# Bearing Capacity of Shallow Foundations on Unsaturated Silty Soils
## 1 Introduction
#### Estimation of UBC of Foundations on Unsaturated Soils
\[q_{ult}=\left\{c^{\prime}+(u_{a}-u_{w})_{b}tan\mathcal{O}^{\prime}+\left[(u_{a }-u_{w})-(u_{a}-u_{w})_{b}\right]tan\mathcal{O}^{b}\right\}N_{c}+\frac{1}{2}B \gamma N_{\gamma} \tag{5}\]  
where \(c^{\prime}\) is the effective cohesion; \(\mathcal{O}^{\prime}\) is the effective angle of internal friction; \(\mathcal{O}^{b}\) is the angle indicating the rate of increase in shear strength relative to the MS; \((u_{a}-u_{w})_{b}\) is the AEV/bubbling pressure; \((u_{a}-u_{w})\) is the MS value; B is the width of the foundation; \(N_{c}\) and \(N_{\gamma}\) are the UBC coefficients of [PERSON] (1943) [1] and [PERSON] (1993) [45], respectively. Since the behavior of \(\mathcal{O}^{b}\) is non-linear beyond the AEV, this method can estimate the UBC within an acceptable range depending on the suction range. A bilinear envelope [46] can be used to simulate the nonlinear relationship between the SS and the MS, as shown in Figure 1. As can be seen from Figure 1, the contribution of MS to the SS is \(\mathcal{O}^{b}=\mathcal{O}^{\prime}\) for suction values less than the AEV. This region, where the SS increases linearly with the MS, is called the linear region. Beyond the AEV, the contribution of \(\mathcal{O}^{b}\) gradually decreases despite the increase in MS; therefore, \(\mathcal{O}^{b}\) is less than \(\mathcal{O}^{\prime}\). In this region, which is called the nonlinear region, the SS of unsaturated soils changes nonlinearly with the MS [46; 47]. Such a behavior is consistent with the reason explained using Figure 1 for the SS of unsaturated soils. The red line here shows the contribution of \(\mathcal{O}^{b}\), to the SS at MS values greater than the AEV. [PERSON] and [PERSON] (2007) [14] developed the original UBC equation for continuous footings which was based on the [PERSON] (1943) [1] equation for unsaturated soils and reference the UBC model of [PERSON] et al. (1997) [12], and the nonlinear shearing resistance model proposed by [PERSON] et al. (1996) [48] (Equation (6)). [PERSON] and [PERSON] (2007) [14] also suggested that by using the measured shear strength angle value increased by 10% (1.1\(\phi^{\prime}\)), a more accurate estimation of the UBC value can be obtained for both saturated and unsaturated conditions. \[q_{ult}=\Big{\{}\Big{[}c^{\prime}+(u_{a}-u_{w})_{b}tan\mathcal{O}^{\prime}-S^ {\Psi}tan\mathcal{O}^{\prime}\Big{]}+(u_{a}-u_{w})_{AVE}S^{\Psi}tan\mathcal{O }^{\prime}\Big{\}}N_{c}\xi_{c}+\frac{1}{2}B\gamma N_{\gamma}\xi_{\gamma} \tag{6}\]  
where \(c^{\prime}\) is the effective cohesion; \(\mathcal{O}^{\prime}\) is the effective angle of internal friction; \(S\) is the degree of saturation; \(\Psi\) is the fitting parameter; \((u_{a}-u_{w})_{b}\) is the AEV; \((u_{a}-u_{w})_{AVE}\) is the average MS value; \(N_{c}\) and \(N_{\gamma}\) = are [PERSON] (1943) [1] and [PERSON] (1993) [45] bearing capacity coefficients, respectively, and \(\xi_{c}\) and \(\xi_{\gamma}\) = [PERSON] (1973) [44] shape factors. [PERSON] and [PERSON] (2007) [14] developed a correlation between the UBC fitting parameter (\(\Psi\)) and the plasticity index (\(I_{p}\)) based on the results of the study performed on five soils (Figure 2). All three of the sandy soils examined in the study had a fitting parameter (\(\Psi\)) = 1; however, it was noted that other fine-grained soils needed greater values. Equation 7 explains the link between the UBC fitting parameter (\(\Psi\)) and the \(I_{p}\) based on the findings of the research conducted on five soils. \[\Psi=-0.0031I_{p}^{2}+0.34I_{p}+1 \tag{7}\]  
Figure 1: The effect of MS value on SS [46]. In this study, a new equation is proposed in which the UBC of foundations on unsaturated silty soils is determined by adopting the total stress analysis philosophy presented above, with a bearing capacity fitting parameter (\(S^{\Psi}\)) based on the SS determined via the UCS tests and the SSD/MS value. Various models have been proposed in the literature to estimate the UBC of unsaturated soils (see Table 1).