# Cost-Based Optimum Design of Reinforced Concrete Retaining Walls Considering Different Methods of Bearing Capacity Computation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}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., [7,8,9,10]) 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.

## 2. Design of Retaining Walls

_{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.

_{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].

_{all}) is defined as follows [20]:

_{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.

_{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.

_{a}and p

_{p}, respectively), and they are computed as follows [21]:

_{a}and k

_{p}are the Rankine active and passive earth pressure coefficients, respectively; D′ is the buried depth of shear key; γ

_{rs}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; γ

_{bs}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]:

_{rs}are the slope and internal friction angle of the back fill, respectively; k

_{p}is computed as follows [21]:

_{bs}(in degrees) is the internal friction angle of soil in front of the toe and beneath the base slab.

#### 2.1. Geotechnical Stability Demands

_{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:

_{R}is the sum of the resisting forces against the sliding and ∑F

_{D}is the sum of the horizontal driving forces.

_{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]:

_{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]:

_{c}, N

_{q}, and N

_{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.

#### 2.2. Structural Requirements

_{m}is known as the nominal strength coefficient (equal to 0.9 [20]); A

_{s}is the cross-sectional area of the steel reinforcement; f

_{y}is the yield strength of steel; d is the effective depth of the cross section; and a is the depth of the compressive stress block. The shear strength is estimated by [20]:

_{v}is the nominal strength coefficient (equal to 0.75 [20]); f

_{c}is the specified compressive strength of concrete; and b is the width of the cross section.

#### 2.2.1. Flexural Moment and Shear Force Demands of Stem

_{s}from the intersection of the stem with the foot slab, defined as d

_{s}= X

_{3}− C

_{C}, where C

_{C}is the concrete cover. The flexural moment and shear force demands of the stem at its critical section are computed by the following equations:

#### 2.2.2. Flexural Moment and Shear Force Demands of Toe Slab

_{t}from the front face of the stem (d

_{t}= X

_{5}− C

_{C}). The flexural moment and shear force demands of the toe slab at its critical section are computed by the following equations [6]:

_{2}is the soil pressure intensity at the foot of the front face of the stem; γ

_{c}is the unit weight of concrete; l

_{toe}is the length of the toe slab; and q

_{dt}is the soil pressure at a distance d

_{t}from the foot of the stem front face.

#### 2.2.3. Flexural Moment and Shear Force Demands of Heel Slab

_{h}from the foot of the stem back face (d

_{h}= X

_{5}− C

_{C}). The flexural moment and shear force demands of the heel slab at its critical section are computed by the following equations [6]:

_{bs}is the maximum load due to triangular backfill soil weight at the top of the heel slab; q

_{1}is the soil pressure intensity at the foot of the stem back face; l

_{heel}is the length of retaining wall heel; W

_{bsdh}is the load resulting from triangular backfill soil weight; and q

_{dh}is the intensity of soil pressure at a distance d

_{h}from the stem back face.

## 3. Optimization Problem

#### 3.1. Formulation

**x**) is the objective function; g

_{i}(

**x**) is the i-th constraint; m and n are the total number of constraints and design variables, respectively; R

^{d}is a given set of discrete values from which the individual design variables x

_{j}can take values. In this paper, an exterior penalty function method was used to transform the constrained structural optimization problem into an unconstrained one, as follows [26]:

_{r}and g

_{r,all}are the r-th constraint and its allowable value, respectively; and r

_{p}is a positive penalty parameter, which in this study was assumed to be equal to 25.

#### 3.2. Objective Function

_{RCRW}of the RCRW was considered the single objective function of the optimization problem, to be minimized. The objective function was defined as follows:

_{c}and C

_{st}are the cost per unit volume of concrete and the cost per unit weight of steel reinforcement, respectively; V

_{c}and W

_{st}are the volume of concrete and weight of steel per unit length of the retaining wall, respectively. It is worth noting that the cost of the formwork, casting concrete, vibration, and generally all related labor costs were taken into consideration in the parameter C

_{c}. In addition, the earthwork cost was not considered in the calculation of the total cost of the RCRWs.

#### 3.3. Design Constraints

_{O}, FS

_{S}, and FS

_{B}are the safety factor demands against overturning, sliding, and bearing capacity failure modes, respectively, and FS

_{O,all}, FS

_{S,all}, and FS

_{B,all}are their allowable values.

