#
White-Tailed Eagle Algorithm for Global Optimization and Low-Cost and Low-CO_{2} Emission Design of Retaining Structures

^{1}

^{2}

^{3}

^{4}

^{5}

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## Abstract

**:**

_{2}emissions. The results of the experiments and comparisons reveal that the WEA is a high-performance algorithm that can effectively explore the decision space and outperform almost all comparative algorithms in the majority of the problems.

## 1. Introduction

- Multiple potential solutions communicate information regarding the search space, resulting in unexpected leaps to the most promising area of the space;
- Several potential solutions collaborate to prevent finding the best solution locally;
- As opposed to single-solution algorithms, population-based metaheuristics allow for more exploration.

_{2}) emissions [11,12]. The main binder used in concrete is Portland cement, and a large amount of CO

_{2}is produced during its manufacturing. The interest in the optimization of concrete structures by taking into account CO

_{2}emissions reduction is justified because the cement industry is responsible for 5% of the world’s greenhouse gas emissions [13]. Therefore, incorporating design criteria to reduce embedded CO

_{2}emissions in reinforced concrete (RC) structures seems essential. Paya-Zaforteza et al. [14] conducted an optimization study comparing CO

_{2}efficiency and the total cost of RC building frames using the well-known simulated annealing (SA) algorithm. Nelson [15] developed a hybrid big bang-big crunch algorithm for multi-objective optimization of CO

_{2}emissions and design cost of reinforced concrete beams. Camp and Assadollahi [16] employed a hybrid big bang-big crunch algorithm for the optimum design of reinforced concrete footings considering CO

_{2}emissions and construction cost. Yepes et al. [17] developed a hybrid glowworm swarm optimization algorithm to optimize total cost and CO

_{2}emissions of concrete road bridges with a double U-shape cross-section.

_{2}emission as the objective function. Therefore, the main contributions of this work can be summarized as follows:

- An effective optimization approach, namely the white-tailed eagle algorithm (WEA) has been developed for global optimization problems;
- The performance of the WEA for numerical function optimization is evaluated on 13 frequently used benchmark functions and compared to other optimization algorithms;
- To verify the effectiveness of the proposed method for the solution of real-world problems, the new method is applied to retaining wall optimization under static and seismic loads;
- In the optimum design of the retaining walls, total construction cost as well as total CO
_{2}emissions are considered objective functions; - A sensitivity analysis is performed to determine the impact of the horizontal acceleration coefficient on the construction cost and CO
_{2}emissions of the structure.

## 2. Related Works

#### 2.1. Swarm Intelligence Algorithms

#### 2.2. Evolutionary Algorithms

#### 2.3. Physics-Based Algorithms (PhA)

#### 2.4. Human-Based Algorithms

## 3. White-Tailed Eagle Algorithm (WEA)

#### 3.1. Inspiration and Behavior of White-Tailed Eagles

#### 3.2. Optimization Algorithm

_{1}and r

_{2}are random numbers in the range of (0, 1).

Algorithm 1. White-Tailed Eagle Algorithm (WEA) |

Determine the parameters N, t_{Max} Generate initial population of eagles using Equation (1) Evaluate eagles’ fitness Rank the eagles based on their fitness Consider the best eagle as ${E}_{Best}$ t = 1 while t < t_{Max} Update the position of each eagle based on Equation (2) Move each eagle toward the prey using Equation (3) Check if any eagle goes beyond the search space limit adjusts it Evaluate eagles’ fitness Rank the eagles based on their fitness Update ${E}_{Best}$ t= t +1 end whileOutput the best solution |

## 4. Comparative Analysis of the WEA

_{Max}). Actually, the termination criterion of the algorithm is the maximum number of iterations.

#### 4.1. Exploitation Validation

_{1}, F

_{2}, F

_{3}, F

_{6}). In comparison to other optimization methods, the obtained results suggest that the novel approach has a vast potential search capacity.

