Performance evaluation of a shallow foundation built on structured clays under seismic loading

Abstract

Due to the increased need of storage, larger and higher structures are being built all over the world, thus requiring a more careful evaluation of the mechanical performance of their foundation deposits both in terms of bearing capacity and compressibility behaviour. The design of such structures and their serviceability and stability is largely governed by the effects of the dynamic loading conditions principally because of their significantly elevated risk in seismic prone zones. In this paper, numerical analyses using an advanced constitutive model, able to account for the initial soil structure and its progressive degradation, have been performed to investigate the seismic response of a silo foundation built on structured clays. The proposed analyses involve the use of a fully-coupled finite element approach. For the dynamic simulations, three different input motions have been selected form earthquake databases according to the seismic hazard study of the specific site. The results of the silo dynamic response are illustrated in terms of signal amplification, permanent excess pore water pressures, accumulated displacements and structure induced degradation during and after the seismic loading. The dynamic behaviour of the footing indicates that extreme earthquake events can induce large destructuration in natural clays, leading to ground settlements up to twice the observed ones under static loads, which need to be properly accounted for in the design. This suggests that there are significant advantages in using advanced models which recognise the existence of initial soil structure and its subsequent damage due to the applied dynamic loads.

Appendices

Appendix 1: Numerical formulation of the finite element code

In the context of FE analysis, assuming that the relative velocity of the fluid phase is negligible, the system of ordinary equation that results from the $u-p$ formulation can be written as follows (Biot 1941; Zienkiewicz et al. 1999):

$$\begin{aligned} \left\{ {\begin{array}{l} {[M]}\ddot{\mathbf{u}}+[C]\dot{\mathbf{u}}+[K]\mathbf{u}-[Q]\mathbf{p}=\mathbf{f}^{s} \\ {[Q]}^{T}\dot{\mathbf{u}}+[S]\dot{\mathbf{p}}+[H]\mathbf{p}=\mathbf{f}^{p} \\ \end{array}} \right. \end{aligned}$$

(1)

where u is the solid phase displacement vector and p is the pore fluid pressure vector, [$M$] is the mass matrix, [$K$] is the stiffness matrix, [$C$] is the viscous damping matrix, [$Q$] is the coupling matrix between the motion and flow equations, [$H$] is the permeability matrix, [$S$] is the compressibility matrix, $\mathbf{f}^{p}$ is the force vector for the fluid phase and $\mathbf{f}^{s}$ is the force vector for the solid phase. Frequency dependent viscous damping is included via the Rayleigh damping matrix (Clough and Penzien 1993):

$$\begin{aligned}{}[C]=\alpha [M]+\beta [K] \end{aligned}$$

(2)

where the factors $\alpha$ and $\beta$ are related to the modal damping coefficients according to the relationship:

$$\begin{aligned} \left\{ {{\begin{array}{l} \alpha \\ \beta \\ \end{array} }} \right\} =\frac{2D}{\omega _m +\omega _n }\left\{ {{\begin{array}{l} {\omega _m \omega _n } \\ 1 \\ \end{array} }} \right\} \end{aligned}$$

(3)

These coefficients can be calculated by selecting one value of damping ratio $D$ and two frequencies, $\omega _{m}$ and $\omega _{n}$, outside which damping is larger than the selected value.

The algebraic counterparts of Eq. (1) are obtained by applying a time-integration scheme. Assuming that the values of displacements, pore pressures and their time derivatives $\left\{ {\mathbf{u}_n ,\dot{\mathbf{u}}_n ,\ddot{\mathbf{u}}_n ,\mathbf{p}_n ,\dot{\mathbf{p}}_n } \right\}$ have been obtained at time $t_{n}$, the integration consists of updating $\left\{ {\mathbf{u}_{n+1} ,\dot{\mathbf{u}}_{n+1} ,\ddot{\mathbf{u}}_{n+1} , \mathbf{p}_{n+1} ,\dot{\mathbf{p}}_{n+1} } \right\}$ at the next time step $t_{n+1}$ according to the Generalised Newmark scheme (Katona and Zienkiewicz 1985). In particular, for the solid phase:

