Seismic codes based equivalent nonlinear and stochastic soil structure interaction analysis

The nonlinear stress-strain behaviour of a soil can be expressed in two ways, with an increase in the shear strain amplitude: (i) degradation of shear modulus G and (ii) an increase in the damping coefficient ξ.

The soil nonlinearity can be taken into account by many models.^{[20,21,22,23,24,25]} In the equivalent linear method, the nonlinear stress-strain behaviour of a soil is approximated by a secant stiffness G_sec and an equivalent damping ξ_eq that are compatible with the strain in the soil induced by the ground shaking. Okada et al.^[26] studied the seismic pile response of a structure-pile-soil system to evaluate the strain-dependent nonlinearity of the ground using an equivalent linearization method.

In the present study, the shear modulus and damping coefficient are modified using a linearized iterative method based on design response spectra (Sa) for four seismic codes (RPA99-2003,^[17] EC8-2004,[5] IBC-2015[18] and IS-1893-2002^[19]). All the steps of the method are summarized in Appendix 1 according to Chowdhury and Dasgupta.^[27]

The soil properties can be delineated using deterministic or probabilistic model. They greatly depend on the soil deposition conditions and loading history, and may be derived from a common set of in situ or laboratory test data.^[28,29] In other words, soil is nonhomogeneous and its formation is an outcome of a variety of random geological processes,^[30] and consequently, the stochasticity of soil properties remains largely unknown. Several methods are used to treat stochastic problems where Monte-Carlo simulation (MCS)^[31,10] provide a robust conceptually simple way to account rationally for various uncertainties.^[32,33,34] The most used probability distribution function to estimate uncertainty in geotechnical properties for performance based earthquake engineering are uniform, normal, lognormal, gamma and exponential.^[35] The lognormal distribution is commonly used to model certain types of data that appear in several fields of engineering. Since this includes most, if not all, engineering (mechanical) systems, particular properties of the lognormal random variable (such as non-negativeness and skewness) and of the lognormal hazard function (which increases initially and then decreases) make lognormal distribution a suitable fit for some engineering data sets. However, the lognormal distribution can have widespread application; a generalized lognormal distribution can be used to provide better fits for many types of experimental or observational data.^[36]

On other hand, structural engineering design is replete with uncertainties, some of which are obvious and some of which many engineers may never have considered as stated by Bulleit.^[37] Otherwise, most shallow foundations supporting structures are generally not circular in shape. So equivalent circular foundations, where foundation radii are computed separately for translational (r_a) and rotational (r_m) deformation modes,^[38] are used.

In addition, the effective height of the structure (h) shall be taken as 0.7 times the structure height for multi-story structures, although in reality, there are irregular structures in plan and/or elevation (Fig. 1a). Thus, the equivalent foundation radius and the effective height of the structure may be assumed subject to uncertainties and may be considered as random variables.^[11]

Figure 1

In this study, the effects of soil nonlinearity by correction of the shear modulus and the damping ratio together with soil stochasticity are considered. Several thousands of random draws are generated for soil according to the lognormal distribution with a coefficient of variation (COV) for G taken as 20% according to the literature.^{[39,40,41,42]}

The lognormal distribution is written as follows: (1-a)f(x)={1x.2.π.σln.xexp(−12.σln.x2(logx−μln.x)2),for.0.〈x〈∞0otherwise\matrix{ {f\left( x \right) = \left\{ {{1 \over {x.\sqrt {2.\pi .{\sigma _{\ln .x}}} }}\exp \left( { - {1 \over {2.{\sigma _{\ln .x}}^2}}{{\left( {\log x - {\mu _{\ln .x}}} \right)}^2}} \right),} \right.} & {for.0.\left\langle x \right.\left\langle \infty \right.} \cr 0 & {otherwise} \cr } Mean and variance are given as: (1-b)E(X)=exp(μln.x+0.5.σln.x2),V(X)=exp(2μln.x+2.σln.x2)−exp(2μln.x+σln.x2)\matrix{ {E(X) = \exp \left( {{\mu _{\ln .x}} + 0.5.{\sigma _{\ln .x}}^2} \right),} \cr {V(X) = \exp \left( {2{\mu _{\ln .x}} + 2.{\sigma _{\ln .x}}^2} \right) - \exp \left( {2{\mu _{\ln .x}} + {\sigma _{\ln .x}}^2} \right)}} Where, μ_ln,x and σ_ln,x are the lognormal mean value and standard deviation, respectively.