#### 4.2. Exploration Verification

_{8}to F

_{13}) in this study using the presented method. Table 5 show that the Best and Mean values obtained by the WEA for all problems are much better than those obtained by the other methods. Additionally, when compared to the other strategies, the findings indicate that the WEA is a more reliable method in terms of standard deviation.

#### 4.3. Convergence Ability

## 5. Retaining Structure Analysis

_{AE}and P

_{PE}, respectively. H stands for the wall’s overall height, β for the angle of backfill slope, D for the soil depth, q for the surcharge load, and q

_{min}and q

_{max}for the contact pressure’s minimum and maximum values.

_{V}and K

_{h}represent the vertical and horizontal acceleration constants:

## 6. Optimization of Retaining Structure

^{L}and X

^{U}, respectively. The design variables, the objective function, and the design constraints related to the optimization of retaining structures are presented in the following.

#### 6.1. Objective Function

_{2}emission and construction price of the structure that is subject to both static and dynamic loads are taken into consideration as objective functions. Therefore, the goal is to reduce the value of one of these two objective functions. The volume of concrete, excavation, compacted backfill, formwork, and reinforcing steel are considered by both objective functions. The following equation shows the structure’s overall cost:

_{c}, V

_{e}, and V

_{b}represent the volumes of concrete, excavation, and backfill, and W

_{st}is the steel bars weight. A

_{f}displays the formwork area. The unit prices of excavation (C

_{e}), formwork (C

_{f}), reinforcement (C

_{s}), backfill (C

_{b}), and concrete (C

_{c}), are presented in Table 6 [12].

_{2}emissions of retaining walls:

#### 6.2. Design Variables

_{1}represents the heel’s width, X

_{2}stand for the top stem thickness, X

_{3}for the bottom stem thickness, X

_{4}for the toe’s width, and X

_{5}represents the base slab’s thickness. S

_{1}represents the stem’s vertical reinforcement, S

_{2}represents the toe’s horizontal reinforcement, and S

_{3}represents the heel’s horizontal reinforcement.

#### 6.3. Design Constraints

- Overturning Stability Constraint:$$\frac{\mathrm{total}\mathrm{resistant}\mathrm{moments}}{\mathrm{total}\mathrm{overturning}\mathrm{moments}}\ge F{S}_{Odesign}{}_{}$$
_{Odesign}is prescribed factors of safety against overturning. - Sliding stability constraint:$$\frac{totalhorizontalresistantforces}{totalhorizontaldrivingforces}\ge F{S}_{Sdesign}{}_{}$$
_{Sdesign}is prescribed factors of safety against sliding. - Bearing capacity constraint:$$\frac{{q}_{u}}{{q}_{max}}\ge F{S}_{Bdesign}$$
_{u}is the ultimate bearing capacity obtained by the Meyerhof Bearing Capacity Theory [90]; q_{max}is the maximum applied bearing stress. The maximum and minimum contact pressure are defined in the following equation:$${q}_{max,min}=\frac{\sum V}{B}\left(1\pm \frac{6e}{B}\right)$$ - No tension at the foundation:$${q}_{min}\ge 0$$
- Moment capacity of toe, heel and bottom of stem:$${M}_{u}\le 0.9{A}_{s}{f}_{y}\left(d-0.5\times \frac{{A}_{s}{f}_{y}}{0.85{f}_{c}b}\right)$$
_{u}is the ultimate bending moment, f_{c}is the compressive concrete strength, and f_{y}is the steel yield strength. - Shear capacity of toe, heel, and stem:$${V}_{u}\le \frac{1}{6}0.75\sqrt{{f}_{c}}bd$$
_{u}is ultimate shearing force - Limitation of flexural reinforcement:$${\rho}_{min}\le \rho \le {\rho}_{max},\rho =\frac{{A}_{s}}{bd},{\rho}_{min}=\frac{1.4}{fy},{\rho}_{max}=\left(\frac{{0.85}^{2}{f}_{c}}{{f}_{y}}\right)\left(\frac{600}{600+{f}_{y}}\right)$$

## 7. Model Validation

## 8. Model Application and Parametric Study

_{2}emission and construction cost. The results of the cost and emission optimization are compared with each other. To investigate the effect of the horizontal and vertical acceleration coefficients on the total cost and CO

_{2}emission of the wall, a set of six different combinations of K

_{h}and K

_{V}have been considered as presented in Table 9.

