**1. Introduction**

Reinforced concrete retaining walls (RCRWs) are referred to as structures that withstand the pressure resulting from the difference in the levels caused by embankments, excavations, and/or natural processes. Such situations frequently occur in the construction of several structures, such as bridges, railways, and highways. Due to the frequent application of RCRWs in civil engineering projects, minimizing the construction cost of such structures is an issue of crucial importance.

The satisfaction of both geotechnical and structural design constraints is a key component in the design of RCRWs. In most cases, primary dimensions are initially estimated based on reasonable assumptions and the experience of the designer. Then, in order to reach a cost-effective design while satisfying the design constraints, the design variables (particularly the wall dimensions) need to be revised by using a trial-and-error process, which makes it rather grueling. On the other hand, there is no guarantee that the final design will be the best possible one. To eliminate this problem, which can hinder the designer from reaching a cost-effective solution, and by considering the advances in computational technologies during the recent decades, it makes sense to express the design in the form of a formal optimization problem.

The design optimization of RCRWs has received significant attention during the last two decades. Some of the pertinent works are briefly investigated herein. As a benchmark work, Saribas and Erbatur [1] used a nonlinear programming method and investigated the sensitivity of the optimum solutions to parameters such as backfill slope, surcharge load, internal friction angle of retained soil, and yield strength of reinforcing steel. The simulated annealing (SA) algorithm has been also applied to minimize the construction cost of RCRWs [2,3]. Camp and Akin [4] developed a procedure to design cantilever RCRWs using Big Bang–Big Crunch optimization. They captured the effects of surcharge load, backfill slope, and internal friction angle of the retained soil on the values of low-cost and low-weight designs with and without a base shear key. Khajehzadeh et al. [5] used the particle swarm optimization with passive congregation (PSOPC), claiming that the proposed algorithm was able to find an optimal solution better than the original PSO and nonlinear programming. In their work, the weight, cost, and CO2 emissions were chosen as the three objective functions to be minimized. Gandomi et al. [6] optimized RCRWs by using swarm intelligence techniques, such as accelerated particle swarm optimization (APSO), firefly algorithm (FA), and cuckoo search (CS). They concluded that the CS algorithm outperforms the other ones. They also investigated the sensitivity of the algorithms to surcharge load, base soil friction angle, and backfill slope with respect to the geometry and design parameters. Kaveh and his colleagues (e.g., [7–10]) optimized the RCRWs using nature-inspired optimization algorithms, including charged system search (CSS), ray optimization algorithm (RO), dolphin echolocation optimization (DEO), colliding bodies of optimization (CBO), vibrating particles system (VPS), enhanced colliding bodies of optimization (ECBO), and democratic particle swarm optimization (DPSO). Temur and Bekdas [11] employed the teaching–learning-based optimization (TLBO) algorithm to find the optimum design of cantilever RCRWs. They concluded that the minimum weight of the RCRWs decreases as the internal friction angle of the retained soil increases, and increases with the values of the surcharge load. Ukritchon et al. [12] presented a framework for finding the optimum design of RCRWs, considering the slope stability. Aydogdu [13] introduced a new version of a biogeography-based optimization (BBO) algorithm with levy light distribution (LFBBO) and, by using five examples, it was shown that this algorithm outperforms some other metaheuristic algorithms. In this work, the cost of the RCRWs was used as the criterion to find the optimum design. Nandha Kumar and Suribabu [14] adopted the differential evolution (DE) algorithm to solve the design optimization problem of RCRWs. The results of sensitivity analysis showed that width and thickness of the base slab and toe width increases as the height of stem increases. Gandomi et al. [15] studied the importance of different boundary constraint handling mechanisms on the performance of the interior search algorithm (ISA). Gandomi and Kashani [16] minimized the construction cost and weight of RCRWs analyzed by the pseudo-static method. They employed three evolutionary algorithms, DE, evolutionary strategy (ES), and BBO, and concluded that BBO outperforms the others in finding the optimum design of RCRWs. More recently, Mergos and Mantoglou [17] optimized concrete retaining walls by using the flower pollination algorithm, claiming that this method outperforms PSO and GA.

