# Bearing Capacity of Shallow Foundations on Unsaturated Silty Soils
## 1 Introduction
#### Estimation of UBC of Foundations on Unsaturated Soils  
Several methods have been proposed in the literature for estimating the UBC of foundations on unsaturated soils. These methods mainly involve modifying the equations proposed by [PERSON] [1] and [PERSON] [2] to take into account the effect of matric suction. Research has shown that foundations on unsaturated soils typically fail through the punching shear failure mode [11; 12; 15; 41]. In the case of cohesive soils, it is assumed the a constant pore water pressure under undrained conditions can provide accurate results. [PERSON] et al. [42] also found that in-place plate bearing tests typically occur under constant water content condition, and that assuming a constant value for the effective stress parameter does not lead to errors in the results. [PERSON] and [PERSON] [18] proposed an equation (Equation (1)) for estimating the UBC of shallow foundations on the surface of unsaturated fine-grained soil, based on the assumption of an undrained loading condition. \[q_{ult(unsat)}=s_{unsat}\times\xi\times N_{cs} \tag{1}\]  
where \(q_{ult(unsat)}\) = UBC of the foundation on fine-grained soil; \(s_{unsat}\) = SS of unsaturated soil based on unconfined compressive strength (UC); \(\xi\) = shape factor; and \(N_{cs}\) = UBC factor under undrained loading conditions (i.e., 5,14). In the same study, [PERSON] and [PERSON] (2013) [18] have also demonstrated that the SS of an unsaturated soil is equal to the sum of the saturated SS and the increase in SS produced by the MS (Equation (2)). In addition, it has been reported that the total SS of the unsaturated cohesive soil (including the contribution of the MS to the SS) can be obtained with the UC, so there is no need to measure the MS value of the cohesive soils [18]. \[s_{unsat}=\left[s_{sat}+f(u_{a}-u_{w})\right]=\left[\frac{q_{u(unsat)}}{2}\right] \tag{2}\]  
where \(s_{unsat}\) is the SS of the soil based on the UC; \(f(u_{a}-u_{w})\) is the increase in strength with MS contribution; and \(q_{ult(unsat)}\) is the UC of the soil. [PERSON] and [PERSON] (2013) [18] demonstrated that the UC of fine-grained soils, and mainly of clay soils, can be used to predict the UBC of shallow square foundations (Equation (3)). A shape factor, \(\xi_{unsat}\) = 1 + 0.2B/L (where B and L corresponds to the foundation width and length, respectively) established by [PERSON] (1963) [43] and [PERSON] (1973) [44] for undrained circumstances, was also employed in the proposed equation:  
\[q_{ult(unsat)}=\left[\frac{q_{u(unsat)}}{2}\right]\times\left[1+0.2\frac{B}{L} \right]\times N_{unsat} \tag{3}\]  
where \(B\) is the width of the foundation; \(L\) is the length of the foundation; and \(N_{unsat}\) is the UBC factor in unsaturated soils, which was obtained using back analysis by [PERSON] and [PERSON] (2013) [18]. In addition, for the estimation of the UBC of foundations on coarse-grained unsaturated soils under drained loading conditions, equations based on the modification of the UBC equation (Equation (4)) proposed by [PERSON] (1943) [1] for strip foundations on saturated soils is available in the literature [12; 13; 15; 26; 29; 30; 32; 34]. \[q_{ult}=K_{1}c^{\prime}N_{c}+qN_{q}+K_{2}\gamma BN_{\gamma} \tag{4}\]  
where \(c^{\prime}\) is the effective cohesion; \(q\) is the effective vertical stress; \(B\) is the width of foundation; \(K_{1}\) and \(K_{2}\) are shape factors; and \(N_{c}\), \(N_{q}\) and \(N_{\gamma}\) are UBC factors. [PERSON] et al. (1997) [12] modified [PERSON]'s (1943) [1] effective stress approach and proposed a model that can be used to estimate the UBC of a shallow foundation on unsaturated soils, taking into account the effect of MS (Equation (5)). The model covers the situation in which both pore air and water pressure are effective during the loading phase. The model employs a fixed \(\mathcal{O}^{b}\) for suction values greater than the air-entry value (AEV) of the soil.