Equivalent Linear Site Response Analysis of Partially Saturated Sand Layers

Transcription

1 Equivalent Linear Site Response Analysis of Partially Saturated Sand Layers M. Ghayoomi & M. Mirshekari University of New Hampshire, Durham, New Hampshire, USA ABSTRACT: Suction can change the dynamic material properties of soil including the small-strain and straindependent shear modulus and damping. These changes influence the seismic site response of unsaturated soil layers by affecting the wave propagation mechanisms, acceleration amplification and de-amplification, and seismically induced settlements. The higher shear modulus of unsaturated soils due to the presence of interparticle suction forces could result in a different site response. To study this behavior, a set of equivalent linear site response analysis were performed using DEEPSOIL program. A 3-m layer of sand with different constant suction profiles was modeled under Northridge earthquake motion with different scale factors. Bishop s single-parameter effective stress approach was implemented to incorporate suction into the effective stress equation. The site response was different in soils with various suction levels especially in shallow ground. INTRODUCTION Partially saturated soil may differ in seismic site response compared to dry or saturated soils. Suction in partially saturated soil increases the effective stress (Lu and Likos 6), which affects the dynamic material properties of soil such as shear modulus, shear wave velocity, damping, and unit weight. This effect would be significant in arid regions or in shallow ground. As a result, the degree of saturation (suction) is expected to influence the seismic wave propagation mechanism and site response. Recent studies have confirmed the effect of the degree of saturation on the seismic response of partially saturated soils (Yang & Sato, Yang 6, D Onza et al. 8, Ghayoomi & McCartney ). However, the current state of practice for seismic site response analysis relies on procedures that ignore the direct effect of the degree of saturation. Although the shear wave velocity is incorporated in current design guidelines its variation due to seasonal fluctuation of water table is not considered. In addition, current seismic design philosophy is based on the most conservative scenario in stress and deformation, which is predicted to be either dry or fully saturated soil. This is due to the presence of interparticle suction forces in partially saturated soil, which is believed to lead to a stiffer behavior. However, partially saturated soils with higher shear wave velocity may not be conservative with regards to site response analysis and surface-to-base motion intensity amplification. The objective of this paper is to pave the path for considering the effect of partial saturation on the site response analysis. A modified Bishop s effective stress equation was employed to define the soil dynamic properties. Then, the estimated values were implemented in an equivalent linear site response analysis using DEEPSOIL software. BACKGROUND. Site response analysis Local site conditions demonstrated a paramount effect on the seismic design of surface structures. Amplification or de-amplification of seismic waves originated from bedrock and propagated through the soil layer could produce severe damage to the buildings and infrastructure. Changes in the intensity and frequency content of the earthquake motion due to mechanical and environmental variations in the soil layer are considered site effects. The behavior of soil during seismic events, even at small strains, is nonlinear. Idriss & Sun (99) proposed an equivalent linear approach to estimate the material dynamic properties implemented in a program called SHAKE9. This approach is an iterative procedure to find the strain-dependent shear modulus and damping ratio for different soil sublayers. Alternatively, direct numerical integration in time or frequency domain analysis may be used with nonlinear soil properties. None of the available site response analysis programs consider the effect of

