Effect of Pore Pressure on Seismic Soil Response: 1D-3 Components Numerical Modelling

During strong quakes, the propagation of seismic waves in soil layers involves nonlinearities changing with the excitation level. Furthermore, the influence of the pore pressure cannot be neglected for saturated soils. Starting from a FEM formulation describing 1D propagation and three-dimensional loading ("1D3 components approach"), the influence of the fluid is accounted for through a relation between the pore pressure and the work of the three-dimensional stress state initially proposed by Iai. It has been validated through comparisons to laboratory tests. Threecomponents synthetic motions have also been considered and the results are satisfactory both in terms of accelerations as well as pore pressure build-up. Ongoing researches investigate the seismic response of realistic soil columns during 4 actual quakes (Superstition Hills, 1987, USA, Mw=6.7; Tohoku, 2011, Japan, Mw=9.1; Kushiro, 1993, Japan, Mw=7.8; Emilia, 2012, Italy, Mw=5.9). Introduction The soil response to strong seismic excitations should account for the time variations of the stresses and strains. Various one-dimensional models have been proposed in previous researches (Joyner and Chen, 1975; Pyke, 1979; Bonilla, 2000; Hartzell et al., 2004; Phillips and Hashash, 2009; Segalman and Starr, 2008; Delépine et al., 2009). The rheological description often raises the need for tens of parameters difficult to estimate in large sedimentary basins. The nonlinear constitutive law proposed in this paper (Santisi et al., 2012) is based on the definition of several plasticity surfaces involving a limited number of parameters that could be estimated from simple laboratory tests (shear modulus reduction for dry soils, determination of a “liquefaction front” for saturated soils). In particular, the role of water is accounted for through a “liquefaction front” approach initially proposed by Towhata & Ishihara (1985) and Iai et al. (1990a,b). This model is based on the identification of an empirical relationship between the pore pressure and the work of the shear stress derived from laboratory tests (Bonilla, 2000). Dr, Dept GERS, IFSTTAR, 14-20 bd Newton, Champs sur Marne, France, viet-anh.pham@ifsttar.fr Dr, Dept GERS, IFSTTAR, 14-20 bd Newton, Champs sur Marne, France, fabian.bonilla@ifsttar.fr Dr, Dept GERS, IFSTTAR, 14-20 bd Newton, Champs sur Marne, France, luca.lenti@ifsttar.fr Dr, Dept GERS, IFSTTAR, 14-20 bd Newton, Champs sur Marne, France, jean-francois.semblat@ifsttar.fr This constitutive law for saturated soils has been implemented in a finite element code in order to estimate the seismic response of a 1D soil profile accounting for the simultaneous propagation of the three components of the seismic motion. The influence of the pore pressure will thus be analysed through the comparison of various examples. Modeling 3C Seismic Wave Propagation The three motion components are propagated vertically in a multilayered soil profile (Figure 1). The soil profile includes several homogeneous horizontal layers (xy plane) with various thicknesses. The layered soil is discretized through quadratic 3 noded elements. Since the full 3C polarization is considered, both pressure and shear waves are propagated along the z direction. Figure 1. Spatial discretization of the horizontally multilayered soil. The weak formulation of the equation of motion for a soil column considering a nonlinear constitutive law and a spatial discretization may be written as follows: [M]�D̈� + [C]�Ḋ� + {Fint} = {F} (1) In Equation (1), [MM] is the mass matrix; �Ḋ� and �D̈� are the first and second time derivatives of the displacement vector {DD} respectively; {FFiiiiii} is the vector of internal forces and {F} is the vector of external loadings; [CC] being a matrix depending on the boundary conditions. The system of horizontal soil layers is bounded at top (z=H) by a free surface and at the bottom by the bedrock with elastic boundary conditions considering a prescribed displacement ub. The equation of motion may be more conveniently expressed considering the relative displacements {XX} with respect to the base of the soil column: {X} = {D} − {I}ub (2) Equation (1) then becomes: [M]�Ẍ� + [K]{X} = −[M]{I}üb (3) The vector {II} is a unit column matrix and [KK] is the tangent stiffness matrix. The nonlinear behaviour leads to a time-varying matrix [KK]. We consider Newmark algorithm to solve Equation (2) in the time domain (displacements increments are denoted {∆XX}). This method is detailed in Santisi et al. (2012). Proposed Constitutive Law To model three-components wave propagation in layered soils, we need a 3D constitutive law. In this work, the proposed approch combines two different models: a nonlinear constitutive law for the soil with a three-dimensional stress state (MPii model, Iwan (1967), Segalman and Starr (2008)) and a model accounting for the dependency of the pore pressure on the work of the shear stress proposed by Iai et al. (1990a,b). Iwan’s model is first considered to compute the total stresses. Then, through Iai’s model, the total stresses are corrected to estimate the effective stresses by evaluating the shear work on a given soil volume at each time step and using the empirical relations making the link with the pore pressure. Constitutive Law for Dry Soil The MPii model (Iwan, 1967; Joyner et Chen, 1975; Joyner, 1975; Santisi et al, 2012; Segalman and Starr, 2008) accounts for the nonlinear hysteretic behaviour of the soil considering a hardening elastic-plastic approach through a series of plasticity surfaces. For a one-dimensional model, one considers a series of rheological cells combining a linear spring and a friction unit (Figure 2). The friction unit i remains locked until the stress reaches the value Yi. The spring constants, Gi, are chosen in order to recover the stress-strain behaviour observed in the laboratory. Figure 2. 1D rheological model originally proposed by Iwan (1967). For a three-dimensional problem, Iwan (1967) extended the classical theory of incremental plasticity (Fung, 1965). Instead of a single plasticity surface, we consider a family of plasticity surfaces. The 3D stress-strain “MPii” relation may then be expressed as (Santisi et al., 2012):