# Bearing Capacity of Ring Foundations on Sand Overlying Clay
## 5 Results and Discussion
### Failure Mechanisms  
With \(R_{i}/R_{0}=0.5\), \(\varphi=35^{\circ}\), and \(H/R_{0}=0.5\), Figure 8 illustrates the failure mechanisms for four different magnitudes of \(c_{u}/(\gamma R_{0})\), namely 0.5, 1, 3, and 5. For \(c_{u}/(\gamma R_{0})=0.5\) and 1, two rupture lines develop from both external and internal edges of the ring footing to the bottom of the sand layer, and the log-spiral lines extend from the underlying clay to the upper sand layer. On the other hand, for \(c_{u}/(\gamma R_{0})=3\) and 5, the failure patterns are confined only within the sand layer, with a non-plastic triangular wedge below the ring footing base, and the slip lines extend to the free surface. This is similar to the classical shallow footing mechanism.  
Figure 9 presents the variation in failure mechanisms with the variations in magnitudes for \(\varphi\) from 25\({}^{\circ}\) to 45\({}^{\circ}\), with \(R_{\rm{\ddot{H}}}\)/\(R_{0}=0.5\), \(c_{\rm{\ddot{u}}}\)/(\(\gamma R_{0}\)) = 1, and \(H\)/\(R_{0}=0.75\). For \(\varphi=25^{\circ}\) and 30\({}^{\circ}\), the failure modes are developed only within the sand layer. On the other hand, for \(\varphi=35^{\circ}\), 40\({}^{\circ}\), and 45\({}^{\circ}\), the slip lines of the sand block extend from the external edge of the ring footing to the upper surface of the clay layer, and the rupture lines develop from the clayey strata to the upper sand layer. It should be noted that the size of the plastic zone decreases as the value of \(\varphi\) decreases. Therefore, as the value of \(\varphi\) increases, the proportion of the collapse mechanism in the clay layer increases, giving rise to the possibility of punching shear failure [3,7].  
Figure 8: Upper bound collapse mechanisms for \(R_{\rm{\ddot{u}}}\)/\(R_{0}=0.5\), \(\varphi=35^{\circ}\), and \(H\)/\(R_{0}=0.5\) with (a) \(c_{\rm{\ddot{u}}}\)/(\(\gamma R_{0}\)) = 0.5, (b) \(c_{\rm{\ddot{u}}}\)/(\(\gamma R_{0}\)) = 1, (c) \(c_{\rm{\ddot{u}}}\)/(\(\gamma R_{0}\)) = 3, and (d) \(c_{\rm{\ddot{u}}}\)/(\(\gamma R_{0}\)) = 5.  
Figure 9: _Cont._  
Figure 10 shows the failure mechanisms for different \(R_{i}/R_{0}\) with \(\varphi=35^{\circ}\), \(c_{u}/(\gamma R_{0})=2\), and \(H/R_{0}=0.75\). For the cases of \(R_{i}/R_{0}=0\), 0.25, and 0.33, the failure modes are similar to Figure 9c-e. It needs to be mentioned that the failure surface in the sand layer inclines to the center line of the footing. From Figure 10d for \(R_{i}/R_{0}=0.5\) and Figure 10e for \(R_{i}/R_{0}=0.75\), it can be seen that the clayey stratum does not affect the collapse mechanisms. In addition, the size of the failure zone decreases with an increase in the value of \(R_{i}/R_{0}\), which also can be observed in Figure 10. Several failure modes for different \(H/R_{0}\) with \(\varphi=35^{\circ}\), \(c_{u}/(\gamma R_{0})=2\), and \(R_{i}/R_{0}=0.5\) are illustrated in Figure 11. A critical thickness can be clearly seen. Interestingly, the failure mechanics for \(H/R_{0}=1\) and 3 are exactly the same. Therefore, it can be concluded that the collapse mechanics do not change when the thickness of the sand layer exceeds the critical depth.