2 G max /F(e) (MPa) Volumetric water content (%) partial saturation and changes in suction due to seasonal variation on the shear wave velocity profile and indirectly on site response. D Onza et al. (8) predicted that neglecting the effect of partial saturation may lead to error in amplification estimates and change the natural frequency of the soil layer. Based on numerical results by Yang & Sato (), even a slight decrease of complete saturation may lead to an extensive influence on both reflected and transmitted waves at the layer interface.. Dynamic properties of partially saturated soils Small strain shear modulus (G max ) is typically calculated using the empirical equations as a function of void ratio (e), mean effective stress (σ m ), and over consolidation ratio (OCR) (Hardin & Black 966, Seed & Idriss 97, Iwasaki et al. 978). It is also directly related to the shear wave velocity (G max =ρv s ), which is the main factor in determining the site class in design provisions or in site response analysis. Different site response analysis approaches incorporate the shear modulus reduction curves (Ishibashi & Zhang 993, Darendeli & Stokoe ) to estimate the strain-dependent shear modulus and damping values based on the intensity of shaking. Thus, alteration of dynamic material properties due to degree of saturation variation in the soil profile affects the evaluation of site response. Most studies on the impact of partial saturation (suction) on the dynamic response of soils have been focused on the measurement of Gmax or V s in various degrees of saturation using bender elements or resonant column tests (Marinho et al., 5, Ghayoomi & McCartney, Mancuso et al., Mendoza et al. 5). The common trend among these studies has been the influence of suction increasing the shear modulus. However, Qian et al. (99) and Khosravi et al. () showed a peak modulus at middle-range degrees of saturation in sands. The normalized G max results from bender element tests (BE) inside a geotechnical centrifuge and resonant column tests (RC) on fine, uniform sand is shown in Figure RC(σ n =kpa) S r BE(σ n =78kPa) BE(σ n =4.5kPa) RC(σ n =kpa) Figure. Effect of saturation on G max of partially saturated sand (Ghayoomi & McCartney ). F(e): Void ratio function. Khosravi & McCartney () used a modified version of Bishop s effective stress equation to develop a unified effective stress-based equation to estimate G max applicable to all degrees of saturation. Ghayoomi et al. (3) showed how this approach may be combined with empirical methodologies for shear modulus reduction curves to define the straindependent material properties. Aside from shear modulus, Michaels (6), through viscoelastic analyses, predicted higher damping ratio for lower suction values under the same excitation frequency. 3 MATERIAL PROPERTIES F-75 Ottawa sand was selected for this study because it has relatively high saturated permeability, while still having fine enough soil particles to retain water suctions up to kpa (Ghayoomi et al. ). The geotechnical properties of the fine silica, uniformly graded sand used in this study are summarized in Table and the grain size distribution and obtained from the sieve analysis is available in Ghayoomi et al. (). The Soil-Water-Characteristics-Curve (SWRC) correlating suction and degree of saturation and the van Genuchten s (98) SWRC fitting parameters are shown in Figure. Table. Geotechnical properties of F-75 Ottawa sand Property Description Grain shape Rounded Specific gravity, G s.65 C u.7 D 5.7 mm e min, e max.49,.8 K saturated 6-4 cm/s ϕ 35º α =.4 N = 7 θ s =.395 θ r =.4 van Genuchten (98) SWRC model SWRC data for F-75 Ottawa sand Suction (kpa) Figure. SWRC curve of Ottawa F-75 Sand 4 MODEL PREPARATION AND ANALYSIS A 3-m layer of sand with a uniform relative density of 4% (e=.66) was modeled in DEEPSOIL software. The sand layer was divided into 3 sub-layers and the dynamic material properties were calculated

3 Suction Stress (kpa) at the center of each sub-layer. The thickness of layers increases in depth (from.5 to m) where the material properties become less sensitive to depth. Five different models with uniform suction (degree of saturation) profiles were prepared for 5 different degrees of saturations, S, including,.,.4,.6,.8, where S=.87 would have led to the peak suction stress. A Uniform saturation profile was chosen because it is simple, suitable for future physical modeling (Ghayoomi et al. ), and convenient to see the effect of degree. The fully saturated profile was not studied because of its different mechanisms in liquefied and non-liquefied cases. A uniform unit weight based on the soil water content (degree of saturation) was assigned to each model. Lu et al. () combined Bishop s (959) effective stress equation and van Genuchten s (98) SWRC model to correlate suction, degree of saturation, and the effective stress. This single-saturation parameter equation was implemented to estimate the vertical effective stress for different suction values as shown in Equation. S ( S u r ' () a S r where σ is the vertical effective stress, σ is the total stress, u a is the pore air pressure, S is the degree of saturation, S r is the residual degree of saturation, and ψ is the suction. The variation of the suction stress with the degree of saturation (i.e. the second term in Equation ) is shown in Figure 3. The maximum shear modulus, G max, was estimated using Ghayoomi & McCartney () equation, which was shown to be consistently valid for F-75 Ottawa sand with 4% relative density at various degrees of saturation, as shown in Equation. ' (. 5 G max 9. 6 P () m a where P a is the atmospheric pressure, σ m is the mean effective stress, and G max and σ m are in MPa and kpa, respectively. The shear modulus would increase consistently with increasing the degree of saturation despite the expected peak value. This shows a dominant contribution of the unit weight and total stress on the effective stress, and as a result, on the shear modulus. This is not necessarily in contrast with Figure that showed the effect of the degree of saturation on G max under the same confining stress. However, peak G max values were calculated in shallower depth with lower total stresses. However, the trend might be different with soils with finer particles that can retain very high suction values. The small-strain damping, D s,min, was calculated using the equation proposed by Menq (3) and shown in Equation D ',min.55 s Cu D5 (3) Pa where C u is the soil coefficient of uniformity and D 5 is the median grain size. Since strong ground motion records induce larger shear strains into the soil layer, understanding the strain-dependent shear modulus and damping ratio of the soil is essential in site response analysis Degree of Saturation, S Figure 3. Suction stress variation with the degree of saturation. Menq (3) proposed the following equations for shear modulus reduction and damping obtained from sets of torsional shear tests on different types of sands. G G a max r a.86. log ' P (4). C r.6 u. ( ' Pa a G D b. D Ma sin g Ds,min G (5) max where G and D are the strain-dependent shear modulus and damping, γ is the shear strain, γ r is the reference modulus, b is a scaling coefficient that depends on number of cycles and typically is around.6, and D masing the material damping determined from the Masing behavior. Equations -5 were consistently used for different degrees of saturation given that the effective stress was determined from Equation. Frequency domain, equivalent site response analysis was performed on the five models. Equation with a single-parameter effective stress formula was used to incorporate suction in the analysis. In addition, the shear modulus reduction and damping curves were modified slightly at large strain values in order to keep the friction angle and soil strength remain constant in depth (Hashash et al. ). Northridge earthquake motion, 99, recorded at WPI station was selected as the reference earthquake for this study because of its wide-band frequency content. The motion was filtered and baseline corrected to satisfy the shake table limitation for future experiments. In order to consider different earthquake intensities, the reference motion with Peak Ground Acceleration (PGA) of.45g was scaled to PGA b =.g,.3g and.6g, as shown in Figure 4. In addition, the 5% damped spectral acceleration and

