# Settlement Prediction of Footings Using \(V_{s}\)
## 2 Development of Settlement Prediction Method Based on \(V_{s}\)
### Stress Distribution Beneath Footings  
As mentioned earlier, the stress transmitted into the soil media caused by the footing load can be derived from load transfer mechanisms, such as the strain influence factor (\(I_{z}\)) or elasticity theory using [PERSON]'s equation. Figure 1 shows \(I_{z}\) profiles for square footing (length to breadth ratio, \(L/B=1.0\)) and strip footing (\(L/B\geq 10.0\)). [PERSON] et al. [6] proposed a simple triangular \(I_{z}\) with a peak value occurring at 0.5 \(B\) depth, and the value converges to zero at 2.0 \(B\) depth for square footing, while the peak and zero values appear at 1.0 \(B\) and 4.0 \(B\) depth, respectively, for strip footing. Further, in [PERSON]'s \(I_{z}\) profile, the peak value increases with increasing applied pressure. This is used when considering the nonlinear characteristics of soil because the fixed modulus calculated from CPT-\(q_{c}\) does not consider the modulus degradation with respect to stress (or strain) level. Without changing the modulus, [PERSON] et al. [6] intended to increase the applied stress within a soil mass to take into account the nonlinear deformation behavior by increasing the peak value of the \(I_{z}\). In the developed method, secant Young's modulus changes based on the nonlinear property of soil, which depends on the initial stress state (i.e., \(V_{S}\)) and design load. On the contrary, in [PERSON]'s method, a fixed modulus converted from CPT-\(q_{c}\) is utilized while the nonlinear soil property is considered by the variation of \(I_{z}\). Thus, in the newly developed method, the \(I_{z}\) should be invariable regardless of the design load so that the nonlinear property of soil is not considered twice in the modulus determination. In fact, [PERSON]'s triangles were proposed for convenience in the application, and their simple shape was determined to accord with the peak value changing with the design load. Although [PERSON]'s triangle has an advantage in engineering practice due to its simplicity, definite error has been reported because of the simplicity. [PERSON] [17] reported that clipping the \(I_{z}\) arbitrarily below \(z=2\)\(B\) for square footing results in 18% unconservative error. As mentioned before, [PERSON]'s method assumes linear elasticity of soil deformation. Similarly, \(I_{z}\) used in [PERSON]'s method can be determined based on the elasticity theory after substituting the triangular \(I_{z}\). In Figure 1, the strain influence factors derived from elasticity theory are shown. The unit vertical strains in cylindrical coordinates could be calculated from the constitutive relationship of [PERSON]'s law:  
\[\varepsilon_{z}=\frac{1}{E}[\Delta\sigma_{z}-2\
u\Delta\sigma_{r}] \tag{7}\]  
where \(\Delta\sigma_{z}\) is the change in vertical stress, and \(\Delta\sigma_{r}\) is the change in radial stress. The vertical stress increment on the axis passing through the center of a circular foundation is derived by integrating the Boussinesq's point loads over a distributed area:  
\[\Delta\sigma_{z}=q\left[1-\frac{1}{[1+(a/z)^{2}]^{1.5}}\right] \tag{8}\]  
where \(q\) is applied stress on the foundation, \(a\) is half of the foundation's diameter (\(d/2\)), and \(z\) is the depth from the foundation base. Similarly, the expression for the radial stress increment is:  
\[\Delta\sigma_{r}=q\left[\frac{1}{2}+\
u-\frac{(1+\
u)}{[(a/z)^{2}+1]^{0.5}}+ \frac{1}{2[(a/z)^{2}+1]^{1.5}}\right] \tag{9}\]  
The above equations can be applied to a circular foundation resting on an elastic, homogeneous, and an isotropic half-space medium. Thus, Young's modulus can be replaced with a constant such as 1, and the vertical strain is determined by the difference between vertical and radial stresses in Equation (7). On the other hand, the \(I_{z}\) addressed in Equation (1) can be calculated as the product of Young's modulus and the vertical strain (i.e., \(I_{z}=E\varepsilon\)) assuming a unit stress increment of 1.