_{2}objectives are presented in Table 10 and Table 11, respectively.

_{h}to 0.1 (case 2), the price will rise by 4.7%. When K

_{V}is equal to 0.1 (case 3), the best cost slightly decreases, as it was predicted from Equation (1). In addition, by increasing K

_{h}from 0 to 0.2 (case 4), the best cost increases by 12%, approximately. In cases 5 and 6, by increasing the value of K

_{V}to 0.2, the construction cost will be decreased by up to 1.5%. According to these results, ignoring the K

_{V}is acceptable under the general seismic optimization conditions for the retaining structure.

_{2}optimization, the results of Table 11 reveal that the total amount of CO

_{2}emissions will be increased by up to 4.5% and 11.1%, whereas the horizontal acceleration coefficient varies from zero to 0.1 and 0.2, respectively.

_{2}emission when the vertical acceleration coefficient is equal to zero. Figure 6 and Figure 7 show the parabolic curves of cost and CO

_{2}emission versus different wall heights, respectively.

_{h}is equal to 0.2. The construction cost adjusts to Cost = 64.25H

^{2}− 234.8H + 755.2 with R

^{2}= 0.9994 for K

_{h}= 0.0 and it is equal to Cost = 130.2H

^{2}− 770H + 1951 with R

^{2}= 0.9953 for K

_{h}= 0.2. The results of CO

_{2}emissions presented in Figure 7 are also comparable with the results of the cost objective function. As shown in this figure, by increasing the height of the wall to 11m, the differences between the amount of CO

_{2}emission increases by up to 60%.

_{2}emission and low-cost designs of a wall with a height of 3 m for various values of K

_{h}as the friction angle of the retained soil varies from 28 to 36 degrees. Over this range, for K

_{h}equal to 0.0 and 0.2, both the low-cost and low- CO

_{2}emission designs decrease by approximately 19% as the friction angle increases.

_{2}emission and low-cost designs of a 6 m-height wall for various values of the friction angle of the retained soil. As shown in these figures, the low-cost designs decrease by 26% as the friction angle increases, whereas the low-CO

_{2}emission designs decrease by 30%. These findings indicate that, by increasing the height of the wall, the effect of the friction angle on the optimum design becomes more significant.

## 9. Conclusions and Further Research

_{2}emission designs of retaining structures. This approach mimics the natural behavior of white-tailed eagles. The optimization processes of the WEA are divided into two main phases: exploring the search space effectively and exploiting within a converged search space based on the position of the best eagle (i.e., the best position obtained so far). Several experiments are used to validate the new method’s performance. To study the exploitation, exploration, and convergence speed of the proposed algorithm, a set of diverse benchmark functions were examined. In addition, the findings were compared against GSA, SCA, TSA, and GWO, four well-known and recently created algorithms. According to the findings of the presented study, the following conclusions are obtained:

- The major features of the WEA include its simplicity with just two main parameters, which are ease of coding and ease of implementation;
- Based on the statistical outcomes of the benchmark test problems, the WEA could produce either superior or relatively close results to other well-known competitors;
- Among thirteen considered benchmark problems, the new WEA reached the global optimum for six problems and in early iterations, indicating the robustness of the new method;
- The performance of the new algorithm for optimizing retaining structures subjected to both static and dynamic loading conditions indicates that the WEA design is nearly 5% less expensive than the previous approach;
- The numerical investigations show that, when compared to the other techniques, the newly proposed algorithm for the optimization of retaining structures is quite reliable and effective;
- Finally, seismic optimization results reveal that by increasing the horizontal acceleration coefficient to 0.2, the best cost and best CO
_{2}emission designs will be increased by up to 12% and 11.1%, respectively.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Function | Range | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | n (Dim) |
---|---|---|---|