The impedance functions for horizontal and rotational motions are determined for a circular foundation of radius r₀ on the surface of a soil layer resting on a half-space.

The results in terms of impedance functions (written in terms of real and imaginary parts versus the dimensionless frequency) for horizontal and rotational vibration modes are shown in Fig. 3. This figure shows the influence of the correction of shear modulus and damping coefficient on the real and imaginary part of impedance functions (spring and damping coefficients). It appears from this figure that the correction of the shear modulus and the damping coefficient due to the imposed seismic motion is very remarkable. Both real and imaginary parts of impedance functions changing as G and ξ values are corrected.

Figure 3

The change is apparent for all the frequencies from 0 to 8. This result is consistent with the results of other researchers like those of Lutes et al.,^[39] when they used a non-classical modal-analysis-based formulation to quantify the variation of stochastic response in an SSI system with both uncertain soil properties in the soil–foundation system and uncertain structural properties in the superstructure on structural response for seismically excited soil-structure interacting (SSI) systems, those of Mittal et al.,^[54] when carrying out a parametric study on the effect of seismic zone on structural response of tall chimney by incorporating strain dependent shear modulus and determining the effect of variability in soil type and those of Zafarkhan and Dehkordi^[55] while presenting the effect of variability in soil type (different shear modulus) and structure height on soil-structure system responses.

The displacements of the structure u (ω) and the mass with respect to the free field (u₀ (ω) + hθ₀ (ω) + u (ω)) and the total displacement of the base (u_g (ω) + u₀ (ω)) are plotted versus the excitation frequency normalized by that of the structure ω/ω_s in Fig 4. The displacements before and after the correction of G and ξ are compared.

Figure 4

The displacement relative to the free field of the structure shows a decrease of about 33%, while the mass displacement and the total displacement of the base increase in absolute values by about 4% and 6%, respectively. But on average, all the displacements show the same dimensionless frequency shift of 18%. Fattah et al.^[56] studied the effect of sand density and whether the sand is dry or saturated on its dynamic response for the foundation-soil system. They found that for dry and saturated conditions, the maximum amplitude of displacement decreases with increasing the relative density of sand and contact area of footing and increase with increasing the amplitude of loading. The maximum displacement amplitude response of the foundation resting on dry sand models is more than that on the saturated sand by about 5.0–10%.

This variation explains that the structure becomes more rigid and the soil more flexible and shows the interest of taking into account the nonlinear soil behaviour in SSI analysis. The figure clearly shows that SSI alters the frequency content of the structure response. Also, the figure reported herein shows that SSI is more significant when soil nonlinearity case will take place during earthquake compared to the linear case (G and ξ are constants), where the nonlinearity reduces the stiffness of the soil (reduce shear modulus) and increases the hysteretic damping. Similar observation has been made by Li et al.^[57] when studying the effect of different soil properties on structure response, and Farghaly and Ahmed^[2] in the case of seismic loading. Fattah et al.[58] found that the displacement amplitude of vibration response of foundation system placed on the surface of dry dense sand models is less than that of dry loose sand models; according to the authors, these results are attributed to the increase in the stiffness and the modulus of elasticity of dense sandy soil that makes the soil stiffer and resist vibrations. In 2017, Fattah et al.^[59] found that the amplitude of displacement of a single pile embedded in saturated sandy soil due to vertical vibration is more than that of the dry soil, and Fattah et al.^[60] explained that this trend is due to the increase in the pore water pressure during dynamic load that causes reduction in the inter-particle forces between the solid particles of the soil skeleton, hence causing an increase in displacement response.

