# Study on the Influence of Groundwater Variation on the Bearing Capacity of Sandy Shallow Foundation
## 2 Experimental Tests
### Analysis of Results
It is a formula of soil damage put forward by the French scientist [PERSON], who summarized the phenomenon and influencing factors of soil damage. The shear strength of soil depends on the properties of rock particles which are related to normal stress, internal friction angle and cohesion of soil [33; 34; 35]. Later, [PERSON] continued [PERSON]'s research and put forward the destruction of the material of shear failure theory. The function defined the Mohr failure envelope curve in the general case. The Mohr failure envelope can be expressed by the Coulomb formula, namely the soil shear strength and normal stress into linear function relation [36; 37]. The constitutive relation that was selected for this model is Mohr-Coulomb criterion of rock failure because it is a classical theory in geotechnical engineering and has been widely verified. The Mohr-Coulomb constitutive relation requires fewer parameters and can reduce the error caused by finite element calculation. For any stress surface considered in rock and soil under general stress conditions, its ultimate shear strength can usually be expressed by [PERSON]'s law:  
\[\tau_{n}=\sigma_{n}\text{tan}\varphi+c \tag{2}\]  
In the spatial stress state, it can be expressed as follows:  
\[\sigma_{1}=\sigma_{3}\text{tan}^{2}(45^{\circ}+\frac{\varphi}{2})+2\text{tan}( 45^{\circ}+\frac{\varphi}{2}) \tag{3}\]  
In the formula, \(\sigma\) is the shear strength index of the soil sand, and \(\varphi\) is the angle of the internal friction. The above equation is the shear failure criterion of rock and soil. It can be called the failure yield surface equation [38]. The failure surface is the limit surface of the yield surface. Therefore, the yield surface can be expressed as follows:  
\[F=-\frac{1}{3}I_{1}\text{sin}\varphi+(\text{cos}\theta_{\sigma}-\frac{1}{3} \text{sin}\theta_{\sigma}\text{sin}\varphi)\sqrt{I_{2}}-c\text{cos}\varphi=0 \tag{4}\]  
In the formula, \(-\frac{\pi}{6}\leq\theta_{\sigma}\leq\frac{\pi}{6}\). The problem is simplified as shown in Figure 6. The upper sandy soil is divided into three zones, which are Zones A, B and C. The sandy soil in Zone A is all unsaturated soil. Zone B is the mixed region of the saturated soil and unsaturated soil and contains the whole range of water level fluctuation. The sand soil in Zone C is saturated soil. The final problem is to determine the specific location and soil thickness of Zones A, B and C.  
Among the three regions, Zone B is the most important, because it contains all the fluctuation of water level and is the key area to study the influence of water-level fluctuation on the foundation bearing capacity. The soil saturation in this region is very complex and is the most difficult part of this problem. The foundation problem with the dynamic change of groundwater level over a large area can be regarded as a one-dimensional problem, which can be directly described by the traditional consolidation governing equation:  
\[Y(z)c_{\rm vs}\frac{\partial^{2}u}{\partial z^{2}}=\frac{\partial u}{\partial t} \tag{5}\]  
In the formula, \(z\) represents the position variable along the depth of the soil layer, f is the time variable, \(u\) represents the excess pore pressure with depth \(z\) and time, and \(c_{\rm ps}\) is the consolidation or rebound coefficient of the calculation model. In this paper, the possible changes of the foundation permeability coefficient and volume compression coefficient in the process of dynamic change of groundwater level are not taken into consideration temporarily. To simplify the calculation, \(c_{\rm ps}\) is directly taken as the consolidation coefficient and a constant. The mathematical expression of the dynamic change of groundwater head pressure over time is used to describe the top boundary conditions of the calculation model, which can be expressed as follows:  
\[u|z=0=f(t) \tag{6}\]  
The boundary condition of impervious boundary can be expressed as follows:  
\[\frac{\partial u(H,t)}{\partial z}=0 \tag{7}\]  
In the formula, \(H\) is the total thickness of the model foundation.