## 2 Derivation of the Strain Influence Factor Charts  
Boussinesq [25] solved the problem of stresses produced at any point in a homogenous, elastic and isotropic medium as the result of a point load applied on the surface of an infinitely large semi-space. Following the notation of Figure 1, the increase in normal stresses caused by the point load \(P\) is:  
\[\Delta\sigma_{x}=\frac{P}{2\pi}\Bigg{(}\frac{3x^{2}z}{L_{1}^{5}}-(1-2\
u) \Bigg{(}\frac{x^{2}-y^{2}}{L_{1}r^{2}(L_{1}+z)}+\frac{y^{2}z}{L_{1}^{3}r^{2}} \Bigg{)}\Bigg{)} \tag{1}\]  
\[\Delta\sigma_{y}=\frac{P}{2\pi}\Bigg{(}\frac{3y^{2}z}{L_{1}^{5}}-(1-2\
u) \Bigg{(}\frac{y^{2}-x^{2}}{L_{1}r^{2}(L_{1}+z)}+\frac{x^{2}z}{L_{1}^{3}r^{2}} \Bigg{)}\Bigg{)} \tag{2}\]  
\[\Delta\sigma_{z}=\frac{3P}{2\pi}\frac{z^{3}}{\left(r^{2}+z^{2}\right)^{\frac{5} {2}}}, \tag{3}\]  
where, \(L_{1}=\sqrt{x^{2}+y^{2}+z^{2}}\), \(r=\sqrt{x^{2}+y^{2}}\) and \(\
u\) is the Poisson's ratio [26]. The increase in the normal stress parallel to the \(i\)-axis due to loading over a whole surface area can be found by integrating the respective stress increase due to point loading over this area:  
\[\Delta\sigma_{i,S}=\int\limits_{x_{1}}^{x_{2}}\int\limits_{y_{1}}^{y_{2}}\Delta \sigma_{i}(x,y,z). \tag{4}\]Note that the differentials of the variables \(x\) and \(y\) (i.e., \(dx\) and \(dy\)) are included in \(\Delta\sigma_{i}\) as the point force at an arbitrary point (\(x\),\(y\)) on the surface of the semi-infinite mass domain is \(P\)=\(qdydx\) (recall Equations (1)-(3); \(q\) is the distributed load over the \(dx\times dy\)). The unit vertical strain can then be calculated from the constitutive relationship of [PERSON]'s law:  
\[\varepsilon_{z}=\frac{1}{E}\Big{[}\Delta\sigma_{z,S}-\
u\Big{(}\Delta\sigma_{x,S}+\Delta\sigma_{y,S}\Big{)}\Big{]}. \tag{5}\]  
(\(E\) is the modulus of elasticity of the compressible medium). The latter can be rewritten as:  
\[\varepsilon_{z}=qI_{z}/E, \tag{6}\]  
where, \(I_{z}\) is the non-dimensional strain influence factor:  
\[I_{z}=\frac{\Delta\sigma_{z,S}-\
u\Big{(}\Delta\sigma_{x,S}+\Delta\sigma_{y,S }\Big{)}}{q}. \tag{7}\]  
The settlement, \(\rho\), at depth \(z_{\alpha}\) corresponding to a layer extending from \(z_{\alpha}\) to a lower depth \(z_{b}\) derives from the integration of Equation (6) between these depths (apparently, at the foundation level, \(z_{\alpha}\) equals zero):  
\[\rho=\int\limits_{z_{a}}^{z_{b}}\varepsilon_{z}dz=q\int\limits_{z_{a}}^{z_{b}} \frac{I_{z}}{E}dz=q\sum\limits_{z_{a}}^{z_{b}}\frac{I_{z}\Delta z}{E}. \tag{8}\]  
Thus, the problem is reduced to defining the proper \(I_{z}\) factor reflecting the shape of foundation in plan view. Using the above equations with the integration limits summarized in Table 1, the \(I_{z}\) versus normalized depth charts for circular and rectangular footings as well as for embankment loadings have been drawn (see Figures 2-7). More specifically, charts (a) to (d) of Figure 2 refer to distance equal to zero (i.e. at the center), \(R\)/3, \(2R\)/3 and \(R\) (i.e. at the perimeter) from the center of flexible circular footings, whilst charts (e) and (f) refer to rigid circular footing on clay and sand, respectively. The case of \"clay\" is distinguished from the case of \"sand\" by the contact pressure distribution; these distributions are shown in Figure 8. More specifically, a contact pressure distribution of parabolic form, where the minimum and maximum pressure value is found at the center and edges of footing, respectively, is more suitable for clays. For smooth rigid footings on clean sand, the minimum and maximum contact pressure are observed at the edges and the center of the footing, respectively [2]. Figures 3 and 4 refer to the center and corner of \(L\)/\(B\) = 1, 2, 3, 4, 5 and 10 rectangular footings, respectively, where \(B\) and \(L\) are the width and length of footing.