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Cantilever Soldier Pile Design: The Multiobjective Optimization of Cost and CO_{2} Emission via Pareto Front Analysis

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## Abstract

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_{2}emission considering the change in the excavation depth, the shear strength parameters of the foundation soil strata, and the unit costs and unit emission amounts of structural materials. Considering this aim, the harmony search algorithm was used as a tool to achieve the integrated effects of the solution variants. The lateral response of the soil mass was determined based on the active Rankine earth pressure theory and the design process was shaped according to the beams on the elastic foundation soil assumption. Moreover, the specification envisaged by the American Concrete Institute (ACI 318-11) was used to control the structural requirements of the design. Pareto front graphs and also design charts were created to achieve the eco- and cost optimization, simultaneously, for the design with arbitrarily selected cases to compare the results of the multiobjective analysis to minimize both the cost and the CO

_{2}emission.

## 1. Introduction

_{2}emission values are simultaneously minimized by using multiobjective HS and GA algorithms. Moreover, investigations are conducted to measure the differentiation abilities of the design variants to achieve the sustainable design objective, such as the shear friction angle of the foundation soil strata, the unit costs of the construction materials, and the unit CO

_{2}emission values of the construction materials on the CSP wall design. Pareto-optimal solutions are obtained with the investigation of the trade-off relations among the design objectives. Moreover, comparative analyses are conducted to search for the effectiveness of the used algorithm on the Pareto-optimal analyses of CSP wall designs. As a result of the study, in light of the mentioned aims, design charts and tables are arranged to easily obtain an environmentally friendly and lower-cost design at the same time.

## 2. Design and Methodology

#### 2.1. The Design of Cantilever Soldier Pile Walls

_{pnt}is the depth of the penetration of the pile, ΣL illustrates the length of the pile that is determined by the sum of the excavation and penetration depths of the pile, and D is the diameter of the pile. Figure 1b illustrates the stress distribution along with both active and passive sides of the pile depending on beams on the elastic soil. In Figure 1b, q

_{a}defines the surcharge loading that is converted to F

_{qa}as a lateral surcharge force. In addition, F

_{sa}and F

_{p}define the lateral soil reaction forces generated in the active and passive states, respectively. Φ and γ represent the shear strength angle (friction angle) and the unit weight of the envisaged frictional soil strata, respectively.

_{a}and K

_{p}, respectively (Equations (1) and (2)).

_{0}. E represents the elasticity modulus and I is the inertia of the pile material. In Figure 1c, a single lateral force, F

_{0,}and in Figure 1d, a moment M

_{0,}is applied at the edge of the beam depending on the beams on the elastic soil approach. The change in the system deflection, rotation, bending moment, and shear force can be calculated with the use of Equations (3)–(6), respectively. The terms a

_{wp}, a

_{wm}, a

_{tp}, a

_{tm}, a

_{mp}, a

_{mm}, a

_{vp}, and a

_{vm}characterize the coefficients which refer to deflection, rotation, bending moment, and shear force, respectively, that were proposed by Celeb and Kumbasar [21].

_{0}if the soil reaction coefficient k·(k = k

_{0}·z) remains constant along the depth z. In addition, Equation (8) is preferred to be used in a case if the coefficient of soil reaction k·(k = k

_{0}′z/D) is increasing linearly along with the depth. The length of the penetration of CSP is obtained mathematically by the multiplication of the characteristic length of the CSP with “π” to behave like a beam that is supported by an elastic soil stratum.

#### 2.2. Multiobjective Optimal Design of Cantilever Soldier Piles

_{2}emission and total cost values to reach a sustainable design. HS was used to control the minimization process of cost and CO

_{2}emission through the attainment of geotechnical stability conditions and the supplementation of structural design requirements. A multivariant analysis was also performed to compare the effects of design parameters on the design loop and the effectiveness of the algorithms. As a result, the applicability of the HS algorithm for Pareto-front analysis was controlled depending on the variants of the analysis.