${F}_{1}\left(X\right)={{\displaystyle \sum}}_{i=1}^{n}{x}_{i}^{2}$ | ${\left[-100,100\right]}^{n}$ | 0 | 30 |

${F}_{2}\left(X\right)={{\displaystyle \sum}}_{i=1}^{n}\left|{x}_{i}\right|+{{\displaystyle \prod}}_{i=1}^{n}\left|{x}_{i}\right|$ | ${\left[-10,10\right]}^{n}$ | 0 | 30 |

${F}_{3}\left(X\right)={{\displaystyle \sum}}_{i=1}^{n}{\left({{\displaystyle \sum}}_{j=1}^{i}{x}_{j}\right)}^{2}$ | ${\left[-100,100\right]}^{n}$ | 0 | 30 |

${F}_{4}\left(X\right)=\underset{i}{\mathrm{max}}\left\{\left|{x}_{i}\right|,1\le i\le n\right\}$ | ${\left[-100,100\right]}^{n}$ | 0 | 30 |

${F}_{5}\left(X\right)={{\displaystyle \sum}}_{i=1}^{n-1}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]$ | ${\left[-30,30\right]}^{n}$ | 0 | 30 |

${F}_{6}\left(X\right)={{\displaystyle \sum}}_{i=1}^{n}{\left(\left[{x}_{i}+0.5\right]\right)}^{2}$ | ${\left[-100,100\right]}^{n}$ | 0 | 30 |

${F}_{7}\left(X\right)={{\displaystyle \sum}}_{i=1}^{n}i{x}_{i}^{4}+random\left[0,1\right)$ | ${\left[-1.28,1.28\right]}^{n}$ | 0 | 30 |

Function | Range | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | n (Dim) |
---|---|---|---|

${F}_{8}\left(X\right)={{\displaystyle \sum}}_{i=1}^{n}-{x}_{i}\mathrm{sin}\left(\sqrt{\left|{x}_{i}\right|}\right)$ | ${\left[-500,500\right]}^{n}$ | 428.9829 × n | 30 |

${F}_{9}\left(X\right)={{\displaystyle \sum}}_{i=1}^{n}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]$ | ${\left[-5.12,5.12\right]}^{n}$ | 0 | 30 |

${F}_{10}\left(X\right)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}^{2}}\right)-\phantom{\rule{0ex}{0ex}}\mathrm{exp}\left(\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+e$ | ${\left[-32,32\right]}^{n}$ | 0 | 30 |

${F}_{11}\left(X\right)=\frac{1}{4000}{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}^{2}-{{\displaystyle \prod}}_{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | ${\left[-600,600\right]}^{n}$ | 0 | 30 |

${F}_{12}\left(X\right)=$ | ${\left[-50,50\right]}^{n}$ | 0 | 30 |

$\frac{\pi}{n}\left\{10\mathrm{sin}\left(\pi {y}_{1}\right)+{{\displaystyle \sum}}_{i=1}^{n-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {y}_{i+1}\right)\right]+{\left({y}_{n}-1\right)}^{2}\right\}\phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \sum}}_{i=1}^{n}u\left({x}_{i},10,100,4\right)$ | |||

${y}_{i}=1+\frac{{x}_{i}+4}{4}$$,u\left({x}_{i},a,k,m\right)=\left\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{m}{x}_{i}a\\ 0a{x}_{i}a\\ k{\left(-{x}_{i}-a\right)}^{m}{x}_{i}-a\end{array}\right.$ | |||

${F}_{13}\left(X\right)=0.1\left\{{\mathrm{sin}}^{2}\left(3\pi {x}_{1}\right)\phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(3\pi {x}_{i}+1\right)\right]\phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\left({x}_{n}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(2\pi {x}_{n}\right)\right]\right\}+{{\displaystyle \sum}}_{i=1}^{n}u\left({x}_{i},5,100,4\right)$ | ${\left[-50,50\right]}^{n}$ | 0 | 30 |