_{s,min}and A

_{s,max}are the minimum and maximum allowable area of steel reinforcement in accordance with the code, and A

_{s}is defined as the cross-sectional area of steel reinforcement in each section. The subscript st refers to all sections of the RCRW including stem, heel, toe and shear key, and similarly subscripts 5 to 8 are used for the stem, toe, heel and shear key, respectively. The same is also the case for subscripts 9 to 12.

_{n}and V

_{n}are the moment and shear nominal capacity, and M

_{d}and V

_{d}are the moment and shear demands, respectively. M

_{n}and V

_{n}are the flexural moment and shear strength formulated before. The subscripts 13 to 16 are for stem, toe, heel, and shear key. The same is true for subscripts 17 to 20.

_{db}against the allowable space is checked. If the available space is not enough, a hook is added to achieve the additional development length. In this case, a minimum hook development length l

_{dh}and minimum hook length of 12d

_{bh}(d

_{bh}is the diameter of the hooked bar) should be satisfied. The following limitations are considered for stem, toe, heel, and shear key in the design, respectively:

_{23}to g

_{26}as described in Equations (35)–(38) were actually not considered as design constraints in the present study. In fact, during the optimization process, the required development lengths were simply considered and their role in steel cost was computed and added to the construction cost. The cost of shrinkage reinforcement was also added to the total construction cost. Another important point is that the other constraints on the arrangements of steel bars in the wall sections, such as the number of allowable bars, bar size, and bar spacing, were all considered in the optimum design, as can be seen in Table 2.

#### 3.4. PSO Algorithm

**X**

_{i,k}and

**V**

_{i,k}are the current position vector and velocity vector at iteration k, respectively;

**P**

_{i,k}is the best position that the particle has visited;

**P**

_{g,k}is the global best position obtained so far by all the particles in the population; r

_{1}and r

_{2}are random numbers drawn from a uniform distribution in the range of [0, 1]; c

_{1}and c

_{2}are constants, called cognitive and social scaling parameters, and are usually in the range of [0, 2]; w

_{k}known as the inertia weight, has a pivotal role in updating the position and the velocity vectors. In fact, this parameter is used to stabilize the motion of the particles, making the algorithm converge more quickly. In this paper, a linear weight-updating rule was implemented as follows:

_{max}and w

_{min}are the upper and lower bounds of the current weight w

_{k}; and k

_{max}is the maximum number of iterations used. Figure 3 shows a schematic view for the PSO algorithm, i.e., how a particle’s position is updated from one iteration to another toward finding the global optimum.

## 4. Methodology

_{1}to X

_{8}and R

_{1}to R

_{4}) was generated. Then, the PSO algorithm was used to solve the optimum design problem defined above. All the process of optimum design was implemented via a code written in MATLAB [18]. The flowchart of the design optimization process is shown in Figure 4. The next section investigates the effects of the above three methods on the optimum design of RCRWs.

## 5. Design Examples

#### 5.1. Assumptions

_{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.

_{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

_{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.

_{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.

_{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, yet 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.

#### 5.3. Comparative Study

#### 5.4. Effect of Backfill Slope

#### 5.5. Effect of Surcharge Load

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Method | Equation | Condition |
---|---|---|

Meyerhof [22] | ${N}_{\mathrm{q},\mathrm{M}}={e}^{\pi \mathrm{tan}{\varphi}_{\mathrm{bs}}}{\mathrm{tan}}^{2}\left(45\xb0+\frac{{\varphi}_{\mathrm{bs}}}{2}\right)$ | ${\varphi}_{bs}$ in degrees |

${N}_{\mathrm{c},\mathrm{M}}=\left({N}_{\mathrm{q},\mathrm{M}}-1\right)\mathrm{cot}{\varphi}_{\mathrm{bs}}$ | - | |

${N}_{\mathsf{\gamma},\mathrm{M}}=\left({N}_{\mathrm{q},\mathrm{M}}-1\right)\mathrm{tan}\left(1.4{\varphi}_{\mathrm{bs}}\right)$ | - | |

