Dynamic Problem for the Deformation of Saturated Soil in the Vicinity of a Vibrating Pile Toe

Abstract

A numerical study conducted recently by the authors showed that the vibration of a pile in saturated granular soil leads to the formation of a zone with nearly zero effective stresses (liquefaction zone) around the pile toe. The dynamic problem was solved with the finite-element program Abaqus/Standard using a hypoplasticity model for soil with the assumption of zero soil permeability and without a mass force. A question which still remained open was the influence of the soil permeability and the gravity force on the solutions. In the present study, the problem is solved with nonzero permeability and gravity, and the solutions are compared with those obtained earlier. For this purpose, a user-defined element has been constructed in Abaqus to enable the dynamic analysis of a two-phase medium with nonzero permeability. The solutions show that high permeability and gravity do not prevent the formation of a liquefaction zone around the pile toe in spite of the fact that a build-up of the pore pressure is inhibited by the pore pressure dissipation.

Appendix: Finite-Element Discretization and Time Integration

The unknown functions u and p are approximated with the help of shape functions $N^u_I,N^p_J$:

$$\begin{aligned} \mathbf{u}=N^u_I \mathbf{u}_I, \quad p=N^p_J p_J, \end{aligned}$$

(9)

where $\mathbf{u}_I$ is the nodal displacement vector at node I, and $p_J$ is the nodal value of the pore pressure at node J. The same shape functions are used for the test functions:

$$\begin{aligned} \delta \mathbf{u}=N^u_I \delta \mathbf{u}_I, \quad \delta p=N^p_J \delta p_J. \end{aligned}$$

(10)

Substituting (9), (10) into (6), (7) yields a system which contains the nodal variables $\mathbf{u}_I,p_J$ and their time derivatives:

$$\begin{aligned}&\int \limits _\Omega \left( {\varvec{\sigma }}\cdot \mathrm{grad}\,N^u_I -\varrho N^u_I \mathbf{g} \right) {\text {d}}\Omega -p_J \int \limits _\Omega N^p_J\,\mathrm{grad}\,N^u_I\, {\text {d}}\Omega \nonumber \\&+\, \ddot{\mathbf{u}}_K \int \limits _\Omega \varrho N^u_I N^u_K {\text {d}}\Omega -\int \limits _{\Gamma _t}\mathbf{t}^{\text {tot}} N^u_I {\text {d}}\Gamma = 0, \end{aligned}$$

(11)

$$\begin{aligned}&\dot{p}_L \int \limits _\Omega \frac{n}{K_f}\,N^p_L N^p_J {\text {d}}\Omega + \dot{\mathbf{u}}_K \cdot \int \limits _\Omega N^p_J\,\mathrm{grad}\,N^u_K\, {\text {d}}\Omega \nonumber \\&+\, p_L \int \limits _\Omega \frac{k}{g\varrho _f}\,\mathrm{grad}\,N^p_L \cdot \mathrm{grad}\,N^p_J\,{\text {d}}\Omega - \int \limits _\Omega \frac{k}{g}\, \mathbf{g} \cdot \mathrm{grad}\,N^p_J\,{\text {d}}\Omega \nonumber \\&+\, \ddot{\mathbf{u}}_I \cdot \int \limits _\Omega \frac{k}{g}\,N^u_I \mathrm{grad}\,N^p_J \, {\text {d}}\Omega - \int \limits _{\Gamma _q} q N^p_J {\text {d}}\Gamma = 0. \end{aligned}$$

(12)

Let $U_t$ be a column vector of the nodal variables $u_{Ii},p_J$ at time t, and $\dot{U}_t,\ddot{U}_t$ be its time derivatives. With these vectors being known, Abaqus solves a system

$$\begin{aligned} F\left( U_{t+\Delta t} \right) =0 \end{aligned}$$

(13)

for $U_{t+\Delta t}$, where a column vector F is a (generally nonlinear) function of $U_{t+\Delta t}$ to be specified by the user. The function F also involves $\dot{U}_t$ and $\ddot{U}_t$. The solution proceeds iteratively. If $U^i_{t+\Delta t}$ is an ith approximation, the next approximation is

$$\begin{aligned} U^{i+1}_{t+\Delta t}=U^i_{t+\Delta t}+C^{i+1}, \end{aligned}$$

(14)

where the correction term $C^{i+1}$ is found from the linear system

$$\begin{aligned} A\left( U^i_{t+\Delta t}\right) C^{i+1}=-F\left( U^i_{t+\Delta t}\right) \end{aligned}$$

(15)

with a matrix $A=\partial F/\partial U_{t+\Delta t}$. Equation (15) is obtained by expanding (13) in a Taylor series about $U^i_{t+\Delta t}$.

