SIMPLIFIED EQUIVALENT LINEAR AND NONLINEAR SITE RESPONSE ANALYSIS OF PARTIALLY SATURATED SOIL LAYERS

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1 SIMPLIFIED EQUIVALENT LINEAR AND NONLINEAR SITE RESPONSE ANALYSIS OF PARTIALLY SATURATED SOIL LAYERS M. Mirshekari and M. Ghayoomi, Ph.D., A.M.ASCE Research Assistant, University of New Hampshire, Dept. of Civil Engineering Assistant Professor, University of New Hampshire, Dept. of Civil Engineering ABSTRACT Dynamic properties of soils including small-strain shear modulus (G max ), shear modulus reduction function (G/G max ), and damping (D) are affected by changes in the degree of saturation. Inter-particle suction forces in partially saturated soils result in higher effective stress values, which in turn, vary the dynamic soil properties. These alterations could lead to different wave propagation mechanisms, acceleration amplification patterns, and seismically induced settlements. This paper aims to identify the challenges involved in nonlinear seismic site response analysis of partially saturated soils by looking at the response of -m sand and silt layers with different constant suction profiles. A set of frequency domain equivalent linear and nonlinear site response analysis under scaled Northridge earthquake motion was performed. A modified version of Bishop s effective stress equation for partially saturated soils has been utilized to calculate the dynamic soil properties (i.e. G max, G/G max, and D). Specifically, surface-to-base intensity amplifications (Peak Ground Amplifications and Arias Intensities), spectral accelerations, and lateral deformation profiles of the sand and silt layers with different suction profiles were generated and compared. The insight gained from this study was used to plan and design more complex nonlinear Finite Element site response analysis. INTRODUCTION Seismic response of partially saturated soil layers may vary from that of dry or fully saturated soil layers (Ghayoomi et al. ). Inter-particle suction stresses increase the effective stress in the soil particles (Lu and Likos 6) which, in turn, is a key parameter to evaluate dynamic properties of soil, such as small-strain shear modulus, non-linear shear modulus, damping, and shear wave velocity. This difference in dynamic material properties, consequently, leads to different wave propagation mechanisms. Hence, seismic site response evaluated as a function of degree of saturation would be valuable where the degree of saturation of the soil layer could change due to the seasonal effects, rainfall amount, soil type, and stratification. Although the effect of degree of saturation on the seismic site response has been recently shown (Yang & Sato, Yang 6, D Onza et al. 8, Ghayoomi & McCartney ), it has not been employed in the Civil Engineering practice yet. Current state-of-the-practice in site response analysis considers soil in either dry or fully saturated conditions neglecting the partial saturation. Even procedures requiring

2 the shear wave velocity profile in depth ignore its seasonal variability due to fluctuation of water table level. The objective of this paper is to study the influence of different degrees of saturation on the seismic site response and investigate the challenges associated with it. These challenges include simplified assumptions in modeling approach and material properties during site response analysis of unsaturated soils. Site response analyses were performed using DeepSoil software. Two soils with different fine contents were chosen for this study representing wide range of suction stresses within the soil layer: i.e. Ottawa sand and Bonny Silt. Dynamic material properties were estimated using a single-parameter effective stress equation. Then, a set of nonlinear site response analyses were performed under various initial and loading conditions. BACKGROUND Subsurface soil properties play a major role in wave propagation mechanisms within the ground. Different mechanisms may cause amplification or de-amplification of seismic waves which, in turn, may lead to different seismic demands imposed on surface structures. Hence, a precise estimation of site response would be highly dependent on the material properties and is crucial for design of surface structures. Site response might be evaluated in both frequency and time domain using wave propagation analysis. In last decades, several numerical software have been developed to perform site response analysis (e.g. SHAKE by Idriss & Sun 99, DESRA by Lee & Finn 