4 Arias Intensity (m/s) S a (g) Depth (m) Acceleration (g) Depth (m) Arias Intensity time histories of the motions are shown in Figure 5. These three scaled base motions were applied to each of the five soil profiles and the results were analyzed separately. Then, the acceleration time histories of induced motions were recorded at the center of each layer throughout the depth PGA=.6g PGA=.3g PGA=.g Time (s) Figure 4. Acceleration time histories of three applied earthquake motion. (a) T (s) 3 PGA=.6g PGA=.3g PGA=.g Time (s) (b) Figure 5. (a) 5% damped spectral accelerations; (b) Arias Intensity time histories of the applied base motions 5 DATA ANALYSIS 5. Intensity amplification Peak Ground Acceleration (PGA) is the main intensity index parameter used to represent an earthquake motion. It reflects the maximum absolute acceleration value of the motion. The intensity and frequency content of the motion changes as it propagates from the bedrock to the soil surface. Accordingly, the PGA of the motion alters from the bedrock to the soil surface. PGA amplification factor (F PGA ) representing the ratio of the PGA of the motion in depth to the PGA of the bedrock motion is practically used in site response analysis, F PGA =PGA m /PGA b. The F PGA variations with depth for different degrees of saturation for PGA b of.,.3, and.6g are shown in Figure 6a, b, and c, respectively. (a) S= S=. S=.4 S=.6 S= F PGA F PGA.5.5 F PGA (b) (c) Figure 6. PGA Amplification Factor (F PGA ) profiles in depth for different degrees of saturation under the earthquake motion with (a) PGA b =.g; (b) PGA b =.3g, (c) PGA b =.6g. The F PGA in shallow ground under the low intensity earthquake, i.e. PGA b =.g, decreases as the degree of saturation changes from.8 to (Figure 6a). This is attributed to the higher shear modulus calculated in higher degrees of saturation that led to a stiffer response. Although the soil with higher water content is expected to result in higher inertial shear stress and shear strain the effect of the modulus seems to be more dominant in motion amplification. In contrary, the soil profile with S=. resulted in slightly lower F PGA, which might be due to higher induced inertial stress in wetter soil. Nevertheless, the impact of degree of saturation on F PGA in deeper ground is almost zero mostly due to minimal effects of suction and water content on soil properties. Further, the abovementioned trends became insignificant in medium intensity motions, i.e. PGA b =.3g, as shown in Figure 6b. This could be due to lower sensitivity of the large strain modulus and damping to suction, based on the current definitions. The irregular amplification pattern observed in Figure 6c indicates that the equivalent linear analysis is not accurately predicting the site response under high intensity shaking, i.e. PGA b =.6g. The Arias Intensity (I a ) time history showing the shaking energy buildup during an earthquake motion is defined as (Arias 97):