${F}_{\mathrm{cd},\mathrm{M}}=1+0.2\sqrt{{k}_{\mathrm{p}}}\frac{D}{B-2e}$ | ||

${F}_{\mathrm{qd},\mathrm{M}}={F}_{\mathsf{\gamma}\mathrm{d},\mathrm{M}}=1+0.1\sqrt{{k}_{\mathrm{p}}}\frac{D}{B-2e}$ | ${\varphi}_{bs}>{10}^{\circ}$ | |

${F}_{\mathrm{cs},\mathrm{M}}={F}_{\mathrm{qs},\mathrm{M}}={F}_{\mathsf{\gamma}\mathrm{s},\mathrm{M}}=1$ | - | |

${F}_{\mathrm{ci}}={F}_{\mathrm{qi}}={\left(1-\frac{\theta}{{90}^{\circ}}\right)}^{2}$ | $\theta $ in degrees | |

${F}_{\mathsf{\gamma}\mathrm{i}}={\left(1-\frac{\theta}{{\varphi}_{\mathrm{bs}}}\right)}^{2}$ ${\theta}^{\xb0}={\mathrm{tan}}^{-1}\left(\frac{{p}_{a}\mathrm{cos}\beta}{{\displaystyle \sum V}}\right)$ θ is defined as angle of resultant measured from vertical direction | for ${\varphi}_{\mathrm{bs}}>0$ | |

Hansen [23] | ${N}_{\mathrm{q},\mathrm{H}}={N}_{\mathrm{q},\mathrm{M}}$ | - |

${N}_{\mathrm{c},\mathrm{H}}={N}_{\mathrm{c},\mathrm{M}}$ | - | |

${N}_{\mathsf{\gamma},\mathrm{H}}=1.5\left({N}_{\mathrm{q},\mathrm{H}}-1\right)\mathrm{tan}{\varphi}_{\mathrm{bs}}$ | - | |

${F}_{\mathrm{cd},\mathrm{H}}=1+0.4\frac{D}{B-2e}$ | $\frac{D}{B-2e}\le 1$ | |

${F}_{\mathrm{cd},\mathrm{H}}=1+0.4{\mathrm{tan}}^{-1}\left(\frac{D}{B-2e}\right)$ | $\frac{D}{B-2e}>1$ | |

${F}_{\mathrm{qd},\mathrm{H}}=1+2\mathrm{tan}{\varphi}_{\mathrm{bs}}{\left(1-\mathrm{sin}{\varphi}_{\mathrm{bs}}\right)}^{2}\frac{D}{B-2e}$ | $\frac{D}{B-2e}\le 1$ | |

${F}_{\mathrm{qd},\mathrm{H}}=1+2\mathrm{tan}{\varphi}_{\mathrm{bs}}{\left(1-\mathrm{sin}{\varphi}_{\mathrm{bs}}\right)}^{2}{\mathrm{tan}}^{-1}\left(\frac{D}{B-2e}\right)$ | $\frac{D}{B-2e}>1$ | |

${F}_{\mathsf{\gamma}\mathrm{d},\mathrm{H}}=1$ | - | |

${F}_{\mathrm{cs},\mathrm{H}}={F}_{\mathrm{qs},\mathrm{H}}={F}_{\mathsf{\gamma}\mathrm{s},\mathrm{H}}=1$ | - | |

${F}_{\mathrm{ci},\mathrm{H}}={F}_{\mathrm{qi},\mathrm{H}}-\left(\frac{1-{F}_{\mathrm{qi},\mathrm{H}}}{{N}_{\mathrm{q},\mathrm{H}}-1}\right)$ | ${\varphi}_{\mathrm{bs}}\ne 0$ | |

${F}_{\mathrm{qi},\mathrm{H}}={\left(1-\frac{0.5{p}_{\mathrm{a}}\mathrm{cos}\beta}{{\displaystyle \sum V+\left(B-2e\right){c}_{\mathrm{bs}}\mathrm{cot}{\varphi}_{\mathrm{bs}}}}\right)}^{5}$ | ${\varphi}_{\mathrm{bs}}\ne 0$ | |

${F}_{\mathsf{\gamma}\mathrm{i},\mathrm{H}}={\left(1-\frac{0.7{p}_{\mathrm{a}}\mathrm{cos}\beta}{{\displaystyle \sum V+\left(B-2e\right){c}_{\mathrm{bs}}\mathrm{cot}{\varphi}_{\mathrm{bs}}}}\right)}^{5}$ | ${\varphi}_{\mathrm{bs}}\ne 0$ | |

