## 5 Development of Pile-Wall Deflection Model  
For the practical design of a pile-wall in the footing-soil-pile system (FSPS), two empirical equations are developed to predict the pile lateral deflections at the toe (\(\delta_{\text{LT}}\)) and head (\(\delta_{\text{LH}}\)). The predicted deflections consider the induced lateral pressure caused by the ultimate load subjected on an adjacent footing. Also, as a condition to the use of developed equations, the footing-pile distance (\(\text{B}_{\text{fp}}\)) should be equal or less than the critical distance (\(\text{B}_{\text{cr}}\)) calculated using Equation (1). These two equations are derived using multiple linear regression by Minitab 16.0 software [39]. Based upon the geometric and soil strength properties of the FSPS, eight non-dimensional terms are used as the predictors in the model: \(\text{B}_{\text{f}}/\text{L}_{\text{f}}\), \(\text{B}_{\text{f}}/\text{B}_{\text{p}}\), \(\text{B}_{\text{f}}/\text{B}_{\text{fp}}\), \(\text{B}_{\text{p}}/\text{B}_{\text{fp}}\), \(\text{YB}_{\text{f}}/\text{c}\), \(\gamma\text{B}_{\text{p}}/\text{c}\), \(\gamma\text{B}_{\text{fp}}/\text{c}\), and \(1/\tan\phi\). Four terms are found to be significant for predicting pile toe deflection shown in Equation (2), while six terms are found to be significant for predicting pile head deflection shown in Equation (3). Terms with a \(p\)-value less than 0.05 are omitted from the regression results, as recommended by [PERSON] et al. [40]. The values of adjusted R-squared, R-Sq (adj), are 94.3% and 86% for Equations (2) and (3), respectively. \[\frac{\text{B}_{\text{P}}}{\delta_{\text{LT}}}=-72.42+176.65\frac{\gamma\text{ B}_{\text{f}}}{\text{c}}+177.25\frac{\gamma\text{B}_{\text{P}}}{\text{c}}-24.24 \frac{\gamma\text{B}_{\text{fp}}}{\text{c}}+13.51\frac{1}{\tan\phi} \tag{2}\]  
\[\frac{\text{B}_{\text{P}}}{\delta_{\text{LH}}}=-27.73+28.77\frac{\text{B}_{ \text{f}}}{\text{L}_{\text{f}}}-44.73\frac{\text{B}_{\text{P}}}{\text{B}_{ \text{fp}}}+46.45\frac{\gamma\text{B}_{\text{f}}}{\text{c}}+77.48\frac{\gamma \text{B}_{\text{P}}}{\text{c}}-7.35\frac{\gamma\text{B}_{\text{fp}}}{\text{c }}+3.38\frac{1}{\tan\phi} \tag{3}\]  
where \(\delta_{\text{LT}}\) = lateral deflection of pile toe; \(\delta_{\text{LH}}\) = lateral deflection of pile head; \(\text{B}_{\text{p}}\) = pile width; \(\gamma\) = wet unit weight of soil; \(\text{c}\) = unit cohesion of soil; \(\text{B}_{\text{f}}\) = footing width; \(\text{L}_{\text{f}}\) = footing length; \(\text{B}_{\text{fp}}\) = footing-pile distance, which is equal or less than the critical distance (\(\text{B}_{\text{cr}}\)); and \(\phi\) = internal friction angle of soil. All terms in Equations (2) and (3) are non-dimensional; therefore, the units used must be consistent in each term. The ultimate load (\(\text{q}_{\text{u}}\)) subjected on the adjacent footing is calculated using the empirical equation developed by [PERSON] et al. [20], as shown in Equation (4). \[\frac{\text{q}_{\text{u}}}{\text{Q}_{\text{u}}}=1.0409+0.0949\frac{\text{B}_{ \text{f}}}{\text{L}_{\text{f}}}-0.2821\frac{\text{B}_{\text{f}}}{\text{B}_{ \text{fp}}}+0.5529\frac{\text{B}_{\text{P}}}{\text{B}_{\text{fp}}}-0.5961 \frac{\gamma\text{B}_{\text{f}}}{\text{c}}+0.0761\frac{\gamma\text{B}_{ \text{fp}}}{\text{c}}-0.0126\frac{1}{\tan\phi} \tag{4}\]  
where \(\text{Q}_{\text{u}}\) = ultimate bearing capacity of footing without adjacent pile-wall, which is calculated by any approved (classical) equation for an individual footing. The relationships between the observed values, which are generated by three-dimensional finite element analysis, and the predicted values, which are generated by Equations (2) and (3), for the lateral deflections of the pile toe and head are shown in Figures 11 and 12, respectively. Figures 11 and 12 show a considerable agreement between each one of Equations (3) and (4) and the finite element analysis results.