# Bearing Capacity of Ring Foundations on Sand Overlying Clay
## 3 Finite Element Limit Analysis  
The FELA method combines the upper and lower bound limit theorems of classical plasticity with the concept of finite elements, to deduce the failure load directly, which is developed by [PERSON] and [PERSON] [37,38] and [PERSON] et al. [39,40]. The technique can model anisotropy, heterogeneity, complicated loading conditions, complex boundary shapes, and arbitrary geometries, as demonstrated by [PERSON] [41]. Compared to the conventional displacement FEA, the FELA method does not require performing a complete incremental analysis and the load-settlement curve to determine the limit load, showing high efficiency for the geotechnical stability problem analysis. In this study, the ultimate bearing capacity of ring footings on layered sand-clay media is computed by using the software OptumG2 [36], developed by the method of FELA. In the following section, the present numerical model of FELA in OptumG2 is summarized.  
Figure 2 depicts a numerical model of the ring footing on sand overlying clay profile in OptumG2. Since the geometry of the problem is axis-symmetric, only half of the model domain is considered. In order to minimize boundary effects, the width and depth of the domain are set to \(15R_{0}\) and \(10R_{0}\), respectively. The external radius (\(R_{0}\)) is kept at 3 m, and five different values of the internal radius \(R_{i}\), namely 0, 0.75, 1, 1.5, and 2.25 m, are chosen. The bottom boundary of the problem is fixed, while the vertical boundaries of the model are only constrained in the horizontal direction.  
The footing is considered to be a weightless rigid material, and the interface between the footing and the sand layer is assumed to be perfectly rough. The behavior of sands is modeled by using the  
Figure 1: Problem definition of a rigid ring footing on sand overlying clay.  
Figure 2: Numerical model of the problem in OptumG2.  
linear elastic perfectly plastic Mohr-Coulomb material with an associated flow rule. The clay is treated as a linearly elastic-perfectly plastic material, obeying the Tresca criterion with an associated flow rule. Following the previous study by [PERSON] et al. [16], the Young's modulus (\(E\)) of the sand layer and the clayy strata are set to 200 and 50 MPa, respectively. The Poisson's ratio (\(\
u\)) used is 0.495 and 0.3 for clays and sands, respectively. The coefficient of lateral earth pressure (\(K_{0}\)) of the clay is set to 1, and (\(1-\sin\,\varphi\)) for sands. It should be mentioned that the values of the elastic parameters and \(K_{0}\) have a negligible effect on the ultimate bearing capacity of foundations [42,43]. The unit weight of the sand and the clay are both set to \(\gamma=16\) kN/m\({}^{3}\), as mentioned by [PERSON] and [PERSON] [30].  
In the numerical simulations, both upper and lower bound limit analyses are considered. In the UB limit analysis, the footing, the sand, and the clay are discretized into a six-noded triangular element, and discretized into a three-noded triangular element in the LB limit analysis. Automatic mesh adaptivity is used for all operations, and five adaptive steps are employed based on the software manual. A value of 1000 elements is selected in the first adaptivity iteration, and the elements increase to approximately 6000 in the final mesh. In order to improve the upper and lower bound solutions, the tool of mesh fan 5\({}^{\circ}\) in the corner of ring footings is applied.