Algorithm (Year) | Parameter | Value |
---|---|---|

WEA (2022) | Number of eagles | 50 |

Iterations’ Number | 1000 | |

GSA (2009) | Agent’s Number | 50 |

Gravitational constant | 100 | |

Iterations’ Number | 1000 | |

GWO (2014) | Agent’s Number | 50 |

Control parameter | [2,0] | |

Iterations’ Number | 1000 | |

SCA (2016) | Agent’s Number | 50 |

Number of elites | 2 | |

Iterations’ Number | 1000 | |

TSA (2020) | Agent’s Number | 50 |

Iterations’ Number | 1000 |

Fun. | Index | WEA | TSA | SCA | GSA | GWO |
---|---|---|---|---|---|---|

F_{1} | Min | 0.00 | 5.238 × 10^{−61} | 1.613 × 10^{−7} | 1.128 × 10^{−17} | 2.513 × 10^{−61} |

Max | 0.00 | 1.218 × 10^{−54} | 2.931 × 10^{−3} | 3.243 × 10^{−17} | 3.754 × 10^{−58} | |

Avg | 0.00 | 8.245 × 10^{−56} | 2.298 × 10^{−4} | 2.276 × 10^{−17} | 4.817 × 10^{−59} | |

Med | 0.00 | 7.221 × 10^{−58} | 1.887 × 10^{−5} | 2.105 × 10^{−17} | 1.132 × 10^{−59} | |

SD | 0.00 | 2.520 × 10^{−55} | 7.875 × 10^{−4} | 5.921 × 10^{−18} | 1.144 × 10^{−58} | |

F_{2} | Min | 0.00 | 1.029 × 10^{−35} | 1.485 × 10^{−9} | 1.473 × 10^{−8} | 8.412 × 10^{−36} |

Max | 0.00 | 3.321× 10^{−32} | 9.796 × 10^{−6} | 3.419 × 10^{−8} | 5.295 × 10^{−34} | |

Avg | 0.00 | 2.233 × 10^{−33} | 1.732 × 10^{−6} | 2.465 × 10^{−8} | 8.421 × 10^{−35} | |

Med | 0.00 | 3.224 × 10^{−34} | 5.342 × 10^{−7} | 2.497 × 10^{−8} | 5.891 × 10^{−35} | |

SD | 0.00 | 6.133 × 10^{−33} | 2.316 × 10^{−6} | 3.898 × 10^{−9} | 9.789 × 10^{−35} | |

F_{3} | Min | 0.00 | 2.575 × 10^{−32} | 70.8285 | 102.955 | 1.311 × 10^{−19} |

Max | 0.00 | 2.452 × 10^{−17} | 267.0 | 468.616 | 3.499 × 10^{−13} | |

Avg | 0.00 | 8.182 × 10^{−19} | 789.1620 | 245.469 | 1.488 × 10^{−14} | |

Med | 0.00 | 1.871 × 10^{−24} | 619.4506 | 221.115 | 2.132 × 10^{−17} | |

SD | 0.00 | 4.468 × 10^{−18} | 746.2287 | 100.102 | 6.612 × 10^{−14} | |

F_{4} | Min | 6.02 × 10^{−224} | 3.318 × 10^{−8} | 1.2610 | 2.312 × 10^{−9} | 9.716 × 10^{−16} |

Max | 3.82 × 10^{−218} | 6.419 × 10^{−5} | 35.6743 | 5.123 × 10^{−9} | 2.332 × 10^{−13} | |

Avg | 6.27 × 10^{−219} | 1.222 × 10^{−5} | 9.3080 | 3.221 × 10^{−9} | 1.872 × 10^{−14} | |

Med | 7.98 × 10^{−220} | 2.110 × 10^{−6} | 6.9806 | 3.191 × 10^{−9} | 6.412 × 10^{−15} | |

