# Settlement Prediction of Footings Using \(V_{s}\)
## 2 Development of Settlement Prediction Method Based on \(V_{s}\)
### Modulus Determination of the Soil Layer  
Many of the proposed stress-strain equations are limited in terms of the function of shear strain. In contrast, [PERSON] (1998) reported that hyperbolic equations can be rewritten as a function of shear stress rather than shear strain to describe the nonlinear stress-stain relationship. [PERSON] & [PERSON] [PERSON] and [PERSON] (1998) proposed two additional model parameters \(f\) and \(g\) to have flexibility in changing the shape of the stress-strain curve and to adjust the model to approach failure at finite strain. The proposed equation is given as:  
\[\frac{G}{G_{o}}=1-f\left(\frac{\tau}{\tau_{max}}\right)^{g} \tag{2}\]  
where \(G\) is the secant shear modulus, \(G_{o}\) is the small-strain shear modulus, \(\tau\) is shear stress and \(\tau_{max}\) is the shear strength.  
Previous studies show that this modified hyperbolic model can be used in load-settlement predictions, simple hand calculation [PERSON] and [PERSON] (1998), and finite element studies [PERSON] and [PERSON] (1998); [PERSON] and [PERSON] (1998). It is convenient to adopt this modified hyperbolic model for the estimating footing's settlement because the transmitted stress into the soil media caused by the footing load (or design load) can be derived by load transfer mechanisms, such as the strain influence factor or elastic theory using [PERSON]'s equation. Thus, the modulus reduction can be considered as a function of the mobilized stress level (i.e., G/G\({}_{o}\) versus \(\tau/\tau_{max}\)), and it is more advantageous than using strain.  
While the footing is loaded, the confining stress increases in the underlying soil and this effect can be considered as a change in \(G_{o}\) as:  
\[G_{o}(\sigma_{2})=\left(\frac{\sigma_{2}}{\sigma_{1}}\right)^{n}G_{o}(\sigma_ {1}) \tag{3}\]  
where \(\sigma_{1}\) is the initial effective confining stress, and \(\sigma_{2}\) is the effective confining stress with the footing load. The stress state-dependent shear modulus in Equation (3) can be used in the settlement prediction procedure to reflect a reasonable stress increment in the soil; this approach was validated from previoussettlement predictions [9; 15; 16]. The initial \(G_{o}\) can be directly calculated from \(V_{S}\) and mass density (\(\rho\)) according to the following equation:  
\[G_{o}=\rho V_{s}^{2} \tag{4}\]  
Assuming constant [PERSON]'s ratio, \(\
u\), regardless of strain level, the modified hyperbolic model in Equation (2) can be rewritten as the axial stress-stain formula:  
\[\frac{E}{E_{o}}=1-f\bigg{(}\frac{\Delta\sigma_{v}}{\sigma_{max}}\bigg{)}^{S} \tag{5}\]  
where \(E\) is secant [PERSON]'s modulus, \(E_{o}\) is small-strain [PERSON]'s modulus, and \(\Delta\sigma_{v}\) and \(\sigma_{max}\) are the vertical deviator stress and ultimate bearing capacity, respectively. The secant modulus, \(E\), in [PERSON]'s method can be evaluated by the small-strain shear modulus directly calculated from the \(V_{S}\) and the nonlinear stress-strain equation. Combining the modulus reduction by the stress (or strain) level in Equation (5) with the increasing modulus by effective confining stress increment in Equation (3), the design modulus can be determined as:  
\[E=E_{o}\bigg{(}\frac{\sigma_{ov}+\Delta\sigma_{v}}{\sigma_{ov}}\bigg{)}^{n} \bigg{(}1-f\bigg{(}\frac{\Delta\sigma_{v}}{\sigma_{max}}\bigg{)}^{S}\bigg{)} \tag{6}\]  
where \(E_{o}\) is the initial small strain Young's modulus, \(\sigma_{ov}\) is the theoretically calculated vertical effective stress and \(\Delta\sigma_{v}\) is the vertical stress increment by footing load. Based on Equation (6), the stiffness of soil layers can be calculated considering both the modulus increment by increasing effective confining stress and modulus reduction by stress or strain level using \(V_{S}\) measurements directly.