#### 2.2.1. Harmony Search Algorithm (HS)

_{2}gas to the atmosphere. In the next step of HS, a distinctive harmony element is produced. HMCR and PAR parameters are used for generating a new harmony element similar to a single-objective HS. The HMCR value equals 0.7, and the PAR value equals 0.2. The main difference, in this case, is a distinct element is generated that is different from any harmony elements in the memory to eliminate dealing with duplicated elements. After the element is generated as a new harmony, fitness values are calculated for this new element. Generating the Pareto front is an additional procedure in multiobjective algorithms. As shown in Figure 3, Pareto front elements are dominant elements in the population taking into account whole objectives, i.e., in this study, an element in the Pareto front is cheaper and emits less CO

_{2}compared to every non-Pareto element. To determine the worst element in the harmony memory, distances from non-Pareto elements to the Pareto front are calculated. Figure 3 shows the Pareto front with line parts between Pareto elements. In addition, the curve of the Pareto front is divided into four parts of the lines. To calculate the distance from a non-Pareto element to the Pareto front, firstly, all perpendicular distances (i.e., h

_{1}, h

_{2}, h

_{3}, and h

_{4}values in Figure 4) between these line parts ${l}_{i}$ (i.e., the line segments between Pareto elements in Figure 3) and the non-Pareto element are calculated. Then, the minimum value of these perpendicular distances is asserted as the distance from this non-Pareto element to the Pareto front. The element with a maximum distance to the Pareto front is labeled as the worst element in HM. The fitness function returns fitness values as a vector containing two elements, as shown in Equation (1). After the Pareto front is generated, the stop criterion is checked. In this study, the upper limit of the iteration number equals 10,000. If the total cycle count does not reach the upper limit, the HS continues. It may model the bases of the selection of iteration numbers by focusing on the study by Geem et al. [9], which shows that after 1095 iterations, the optimization finds the same solution.

#### 2.2.2. The Variables and the Constraints of the Cantilever Soldier Pile Design Problem

## 3. Parametrical Analysis

^{3}. The analyses are also focused on the holistic solution of cost and CO

_{2}emission for different depths of excavation. The depth of excavations is selected beginning from 4 and reaching 12 m based on the construction limits defined in the national and international references [23,24]. The characteristic length of the CSP can be determined by the use of the Ks, which is assumed to be 200 MN/m

^{3}in this study. The amount of the surcharge loading is assumed constant, at 10 kPa. The integrated alteration of relationships was investigated by the use of different unit costs and unit CO

_{2}emission values of the structural materials. The unit cost of the concrete per m

^{3}was assumed as USD 50, 75, 100, 125, and 150, respectively, and the unit cost of the steel of the reinforcing bars per ton was defined as USD 700, 800, 900, 1000, and 1100, respectively. The constants and variables of the CSP wall design are also given in Table 3. In addition, some cases are modeled with the assumption that the CO

_{2}emission values are selected to be different based on two different literature sources. Yeo and Potra [25] used 376 kg as the CO

_{2}emission amount of concrete and 352 kg as the CO

_{2}emission amount of steel for the recycled type of steel. Paya et al. [26] suggested using a CO

_{2}emission amount of steel of 3010 kg and a CO

_{2}emission amount of 143.48 kg for the concrete.

_{2}emissions approximately exhibit the upper and lower limits of the available values envisaged within the literature sources. These unit amounts of CO

_{2}emission values for both steel and concrete were used to arrange four different emission couples. The couples of CO

_{2}emission values are shown in Table 4. Couple 3 represents the lower limits and couple 4 represents the upper limits of material emission values.

_{c}and C

_{s}are concrete and steel cost prices, and a cost couple is picked in order from the 9th and 10th rows in Table 3. Different shear strength angles “ϕ” are selected by using the third loop. Additionally, in the last loop, each of the emission couples is selected in order from Table 4. By these loops, conditions are generated. For each condition, a Pareto front of optimum designs is obtained by using each genetic algorithm and harmony search algorithm. The term “Generate optimum results based on the condition” includes the whole optimization process (by using both GA and HS methods) that finds optimum pile designs under the related condition.