SD | 0.00 | 1.717 × 10^{−5} | 8.0720 | 7.398 × 10^{−10} | 4.886 × 10^{−14} | |

F_{5} | Min | 22.441 | 25.6273 | 27.3230 | 25.745 | 25.2273 |

Max | 22.945 | 29.5430 | 49.5110 | 220.911 | 28.7294 | |

Avg | 22.646 | 28.4422 | 29.9106 | 42.2647 | 26.9256 | |

Med | 22.624 | 28.8115 | 29.0097 | 26.1443 | 27.1173 | |

SD | 0.163 | 0.7616 | 4.1508 | 45.4674 | 0.8418 | |

F_{6} | Min | 0.00 | 2.0585 | 3.4070 | 9.669 × 10^{−18} | 0.2466 |

Max | 0.00 | 4.7791 | 4.4435 | 8.712 × 10^{−16} | 1.2619 | |

Avg | 0.00 | 3.6724 | 4.0360 | 3.123 × 10^{−17} | 0.6376 | |

Med | 0.00 | 3.5615 | 4.0572 | 2.889 × 10^{−17} | 0.7452 | |

SD | 0.00 | 0.6918 | 0.2954 | 6.214 × 10^{−18} | 0.3353 | |

F_{7} | Min | 9.764 × 10^{−6} | 6.711 × 10^{−4} | 0.0015 | 0.0061 | 1.492 × 10^{−4} |

Max | 1.459 × 10^{−4} | 0.0036 | 0.0431 | 0.0462 | 2.132 × 10^{−3} | |

Avg | 5.385 × 10^{−5} | 0.0018 | 0.0116 | 0.0237 | 7.885 × 10^{−4} | |

Med | 5.271 × 10^{−5} | 0.0018 | 0.0078 | 0.0222 | 7.111 × 10^{−4} | |

SD | 3.772 × 10^{−5} | 7.726 × 10^{−4} | 0.0101 | 0.0098 | 4.711 × 10^{−4} |

Fun. | Index | WEA | TSA | SCA | GSA | GWO |
---|---|---|---|---|---|---|

F_{8} | Min | −1.242 × 104 | −7.776 × 103 | −5.341 × 103 | −3.713 × 103 | −8.964 × 103 |

Max | −1.182 × 104 | −5.324 × 103 | −3.449 × 103 | −2.122 × 103 | −4.888 × 103 | |

Avg | −1.204 × 104 | −6.598 × 103 | −4.143 × 103 | −2.654 × 103 | −6.161 × 103 | |

Med | −1.193 × 104 | −6.599 × 103 | −3.886 × 103 | −2.854 × 103 | −6.155 × 103 | |

SD | 88.432 | 600.1324 | 341.645 | 359.543 | 848.243 | |

F_{9} | Min | 0.00 | 77.7761 | 1.0560 × 10^{−6} | 8.9546 | 0.00 |

Max | 0.00 | 254.9883 | 51.4451 | 21.8891 | 10.0548 | |

Avg | 0.00 | 151.4539 | 5.9694 | 15.6209 | 0.8853 | |

Med | 0.00 | 149.6596 | 9.3391 × 10^{−4} | 15.9193 | 0.00 | |

SD | 0.00 | 35.8717 | 12.2476 | 3.1043 | 2.4438 | |

F_{10} | Min | 8.882 × 10^{−16} | 1.5099 × 10^{−14} | 1.5579 × 10^{−5} | 2.612 × 10^{−9} | 1.321 × 10^{−14} |

Max | 4.441 × 10^{−15} | 4.3125 | 20.2198 | 4.325 × 10^{−9} | 2.314 × 10^{−14} | |

Avg | 2.664 × 10^{−15} | 2.4095 | 14.3622 | 3.513 × 10^{−9} | 1.623 × 10^{−14} | |

Med | 2.664 × 10^{−15} | 2.9381 | 20.1275 | 3.524 × 10^{−9} | 1.445 × 10^{−14} | |

