## 3 Suitability of the \"Characteristic Point\" Concept in a Stress-Strain Analysis Framework  
The charts of Figure 3 could be used for the calculation of the settlement of flexible rectangular footings at the so-called \"characteristic point\", \(\rho_{\text{C}\text{H}\text{a}\text{r}}\)[27; 28]. The settlement at the characteristic point of a flexible footing is considered to be the same as the uniform settlement of the respective rigid footing, \(\rho_{R}\). According to [PERSON] [27], the characteristic point of circular footings lies at distance 0.845R from the center, whilst it lies approximately at distance 0.74\(B\)/2 and 0.74\(L\)/2 from the center of the \(B\times L\) rectangular footings; \(B\) and \(L\) are the width and length of the footing, respectively, whilst \(R\) is the radius of the circular footing. As shown in Figure 10a, the settlement at the characteristic point of flexible footings effectively approximates the settlement of the respective rigid footings, but only for Poisson's ratio values of soil smaller than 0.45. As \(\
u\) approaches 0.5, an abrupt reduction in the \(\rho E\)/\(\eta B\) value is observed (see Figure 10b). Such behavior is not observed for the respective \(\rho E\)/\(\eta B\) values of the flexible footing. The settlement of rigid rectangular footings was calculated using the three-dimensional finite element program nestl3d developed by Professors [PERSON] and [PERSON] (the program in question is freely available at [[http://random.engmath.dal.ca/rfem/](http://random.engmath.dal.ca/rfem/)]([http://random.engmath.dal.ca/rfem/](http://random.engmath.dal.ca/rfem/)) [29]). In this parametric analysis, various cases were considered, i.e. rigid rectangular footings with \(L\)/\(B=1\), 1.8, 3.2, 5.6 and 10 over homogenous elastic medium with \(\
u=0\), 0.1, 0.2, 0.3, 0.4, 0.45 and 0.499, and normalized thickness (\(H\)/\(B\)) of the medium in question ranging from 1 to \(z\)/\(B\) with 0.5 interval (plus the case of \(2z\)/\(B\)); \(H\) is the thickness of the compressible medium. The footing width, \(B\), was kept constant and equal to 1 m for all 210 cases analyzed. Eight-noded cubic elements of edge 0.1 m were used. The settlement at the characteristic point was calculated based on the theory of elasticity [30] using Wolfram Mathematica.  
As shown in the \(I_{z}\) vs. normalized depth charts of Figures 11 and 12 for circular and strip footings, respectively, the case of rigid footings deviates greatly from the respective one referring to flexible footings at the characteristic point and despite the fact that the areas bounded by the two curves in each chart are (approximately) equal. This means that the \"characteristic point\" concept, although suitable for homogenous soils (at least for \(\
u\) values smaller than 0.45), is not suitable for non-homogenous soils, as it largely overestimates the settlement near the footing (for \(z\)/\(B<0.35\)) and underestimates it at greater depths (for \(z\)/\(B>0.35\)); \(B\) is the footing width. The contact pressure distribution used for the case of rigid circular footings is given in Figure 8a, whilst the contact pressure distribution for the case of rigid strips is given below [31]:  
\[\sigma_{c}=\frac{2}{\pi}\frac{b}{\sqrt{b^{2}-x^{2}}}, \tag{10}\]  
where \(b\) is the half width of the strip while \(x\) measures from the centerline of the stip.