978, DMOD by Matasovic 993, and DeepSoil by Hashash ). Despite the advances made by these efforts to predict the nonlinear complicated behavior of ground layers under earthquake excitations, none of them directly account for the effect of suction variation on the site response. It has been shown, numerically and analytically, that partial saturation of the soil layers has a considerable influence on the site response (Yang 6, D Onza et al. 8).Yang & Sato () showed that a small variation in degree of saturation of a completely saturated soil may lead to a different wave propagation mechanism at layer interfaces. Based on parametric study performed by D Onza et al. (8), ignoring the influence of degree of saturation in soil may result to misleading amplification parameters and natural frequency of soil layer. MATERIAL PROPERTIES F-75 Ottawa sand, a fine uniformly graded sand (USCS symbol is SP, γ d =5.49 kn/m 3, G s =.65, C u =.7, C c =., ϕ = 35 ), and Bonny silt, silty sand from near Bonny dam in Colorado (USCS symbol is ML, γ d =6.5 kn/m 3, G s =.6, C u = 9.7, C c = 8.59, ϕ = 9 ), were selected for this study. F-75 Ottawa sand has relatively high permeability, about 6-6 m/s at the desired density of this analysis. Considering Soil Water Retention Curves (SWRC) of Ottawa sand, however, it still has fine enough soil particles to retain water suctions up to kpa. Bonny silt is less permeable (K saturated =.74-8 m/s) and was selected to simulate soils with higher suction levels (e.g. 7 kpa). SWRC curve of Ottawa sand with 45% relative density at low confining stress was obtained by performing hanging column test and van Genuchten fitting parameters were estimated based on the tests results (α=.5, N=9, S r =.8).

3 Se (%) Suction Stress (kpa) Van Genuchten SWRC parameters of Bonny silt was obtained from the data reported by Khosravi and McCartney () for kpa net normal stress (α=.36, N=5, S r =.6). SWRC curves associated with both materials are demonstrated in Figure. Void ratios of.66 and.53 were employed in the analyses for Ottawa sand and Bonny silt, respectively Suction (kpa) Figure. SWRC of Ottawa sand (hanging column test) and Bonny silt (Khosravi and McCartney, ) Suction stress vs. effective degree of saturation for the tested soils NUMERICAL MODELING PROCEDURE -m deep sand and silt layers were modeled using DeepSoil software. Soil layers, then, were divided into 3 sub-layers. Dynamic properties of soil were assumed to be constant within each layer and calculated for the center of each layer. Thickness of the layers increased gradually in depth (.5 m thickness for the first two top layers followed by three layers of.5m thick, and then m thickness for the rest of the layers). To have a more clear vision of the changes of the motion characteristics among the dry and unsaturated cases, unsaturated soils with maximum suction stress values, were modeled in this study. Thus, the degrees of saturation of 9% (S e =89%) and 8% (S e =78%) were chosen for Ottawa sand and Bonny silt, respectively. Degree of saturation is assumed to be constant within the depth enabling the parametric evaluation of its effect on site response (Ghayoomi et al. ). Fully saturated soil layers were not evaluated as they have different response mechanism as a result of seismically induced excess pore water pressure and potential liquefaction event. The effective stress formula proposed by Lu et al. () was employed, which incorporates van Genuchten (98) parameters in Bishop s single-parameter effective stress equation to relate effective stress, net normal stress, and matric suction (Equation ). S S r S u S ; S ; S S r, m / n ' () a e e S r n m r ( ( ) ) Where σ is the vertical effective stress, σ is the total stress, u a is the pore air pressure, S e is the effective degree of saturation, S is the degree of saturation, S r is the residual degree of saturation, ψ is the matric suction which correlates with degree of saturation using SWRC fitted equations, and α, n, and m are the fitting parameters. The 5 5 Ottawa Sand Bonny Silt 5 Se (%)

