# Bearing Capacity of Ring Foundations on Sand Overlying Clay
## 5 Results and Discussion  
The values of \(p_{u}\)/(\(\gamma R_{0}\)) for ring footings on sand overlying clay from FELA are depicted graphically in Figures 3-6. The maximum error of the upper and lower bound analyses is restricted within 9.6%, and the smaller values are obtained at small \(H\)/\(R_{0}\) ratios. Hence, average values of the upper and lower bound bearing capacity \(p_{u}\)/(\(\gamma R_{0}\)) were determined and are employed in the following discussions. In addition, Equation (3) can be used to compute the error between UB and LB solutions. \[Error=\frac{2(LIB-LB)}{(LIB+LB)}\times 100\% \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(R_{0}\) (m)} & \multirow{2}{*}{\(H\)/\(R_{0}\)} & \multirow{2}{*}{\(\alpha\)/\(\gamma R_{0}\)} & **Present** & **Present** & **[PERSON] and [PERSON]** & **[PERSON] et al.** & **[PERSON] et al.** \\  & & & **Study \({}^{\mathrm{a}}\)** & **Study \({}^{\mathrm{b}}\)** & **(2015)** & **(1997)** \({}^{\mathrm{c}}\) & **(1998)** \({}^{\mathrm{d}}\) \\ \hline \multirow{2}{*}{\(1.5\)} & 0 & 1.61 & 9.70 & 9.76 & 9.71 & 10.00 & 10.71 \\  & & 3.01 & 18.14 & 18.24 & 18.09 & – & 19.27 \\  & & 5.15 & 30.03 & 31.21 & 30.97 & – & 32.88 \\  & & 6.26 & 37.71 & 37.94 & 37.59 & – & 39.12 \\  & 1 & 0.75 & 11.82 & 12.04 & 14.07 & – & 10.69 \\  & & 3.05 & 26.47 & 27.14 & 29.78 & 28.90 & 31.57 \\  & & 5.19 & 36.60 & 37.66 & 44.17 & – & 52.45 \\  & & 6.26 & 41.08 & 42.27 & 51.30 & – & 62.61 \\  & 2 & 0.80 & 33.98 & 34.85 & 46.92 & 31.95 & 25.23 \\  & & 1.20 & 40.42 & 41.51 & 55.79 & 37.01 & 31.98 \\  & & 3.21 & 60.34 & 62.40 & 78.14 & 58.30 & 58.43 \\  & & 5.15 & 73.29 & 76.05 & 96.86 & 73.30 & 87.12 \\  & 3 & 2.45 & 106.62 & 111.14 & 109.02 & – & 84.65 \\  & & 3.54 & 111.53 & 125.53 & 118.52 & 103.00 & 103.75 \\  & & 5.84 & 114.89 & 125.80 & 131.73 & 141.00 & 152.05 \\  & 4 & 1.63 & 114.67 & 124.82 & 131.13 & 129.31 & 112.54 \\  & & 1.85 & 115.01 & 125.19 & 134.62 & 133.80 & 117.04 \\  & & 3.67 & 114.53 & 124.93 & 132.46 & 125.00 & – \\ \hline \hline \end{tabular} Note: \({}^{\mathrm{a}}\) upper bound solution; \({}^{\mathrm{b}}\) lower bound solution; \({}^{\mathrm{c}}\) using centrifuge test; \({}^{\mathrm{d}}\) limit equilibrium method.  
\end{table} 
Table 4: Comparison of \(p_{u}\)/(\(\gamma R_{0}\)) for a rough circular foundation on sand overlying clay (\(\varphi=40^{\circ}\)). Figure 3: _Cont._  
Figure 4: _Cont._  
Figure 4: Variation in the normalized bearing capacity, \(p/(\gamma R_{0})\), with \(R_{i}/R_{0}\) and \(\varphi\) for \(c_{u}/(\gamma R_{0})=0.5\) with (**a**) \(H/R_{0}=0.5\), (**b**) \(H/R_{0}=1\), (**c**) \(H/R_{0}=2\), and (**d**) \(H/R_{0}=4\). it can be seen that the depth of the failure mechanisms decreases with increasing the value of \(c_{ul}\)/(\(\gamma R_{0}\)), and eventually does not exceed the sand layer. This finding is similar to the previous studies conducted by [PERSON] et al. [3] for strip footings and by [PERSON] and [PERSON] [30] for circular foundations. However, for the cases of \(\varphi=35^{\circ}\) with \(R_{i}\)/\(R_{0}\leq 0.25\) and \(\varphi=40^{\circ}\) to \(45^{\circ}\) with \(R_{i}\)/\(R_{0}\leq 0.5\), as the value of \(c_{ul}\)/(\(\gamma R_{0}\)) increases, the bearing capacity increases continuously, and the rate increase in bearing capacity decreases. It is implied that when the value of \(\varphi\) increases, the bearing capacity reaching a constant value needs a larger magnitude for \(c_{ul}\)/(\(\gamma R_{0}\)). This observation can be explained by the failure patterns for different values of \(\varphi\), as presented in Figure 9, indicating that the failure mechanisms extend from the sand layer to the underlying clay when the value of \(\varphi\) increases. Thus, for the sand layer with larger values of \(\varphi\), it requires a larger magnitude of \(c_{ul}\)/(\(\gamma R_{0}\)) to constrain the failure mechanisms within the sand layer, when the bearing capacity becomes independent of \(c_{ul}\)/(\(\gamma R_{0}\)). A similar tendency can also be observed in the work by [PERSON] and [PERSON] [30].