## 4 Result and Discussion
The unit cost of the concrete is selected as $50. It can be seen from Figure 2 that the increase of the depth of the excavation enlarges the pile diameter and extends the pile length. The unit cost of the cantilever soldier pile is calculated as the indirect function of pile diameter and length; therefore, the cost is raised with the deepening of excavation, as expected. The change of the cantilever soldier pile diameter, length, and cost exhibits same trend as the reference case for all the fictionalized other cases. The increased ratios of all the dependent variables are relatively smaller for shallow excavations, but exhibit a raising tendency based on the increment of depth.  
In Figure 3, the effects of the change of soil strength angle are shown for a specific reference case. The shear strength angle is changed between 27\({}^{\circ}\) and 38\({}^{\circ}\) to represent the sand formation's relative density beginning from a loose to a dense state. The depth of excavation is assumed to be constant (\(h\) = 9 m). The surcharge load application is not taken into consideration, and the unit weight of the soil is assumed to be 18 kN/m\({}^{3}\). Figure 3a represents the change of pile length, Figure 3b shows the change of pile diameter, and Figure 3c represents the cost change. The increase of the shear strength angle is directly proportional to the increase of soil strength. Therefore, the rise of the shear strength angle straightens the length of the pile and embedment depth. The relative decrease of pile length is approximately 9% in such a case that the shear strength angle increases from 27\({}^{\circ}\) to 38\({}^{\circ}\). Then, the relative decrease of pile diameter is approximately 25% in such a case that the shear strength angle increases from 27\({}^{\circ}\) to 38\({}^{\circ}\). Therefore, it can be said that the increase of soil the shear strength angle affects the pile diameter more than the length of the pile. However, in geotechnical applications, generally, a standard diameter is preferred to be used in order to reduce the equipment cost. That's why if the pile diameter is selected as a constant value for all the excavation depths, the length of the pile will be directly affected by this change more than the change of the internal friction angle. The cost of the unit pile is influenced from the change of the shear strength angle directly. The rise of the strength of the soil leads to decreasing the costs by approximately 46%. In such a case that the soil 44 shear strength angle increases from 38\({}^{\circ}\) from 27\({}^{\circ}\), the cost of the construction has decreased to $1010 from $1873.  
Figure 2: Change of the diameter of cantilever soldier pile (**a**), the length of pile (**b**), and the unit cost of concrete (**c**) via excavation depth.  
Figure 4 is acquired for the decrease of excavation depth of 6 m. All the other parameters are selected as the same as the case calculated for Figure 3. The relative decrease of pile length is approximately 9% when the increase of the shear strength angle is from 27\({}^{\circ}\) to 38\({}^{\circ}\) again for 6 m excavation depth. However, the relative decrease of pile diameter in such a case that the increase of the shear strength angle is from 27\({}^{\circ}\) to 38\({}^{\circ}\), is approximately 17%. Similarly with Figure 3, the rise of the strength of the soil leads to the costs decreasing by approximately 46%. In such a case that the soil shear strength angle increases to 38\({}^{\circ}\) from 27\({}^{\circ}\), the cost of the construction has decreased to $170 from $315. As a result, the decrease of excavation depth reduces the influence ratio of the change of the shear strength angle. Besides, this reference case is analyzed for 3 m excavation depth, but there is no noticed change of pile length or dimension due to the change of soil shear strength angle.  
The change of soil shear strength angle is also evaluated according to the change of excavation length. Figure 5 is illustrated to understand the change of pile length (a), pile diameter (b), and unit pile cost (c), respectively. It can be directly seen from the comparison of Figure 5a,b that the change of shear strength angle has a greater effect on the pile diameter than the pile length, especially for deeper excavations. This condition causes encountering unexpected pile lengths for deep excavation projects. In Figure 5, the main dimension change begins to occur at the depth of 5 m. In the context of this study, according to the worked parameters, it can be a general approach that 5 m excavation depth is assumed to be a limit length of the change of the design. It is a noticeable point that for loose sands that have a shear strength angle between 27\({}^{\circ}\) and 30\({}^{\circ}\) according to [PERSON] [14], the optimization analysis can't find an appropriate cross-section, which is ensuring both geotechnical and structural design requirements.  
Figure 4: The effects of the change of shear strength angle to the diameter of the cantilever soldier pile (**a**), the length (**b**), and the unit cost of pile (**c**) (for h = 6 m excavation depth).  
Figure 3: The effects of the change of the shear strength angle to the diameter of the cantilever soldier pile (**a**), the length (**b**), and the unit cost of the pile (**c**) (for h = 9 m excavation depth).  
For this reason, approximately 10 m depth can be assumed to be the upper bound of the applicable excavation with the construction of cantilever soldier piles.  
