# Immediate Settlement of Ring Footings Resting on Inhomogeneous Finite Stratum
## 4 Results and Discussion  
Figures 4-8 show the numerical solutions for uniformly loaded ring footings, as related to inner-to-outer radius ratio \(R_{i}/R_{0}\), Poisson's ratio \(\
u_{s}\), footing stiffness \(K\), finite layer thickness ratio \(H/D\), and modulus inhomogeneity \(nD/E_{0}\). The results are presented in the familiar form of the settlement influence factors at the points investigated: in the perfectly flexible footing case (\(K=0.001\)), the influence charts for the inner and outer settlement of a footing are given; in the perfectly rigid footing case (\(K=100\)), the influence charts for inner settlement (representing any settlement beneath the footing) are provided. This is because a flexible footing imposes non-uniform settlement but a rigid footing induces uniform settlement. Figure 4 shows the variation of the settlement influence factor \(I_{P}\) of perfectly flexible and rigid ring footings on a homogeneous elastic half-space for different combinations of \(\
u_{s}\) and \(R_{i}/R_{0}\). It can be seen that an increase in Poisson's ratio leads to a decrease in \(I_{P}\), regardless of footing stiffness. However, the effect of the [PERSON]'s ratio on the results is independent of \(R_{i}/R_{0}\). Figure 4 also provides a comparison of the \(I_{P}\) values obtained in this study with the theoretical solutions of [PERSON] [9] for flexible ring footings on incompressible elastic medium. It can be observed that the present solutions compare reasonably well with the results of [PERSON] [9]. Meanwhile, it should be noted that the contribution of [PERSON]'s ratio to the settlement of ring footings can be reflected by the product of a correction factor (\(1-{\
u_{s}}^{2}\)) and a value of \(I_{K}\) for \(\
u_{s}=0\). Figure 5 highlights the effects of \(K\) and \(R_{i}/R_{0}\) on the settlement influence factor \(I_{K}\) of ring footings on a homogeneous elastic half-space. It can be seen that at the inner point of the footing (Figure 5a), the \(I_{K}\) values decrease with increasing the values of \(K\), implying that the higher footing stiffness reduces the inner settlement of the footing. At the outer point of the footing (Figure 5b), the \(I_{K}\) values increase with increasing values of \(K\); that is, higher footing stiffness causes a larger outer settlement of the footing. It can be also noticed that for smaller values of \(R_{i}/R_{0}\), the change in \(I_{K}\) with respect to the footing stiffness is significant, irrespective of the point studied. Meanwhile, the values of \(I_{K}\) for the circular footing were compared with those obtained by [PERSON] [20] using integral transform  
Figure 4: Variation of \(I_{P}\) with \(\
u_{s}\) for ring footings on homogeneous infinite soil: (**a**) flexible footing of inner point; (**b**) flexible footing of outer point; (**c**) rigid footing. technique. As shown in Figure 5a, the present \(I_{K}\) values are found to be invariably lower than the solutions of [PERSON] (1975). This result is unsurprising, given the different footing roughnesses (i.e., rough and smooth roughnesses) for which current and [PERSON]'s analyses were adopted, so they are not directly comparable. Figure 6 shows the variation of the settlement influence factor \(I_{H}\) of perfectly flexible and rigid ring footings on homogeneous elastic layer of finite thickness for different combinations of \(H/D\) and \(R_{i}/R_{0}\). It is clear that the \(I_{H}\) value has a general tendency to increase with increasing values of \(H/D\). However, the rate of such increase diminishes for footings with high values of \(H/D\) and \(R_{i}/R_{0}\). Increasing the finite layer thickness ratio to values greater than a critical one does not affect the footing settlement, because a small portion of applied load reaches the underlying rigid base. The critical finite layer thickness ratio is insensitive to the values of \(R_{i}/R_{0}\). For the circular footing, the values of \(I_{H}\) obtained from the present analysis were compared with the results from [PERSON] (1975) and [PERSON] (1975).