Vesic [24] | ${N}_{\mathrm{q},\mathrm{V}}={N}_{\mathrm{q},\mathrm{M}}$ | - |

${N}_{\mathrm{c},\mathrm{V}}={N}_{\mathrm{c},\mathrm{M}}$ | - | |

${N}_{\mathsf{\gamma},\mathrm{V}}=2\left({N}_{\mathrm{q},\mathrm{V}}+1\right)\mathrm{tan}{\varphi}_{\mathrm{bs}}$ | - | |

${F}_{\mathrm{cd},\mathrm{V}}={F}_{\mathrm{cd},\mathrm{H}}$ | - | |

${F}_{\mathrm{qd},\mathrm{V}}={F}_{\mathrm{qd},\mathrm{H}}$ | - | |

${F}_{\mathsf{\gamma}\mathrm{d},\mathrm{V}}={F}_{\mathsf{\gamma}\mathrm{d},\mathrm{H}}$ | - | |

${F}_{\mathrm{ci},\mathrm{V}}={F}_{\mathrm{ci},\mathrm{H}}$ | - | |

${F}_{\mathrm{qi},\mathrm{V}}={\left(1-\frac{{p}_{\mathrm{a}}\mathrm{cos}\beta}{{\displaystyle \sum V+\left(B-2e\right){c}_{\mathrm{bs}}\mathrm{cot}{\varphi}_{\mathrm{bs}}}}\right)}^{2}$ | ${\varphi}_{\mathrm{bs}}\ne 0$ | |

${F}_{\mathsf{\gamma}\mathrm{i},\mathrm{V}}={\left(1-\frac{{p}_{\mathrm{a}}\mathrm{cos}\beta}{{\displaystyle \sum V+\left(B-2e\right){c}_{\mathrm{bs}}\mathrm{cot}{\varphi}_{\mathrm{bs}}}}\right)}^{3}$ | ${\varphi}_{\mathrm{bs}}\ne 0$ |

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

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

Geometric Variable | Description | Lower Bound | Upper Bound |
---|---|---|---|

X_{1} | Width of the base slab | 0.4H | 0.8H |

X_{2} | Toe width | 0.1H | 0.6H |

X_{3} | Thickness at the bottom of the stem | 0.2 m | 0.5 m |

X_{4} | Thickness at the top of the stem | 0.2 m | 0.4 m |

X_{5} | Thickness of the base | 0.2 m | 0.3H |

X_{6} | Distance from the toe to the front face of the shear key | 0.5H | 0.8H |

X_{7} | Width of shear key | 0.2 m | 0.4 m |

X_{8} | Depth of the shear key | 0.2 m | 0.9 m |

Reinforcement Variable | Description |
---|---|

R_{1} | Vertical steel reinforcement of the stem |

R_{2} | Horizontal steel reinforcement of the toe |

R_{3} | Horizontal steel reinforcement of the heel |

R_{4} | Vertical steel reinforcement of shear key |

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

Index Number | Bars Quantity and Size (R) | Bars Cross-Sectional Area (cm^{2})(in Ascending Order) |
---|---|---|

