# New Displacement Method for Free Embedded Cantilever Walls in Sand
### Influence Coefficients of Input Parameters  
The exponent \"a\" in the expression y = x\({}^{a}\) represents the type of relationship between y and x. A value close to 1 for \"a\" indicates a linear relationship, while values smaller or larger than 1 indicate a non-linear relationship. If \"a\" is greater than 0, y increases as x increases, and, if \"a\" is smaller than 0, y decreases as x increases. In essence, \"a\" can be considered as an influence coefficient (I) that signifies the type and strength of the relationship between y and x. However, in Equation (3), there are multiple exponents involved, making it impractical to directly use a single exponent as an influence coefficient. In Equation (3), \(disp_{wall-top}\) is a function of various variables such as \(d\), \(\gamma_{1}\), \(\gamma_{2}\), \(\gamma_{3}\), \(\gamma_{4}\), \(E_{1}\), \(E_{2}\), \(E_{3}\), \(E_{4}\), \(\phi_{1}\), \(\phi_{2}\), \(\phi_{3}\), \(\phi_{4}\), and \(EI\). To determine the influence coefficient between \(disp_{wall-top}\) and \(d\), for example, \(disp_{wall-top}\) is calculated for different values of \(d\) while keeping other variables constant. It is desirable to obtain influence coefficients around 1, so the values of \(d\) and \(disp_{wall-top}\) are normalized using their respective median values. For \(d\) values ranging from 1.6 to 6.4 m, the median \(d\) value is set to 4 m. The normalized values of \(disp_{wall-top}\) and \(d\) are plotted in Figure 14. In this case, the calculated influence coefficient \"a\" is \(-\)3.828. Table 7 displays the median values of the input parameters. It should be emphasized that the calculation of influence coefficients is specific to walls with a height of 4 m and an \begin{table}
\begin{tabular}{c c c c c c c} \hline \(H\) (m) & \(d\) (m) & \(\gamma_{1}=\gamma_{2}=\gamma_{3}=\gamma_{4}\) (kN/m\({}^{3}\)) & \(E_{1}=E_{2}=E_{3}=E_{4}\) (MPa) & I. (MPa) & \(\phi_{1}=\phi_{2}=\phi_{3}=\phi_{4}\) (\({}^{\circ}\)) & \(EI\) (kN-m\({}^{2}\)) \\ \hline
1 & 0.25-0.49 & 15.7 & 5-10 & 0 & 47 & 13,440 \\ \hline \end{tabular}
\end{table} 
Table 6: The input parameters of formula used to predict the displacement measured by [PERSON] and [PERSON] [23]. Figure 13: \(\Delta\mathrm{H/H}\) with respect to d/H from the test by [PERSON] and [PERSON] [23] and the offered formula. embedment depth of 4 m. The values presented in Table 7 represent the medians of the corresponding parameters used in the calculation. The influence coefficients of \(d\), \(\gamma_{1}\), \(\gamma_{2}\), \(\gamma_{3}\), \(\gamma_{4}\), \(E_{1}\), \(E_{2}\), \(E_{3}\), \(E_{4}\), \(\phi_{1}\), \(\phi_{2}\), \(\phi_{3}\), \(\phi_{4}\), and \(E1\) can be calculated in a similar way. The influence coefficients (ICs) on wall displacements are summarized in Figure 15. Among the few parameters that can be controlled in a cantilever wall design, the embedment depth plays a pivotal role in wall-top displacement (WTD) with an influence coefficient of \(1\,\mathrm{I}_{\phi 1}\,|=3.82\). The IC of the soil friction angle on WTD at Zone 2 ranks second with an IC of \(1\,\mathrm{I}_{\phi 2}\,|=2.29\). The IC of the soil friction angle at Zone 1 is smaller than that of Zone 2 with an IC of \(1\,\mathrm{I}_{\phi 1}\,|=1.22\). This difference is attributed to the fact that a larger portion of the potential slip line passes through Zone 2. The IC of the soil friction angle at Zone 3 (\(1\,\mathrm{I}_{\phi 3}\,|=1.57\)) is smaller than that of Zone 2 (\(1\,\mathrm{I}_{\phi 2}\,|=2.29\)) but larger than that of Zone 1 with an IC of \(1\,\mathrm{I}_{\phi 1}\,|=1.22\). The soil friction angle of the wall-front (Zone 4) has an overall smaller IC compared to the zones located at the wall-back (\(1\,\mathrm{I}_{\phi\text{wall-front}}\,|=1\,\mathrm{I}_{\phi 4}\,|=1.81 <1\,\mathrm{I}_{\phi\text{wall-back}}\,|=1\,\mathrm{I}_{\phi 1}\,|+1\, \mathrm{I}_{\phi 2}\,|+1\,\mathrm{I}_{\phi 3}\,|=5.08\)). In other words, the strength of the retained soil is more critical than that of the retaining soil in cantilever walls.