5 Surface/Base Transfer Function S a (g) I a (m/s) S a (g) t I a ( t) a t dt g ( ) (6) where a(t) is the acceleration time history and g is the gravitational acceleration. The Arias Intensity time histories of the top layer at the soil surface for different degrees of saturation follow a similar trend to the F PGA trend, as shown in Figure (a).5. T (s) S= S=. S=.4 S=.6 S= Figure 7. Arias Intensity time histories of surface motions in soil profiles with different degrees of saturation under the earthquake motion with (a) PGA b =.g. 5. Spectral accelerations and frequency content Previous studies revealed the importance of considering different characteristics of an earthquake motion, i.e. intensity, frequency content, and duration, in seismic analysis (Ghayoomi and Dashti 4). As a result, the site response was evaluated regarding the frequency content of the motion. The 5% damped spectral acceleration of the surface motion in soil profiles with different degrees of saturation is shown in Figure 8. The trend is similar to what was observed in Figures 6 and 7 in the dominant natural period (inverse of fundamental frequency), while the differences are not significant in other periods. The surface to base acceleration transfer functions of the soil profiles (i.e. the Fast Fourier Transform ratio of the surface to base motion) are shown in Figure 9. The fundamental natural frequency decreased by increasing the degree of saturation indicating a softer lateral response. This is in contrast with previous figures showing higher amplification and stiffer response due to higher modulus. It might be attributed to higher inertial stress in wetter soil that resulted in lower strain-dependent modulus and softer response. In addition, assuming constant shear strength in soil profiles with different degrees of saturation might have caused this inconsistency. 5.3 Soil Lateral Deformation Soil lateral deformation is directly related to the lateral stiffness and strength. The lateral deformation profiles for different degrees of saturation are shown in Figure. These profiles follow a similar trend to the one in transfer function (Figure 9), indicating a softer response in higher degrees of saturation..4 (b) Figure 8. (a) 5% damped spectral acceleration of surface motions in soil profiles with different degrees of saturation under the earthquake motion with PGA b =.g; (b) Enlarged figure T (s).5 S= S=. S=.4 S=.6 S= Frequency (Hz) Figure 9. Acceleration transfer function in soil layers with different saturation under the earthquake motion with PGA b =.g. 6 CONCLUSION The results of equivalent linear site response analysis indicated that partial saturation and suction have noticeable effects on dynamic material properties and site response. These effects are mainly in shallow soil layers where the suction contributes the most to the effective stress. In addition, the suction stress effects are more evident in the soil layers with very fine particles in which higher suction values can be achieved. The motion intensity amplification (For example in PGA, S a, and Arias Intensity) from the bedrock to the soil surface was greater in higher degrees of saturation. This showed the dominant effect of total stress on material properties. Further, the natural frequency of the soil layers and lateral deformation profiles indicated softer response in higher degrees of saturation. Following limitations in this simplified equivalent linear analysis might have led to the inconsistencies in the observed trends: () the modified suction-dependent effective stress was only

6 Depth (m) considered indirectly in material properties and was not incorporated directly into the solution; () the changes in water content and suction due to the densification was not considered; (3) nonlinear material properties were simplified through the equivalent linear approach and caused irregular results in higher intensity motions; (4) the suction effect was not considered in estimating the shear strength and friction angle of the soil; and (5) low suctions were achieved in the tested sand. As a result, a more sophisticated nonlinear solution and more experimental results are needed to better understand the saturation effects on site response. As an ongoing project, authors are using Finite Element Methods and centrifuge physical modeling to address this problem and the results will be presented in future publications. S= S=. S=.4 S=.6 S= Maximum Deformation (cm) Figure. Lateral deformation profiles with different degrees of saturation under the earthquake motion with PGA b =.g. 7 ACKNOWLEDGEMENT The authors would acknowledge funding by the National Science Foundation through the NSF CMMI grant No REFERENCES Arias, A. 97. A measure of earthquake intensity. Seismic design for nuclear power plants. Hansen RJ, editor. Cambridge, MA. MIT Press; 97. Darendeli, M.B. & Stokoe, K.H.. Development of a new family of normalized modulus reduction and material damping curves, Univ. of Texas, Geotech. Eng. Report GD-. D Onza, F., d Onofrio, A., & Mancuso, C. 8. Effects of unsaturated soil state on the local seismic response of soil deposits. st European Conf. on Unsaturated Soils: Advances in Geo-Engineering, Durham, United Kingdom, Ghayoomi, M., McCartney, J.S., & Ko, H.-Y.. Centrifuge test for seismic compression of partially saturated sands. ASTM Geotechnical Testing Journal. 34(4). -. Ghayoomi, M. & McCartney, J.S.. 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