomenon is esecially evident using the equations by Kausel and Assimaki (22), Figure 6d. Figure 7 shows the calculated shear strain Fourier sectra and the smoothed curves at the surface layer. EQL resulted in lowest sectrum, esecially at high frequencies, due to overestimation of daming at high frequencies. On the contrary, the FDEQL resulted in higher resonse than the NL at high frequencies. The equations by Yoshida et al. (22) alies minimum shear strain at frequencies higher than 5Hz, and hence abrut amlification of the motion can be observed. Kausel and Assimaki (22) s equations alies low shear strain at frequencies higher than 5 Hz, and therefore highly overestimates the resonse..5 (a) Nonlinear (b) Equivalent linear Normalized Stress -.5.5 (c) Yoshida et al. (d) Kausel and Assimaki -.5 - - Strain(%) Figure 6 Stress strain relationshi and skeletons Figure 8 shows the comuted 5% damed acceleration resonse sectra at the surface using both inut ground motions. Using the synthetic motion, rich in high frequency contents, the FDEQL highly overestimated the resonse comared to the EQL or NL. The unrealistic amlification at short eriod range between..4 sec can be observed. When using the recorded motion, use of the equations by Yoshida et al. (22) resulted in a resonse similar to the EQL. Alication of the equations by Kausel and Assimaki (22) resulted in significant overestimation of the comuted resonse.
October 2-7, 28, Beijing, China Normalized Strain Fourier Amlitude.. (a) Nonlinear (b) Equivalent Linear... (c) Yoshida et al. (d) Kausel and Assimaki... Figure 7 Normalized strain Fourier sectrum and smoothed curves (a) Synthetic.5 Sectral Acceleration (g).5.25 (b) Parkfield Nonlinear Equivalent Linear Yoshida et al. Kausel and Assimaki.. Period (sec) Figure 8 5% damed surface acceleration resonse sectrum 4. CONCLUSION D equivalent linear analysis is widely used in ractice due to ease of use and low comutational cost. The main deficiency of the rocedure is constant alication of the shear modulus and daming throughout the analysis
October 2-7, 28, Beijing, China selected at the effective shear strain. The equivalent linear analysis can overestimate the eak ground acceleration due to overestimation of the stiffness at maximum shear strain. Frequency-deendent equivalent linear rocedures were roosed to overcome such limitations and better simulate the nonlinear soil behavior. The rocedure is based on the observation that large strain amlitude is associated with large hysteretic loos and hence low frequency, while small strain amlitudes tyically result in high frequency vibration. Such observation led to develoment of equations of frequency-deendent shear strain. The equations are essentially smoothed curves of shear strain Fourier sectrum. In this study, two equations that relate the frequency with the shear strain are imlemented in an equivalent linear analysis and the accuracies of the equations were tested. The results of the analyses showed that the results using two equations are highly different. It is demonstrated that the rocedure of smoothing can significantly influence the roagated ground motion. At the soil rofile selected in this study, the frequency-deendent equivalent linear analysis showed higher resonses comared to both the equivalent linear and nonlinear analyses. The frequency-deendent rocedure can esecially result in unrealistic amlification of low eriod comonents using a ground motion rich in high frequency contents. The test analyses showed that the frequency deendent algorithm does not always imrove the accuracy of the solution. More in-deth study is warranted to characterize the accuracy and alicability of the frequencydeendent rocedure. REFERENCES Idriss, I. M., "Influence of Local Site Conditions on Earthquake Ground Motions", Proceedings of the 4th U.S. National Conference on Earthquake Engineering, 99,. 55-57 Schnabel, P. B., Lysmer, J. L. and Seed, H. B. (972), SHAKE: A comuter rogram for earthquake resonse analysis of horizontally layered sites, Berkeley, CA Sugito, M., Goda, H. and Masuda, T. (994), Frequency deendent equi-linearized technique for seismic resonse analysis of multi-layered ground, Doboku Gakkai Rombun-Hokokushu/Proceedings of the Jaan Society of Civil Engineers, 493: 3-2, 49-58 Yoshida, N., Kobayashi, S., Suetomi, I. and Miura, K. (22), Equivalent linear method considering frequency deendent characteristics of stiffness and daming, Soil Dynamics and Earthquake Engineering, 22: 3, 25-222 Kausel, E. and Assimaki, D. (22), Seismic simulation of inelastic soils via frequency-deendent moduli and daming, Journal of Engineering Mechanics, 28:, 34-47 Seed, H. B. and Idriss, I. M. (97), Soil moduli and daming factors for dynamic resonse analyses, College of Engineering University of California Berkeley., Berkeley Vucetic, M. and Dobry, R. (99), Effect of soil lasticity on cyclic resonse, Journal of Geotechnical Engineering, 7:, 87-7