_{2}emission values of each design. These Pareto fronts for other conditions are obtained by applying loops in Figure 4. These values constitute a Pareto front given in figure which represents a trade-off between the cost and emission values.

## 4. Results and Discussions

_{2}emission (Figure 6), the total length of the pile and the total cost (Figure 7), the diameter of the pile and the ultimate CO

_{2}emission (Figure 8), the total length of the pile and the diameter of the pile (Figure 9), the total cost and the diameter of the pile (Figure 10), and the total cost and the total length of the pile (Figure 11) are evaluated depending on the lower and upper limits of all the foreseen parameters of the whole design process. The Pareto fronts of these mentioned parameters were evaluated by performing regression analyses and proper design expressions are suggested depending upon the Pareto graphs.

_{CO2}is given against the change in excavation depth. All the axes of the graphs are fixed at special values to ease the comparison of the use of similar limits. The increase in the design parameters leads the analyses to scan more members in the Pareto sets and also enlarge the scanned range of the Pareto sets. The linear-curve-fitting option gives more accurate solutions for the lower limits of the design variables and, also, the polynomial-curve-fitting option gives more applicable results for the upper limits of the design variables to predict the relationship between ΣC and the ΣE

_{CO2}. The coefficient of determination, denoted R

^{2}, has been calculated as approximately bigger than 0.85 for both situations. This result strengthens the applicability of the given mathematical expressions for the design problem of CSP.

_{CO2}is investigated against the change in the excavation depth for both the lower and upper limits of the design variants. The exact solution for the expression of the relation between ΣL and ΣE

_{CO2}is derived with the use of the linear-curve-fitting and polynomial-curve-fitting options, respectively, for the lower and upper boundaries of the design variants. Similar to Figure 6b, the use of design parameters at the upper-limit values causes the Pareto data set to become dispersed and the number of data obtained to increase in Figure 7b. The coefficient of determination, denoted R

^{2}, has been calculated as approximately bigger than 0.96 for both of the situations. This situation enables us to predict the ultimate CO

_{2}emission value directly from the total length of the pile with an admissible accuracy. Therefore, these mentioned simple Pareto expressions can make it possible to design eco-friendly structures when utilized. Herein, if the trend in the acquired relationships for both lower and upper limits is compared, it can be seen that these expressions are only valid for the envisaged conditions. The bigger the design parameters, the more difficult it is to reach the optimal design solutions. Therefore, in the context of this study, numerous relationships are derived in consideration of all the foreseen design variables. In addition, the applicability of the obtained mathematical expressions is controlled against the conducted regression analysis by the coefficient-of-determination values which are determined as approximately equal to or bigger than 0.85 at each time.

_{CO2}is investigated against the change in the excavation depth for both the lower and upper limits of design variants. The coefficient of determination, denoted R

^{2}, has been calculated as approximately bigger than 0.91 for both of the situations. These relations show that although the diameter of the pile seems to be a smaller dimension compared with the length of the pile, it has a direct effect on the emission of CO

_{2}.

^{2}, has been calculated as approximately bigger than 0.98 for both of the situations. In Figure 9a,b, the Pareto-optimal dimensional design can be directly obtained by the use of linear relationships. These graphs can ease the sustainable design of CSP systems by using an integrated usage process with other derived expressions. In such a case, if there is a known shear strength angle of the foundation soil strata, the length of the pile can be expressed with a preadmitted diameter value. Moreover, this identification can be also related to Figure 8 to predict the ultimate CO

_{2}emission value and can be related to Figure 6 to predict the total cost value of the system. This process may be useful in the predesign stage to achieve both eco-friendly and cost-effective solutions. Moreover, Figure 9 is direct proof of the decrement in both the length and diameter of the pile depending on the increase in any design variant.