$$\begin{aligned} \begin{array}{l} \ddot{\mathbf{u}}_{n+1} =\ddot{\mathbf{u}}_n +\Delta \ddot{\mathbf{u}}_n \\ \dot{\mathbf{u}}_{n+1} =\dot{\mathbf{u}}_n +\left[ {\ddot{\mathbf{u}}_n +\beta _1 \Delta \ddot{\mathbf{u}}_n } \right] \Delta t \\ \mathbf{u}_{n+1} =\mathbf{u}_n +\dot{\mathbf{u}}_n \Delta t+0.5\left[ {\ddot{\mathbf{u}}_n +\beta _2 \Delta \ddot{\mathbf{u}}_n } \right] \Delta t^{2} \\ \end{array} \end{aligned}$$

(4)

Similarly for the fluid phase:

$$\begin{aligned} \dot{\mathbf{p}}_{n+1}&= \dot{\mathbf{p}}_n +\Delta \dot{\mathbf{p}}_n \nonumber \\ \mathbf{p}_{n+1}&= \mathbf{p}_n +\left[ {\dot{\mathbf{p}}_n +\beta _1 ^{*}\Delta \dot{\mathbf{p}}_n } \right] \Delta t \end{aligned}$$

(5)

where the coefficients:

$$\begin{aligned} {\begin{array}{l} {\beta _1 \ge 0.5} \\ {\beta _2 \ge 0.5\left( {0.5+\beta _1 } \right) ^{2} } \\ {\beta _1 ^{*}\ge 0.5} \\ \end{array} } \end{aligned}$$

(6)

are typically chosen for unconditional stability of the recurrence scheme (Zienkiewicz et al. 1999). The substitution of the above approximations into Eq. (1) leads to a system of coupled nonlinear equations which are solved iteratively by the FE code using the Newton-Raphson procedure.

Appendix 2: Constitutive model formulation

Figure 18 shows the three characteristic surfaces of the RMW model in the $p:q$ plane and its formulation is summarised in the following. The expression of the reference surface is:

$$\begin{aligned} f_r =\frac{3}{2M_\theta ^2 }{{\varvec{s}}}:{{\varvec{s}}}+\left( {p-p_c } \right) ^{2}-\left( {p_c } \right) ^{2}=0 \end{aligned}$$

(7)

Fig. 18

Characteristic surfaces of RMW model in $p : q$ plane

The bubble surface is written as:

$$\begin{aligned} f_b =\frac{3}{2M_\theta ^2 }\left( {{{\varvec{s}}}-{{\varvec{s}}}_{\bar{{\alpha }}} } \right) :\left( {{{\varvec{s}}}-{{\varvec{s}}}_{\bar{{\alpha }}} } \right) +\left( {p-p_{\bar{{\alpha }}} } \right) ^{2}-\left( {Rp_c } \right) ^{2}=0 \end{aligned}$$

(8)

The structure surface is given by:

$$\begin{aligned} F=\frac{3}{2M_\theta ^2 }\left[ {{{\varvec{s}}}-\left( {r-1} \right) \varvec{\eta }_0 p_c } \right] :\left[ {{{\varvec{s}}}-\left( {r-1} \right) \varvec{\eta }_0 p_c } \right] +\left( {p-rp_c } \right) ^{2}-\left( {rp_c } \right) ^{2}=0 \end{aligned}$$

(9)

where $p_{c}$ is the effective stress which defines the size of the reference surface, $R$ is the size of the bubble, $M_{\theta }$ is a dimensionless scaling function for deviatoric variation of the critical state stress ratio, $\varvec{\eta }_0$ a deviatoric tensor controlling the structure, $r$ is the ratio of the sizes of the structure and the reference surfaces, $p$ and ${\varvec{s}}$ are the mean pressure and deviatoric stress tensor and the symbol ‘:’ indicates a summation of products. Since the model describes the response of the soil skeleton, all stresses are effective stresses (the primes have been dropped for simplicity). The dots over symbols indicate an infinitesimal increment of the corresponding quantity, whereas bold-face symbols indicate tensors.

