## 5 Discussion  
The discussion herein is facilitated through an application example. Let there be a flexible rectangular footing with \(B=2.6\) m and \(L/B=2\). The footing is founded on the surface of a soil medium for which the tip resistance (\(q_{c}\)) of the CPT apparatus is known; see the \(q_{c}\)-\(z\)/\(B\) chart in Figure 13a. The \(q_{c}\) value is available every 0.05 m (that is, \(\Delta z=0.05\) m). [PERSON]'s ratio of soil is \(\
u=0.3\) (assumed to be the same throughout the soil mass). The \(I_{z}\)-\(z\)/\(B\) curve for the Poisson's value in question has also been drawn on the same chart (curve also appearing in Figure 3b). This curve is effectively represented (R\({}^{2}=0.9999\)) by the following polynomial (MS Office Excel was used):  
\[ \begin{array}{c}I_{z}(\
u=0.3;L/B=2)=-0.0294\Big{(}\frac{z}{B}\Big{)}^{6}+0.3 303\Big{(}\frac{z}{B}\Big{)}^{5}-1.4858\Big{(}\frac{z}{B}\Big{)}^{4}+\\ 3.37\Big{(}\frac{z}{B}\Big{)}^{3}-3.8694\Big{(}\frac{z}{B}\Big{)}^{2}+1.6464 \Big{(}\frac{z}{B}\Big{)}+0.5036\end{array} . \tag{13}\]  
The \(I_{z}\)-\(z\)/\(B\) curve for any \(\
u\) value can easily be reproduced using an adequate number of points in a spreadsheet. Additionally, if the soil mass consists of successive layers with different \(\
u\) values, the \(I_{z}\)-\(z\)/\(B\) curve corresponding to the correct \(\
u\) value should be used for each layer. The elastic modulus of soil is often obtained from soil probing tests (e.g., CPT or SPT) relying on an empirical relationship. Indeed, tens of such relationships correlating probe test parameters with \(E\) exist in literature (e.g., [4; 51; 52; 53; 54; 55; 56]). The most common forms of these relationships referring to the CPT and SPT tests are summarized below:  
\[E=\left\{ \begin{array}{l}a_{E}q_{c}\\ a_{E}q_{c}+b_{E}\\ a_{E}(q_{t}-\sigma_{v0})\\ a_{E}N+b_{E}\text{ or }a_{E}(N+b_{E})\\ a_{E}N^{b_{E}}\left(b_{E}=0.5\text{ or }1\right)\\ a_{E}\ln N\end{array} \right\}\text{for the CPT test}, \tag{14}\]  
where \(a_{E}\) and \(b_{E}\) are empirical coefficients, \(N\) is the number of blows required for 12 inches penetration resistance of the soil, \(\sigma_{v0}\) is the vertical geostatic stress and \(q_{t}\) is the corrected cone resistance. The coefficients \(a_{E}\) and \(b_{E}\) are site-specific and depend on the type and relative density of the soil and whether the soil is normally consolidated or overconsolidated, as well as whether it is saturated or not. In the present example, an empirical correlation of the form \(E=a_{E}q_{c}\) has been used. Therefore, for the example examined herein, Equation (8) can be rewritten as follows:  
\[\frac{a_{E}\rho}{q}=\sum_{0}^{2.6B}\frac{I_{z}\Delta z}{q_{c}}. \tag{15}\]  
According to [PERSON] et al. [2], the influence depth of footing is approximately equal to \(z_{t}=\)\(2B(1+\log L/B)\); in this respect, \(2.6B\). The \(I_{z}\Delta z\)/\(q_{c}\)-\(z\)/\(B\) chart is given in Figure 13b for \(z\)/\(B\) up to \(3B\), giving, in essence, the contribution of each soil substratum of thickness \(\Delta z\) to the settlement of footing. The cumulative \(I_{z}\Delta z\)/\(q_{c}\) values with respect to \(z\)/\(B\) are given in Figure 13c. \(z\)/\(B\) is the depth in normalized form. The footing shape is also known to affect the modulus of elasticity of soil and, in turn, the elastic settlement of footing. For considering the effect of the footing shape on the elastic settlement, [PERSON] et al. [24] and later [PERSON] et al. [2] adopted [PERSON]'s [57]\(E_{p}\)/\(E_{t}=1.4\) relationship between the axisymmetric and the plane strain loading case (\(E_{t}\) and \(E_{p}\) are the triaxial and plane strain moduli, respectively; the validity of [PERSON]'s relationship is discussed by the author in in [11]). In this respect, [PERSON] et al.