The change of the unit weight of the soil profile is also investigated within the context of present study. Figure 6 is drawn for 9 m excavation depth to reflect the effect of soil unit weight on the dimension and cost change of the pile. The shear strength angle of the surrounding soil is assumed to be 32\({}^{\circ}\), and surcharge load is not applied. Then, the unit weight of the soil is increased beginning from 14 kN/m\({}^{3}\) to reach 20 kN/m\({}^{3}\) one by one. Figure 6a shows that the increase ratio of the pile length is approximately 7% for a 6 kN/m\({}^{3}\) increase of unit weight. The increase of unit weight of soil leads to widening the pile diameter by nearly 22%. This condition is based on the computation equations of both active and passive lateral forces. The unit weight is the direct multiplier of these equations. The increase of unit weight affects passive lateral forces more than active lateral forces due to the relative difference between active and passive lateral earth coefficients. Active lateral earth pressure is determined by the Rankine Active Earth Pressure Theory with the \(K_{a}=tan^{2}\) (\(45-\Phi/2\)) equation, and passive lateral earth pressure is calculated by Rankine Passive Earth Pressure Theory with \(K_{p}=tan^{2}\) (\(45+\Phi/2\)). The difference of coefficients is directly related to the shear strength angle. However, the shear strength angle is not a direct multiplier of lateral forces. Therefore, it can be said that the change of the unit weight of the soil influences the unit cost of pile more than the change of shear strength angle for the fictionalized cases of this study within the selected parameter limits. The change in the unit cost of the pile can be calculated to be approximately 66% from Figure 6c. In such a case that the unit weight of soil increases to 20 from 14 kN/m\({}^{3}\), the cost of the construction has increased to $1642 from $1017.  
Figure 5: The effects of the change of shear strength angle to the diameter of the cantilever soldier pile (**a**), the length (**b**), and the unit cost of pile (**c**) (for h = 6 m excavation depth) via the excavation depth.  
Figure 6: The effects of the change of soil unit weight to the diameter of cantilever soldier pile (**a**), the length (**b**), and the unit cost of pile (**c**) (for h = 9 m excavation depth).  
The same reference case with common variables is evaluated in such a case that the excavation depth is assumed to be 6 m. Figure 7 shows the change of pile length, diameter, and unit cost, respectively. The change of soil unit weight causes obtaining a similar behavior trend as shown in Figure 6. The increase ratio of the pile length between the lower and upper limit of the envisaged values of unit weight is computed to be 7%. This pile length increment ratio was found to be same for all fictionalized analyses conducted for the evaluation of the unit weight change.  
The change of pile diameter is determined to be approximately 20%, and the unit cost change is calculated to be 64% between the upper ($278) and lower limits of the soil unit weight ($170). The same reference case is applied to the condition that the excavation depth is 3 m. However, similar to the previous analyses performed in the change of the shear strength angle of soil, the change of the soil unit weight has no effect for relatively smaller excavations. In addition to all of these, in representative charts are given in Figure 8 to realize the alteration effect of soil unit weight via the increase of depth. The increase tendency of the pile length and diameter can be classified as the same as those shown in Figure 5. After 4 m excavation depth, the change of pile diameter dominantly governs the relationship between stability and dimensions. This situation can be the proof that the change of diameter has a significant effect on the stability of piled systems. However, different from Figure 5, it is a noticeable point that the optimization algorithm can find an appropriate cross-section for deep excavations, although the soil unit weight increases.  
Figure 8: The effects of the change of soil unit weight to the diameter of the cantilever soldier pile (**a**), the length (**b**), and the unit cost of pile (**c**) (for h = 6 m excavation depth) via the excavation depth.  
Figure 7: The effects of the change of soil unit weight to the diameter of the cantilever soldier pile (**a**), the length (**b**), and the unit cost of pile (**c**) (for h = 6 m excavation depth).  
For the evaluation of the influence of surcharge load on the behavior of cantilever soldier piles, only the load magnitudes are selected as variables, and the other mentioned variable parameters are fixed as specific values. Three reference excavation depths are selected (\(h\) = 9-6-3 m), and a specific soil profile with 18 kN/m\({}^{3}\) unit weight and 32\({}^{\circ}\) shear strength angle is used for the comparison of the effects of surcharge load application adjacent to the top of the retaining pile system. The change of surcharge load that is applied on the active side of the structural system is taken into account by selecting the load values 0, 5, 10, 15, and 20 kPa. In Figure 9, the reference excavation depth is selected as 9 m, and dimensions of the systems are selected according to this situation. The increase of the external loading lengthens and enlarges the pile and raises the unit costs as expected. The length of the pile is increased approximately 5%, and the diameter of the pile is increased 20% by the differentiation of surcharge loading between the upper and lower bounds. Besides this, the increase of surcharge load to 20 kPa raises the material costs 50%, and this ultimate rising happens directly proportional with the increase steps of load. The cost has raised to $2126 from $1422 with the increase of surcharge load to 20 kPa.  
Figure 10 is drawn for 6 m excavation depth to examine the surcharge loading effect for relatively shallow depth.  
In this case, the length of the pile is raised 8%, the pile diameter is increased 28%, and the unit cost is increased 77% due to the increase of external load to 20 kPa. The cost has raised to $423 from $239 with the increase of surcharge load to 20 kPa.