4 parameter m in the above equation may be expressed in terms of n. In the above equation, the term S e ψ is referred to as suction stress (Lu and Likos 6). Suction stress versus effective degree of saturation for the two soils is shown in Figure. Further, the modified effective stress formula was used to find dynamic soil properties (i.e. shear modulus (G) and damping (D)). Shear modulus depends on void ratio (for coarse-grained soils) and over-consolidation ratio (OCR) (for cohesive soils), as well as induced shear strain (Hardin & Black 966, Seed & Idriss 97, Iwasaki et al. 978, Stokoe et al. 995). Following equation was used herein to define small-strain shear modulus of Ottawa sand (Ghayoomi and McCartney ). The equation was shown to be valid for F-75 Ottawa sand with 45% relative density. '. 5 G max 9. 6 P m a () Where P a is the atmospheric pressure and σ m is the mean effective stress. G max and σ m are in MPa and kpa, respectively. The small strain shear modulus of fine material expressed by Sawangsuriya et al. (9) as a function of effective stress and void ratio was used to estimate G max of the Bonny silt in this study. ' n m Gmax Af ( e) (3) Where A and n are fitting parameters and f(e) is used to account for void ratio which is expressed as /( e ). The fitting parameters in Equation 3 were obtained using Khosravi and McCartney () experimental data for Bonny silt. Further, small strain damping (D s,min ) was estimated using Equation 3 (Menq 3). D s,min.55 C. u D.3 5 ' Pa.8 (4) Where C u is the soil coefficient of uniformity and D 5 is the median grain size. In order to estimate soil behavior under a strong ground motion, strain-dependent shear modulus and damping are required. For this purpose, shear modulus reduction formula proposed by Menq (3) was used in this study (Equation 5). G G a max r a.86. log ' P a.6 ' r. Cu Pa Where G is the strain-dependent shear modulus, γ is the shear strain, and γ r is the reference shear strain. It should be noted that after obtaining shear modulus reduction curve using Menq s equation, G/G max values were slightly changed at large strains (higher than.%) to keep the friction angle profile constant within the soil depth (Hashash et al. ). G/G max graphs of dry and unsaturated Bonny silt before and after the correction are shown in Figure. This alteration of G/G max curve may change the relative stiffness of soil layers. For example, as shown in Figure, G/G max for unsaturated soil is higher than that of dry soil before the correction while (5)

5 Acceleration (g) G/Gmax after the correction it becomes less for strain values greater than.3%. In addition, a curve will be fitted through the data points in the nonlinear analysis where G/G max might be further changed for strain values even smaller than.%..... Strain (%) Figure. Correction of G/Gmax graphs for dry and unsaturated Bonny silt Darendeli () adjusted damping equation by applying a scaling factor to Masing damping. This hyperbolic equation may be linked to the small-strain damping using Menq s formula (Equation 6).. Unsaturated, Raw Unsaturated, Corrected Dry, Raw Dry, Corrected G D b. D Ma sin g Ds,min G (6) max Where D is the damping, b is a scaling coefficient that depends on the number of cycles and typically is around.6, and D masing is the material damping determined from the Masing behavior. Filtered and baseline-corrected Northridge earthquake motion, 99, recorded at WPI station was selected as the input motion for this study. Peak Ground Acceleration (PGA) of the earthquake (.45g) was scaled to.g,.3g, and.6g to study the site response with different excitation amplitudes (Figure 3). Sand and silt profiles with five different uniform degrees of saturation, then, were excited numerically by these three scaled earthquake motions. 5% damped spectral acceleration and Arias Intensity time histories of the motions are shown in Figure PGA=.6g PGA=.3g PGA=.g Time (s) Figure 3. Acceleration time histories of scaled Northridge earthquake motion

6 S a (g) Arias Intensity (m/s) The numerical analysis was performed using both frequency-domain Equivalent Linear (EL) and time-domain Nonlinear (NL) approaches. EL analysis assumes shear strains for the soil layer and uses an iterative procedure to estimate the compatible secant properties of the soil and induced shear strains. NL analysis forms and solves a multi degree of freedom equation of motion in time domain. Further information on the EL and NL procedures in DeepSoil can be found in Hashash et al. (). The results associated with PGA b =.3g is presented and discussed in this paper T (s) Time (s) Figure 4. 