By taking a look at the studies so far reported, it can be noticed that there has been no work done in assessing the effect of using different available methods of determining the bearing capacity on the optimum design of the RCRWs. The current study investigates this important issue. In order to model and design the RCRWs, a code is developed in MATLAB [18]. To reach a design with minimum construction cost, an optimization problem is defined and the construction cost is considered as the single objective function to be minimized. The design criteria, including both geotechnical and structural limitations, are considered as the optimization constraints. The wall geometrical dimensions and the amount of steel reinforcement are used as the design variables. The particle swarm optimization (PSO) [19] algorithm is used to find the optimum solution.

### **2. Design of Retaining Walls**

The design of RCRWs includes two sets of variables. The first set consists of the geometrical dimensions of the concrete wall, namely variables *X*1 to *X*8, that are defined as shown in Table 1 and depicted graphically in Figure 1. It can be seen that these variables fully define the geometry of the structure.


**Table 1.** Description of the geometric variables and their lower and upper bounds.

**Figure 1.** Design variables (*X*1 to *X*8 and *R*1 to *R*4), (redesigned based on [4]).

The second set is the amounts of steel reinforcement for four components of the wall—the stem, heel, toe, and shear key. As such, four variables, *R*1 to *R*4, are introduced to represent the steel reinforcement of the different components, as shown in Table 2. In this paper, a total of 223 possible reinforcement configurations were used, resulting from the combinations of using 3–28 evenly spaced bars, with varying sizes (bar diameter) from 10 to 30 mm. It is worth mentioning that the combinations used for the steel reinforcement, as listed in Table 3, are obtained such that the allowable minimum and maximum amount of steel area per unit meter length of the wall are satisfied as per ACI318-14 code [20].

**Table 2.** Description of the reinforcement variables.



**Table 3.** Steel reinforcement combinations (adopted from [4]).

The allowable value of spacing between longitudinal bars (*ds*all) is defined as follows [20]:

$$ds\_{\rm all} = \max\{25mm, d\_{\rm bl}, 1.33d\_{\rm max}\},\tag{1}$$

in which *d*bl and *d*max are the diameter of longitudinal reinforcements and diameter of greatest aggregate of concrete, respectively. For the purposes of steel reinforcement design, ACI318-14 code [20] has been considered.

In the design of RCRWs, geotechnical stability control and the satisfaction of the structural requirements are mandatory. In geotechnical design, the stability of the structure shall be controlled against possible overturning, sliding, and bearing capacity failure modes. On the other hand, in the structural design phase, each component of the structure, including the stem, toe, heel, and shear key, shall be checked against shear and moment demands [21]. Figure 2 shows all the forces acting on the retaining wall. As shown in this figure, *P*a is the resultant force of the active pressure *p*a per unit length of the wall; *Q*s is the resultant force of the distributed surcharge load *q*; *W*C is the weight of all sections of the reinforced concrete wall; *W*S and *W*T are defined as the weight of backfill behind the retaining wall and the weight of the soil on the toe, respectively; *P*p is the resultant force due to the passive pressure (*p*p) on the front face of the toe and shear key per unit length of the wall; *P*b is the resultant force caused by the pressure acting on the base soil; *q*max and *q*min are the maximum and minimum soil pressure intensity at the toe and the heel of the retaining wall, respectively. In this paper, the water level and the seismic actions were not considered in computing the forces acting on the RCRWs. The pressure distributions on the base and retaining soil have also been illustrated in Figure 2.

**Figure 2.** Forces acting on the retaining wall (redesigned based on [6]).

In this study, Rankine theory was used to evaluate the active and passive forces acting on the unit length of the retaining wall (*p*a and *p*p, respectively), and they are computed as follows [21]:

$$p\_{\rm a} = \frac{1}{2} k\_{\rm a} \gamma\_{\rm rs} H'^2 + k\_{\rm a} q H',\tag{2}$$

$$p\_{\rm P} = \frac{1}{2} k\_{\rm P} \gamma\_{\rm bs} D'^2 + 2c\_{\rm bs} \sqrt{k\_{\rm P}} D'^2,\tag{3}$$

where *k*a and *k*p are the Rankine active and passive earth pressure coefficients, respectively; *D* is the buried depth of shear key; γrs is the unit weight of backfill; *H* is the height of the soil located on the embedment depth of the base slab at the edge of the heel; γbs and ff bs are the unit weight and the cohesion of soil in front of the toe and beneath the base slab, respectively. *k*a is computed as [21]:

