# Immediate Settlement of Ring Footings Resting on Inhomogeneous Finite Stratum
## 4 Results and Discussion
As shown, the computed solutions show good agreement with those by [PERSON] and [PERSON] and [PERSON] (1975), although the results of [PERSON] (1975) normally underpredict, resulting in lower values of \(H/D\). Figure 5: Variation of \(I_{K}\) with \(K\) for ring footings on homogeneous infinite soil: (a) inner point; (b) outer point. Figure 6: Variation of \(I_{H}\) with \(H/D\) for ring footings on a homogeneous finite layer overlying rigid base: (a) flexible footing of inner point; (b) flexible footing of outer point; (c) rigid footing. Figure 7 shows the variation of the settlement influence factor \(I_{N}\) of perfectly flexible and rigid ring footings on inhomogeneous elastic half-space for different combinations of \(nD/E_{0}\) and \(R_{i}/R_{0}\). It can be seen that the values of \(I_{N}\) decrease continuously with an increase in \(nD/E_{0}\). At smaller values of \(R_{i}/R_{0}\), the effect of \(nD/E_{0}\) on the results is found to be more predominant. Figure 7 also compares the obtained values of \(I_{N}\) for the flexible and rigid circular footings with the analyses of [PERSON] and [PERSON] [4] using approximate integration technique. It is noted that the present \(I_{N}\) values match well with the solutions of [PERSON] and [PERSON] [4]. From Figures 4-7, it is observed that the ring footing always has lower settlement influence factors than the circular footing, and the settlement influence factor of ring footings decreases with increasing the values of \(R_{i}/R_{0}\). This implies that using narrower ring footings becomes more applicable to reducing the settlement of footings under a given working load. It is also noted that the settlement influence factors of the rigid ring footing fall within the range of the inner and outer settlement influence factors of the flexible footing. The relationships between inner and outer settlements of ring footings are presented in Figure 8, in terms of the footing stiffness \(K\), finite layer thickness ratio \(H/D\), and modulus inhomogeneity \(nD/E_{0}\). Figure 8a indicates that at a given value of \(R_{i}/R_{0}\), the values of \(s_{outer}/s_{inner}\) gradually decrease from 1 at the rigid footing to a certain value at the flexible footing, indicating that a flexible footing imposes non-uniform settlement and a rigid footing induces uniform settlement. It is also observed that the rate of decrease with \(K\) is more extensive for greater values of \(R_{i}/R_{0}\). Also shown in Figure 8a is the \(s_{outer}/s_{inner}\) values suggested by [PERSON] and [PERSON] [4] for circular footings. When compared to the present calculations, the results of [PERSON] and [PERSON] [4] appear to be overpredictive, particulary for lower values of \(K\). Figure 8b demonstrates that as the value of \(H/D\) increases, the value of \(s_{outer}/s_{inner}\) initially decreases to a minimum value at a certain finite layer thickness ratio, and thereafter it increases  
Figure 7: Variation of \(I_{N}\) with \(nD/E_{0}\) for flexible and rigid ring footings on inhomogeneous infinite soil: (**a**) flexible footing of inner point; (**b**) flexible footing of outer point; (**c**) rigid footing. up to a limiting value. This is attributable to the fact that the contribution by the underlying rigid base to the footing settlement increases as the finite layer grows increasingly thin. Figure 8c explains more significant change in \(s_{outer}/s_{inner}\) as the value of \(nD/E_{0}\) increases, suggesting that the variation of modulus with depth has a marked effect on the footing settlement.