SD | 1.872 × 10^{−15} | 1.3920 | 8.9778 | 5.211 × 10^{−10} | 2.643 × 10^{−15} | |

F_{11} | Min | 0.00 | 0.00 | 4.8381 × 10^{−7} | 1.6952 | 0.00 |

Max | 0.00 | 0.0159 | 0.7703 | 10.6642 | 0.0140 | |

Avg | 0.00 | 0.0077 | 0.1368 | 4.2510 | 0.0014 | |

Med | 0.00 | 0.0082 | 0.0032 | 3.5667 | 0.00 | |

SD | 0.00 | 0.0057 | 0.2218 | 2.0234 | 0.0041 | |

F_{12} | Min | 1.571 × 10^{−32} | 0.2738 | 0.2631 | 8.203 × 10^{−2} | 0.0121 |

Max | 1.909 × 10^{−32} | 13.8088 | 5.6300 | 0.1037 | 0.0920 | |

Avg | 1.626 × 10^{−32} | 6.3735 | 0.9568 | 0.0198 | 0.0364 | |

Med | 1.578 × 10^{−32} | 6.7411 | 0.4964 | 1.3512 | 0.0329 | |

SD | 1.086 × 10^{−33} | 3.4586 | 1.1497 | 0.0400 | 0.0201 | |

F_{13} | Min | 1.342 × 10^{−32} | 1.7796 | 1.8452 | 1.291 × 10^{−18} | 0.1006 |

Max | 2.046 × 10^{−31} | 4.1077 | 22.5849 | 0.022 | 1.0416 | |

Avg | 6.44 × 10^{−32} | 2.8976 | 3.4211 | 7.198 × 10^{−4} | 0.5280 | |

Med | 3.075 × 10^{−32} | 2.8914 | 2.3552 | 2.034 × 10^{−18} | 0.5238 | |

SD | 7.528 × 10^{−32} | 0.6436 | 3.9911 | 3.011 × 10^{−3} | 0.2359 |

Item | Notaition | Unit | CO_{2} Emission | Unit Cost |
---|---|---|---|---|

Excavation | V_{e} | m^{3} | 13.16 Kg | 11.41 $ |

Formwork | A_{f} | m^{2} | 31.66 Kg | 37.08 $ |

Reinforcement | W_{st} | kg | 2.82 Kg | 1.51 $ |

Backfill | V_{b} | m^{3} | 27.20 Kg | 38.10 $ |

Concrete | V_{c} | m^{3} | 224.34 Kg | 99.49 $ |

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

Height of stem | m | H | 3.0 |

Internal friction angle of retained soil | degree | φ | 36 |

Internal friction angle of base soil | degree | φ’ | 0.0 |

Unit weight of retained soil | kN/m^{3} | γ_{s} | 17.5 |

Unit weight of base soil | kN/m^{3} | γ’_{s} | 18.5 |

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

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

Cohesion of base soil | kPa | c | 125 |

Depth of soil in front of wall | m | D | 0.5 |

Surcharge load | kPa | q | 20 |

Backfill slop | degree | 𝛽 | 10 |

Concrete cover | cm | d_{c} | 7.0 |

Yield strength of reinforcing steel | MPa | f_{y} | 400 |

Compressive strength of concrete | MPa | f_{c} | 21 |

Shrinkage and temporary reinforcement percent | - | ρ_{st} | 0.002 |

Design load factor | - | LF | 1.7 |

Factor of safety for overturning stability | - | FS_{O} | 1.5 |

Factor of safety against sliding | - | FS_{S} | 1.5 |

Factor of safety for bearing capacity | - | FS_{B} | 3.0 |

Design Variable | Unit | Optimum Values WEA (Current Study) | Optimum Values BB-BC [58] | Optimum Values ISA [91] |
---|---|---|---|---|

heel’s width (X_{1}) | m | 0.65 | 0.8732 | 0.8023 |

top stem thickness (X_{2}) | m | 0.2 | 0.2 | 0.2 |

bottom stem thickness (X_{3}) | m | 0.272 | 0.2678 | 0.2875 |

toe’s width (X_{4}) | m | 0.68 | 0.6017 | 0.7536 |

base slab’s thickness (X_{5}) | m | 0.2722 | 0.2722 | 0.27 |

stem’s vertical reinforcement (S_{1}) | cm^{2}/m | 12 | 12 | 13 |

toe’s horizontal reinforcement (S_{2}) | cm^{2}/m | 8 | 8 | 7 |

heel’s horizontal reinforcement (S_{3}) | cm^{2}/m | 8 | 8 | 7 |

Best Cost | $/m | 68.76 | 70.96 | 73.05 |

Case No. | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 |
---|---|---|---|---|---|---|