$$k\_{\rm a} = \cos \beta \frac{\cos \beta - \sqrt{\cos^2 \beta - \cos^2 \phi\_{\rm rs}}}{\cos \beta + \sqrt{\cos^2 \beta - \cos^2 \phi\_{\rm rs}}},\tag{4}$$

in which β and φrs are the slope and internal friction angle of the back fill, respectively; *k*p is computed as follows [21]:

$$k\_{\rm P} = \tan^2 \left( 45 + \frac{\phi\_{\rm bs}}{2} \right) \tag{5}$$

in which φbs (in degrees) is the internal friction angle of soil in front of the toe and beneath the base slab.

### *2.1. Geotechnical Stability Demands*

A retaining wall may fail due to overturning about the toe, sliding along the base slab, and the loss of bearing capacity of the soil supporting the base [21]. The checks for these three failure modes are described in this section. The safety factor against overturning about the toe is defined as follows:

$$FS\_{\mathcal{O}} = \frac{\sum M\_{\mathcal{R}}}{\sum M\_{\mathcal{O}}} \tag{6}$$

in which *M*R is the sum of the moments tending to resist against overturning about the toe and *M*O is the sum of the moments tending to overturn the wall about the toe [21]. The safety factor against sliding along the base slab is computed as follows:

$$FS\_{\mathbb{S}} = \frac{\sum F\_{\mathbb{R}}}{\sum F\_{\mathbb{D}}} \, \tag{7}$$

where *F*R is the sum of the resisting forces against the sliding and *F*D is the sum of the horizontal driving forces.

The safety factor against bearing capacity failure mode is computed by:

$$FS\_{\rm B} = \frac{q\_{\rm u}}{q\_{\rm max}} \prime \tag{8}$$

in which *q*u is the ultimate bearing capacity of the soil supporting the base slab. The ultimate bearing capacity is the load per unit area of the foundation at which shear failure occurs in the soil. For retaining walls, *q*u is computed as follows [21]:

$$\mathcal{q}\_{\rm u} = \mathcal{c}\_{\rm bs} N\_{\rm c} F\_{\rm cs} F\_{\rm cd} F\_{\rm ci} + q N\_{\rm q} F\_{\rm qs} F\_{\rm qd} F\_{\rm qi} + 0.5 \gamma\_{\rm bs} B' N\_{\rm \gamma} F\_{\rm \gamma s} F\_{\rm \gamma} F\_{\rm \gamma i} \tag{9}$$

where:

$$q = \gamma\_{\rm bs} D\_{\prime} \tag{10}$$

in which *D* is the embedment depth of the toe (base slab). *B* is computed as:

$$B' = B - \mathcal{Z}\varepsilon,\tag{11}$$

where *B* is the width of the base slab. As mentioned earlier, *q*max and *q*min, which are the maximum and minimum stresses occurring at the end of the toe and heel of the structure, respectively, are computed as follows [21]: 

$$q\_{\rm min}^{\rm max} = \frac{\sum V}{B} \left( 1 \pm \frac{6\epsilon}{B} \right) \tag{12}$$

where *e* is defined as the eccentricity of the resultant force, which can be computed as follows:

$$
\sigma = \frac{B}{2} - \frac{\sum M\_{\rm R} - \sum M\_{\rm O}}{\sum V}.\tag{13}
$$

Note that if *e* becomes greater than *B*/6, then some tensile stress will be applied at the soil located under the heel section. In such a case, since the tensile strength of soil is negligible, design and calculations should be repeated.

In Equation (9), the *N* coefficients are used for the modification of the bearing capacity value and the *F* coefficients are used for the modification of the shape, depth, and inclination factors. The bearing capacity factors *N*c, *<sup>N</sup>*q, and *<sup>N</sup>*γ are, respectively, the contributions of cohesion, surcharge, and unit weight of soil to the ultimate load-bearing capacity [21]. Some of these coefficients vary in accordance with the method used, resulting in different values for *q*u. Hence, this issue could affect the final design of the RCRWs. In this paper, the effects of three methods of Meyerhof [22], Hansen [23], and Vesic [24] in computing *q*u were investigated, particularly their effect on the optimized construction cost of the RCRWs. The *N* and *F* coefficients corresponding to each of the three design methods are listed in Appendix A.