5% damped spectral accelerations; Arias Intensity time histories of the applied base motions ANALYSIS Acceleration time histories were captured at the center of each layer. Then, they were used to evaluate the seismic site response. PGA and Arias intensity representing the intensity, 5% damped spectral acceleration showing the frequency content, and soil lateral deformations indicating the soil lateral stiffness were calculated at each layer. Intensity Amplification PGA at each depth is the maximum absolute acceleration of the motion. PGA varies in depth as the earthquake motion propagates through the soil layer. To eliminate the effect of different earthquake amplitudes, PGA amplification factor (F PGA ) was used herein. F PGA is the ratio of PGA of the motion at a certain depth to the PGA of the motion at the bedrock (F PGA =PGA m /PGA b ). F PGA profiles of dry and unsaturated soils under the base PGA of.3g are shown in Figure 5. This figure illustrates the results of both nonlinear and equivalent linear analyses. F PGA values in dry Ottawa sand is slightly less than those in partially saturated condition in both nonlinear and equivalent linear analyses. In Bonny silt, however, more inconsistency in amplification factor is observed in depth. For example, dry Bonny silt experiences higher amplification in shallower depth, but this trend changes in depth. The above mentioned trend was observed in other motions, with different intensities, with much severe irregularities. In addition, there is a substantial difference between the results of nonlinear analyses and those of equivalent linear. Equivalent linear analyses led to much higher amplification factors in both cases of Ottawa sand and Bonny silt. PGA=.6g PGA=.3g PGA=.g

7 I a (m/s) I a (m/s) Depth (m) Depth (m) S=, E.L. S=.9, E.L. S= N.L. S=.9, N.L S=, E.L. S=.8, E.L. S= N.L. S=.8, N.L..5.5 F PGA.5.5 Figure 5. PGA Amplification Factors (F PGA ) variation in depth for dry and unsaturated cases under the earthquake motion with PGA b =.3g for Ottawa sand and Bonny silt F PGA Arias Intensity (I a ) function (Arias 97) can be calculated using Equation 7: t I a ( t) a t dt g ( ) (7) Where I a (t) is the cumulative energy function, a(t) is the acceleration time history and g is the gravitational acceleration. Arias intensity time histories of sand and silt layers under the excitation PGA of.3g are shown in Figure 6 for both EL and NL analyses. The analyses results show higher surface Arias intensities for unsaturated Ottawa sand and dry Bonny silt. This trend is visible in most cases, however there are exceptions; e.g. Arias intensity time histories of Ottawa sand under PGA b =.g. Arias intensity and F PGA, approximately, show the same variation trend for the silt and sand layers. However, this difference is more pronounced in Arias Intensity time histories. The observations herein indicate that PGA is not always the best representative of the motion (Ghayoomi and Dashti 4). Further, the difference between NL and EL analyses is significant S=, E.L. S=.9, E.L. S= N.L. S=.9, N.L. 4 3 S=, E.L. S=.8, E.L. S= N.L. S=.8, N.L Time (s) Time (s) Figure 6. Arias Intensity time histories of surface motions in dry and unsaturated soils under the earthquake motion with PGA b =.3g for Ottawa sand and Bonny silt

8 Depth (m) Depth (m) S a (g) S a (g) Spectral Acceleration The 5% damped spectral acceleration of the surface motion under the PGA b of.3g is shown in Figure 7.The surface spectral accelerations calculated for Ottawa sand are very similar in all cases. Spectral accelerations of dry Bonny silt, however, is consistently higher than those of unsaturated silt. This observation shows a good agreement with the Arias intensity amplification trends..5 S=, E.L. S=.9, E.L. S= N.L. S=.9, N.L..5 S=, E.L. S=.8, E.L. S= N.L. S=.8, N.L T (s) Figure 7. 5% damped spectral acceleration of surface motions in dry and unsaturated soils under the earthquake motion with PGA b =.3g for Ottawa sand and Bonny silt Lateral Deformation T (s) Lateral deformation profile reflects the stiffness of the soil layer and the intensity of induced motion. The lateral deformation profiles of the target soils under PGA b =.3g are shown in Figure 8. Deformations are slightly higher in unsaturated Ottawa sand than in dry sand layer, while response analyses of Bonny silt led to opposite trend as dry silt layer experienced slightly higher deformation. These results show a good agreement with intensity amplification and spectral acceleration results. As a summary, higher intensity amplification in unsaturated sand or in dry silt had led to higher deformation of soil deposits S=, E.L. S=.9, E.L. S= N.L. S=.9, N.L..5.5(x - ) 4 6(x - ) Deformation (m) Deformation (m) Figure 8. Lateral deformation profiles of dry and unsaturated soil layers under the earthquake motion with PGA b =.3g for Ottawa sand and Bonny silt S=, E.L. S=.8, E.L. S= N.L. S=.8, N.L.