In this subsection, the stochastic and the nonlinear effects are considered by means of uncertainty of G and correction of G and ξ, respectively, on the response of the SSI system. A lognormal distribution is used to generate random numbers of shear modulus using Monte Carlo Simulations for coefficient of variation of 20%.

Fig. 5 displays the stochastic as well as the nonlinear displacements of the structure, the mass and the base due to the random variations of shear modulus and the correction of shear modulus and damping, respectively. Fig. 5 shows that the random variations of G greatly affect the response of the structure. It is observed that 20% COV of G reduces the peak structure displacement (16.6%) by half that is reduced by the correction of G and ξ (33.1%).

Figure 5

Additionally, the frequency is shifted about 18% due to the correction of G and ξ, which means that the soil becomes more rigid. However, Fig. 5b indicates that the peak value of mass displacement is reduced by 17.4% due to 20% COV of G compared to the nonlinear as well as to the deterministic ones (initial G). For the displacement of the base (Fig. 5c), for random variations of the shear modulus (COV = 20%), the displacement remains more or less unchanged compared to the initial case but it reduces by 10.4% due to nonlinear effects. Table 1 summarizes the peak SSI responses and the error margin between stochastic and deterministic response and between nonlinear and deterministic (initial G) response.

It can be seen from Table 1 that, for corrected G and ξ, the frequency ratio (ω/ω_s) goes on decreasing about 18% or time period goes on increasing for the three displacements (Figs. 5a to 5c). This trend means that soil becomes more flexible when shear degradation and damping variation are taken into account. Also, when the shear modulus goes on decreasing and damping coefficient increasing the displacements of structure, masse and the base reduce by 33%, 3% and 10%, respectively, compared to SSI responses for deterministic (initial G and ξ).

Usually, the seismic calculation of the structure is performed using the response spectrum with a well-fixed damping ratio. However, in reality, there is not only the soil damping but there is also the foundation damping that includes both effects of energy loss from waves propagating away from a vibrating foundation (radiation damping) and hysteretic action in supporting soil (material damping).

Accordingly, the effects of the foundation damping of flexible base on the design spectrum (Fig. 6) of four design codes (RPA99-2003,^[17] EC8-2004,[5] IBC-2015[18] and IS-1893-2002^[19]) are studied. Furthermore, to further enrich the information, the base shear force of structure is also studied (Fig. 7). The design spectrum and the base shear values are evaluated according to the four cases for each code: (i) fixed base, (ii) SSI effects (initial G and x), (iii) nonlinear SSI (corrected G and x), and (iv) stochastic SSI (random G with COV = 20%). Table 2 in Appendix III summaries the ordinates of design spectra and shear base force for the four codes.

The total soil-structure system damping ratio is calculated as follows:^[61](4)ξ0=ξf+ξ(T¯/T)eff2{\xi _0} = {\xi _f} + {\xi \over {{{\left( {{{\bar T} \mathord{\left/{\vphantom {{\bar T} T}} \right.} T}} \right)}_{eff}}^2}} The flexible-base damping ξ₀ is contributed from both the structural viscous damping ξ and the foundation damping ξ_f consisting generally of radiation and material damping components. T is the fixed base period and T¯\bar T the period of the SSI system. The damping of the flexible base of foundation is a combination of soil damping and radiation such as: (5)ξf=[(T¯/T)2−1(T¯/T)2]ξs+ξrd{\xi _f} = \left[ {{{{{\left( {{{\bar T} \mathord{\left/{\vphantom {{\bar T} T}} \right.} T}} \right)}^2} - 1} \over {{{\left( {{{\bar T} \mathord{\left/{\vphantom {{\bar T} T}} \right.} T}} \right)}^2}}}} \right]{\xi _s} + {\xi _{rd}} The radiation damping ξ_rd of the soil-structure system is calculated based on a combination of translational damping ξ_y and rotational damping ξ_xx: (6)ξrd=(1(T¯/Ty)2)ξy+(1(T¯/Txx)2)ξxx{\xi _{rd}} = \left( {{1 \over {{{\left( {{{\bar T} \mathord{\left/{\vphantom {{\bar T} {{T_y}}}} \right.} {{T_y}}}} \right)}^2}}}} \right){\xi _y} + \left( {{1 \over {{{\left( {{{\bar T} \mathord{\left/{\vphantom {{\bar T} {{T_{xx}}}}} \right.} {{T_{xx}}}}} \right)}^2}}}} \right){\xi _{xx}}

