## 2 Literature Review
### Soil Stiffness  
Measurement of soil stiffness directly in the field can minimize disturbance. The modulus measured in the field corresponds to a very small strain. The in-situ technique is based on measuring the velocity of the stress wave through the ground. The pore fluid response in saturated soil dominates the P-wave or compression wave. As a result, most measured waves are S-waves or shear waves. If the shear wave velocity through the soil deposit is specified as \(\mathrm{V_{s}}\), then the shear modulus (small strain) \(\mathrm{G_{max}}\) is given as [17]:  
\[\mathrm{G_{max}}=\rho\;\mathrm{V_{s}}^{2} \tag{14}\]  
\[\mathrm{G_{max}}=\frac{\gamma}{\mathrm{g}}\;\mathrm{V_{s}}^{2} \tag{15}\]  
where, \(\rho\) = soil density mass, \(\gamma\) = weight of soil, \(\mathrm{G_{max}}\) = shear modulus of seismic conditions, and \(\mathrm{V_{s}}\) = shear wave velocity. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Pile Space**} & \multicolumn{3}{c}{**p-Multiplier, \(\mathrm{p_{m}}\)**} \\ \cline{2-4}  & **Row 1** & **Row 2** & **Row 3 and Upper** \\ \hline
3D & 0.8 & 0.4 & 0.3 \\
5D & 1 & 0.85 & 0.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: p-multiplier according to AASHTO [16]. From the basic theory of elasticity, the modulus of elasticity (E\({}_{\rm s}\)) can be related to the shear modulus of seismic conditions (G\({}_{\rm max}\)) as follows [18]:  
\[\rm E_{s}=2\left(1+v\right)\,G_{\rm max} \tag{16}\]  
where v = poisson's ratio. Researchers from various countries have proposed empirical correlations between soil V\({}_{\rm s}\) values and N-SPT [19; 20; 21; 22]. According to SPT implementation standards, N-SPT is the number of blows required to obtain the last 30 cm of sample tube penetration from a total of 45 cm at each test step. Sample penetration is obtained from a hammer weighing 63.5 kg dropping from a constant free fall height (76 cm) to the anvil mounted on the drill post. The test results cannot be separated from the influence of equipment features and procedures, including the energy content of the hammer. Some of these empirical equations are based on the number of SPT blows with corrected energy (N\({}_{60}\)), and the number of SPT blows corrected with energy and stress (overburden) (N\({}_{1}\))\({}_{60}\)[20; 21]. The main characteristics of each study were the location, soil type, soil stiffness, geological age (if available), type of function used for SPT-V\({}_{\rm s}\) correlation, constant regression coefficient, number of data pairs, and R\({}^{2}\) value. In general, the most suitable function for SPT-V\({}_{\rm s}\) data regression is a function as shown in the following equation [21]:  
\[\rm V_{s}=A\,\times N_{60}{}^{B} \tag{17}\]  
where A and B = constant regression coefficient, N\({}_{60}\) = corrected N-SPT with energy, and V\({}_{\rm s}\) = wave shear velocity at the same depth at which the N-SPT value is measured. [PERSON] and [PERSON] in Hammam and Eliwa [22] stated that the equation for all types of soils, soft soils, and sands based on the number of blows of N-SPT corrected with energy (N\({}_{60}\)) applies to indirect estimates of V\({}_{\rm s}\) as follows:  
\[\rm All\ soils=104.79\,\times\,N_{60}{}^{0.26} \tag{18}\]  
Although it is preferable to determine shear wave velocities directly from field tests, measurement of V\({}_{\rm s}\) is often not economical at all locations [23]. The correlation equation developed between V\({}_{\rm s}\) (m/s) and N-SPT corrected with energy and suitable for all soils by simple regression analysis based on [PERSON] et al. [23] are as follows [20]:  
\[\rm All\ soils=90.75\,\times\,N_{60}{}^{0.304} \tag{19}\]  
\[\rm Sand=96.29\,\times\,N_{60}{}^{0.266} \tag{20}\]  
[PERSON] and [PERSON] [24] conducted a study using a relational database consisting of 416 V\({}_{\rm s}\)-N. The data evaluated refers to soft\(-\)hard clay soils (classified as CH, CL, and SC according to USCS), loose\(-\)very dense silty sand (SM), and loose\(-\)very dense silt (ML).