9 The above presented data clearly indicated the influence of degree of saturation on the seismic site response. The trends associated with this influence depend on the degree of saturation, induced motion, analysis approach, and most importantly the selection of dynamic material properties. For soils with low suction levels (such as Ottawa sand) intensity amplification, spectral acceleration, and deformations are higher for unsaturated soil. This trend is reverse in the soils with high suction level (Bonny silt). Two parameters affect the deformation level of soil layers, induced forces to the soil mass due to earthquake excitation and the stiffness of the soil. Both of these parameters may vary in partially saturated soil conditions. Theoretically, induced forces are higher in unsaturated soil because of the heavier mass getting accelerated by earthquake motion. On the other hand, stiffness increases due to the suction inter-particle forces and consequently higher soil modulus. Thus, deformation of a soil layer is a function of these two parameters. In soils with lower suction levels, the effect of increase in the soil mass is higher than that in stiffness of the soil. This leads to higher deformation in the unsaturated soil layers. However, for finer soils with higher suction levels, the effect of increase in stiffness is higher than the effect of increase in the soil mass which makes the deformations less for unsaturated soils. CONCLUSIONS The equivalent linear and nonlinear site response analyses were performed for dry and partially saturated sand and silt layers with uniform suction profiles. The intensity amplification, spectral acceleration, and lateral deformation, were monitored from the base to the soil surface. The results indicate that site response gets considerably affected by partial saturation condition. However the results are not completely consistent, but general trends may be observed. These trends of variation of motion characteristics in depth depend on the suction level and the induced motion. For soils with higher suction level (e.g. 7 kpa) lateral deformations in dry soils tend to be slightly higher than those in unsaturated soils. In addition, intensity amplification parameters are mostly higher in these soils for dry condition. In the case of soils with low suction level (e.g. kpa) the opposite trend is observed where unsaturated soils show slightly higher amplification and deformation than the dry soils. This could be attributed to the interplay of higher mass and higher stiffness in partially saturated soil and the extent of their influence. Equivalent linear analysis produced significantly higher amplification parameters in comparison with nonlinear analysis for both examined soils. The lateral deformations obtained from equivalent linear analyses are lower in Ottawa sand and higher in Bonny silt, respectively, comparing to the ones in nonlinear analyses. Several issues still need to be addressed to get a valid prediction of site response for unsaturated soils: () suction influence on the wave propagation mechanisms has to be taken into account. In this paper, suction was only modeled by changing the material properties; () the effect of densification could be considered in site response; (3) more accurate estimation of suction- and strain-dependent, dynamic soil properties should be implemented in fully coupled, nonlinear response analysis. An ongoing research is in progress by authors to address the challenges pointed out in this paper through more sophisticated Finite Element analysis and centrifuge physical modelling.

