## 3 Modeling of a Dynamic Rocking Foundation Test  
To demonstrate the capability of the material model to capture the non-linear response of undrained clay and to set the modeling parameters, a dynamic rocking foundation experiment (D9 test at station B East) conducted by [45] is simulated by the FE software ABAQUS, using the aforementioned user material (UMAT) code. The experiment was conducted in a centrifuge device with a \(20\times g\) acceleration. Two rectangular aluminum footings, each carrying a wall on top, and a floor connecting the two walls were placed on top of 8.5 cm saturated stiff San Francisco Bay mud (here referred to as clay). Underneath the clay, there was 13.6 cm saturated dense Nevada sand. A thin layer (1.3 cm) of Monterey sand was placed over the clay (not under the footings). Shaking was applied as a tapered cosine acceleration record (with linearly increasing amplitude) to the bottom of the container, inthe longitudinal direction of the footings. For a detailed description of the test, the reader is referred to [45]. The FE model of the test is established in the prototype scale by multiplying the dimensions of the test model by 20 and is shown in Figure 1. The structure and footings are meshed by linear hexahedral elements with reduced integration. Soil elements are linear hexahedral elements, except the soil elements around the region of contact with footings, which are chosen to be quadratic hexahedral elements to avoid the problem of distortion caused by contact stresses. For the undrained sand layer, the Mohr-Coulomb model available in ABAQUS software is used, with the parameters shown in Table 2. The cohesion and friction angle of the sand are assumed based on the experimental results of [46] for undrained dense sand. The dilation angle is assumed to have very small value of 0.2\({}^{\circ}\) (but not 0\({}^{\circ}\), to avoid numerical problems) to minimize the volume changes of the incompressible undrained soil. It should be noted that total stress approach is considered in the analysis. The Poisson ratio is assumed to be 0.47 (close to 0.5, but not exactly 0.5, to avoid numerical convergence problems) to minimize volume changes of the undrained sand. The modulus of elasticity is calculated from the shear modulus values reported by [47]. The mass density of the sand is the saturated density calculated based on the 90% relative density of the sand. The parameters assumed for the UMAT of clay are summarized in Table 3. The mass density of the saturated clay (San Francisco Bay mud) is taken from [48]. The bulk \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Friction Angle & 29.6\({}^{\circ}\) \\ Dilation Angle & 0.2\({}^{\circ}\) \\ Cohesion & 214.9 kPa \\ Elasticity Modulus & 411.6 MPa \\ Poisson’s Ratio & 0.47 \\ Mass Density & 2051 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material parameters for undrained sand. Figure 1: FE model of the dynamic rocking foundation test. modulus of clay (\(K\)) is calculated from \(K=\rho\left(V_{p}^{2}-\frac{4}{3}V_{s}^{2}\right)\), which can be obtained based on the equations given by [49], with the shear wave and \(P\)-wave velocities (\(V_{s}=135\) m/s and \(V_{p}=1400\) m/s) taken from [50]. It should be noted that the shear modulus (\(G\)) in Table 3 is not the small strain shear modulus; rather, it is a measure of shear stiffness that governs the deformations in elastic range as well as the return-mapping. A small value of 0.01 kPa is assigned to \(H_{\text{iso}}\), since it was observed that the non-linear stress-strain response of the clay is better modeled by kinematic hardening. The parameters \(G\), \(\sigma_{y0}\), \(H_{\text{kin}}\), and \(H_{\text{nl}}\) affect the stress-strain response significantly; therefore, they are calibrated to capture the moment-rotation and settlement response of the foundation after a sensitivity analysis. The undrained shear strength, \(S_{\text{i}0}=100\) kPa, is given by [51] for San Fransisco Bay mud.