1 | 3ϕ10 | 2.356 (lower bound) |

2 | 4ϕ10 | 3.141 |

3 | 3ϕ12 | 3.392 |

4 | 5ϕ10 | 3.926 |

5 | 4ϕ12 | 4.523 |

. | . | . |

. | . | . |

221 | 16ϕ30 | 113.097 |

222 | 17ϕ30 | 120.165 |

223 | 18ϕ30 | 127.234 (upper bound) |

Parameter | Symbol | Value | Unit |
---|---|---|---|

Examples 1–3 | |||

Height of stem | H | 4.0, 5.5, and 7.0 | m |

Steel yield strength | f_{y} | 400 | MPa |

Shrinkage and temperature reinforcement ratio | ρ_{st} | 0.002 | - |

Concrete compressive strength | f_{c} | 21 | MPa |

Concrete cover | C_{C} | 7 | cm |

Unit weight of concrete | γ_{c} | 23.5 | kN/m^{3} |

Unit weight of steel bars | γ_{s} | 78.5 | kN/m^{3} |

Surcharge load | q | 15 | kPa |

Backfill slope | β | 5 | degrees |

Internal friction angle of backfill | φ_{rs} | 36 | degrees |

Unit weight of backfill | γ_{rs} | 17.5 | kN/m^{3} |

Internal friction angle of base soil | φ_{bs} | 39 | degrees |

Cohesion of base soil | c_{bs} | 0 | kPa |

Unit weight of base soil | γ_{bs} | 20 | kN/m^{3} |

Embedment depth of the toe | D | 0.75 | m |

Cost of steel per unit of mass | C_{st} | 0.4 | $/kg |

Cost of concrete per unit of volume | C_{c} | 40 | $/m^{3} |

Variable | Method | Difference (%) | |||
---|---|---|---|---|---|

MM | HM | VM | 100 × (HM − MM)/MM | 100 × (VM − MM)/MM | |

X_{1} | 2.33 | 2.47 | 2.38 | 6.01 | 2.15 |

X_{2} | 0.88 | 1.03 | 1.05 | 17.05 | 19.32 |

X_{3} | 0.37 | 0.34 | 0.32 | −8.11 | −13.51 |

X_{4} | 0.20 | 0.20 | 0.20 | 0.00 | 0.00 |

X_{5} | 0.24 | 0.25 | 0.27 | 4.17 | 12.50 |

X_{6} | 2.01 | 2.01 | 2.05 | 0.00 | 1.99 |

X_{7} | 0.26 | 0.27 | 0.28 | 3.85 | 7.69 |

X_{8} | 0.23 | 0.22 | 0.24 | −4.35 | 4.35 |

R_{1} | 13ϕ12 | 20ϕ10 | 21ϕ10 | 6.84 | 12.18 |

R_{2} | 13ϕ10 | 9ϕ12 | 14ϕ10 | −0.31 | 7.69 |

R_{3} | 13ϕ10 | 12ϕ10 | 9ϕ12 | 0.00 | 8.00 |

R_{4} | 6ϕ12 | 14ϕ10 | 9ϕ10 | 62.04 | 4.17 |

Variable | Method | Difference (%) | |||
---|---|---|---|---|---|

MM | HM | VM | 100 × (HM − MM)/MM | 100 × (VM − MM)/MM | |

X_{1} | 3.20 | 3.24 | 3.16 | 1.25 | −1.25 |

X_{2} | 0.99 | 1.19 | 1.33 | 20.20 | 34.34 |

X_{3} | 0.47 | 0.48 | 0.46 | 2.13 | −2.13 |

X_{4} | 0.21 | 0.20 | 0.20 | −4.76 | −4.76 |

X_{5} | 0.35 | 0.38 | 0.36 | 8.57 | 2.86 |

X_{6} | 2.78 | 2.88 | 2.85 | 3.60 | 2.52 |

X_{7} | 0.28 | 0.26 | 0.29 | −7.14 | 3.57 |

X_{8} | 0.23 | 0.20 | 0.25 | −13.04 | 8.70 |

R_{1} | 13ϕ16 | 22ϕ12 | 13ϕ16 | −4.81 | 0.00 |

R_{2} | 5ϕ16 | 5ϕ20 | 11ϕ14 | 56.25 | 68.44 |

R_{3} | 11ϕ14 | 16ϕ10 | 7ϕ14 | −25.79 | −36.36 |

R_{4} | 12ϕ10 | 7ϕ16 | 7ϕ12 | 49.33 | −16.00 |

Variable | Method | Difference (%) | |||
---|---|---|---|---|---|

MM | HM | VM | 100 × (HM − MM)/MM | 100 × (VM − MM)/MM | |

X_{1} | 4.04 | 3.90 | 3.79 | −3.47 | −6.19 |

X_{2} | 1.34 | 1.51 | 1.76 | 12.69 | 31.34 |

X_{3} | 0.49 | 0.48 | 0.48 | −2.04 | −2.04 |

X_{4} | 0.21 | 0.21 | 0.20 | 0.00 | −4.76 |

X_{5} | 0.45 | 0.49 | 0.56 | 8.89 | 24.44 |

X_{6} | 3.75 | 3.52 | 3.50 | −6.13 | −6.67 |

X_{7} | 0.27 | 0.26 | 0.27 | −3.70 | 0.00 |