10 ACKNOWLEDGMENT The authors would acknowledge funding by the National Science Foundation through the NSF CMMI grant No REFERENCES Arias A. A measure of earthquake intensity. In: Seismic design for nuclear power plants. In: Hansen RJ, editor. Cambridge, MA. MIT Press; 97. D Onza, F., d Onofrio, A., and Mancuso, C. (8). Effects of Unsaturated Soil State on the Local Seismic Response of Soil Deposits. st European Conference on Unsaturated Soils: Advances in Geo-Engineering, Durham, United Kingdom, Ghayoomi, M., McCartney, J.S., and Ko, H.-Y. (). Centrifuge test for seismic compression of partially saturated sands ASTM Geotechnical Testing Journal 34(4).-. Ghayoomi, M. and McCartney, J.S. (). Measurement of small-strain shear moduli of partially saturated sand during infiltration in a geotechnical centrifuge., ASTM Geotechnical Testing Journal 34(5), pg. Ghayoomi, M. and Mirshekari, M. (4), Equivalent Linear Site Response Analysis of Partially Saturated Sand Layers, UNSAT4 conference, Sydney, Australia, -6. Hardin, B.O. and Black, W.L., Sand stiffness under various triaxial stresses. J Soil Mechanics and Foundations Division. 966;9(SM):7 4 Hashash, Y.M.A., Phillips, C., Groholski, D.R., Recent advances in nonlinear site response analysis, 5th International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, San Diego, California, May 4-9,, -. Hashash, Y. (). DEEPSOIL V5., Tutorial and User Manual, UIUC. Idriss, I. M., and J. I. Sun. (99). User's Manual for SHAKE9. Center for Geotechnical Modeling, Department of Civil and Env. Engineering, University of California, Davis. Iwasaki, T., Tatsuoka, F. & Yoshikazu, T. (978). Shear moduli of sands under cyclic torsional shear loading. Soils and Foundations, Vol. 8, No., Khosravi, A., Ghayoomi, M., and McCartney, J.S. () Impact of effective stress on the dynamic shear modulus of unsaturated sand. GeoFlorida, West Palm Beach, Florida, USA. Feb. -4. CD-ROM. Khosravi, A. and McCartney, J.S. () Suction-controlled resonant column device for unsaturatedsoils. ASTM Geotechnical Testing Journal 34(6). pg.in Press. Lee, K. W. and W. D. L. Finn (978). DESRA-: dynamic effective stress response analysis of soil deposit with energy transmitting boundary including assessment of liquefaction potential, Soil Mechanics Series, University of British Columbia, Vancouver, Canada Lu, N. and Likos, W.J. (6). Suction stress characteristic curve for unsaturatedsoil. Journal of Geotechnical and Geo-environmental Engineering, 3(), 3-4. Masing, G. (96) Eigenspannungen und Verfestgung Beim Masing, Proceedings, Second International Congress of Applied Mechanics, pp Matasovic, N. (993). Seismic Response of Composite Horizontally-Layered Soil Deposits. Ph.D. Thesis, University of California, Los Angeles, CA. Menq, F.-Y., (3). Dynamic Properties of Sandy and Gravelly Soils. PhD Dissertation, the University of Texas at Austin. Stokoe, K.H., III, Hwang, S.K., Lee, J.N.K, and Andrus, R.D. (995). Effects of various parameters on the stiffness and damping of soils at small to medium strains Proc. of Int. Symp. on Pre-Failure Deformation of Geomaterials Vol., Van Genuchten, M. 98. A closed form equation for predicting the hydraulic conductivity of unsaturated soils Soil Sci. Soc. Am. J., 58, Yang, J. and Sato, T. (). Effects of Pore-Water Saturation on Seismic Reflection and Transmission from a Boundary of Porous Soils., Bulletin of the Seismological Society of America, 9 (5), Yang, J. (6). Frequency-Dependent Amplification of Unsaturated Surface Soil Layer. ASCE Journal of Geotechnical and Geoenvironmental Engineering, 3(4),