The scalar variable $r$, which is a monotonically decreasing function of both plastic volumetric and shear strain, represents the progressive degradation of the material as follows:

$$\begin{aligned} \dot{r}=-\frac{k}{\left( {\lambda ^{*}-\kappa ^{*}} \right) }\left( {r-1} \right) \dot{\varepsilon }_d \end{aligned}$$

(10)

where $\lambda ^{*}$ and $\kappa ^{*}$ are the slopes of normal compression and swelling lines in the ln$v$ : ln$p$ compression plane (being $v$ the soil specific volume) and $k$ is a parameter which controls the structure degradation with strain. The rate of the destructuration strain $\dot{\varepsilon }_d$ is assumed to have the following form:

$$\begin{aligned} \dot{\varepsilon }_d =\left[ {\left( {1-A} \right) \left( {\dot{\varepsilon }_v^p } \right) ^{2}+A\left( {\dot{\varepsilon }_q^p } \right) ^{2}} \right] ^{1/2} \end{aligned}$$

(11)

where $A$ is a non-dimensional scaling parameter and $\dot{\varepsilon }_q^p$ and $\dot{\varepsilon }_v^p$ are the plastic shear and volumetric strain rate, respectively.

Volumetric hardening rule is adopted in the model, where the change in size of the reference surface, $p_{c}$, is controlled only by plastic volumetric strain rate, $\dot{\varepsilon }_v^p$, given by:

$$\begin{aligned} \frac{\dot{p}_c }{p_c }=\frac{\dot{\varepsilon }_v^p }{\lambda ^{*}-\kappa ^{*}} \end{aligned}$$

(12)

If a stress increment requires movement of the bubble relative to the structure surface, the following kinematic hardening is invoked:

$$\begin{aligned} \dot{\bar{\varvec{\alpha }}}=\dot{\hat{\varvec{\alpha }}}+\frac{\dot{p}_c }{p_c }(\bar{\varvec{\alpha }}-\hat{\varvec{\alpha }})+\dot{\varvec{\mu }}(\varvec{\sigma }_c -\varvec{\sigma }) \end{aligned}$$

(13)

where $\bar{\varvec{\alpha }}$ and $\hat{\varvec{{\alpha }}}=p_c \left[ {r{{\varvec{I}}}+\left( {r-1} \right) \varvec{\eta }_0} \right]$ denote the locations of the centre of the bubble and structure surface respectively, $\varvec{\sigma }_{c}$ is the conjugate stress and $\mu$ is a positive scalar of proportionality. It should be noted that the centre of the structure surface and the deviator of $\hat{\varvec{\alpha }}$ represents the anisotropy of the soil due to structure. The deviator of $\hat{\varvec{\alpha }}$ therefore degrades to zero as $r$ degrades to unity.

The plastic modulus $H$ is assumed to depend on the distance between the current stress and the conjugate stress and is given by:

$$\begin{aligned} H=H_c +\frac{Bp_c^3 }{(\lambda ^{*}-\kappa ^{*})R}\left( {\frac{b}{b_{\max } }} \right) ^{\psi } \end{aligned}$$

(14)

where $H_{c}$ is the plastic modulus at the conjugate stress, $B$ and $\psi$ are two additional material properties, $b=\bar{{{\varvec{n}}}}:(\varvec{\sigma }_c -\varvec{\sigma })$ is the normalised distance between the bubble and the structure surface and $b_{\max } =2(r/R-1)\bar{{\varvec{n}}}:(\varvec{\sigma } -\bar{\varvec{\alpha }})$ is its maximum value.

Finally, the bulk and shear moduli, $K$ and $G$, are assumed to depend linearly on the mean effective pressure $p$:

$$\begin{aligned} K=\frac{p}{\kappa ^{*}}\qquad G=\frac{3(1-2\nu )}{2(1+\nu )}K \end{aligned}$$

(15)

where $\nu$ is a constant Poisson’s ratio.