*5.1. Assumptions*

Three different RCRWs with different heights of 4.0, 5.5, and 7.0 m were considered for all three methods examined (MM, HM, and VM). A reasonable incremental step of 0.01 m was considered for the geometric variables. As shown in Table 3, a set of discrete values was considered for the steel reinforcement. It should be noted that the cross-sectional area of the steel reinforcement bars per unit length of the wall (1 m) was used during the optimization process, therefore, the number of used bars

(*<sup>n</sup>*1 to *n*4) needs to be obtained. All the values assumed for the modeling and design of the RCRWs are listed in Table 4.


**Table 4.** Values of input parameters for Examples 1, 2, and 3.

Because of the stochastic nature of the PSO algorithm, 20 independent runs were conducted for each case study corresponding to each method (MM, HM, and VM). In the PSO algorithm, the population was 20 and the maximum number of iterations was set to 6000; both *c*1 and *c*2 parameters were assumed to be equal to 2; *w*min and *w*max as the minimum and maximum value of the weight *w*k, respectively, were considered to be 0.4 and 0.9.

### *5.2. Results and Discussion*

After implementation of the optimization procedure, the best run among the 20 runs was chosen. The convergence history of all examples as Example 1 with *H* = 4.0 m, Example 2 with *H* = 5.5 m, and Example 3 with *H* = 7.0 m are shown in Figures 5–7, respectively. The convergence plots, which correspond to the best runs, start with a larger value of the cost, which is then minimized by the PSO algorithm at the final iteration. Note that since the optimization procedure was performed for a unit meter length of the wall, all the presented costs are per meter of length of the wall.

The optimum design variables, including the geometrical dimensions and the amounts of steel reinforcement for the three examples, are presented in Tables 5–7. As can be seen, all the results were within their allowable ranges defined in Tables 1 and 3. The last two columns of each table present the difference in the design variables for HM and VM with respect to MM as the chosen benchmark method. Based on the results, it can be noted that in general, the dimensions of *X*1, *X*4, *X*6, *X*7, and *X*8 had relatively low sensitivity to the method used, while the other dimensions, namely *X*2 and *X*5, were quite sensitive. Concerning *X*3, it seems to be sensitive to the method used merely for the case of wall height 4.0 m. The amounts of steel reinforcement (variables *R*1, *R*2, *R*3, and *R*4) were highly dependent on the method used.

**Figure 5.** Convergence history of objective function—Example 1.

**Figure 6.** Convergence history of objective function—Example 2.

**Figure 7.** Convergence history of objective function—Example 3.


**Table 5.** Optimum design variables and comparison of the methods for Example 1—*H* = 4.0 m.

**Table 6.** Optimum design variables and comparison of the methods for Example 2—*H* = 5.5 m.


**Table 7.** Optimum design variables and comparison of the methods for Example 3—*H* = 7.0 m.


To show the convergence of the optimization process to the optimum solution, the best solutions in each iteration corresponding to Example 1 (with *H* = 4.0 m) considering the MM case are shown together in Figure 8. Based on this figure, PSO found the optimum solution in the design space very well. In addition, the optimum shapes of the three examples with consideration of the three methods of MM, HM, and VM are shown in Figure 9. From this figure, it can be seen that the three methods resulted in different designs (geometrical dimensions) for all the optimum RCRWs with different heights.

**Figure 8.** The candidate solutions and the final optimum shape (for *H* = 4.0 m, corresponding to the MM method).

**Figure 9.** Optimal dimensions of the RCRWs for all examples with the three methods.

Figures 10–12 show the demand to capacity ratios of the constraints (all constraints *g*1–3 and *g*5–20, except for the constraint *g*4 on *q*min) for the optimum solutions. As can be seen, all the corresponding values were less than 1.0, indicating that the optimum solutions have satisfied all the design constraints.