Now we show how the time integration of (11), (12) leads to (13). Equations (11), (12) can be written as

$$\begin{aligned} M\ddot{U}-G=0, \end{aligned}$$

(16)

with a constant mass matrix M and a vector G which depends on $\dot{U},U$ and the stress field ${\varvec{\sigma }}$. Note that the elements of M which are multiplied by $\ddot{p}_J$ are equal to zero because Eqs. (11), (12) do not contain $\ddot{p}_J$. According to the Hilber–Hughes–Taylor integration scheme employed in Abaqus/Standard, (16) is written as

$$\begin{aligned} M\ddot{U}_{t+\Delta t}-(1+\alpha )G_{t+\Delta t}+\alpha G_t=0, \end{aligned}$$

(17)

where $\alpha$ is a parameter ($-1/3\le \alpha \le 0$). The implicit time integration scheme for U is

$$\begin{aligned}&U_{t+\Delta t}=U_t+\Delta t \, \dot{U}_t+\Delta t^2 \left[ \left( \frac{1}{2}-\beta \right) \ddot{U}_t+\beta \ddot{U}_{t+\Delta t} \right] , \end{aligned}$$

(18)

$$\begin{aligned}&\dot{U}_{t+\Delta t}=\dot{U}_t+\Delta t \left[ \left( 1-\gamma \right) \ddot{U}_t+\gamma \ddot{U}_{t+\Delta t} \right] , \end{aligned}$$

(19)

Fig. 12

User element

with parameters $\beta =(1-\alpha )^2/4$, $\gamma =1/2-\alpha$. With known $U_t,\dot{U}_t,\ddot{U}_t$, relations (18), (19) give $\dot{U}_{t+\Delta t}$ and $\ddot{U}_{t+\Delta t}$ as functions of ${U}_{t+\Delta t}$. These allows us to write $\ddot{U}_{t+\Delta t}$ and $G_{t+\Delta t}$ in (17) as functions of $U_{t+\Delta t}$. It remains to express ${\varvec{\sigma }}_{t+\Delta t}$ in $G_{t+\Delta t}$ also as a function of $U_{t+\Delta t}$. We write

$$\begin{aligned} {\varvec{\sigma }}_{t+\Delta t}={\varvec{\sigma }}_t+ \Delta {\varvec{\sigma }}, \end{aligned}$$

(20)

where ${\varvec{\sigma }}_t$ is known. The increment $\Delta {\varvec{\sigma }}$ is obtained from the strain increment $\Delta {\varvec{\varepsilon }}$ by the use of the constitutive model, whereas $\Delta {\varvec{\varepsilon }}$ can be expressed through the derivatives of the shape functions and the difference ${U}_{t+\Delta t}-{U}_t$. This finally makes the left-hand side of (17) a function of $U_{t+\Delta t}$ and the known quantities $U_t,\dot{U}_t,\ddot{U}_t,{\varvec{\sigma }}_t$. The differentiation of this function with respect to $U_{t+\Delta t}$ yields the matrix A.

The user-defined element is constructed with the help of the user subroutine UEL. The subroutine is called for each element and receives the nodal values $U^i_{t+\Delta t},\dot{U}^i_{t+\Delta t},\ddot{U}^i_{t+\Delta t}$ of the element as input. The subroutine uses the displacement increments contained in $U^i_{t+\Delta t}-U_t$ to compute the strain increment $\Delta {\varvec{\varepsilon }}$ and calls the user subroutine UMAT to obtain the stress increment $\Delta {\varvec{\sigma }}$ and the material Jacobian $\partial \Delta {\varvec{\sigma }}/ \partial \Delta {\varvec{\varepsilon }}$. The latter is needed for the matrix A. The UEL calculates contributions of the element to the matrix A and the vector F and saves them in the arrays AMATRX and RHS as output. Abaqus collects contributions from all elements, forms global matrix A and vector F and finds a correction vector $C^{i+1}$ by solving (15). The next approximation $U^{i+1}_{t+\Delta t}$ is given by (14), and $\dot{U}^{i+1}_{t+\Delta t},\ddot{U}^{i+1}_{t+\Delta t}$ are calculated from (18), (19).

The user element shown in Fig. 12 is constructed as a two-dimensional quadrilateral u8p4 element for axisymmetric problems with a biquadratic interpolation of the displacements and a bilinear interpolation of the pore pressure.