Figs. 6 and 7 show the reduction in design spectrum and base shear, respectively, due to SSI using Eq. (4) for the three cases (ii, iii and iv) mentioned above, where they are compared to the fixed base case for the four codes.

Figure 6

Figure 7

It appears from Figs. 6a–6d (red line) that when including SSI effects with the foundation damping (flexible base case) of the system, a significant decrease appears in design spectral values. This decrease is of 9%, 10%, 6% and 2% for RPA99-2003,^[17] EC8-2004,[5] IBC-2015[18] and IS-2002^[19] seismic codes, respectively, compared to the fixed base (black line).

Besides, Figs. 6a–6d (green line) indicate that the same structure responds more close to the previous case when the shear modulus is taken as random variable and only the IBC-2015^[18] code shows a remarkable decrease. The design spectral values decrease, compared to the fixed base (black line) for the four codes, by 7%, 9%, 13% and 5%, respectively.

In the last instance, Figs. 6a–6d (blue line) show the effects of the correction of both shear modulus and damping coefficient of the soil on the design response spectra where the correction effect clearly appears by a huge decrease observed in the responses compared to the fixed base and especially for the three codes EC8-2004,^[5] IBC-2015^[18] and IS-1893.[19] The observed reductions in the response spectra for the four codes are 18%, 36%, 37% and 43%, respectively.

On the other hand, Fig. 7 shows the effect of SSI on the base shear for the four codes for the three cases included: SSI effects (initial G and x) in red line, nonlinear SSI (corrected G and x) in blue line, and stochastic SSI (random G with COV = 20%) in green line compared to the fixed base in black line. It is observed that for SSI analysis including flexible base damping, the base shear forces of the structure are noticeably decreased by 9%, 3%, 5% and 4.5%, respectively. In the stochastic case, the results are very close to the previous case results, but only the IBC-2015^[18] code shows remarkable decrease, as 7%, 3%, 11% and 7.5%, respectively, compared to the fixed base for the four codes.

Lastly, the effect of the correction of G and ξ of soil on the base shear force for all codes’ cases, where remarkable decreases of 18%, 9%, 47% and 46%, respectively, are displayed. For the two last codes, results are similar to those obtained by Mittal and Gajinkar^[62] when studying the response of chimney by incorporating strain dependent shear modulus, where they find that the design shear force reduces between 31% to 47% when the soil become more flexible. Whereas Worku,^[63] when studying the importance of inertial SSI on the design spectral and on the shear base for EC8^[5] and NEHRP-2003,^[61] found a reduction between 7% to 39% for two different flexible soil types and four different varying structure heights.

On other hand, Jayalekshmi and Chinmayi^[64] showed that the design base shear forces obtained as per conventional design practice are higher compared to SSI values and decrease with an increase in foundation stiffness, based on a comparative study on seismic provisions of IS-1893^[19] and IBC-2006^[65] codes for a multi-storey reinforced concrete framed buildings. Nevertheless, it is important to note that some of the SSI codes cap the maximum allowed base shear reduction to 30%.