K_{h} | 0.0 | 0.1 | 0.1 | 0.2 | 0.2 | 0.2 |

K_{V} | 0.0 | 0.0 | 0.1 | 0.0 | 0.1 | 0.2 |

Design Variable | Unit | Optimum Values Case 1 | Optimum Values Case 2 | Optimum Values Case 3 | Optimum Values Case 4 | Optimum Values Case 5 | Optimum Values Case 6 |
---|---|---|---|---|---|---|---|

X_{1} | m | 0.5513 | 0.6539 | 0.613 | 0.7923 | 0.7887 | 0.7672 |

X_{2} | m | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |

X_{3} | m | 0.3567 | 0.0.3743 | 0.3686 | 0.3617 | 0.3364 | 0.3387 |

X_{4} | m | 0.7778 | 0.7632 | 0.7778 | 0.7778 | 0.7778 | 0.7778 |

X_{5} | m | 0.2727 | 0.2847 | 0.2821 | 0.2995 | 0.2997 | 0.2921 |

S_{1} | cm^{2}/m | 8 | 8 | 8 | 9 | 9 | 8 |

S_{2} | cm^{2}/m | 8 | 9 | 8 | 11 | 10 | 10 |

S_{3} | cm^{2}/m | 8 | 9 | 10 | 11 | 10 | 10 |

Best Cost | $/m | 572.74 | 599.3 | 593.1 | 641.3 | 631.91 | 622.73 |

Design Variable | Unit | Optimum Values Case 1 | Optimum Values Case 2 | Optimum Values Case 3 | Optimum Values Case 4 | Optimum Values Case 5 | Optimum Values Case 6 |
---|---|---|---|---|---|---|---|

X_{1} | m | 0.6074 | 0.7113 | 0.6835 | 0.862 | 0.7993 | 0.7825 |

X_{2} | m | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |

X_{3} | m | 0.2929 | 0.304 | 0.3064 | 0.2841 | 0.3251 | 0.32 |

X_{4} | m | 0.7631 | 0.7635 | 0.75 | 0.7735 | 0.7777 | 0.7778 |

X_{5} | m | 0.2728 | 0.2775 | 0.2734 | 0.2907 | 0.2928 | 0.2901 |

S_{1} | cm^{2}/m | 10 | 10 | 10 | 11 | 9 | 9 |

S_{2} | cm^{2}/m | 8 | 9 | 9 | 11 | 10 | 10 |

S_{3} | cm^{2}/m | 8 | 9 | 9 | 11 | 10 | 10 |

Best CO_{2} | kg/m | 740.1 | 773.33 | 762.09 | 822.35 | 812.35 | 805.11 |

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## Share and Cite

**MDPI and ACS Style**

Arandian, B.; Iraji, A.; Alaei, H.; Keawsawasvong, S.; Nehdi, M.L.
White-Tailed Eagle Algorithm for Global Optimization and Low-Cost and Low-CO_{2} Emission Design of Retaining Structures. *Sustainability* **2022**, *14*, 10673.
https://doi.org/10.3390/su141710673

**AMA Style**

Arandian B, Iraji A, Alaei H, Keawsawasvong S, Nehdi ML.
White-Tailed Eagle Algorithm for Global Optimization and Low-Cost and Low-CO_{2} Emission Design of Retaining Structures. *Sustainability*. 2022; 14(17):10673.
https://doi.org/10.3390/su141710673

**Chicago/Turabian Style**

Arandian, Behdad, Amin Iraji, Hossein Alaei, Suraparb Keawsawasvong, and Moncef L. Nehdi.
2022. "White-Tailed Eagle Algorithm for Global Optimization and Low-Cost and Low-CO_{2} Emission Design of Retaining Structures" *Sustainability* 14, no. 17: 10673.
https://doi.org/10.3390/su141710673