The constraint on the soil pressure intensity under the heel (*q*min), described as constraint *g*4 in Equation (28), which is one of the fundamental constraints in the retaining walls design, has been excluded from the figures, ye<sup>t</sup> it is examined in detail in Table 8. As described earlier, if *q*min is negative, it means that some tensile stress is developed at the end of the heel, which is undesirable due to the negligible tensile strength of the soil. The final values for *q*min corresponding to each example and method are listed in Table 8. As can be seen, all the values are greater than zero, indicating that no tensile stress appears beneath the heel slab of the RCRWs.

**Figure 10.** Demand to capacity ratio for Example 1.

**Figure 11.** Demand to capacity ratio for Example 2.

**Figure 12.** Demand to capacity ratio for Example 3.

**Table 8.** *q*min (constraint *g*4) values for all three examples and all three design methods.


Next, the construction cost of the optimum designs was examined. The concrete, steel, and total cost for all of the optimum RCRWs with different heights, considering the three methods, are listed in Table 9. As can be seen, for *H* = 4.0 m and *H* = 7.0 m, the MM resulted in the minimum cost compared with the other two methods; also, the cost for HM was less than that for VM. For *H* = 5.5 m, the VM had the minimum cost and the cost for MM was less than that for HM. In addition, based on the results of Table 9 and as also shown in Tables 5–7, the provided amounts of reinforcement varied depending on the method used. Moreover, it was shown that the differences among the methods decreased with increasing the height of the RCRWs.

Figure 13 reveals the variation in cost components (concrete and steel materials) to total cost ratio with respect to the RCRW heights for each method. Based on this figure, it can be concluded that the concrete cost was reduced from 65.64% 52.93% on average with increasing the height, and the steel cost was increased from 34.36% to 47.07% on average with increasing the height.


**Table 9.** Cost of optimum RCRWs for different heights considering the three methods.

**Figure 13.** The contribution of concrete and steel materials in total cost for optimum RCRWs.

### *5.3. Comparative Study*

In order to make a comparison with the literature, a specific example was chosen from the work conducted by Gandomi et al. [6] (Example 2, case 1 of the work [6] for β = 0◦, see corresponding *Table 12* of *that* work). Using the methodology of the present work, all three methods of MM, HM, and VM were considered for the comparison. The results are listed in Table 10. As can be seen in this table, the results obtained for MM, HM, and VM were slightly different to those by Gandomi et al. [6], at least as far as the optimum cost is concerned. This can be attributed to the different analysis method and the different load combinations of ACI318-2014 used. Comparing the results also revealed the accuracy of the code programming used in this paper.


**Table 10.** Comparison of optimum cost obtained in this paper with the work by Gandomi et al. [6].

### *5.4. E*ff*ect of Backfill Slope*

In this section, the effect of the parameter β (backfill slope) on the optimum cost of RCRWs was investigated, considering the three methods of MM, HM, and VM. Herein, this effect was assessed for Example 2—*H* = 5.5 m and the results are presented in Table 11. As can be seen, by increasing β, the cost was increased individually for each method. In addition, there were small differences (accounting for less than 3%) among the MM, HM, and VM methods, except for the case β = 25◦, where the difference was slightly more, at 4.11%.


**Table 11.** Cost (\$/m) of optimum RCRWs for Example 2—*H* = 5.5 m for different β for the three methods.

### *5.5. E*ff*ect of Surcharge Load*

Herein, the e ffect of the parameter *q* (surcharge load) on the optimum cost of RCRWs was investigated, considering the three methods of MM, HM, and VM. Again, Example 2—*H* = 5.5 m was used to investigate the e ffect and the results are presented in Table 12. As shown, the optimum cost increased as *q* increased. In addition, there were small di fferences (accounting for less than 3%) among the MM, HM, and VM methods, except for the case that no surcharge load was applied (i.e., *q* = 0), which led to a di fference of 4.38%.

**Table 12.** Cost (\$/m) of optimum RCRWs for Example 2—*H* = 5.5 m with different *q* for the three methods.