X_{8} | 0.21 | 0.21 | 0.21 | 0.00 | 0.00 |

R_{1} | 19ϕ18 | 11ϕ24 | 19ϕ18 | 2.92 | 0.00 |

R_{2} | 14ϕ12 | 16ϕ12 | 17ϕ12 | 14.29 | 21.43 |

R_{3} | 16ϕ14 | 23ϕ10 | 22ϕ10 | −26.66 | −29.85 |

R_{4} | 5ϕ18 | 22ϕ10 | 7ϕ12 | 35.80 | −37.78 |

Variable | Example 1—H = 4.0 m | Example 2—H = 5.5 m | Example 3—H = 7.0 m | ||||||
---|---|---|---|---|---|---|---|---|---|

MM | HM | VM | MM | HM | VM | MM | HM | VM | |

q_{min} | 16.15 | 26.37 | 17.74 | 29.60 | 31.75 | 23.70 | 40.61 | 27.28 | 9.22 |

Variable | Example 1—H = 4.0 m | Example 2—H = 5.5 m | Example 3—H = 7.0 m | ||||||
---|---|---|---|---|---|---|---|---|---|

MM | HM | VM | MM | HM | VM | MM | HM | VM | |

Concrete ($) | 70.07 | 70.60 | 70.27 | 122.00 | 126.39 | 120.80 | 172.15 | 175.37 | 184.04 |

Steel ($) | 34.96 | 37.34 | 38.19 | 77.07 | 76.64 | 75.89 | 160.45 | 160.19 | 152.04 |

Total ($) | 105.04 | 107.94 | 108.46 | 199.08 | 203.03 | 196.68 | 332.61 | 335.56 | 336.08 |

Diff. (%) to MM for total cost | - | 2.76 | 3.26 | - | 1.99 | −1.20 | - | 0.89 | 1.04 |

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

Work | Method | Cost ($/m) |
---|---|---|

Gandomi et al. [6] | - | 162.37 |

This paper | MM | 154.82 |

HM | 158.13 | |

VM | 150.49 |

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

Method | β (degrees) | |||||
---|---|---|---|---|---|---|

0 | 5 | 10 | 15 | 20 | 25 | |

MM | 193.52 | 199.08 | 200.78 | 215.98 | 224.86 | 231.04 |

HM | 195.11 | 203.03 | 197.55 | 212.83 | 221.07 | 240.54 |

VM | 191.05 | 196.68 | 202.96 | 209.59 | 220.57 | 234.34 |

HM to MM (Diff. %) | 0.82 | 1.98 | −1.61 | −1.46 | −1.69 | 4.11 |

VM to MM (Diff. %) | −1.28 | −1.21 | 1.09 | −2.96 | −1.91 | 1.43 |

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

Method | q (kPa) | |||||
---|---|---|---|---|---|---|

0 | 5 | 10 | 15 | 20 | 25 | |

MM | 168.16 | 180.43 | 189.84 | 199.08 | 209.22 | 211.12 |

HM | 172.47 | 180.88 | 186.93 | 203.03 | 203.69 | 213.82 |

VM | 175.52 | 179.31 | 188.78 | 196.68 | 204.12 | 214.50 |

HM to MM (Diff. %) | 2.56 | 0.25 | −1.53 | 1.98 | −2.64 | 1.28 |

VM to MM (Diff. %) | 4.38 | −0.62 | −0.56 | −1.21 | −2.44 | 1.60 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Moayyeri, N.; Gharehbaghi, S.; Plevris, V.
Cost-Based Optimum Design of Reinforced Concrete Retaining Walls Considering Different Methods of Bearing Capacity Computation. *Mathematics* **2019**, *7*, 1232.
https://doi.org/10.3390/math7121232

**AMA Style**

Moayyeri N, Gharehbaghi S, Plevris V.
Cost-Based Optimum Design of Reinforced Concrete Retaining Walls Considering Different Methods of Bearing Capacity Computation. *Mathematics*. 2019; 7(12):1232.
https://doi.org/10.3390/math7121232

**Chicago/Turabian Style**

Moayyeri, Neda, Sadjad Gharehbaghi, and Vagelis Plevris.
2019. "Cost-Based Optimum Design of Reinforced Concrete Retaining Walls Considering Different Methods of Bearing Capacity Computation" *Mathematics* 7, no. 12: 1232.
https://doi.org/10.3390/math7121232