^{2}, has been calculated as approximately bigger than 0.84 for both of the situations. The Pareto of the ΣC and D relation can be described with a linear-curve-fitting option. However, a striking situation in these analyses is that if the design variables are accepted at the upper-limit values, the Pareto data set obtained for each depth is gradually spreading over a wider range.

^{2}, has been calculated as approximately bigger than 0.86 for both of the situations. In Figure 11, a relation like Figure 10 is obtained. The change in excavation depth leads to the spread of the Pareto data.

_{2}emission couples. In Figure 12, the emission couple 1 is taken into consideration and the lower limit of the unit costs is assumed to be used to search for the effect of the change in Φ. The Pareto graphs were drawn for the evaluation of ΣC and ΣE

_{CO2}against the change in excavation depths. In addition, representative mathematical expressions were obtained for each excavation depth (stated by “a” subdivision of the related figures) and a total expression was derived to represent all the excavation depths and shear-strength-angle changes (stated by “b” subdivision of the related figures). These Pareto graphs were obtained with the use of average values that were determined using the arithmetic means of the Pareto data to reduce the data densities. The change in the shear strength angles is exemplified in the graphs for only one representative excavation depth so as not to complicate the illustrations. As an example, the change in the Φ is shown in Figure 12a for only a 9 m excavation depth. As seen from Figure 12a, that linear relationship can be derived for every single excavation depth with a coefficient-of-determination value minimum of 0.95. Moreover, with the use of a polynomial function, a single integrated expression can state the ΣC and ΣE

_{CO2}relation for different depth and shear strength angles. In addition, the increase in the shear strength angle relatively decreases the total cost and ultimate emission values significantly.

_{2}emission values of the construction materials. Therefore, it is appropriate to compare the obtained numerical values of ΣC and ΣE

_{CO2}with other emission couples. The minimization of the emission values leads to a decrease in the ultimate CO

_{2}emission amount by approximately 250%, but in conjunction with the emission amount, the total cost values are increased in a predictable range.

_{2}emission values of the construction materials. The usage of the upper limits of material emission values for the determination of the Pareto optimization analyses causes an increase in the ultimate emission value in comparison with the usage of the minimum limits of the emissions. However, there are approximately no differences that occurred in the conditions that are analyzed by the use of couple 1, couple 2, and couple 4 in terms of the determined ultimate emissions and total costs. This situation may be the result of a change in the structural design depending on the aim of the minimization process. Therefore, the evaluations have to be conducted in an integrated relationship with the dimensional options.

_{2}emission as expected (Figure 18b).

_{CO2}against the excavation depth. An investigation of the effects of the change in the unit cost of the concrete material is also attempted. According to this purpose, the unit cost of the steel is taken as a constant value of USD 700. In addition, the shear strength angle of the foundation soil strata is assumed to be 30°, and emission the Couple 3 is selected to conduct the analyses. The unit cost of the concrete is envisaged to be USD 50, USD 75, USD 100, USD 125, and USD 150, respectively. To eliminate the illustration of the random distribution of the Pareto data and prevent the disorder of the data spread, upper and lower values of the determined results are used to illustrate the following graphs. Representative mathematical expressions are also derived by the performed regression analysis for both upper and lower limits of the results depending on the change in the excavation depth. As a result, an average mathematical expression is generated based on the upper and lower limits of the analysis results by examining the sufficient applicability according to the coefficient-of-determination value. It is an apparent and expectable situation that the increase in the cost of the concrete raises the total costs of the design in a manner directly proportional to the increment of the excavation depth. The coefficient-of-determination values, denoted as R

^{2}, is calculated as bigger than 0.99 for all the conditions.

_{2}emission values are not changed but the total cost values are directly affected by the rise in the unit cost of the steel if the unit cost of the concrete remains at the upper level of the envisaged values. In addition to all these, the possibility of generating an integrated graph was also investigated with the change in both the unit costs of the concrete and the unit cost of the steel material. Figure 24 represents this mentioned relationship within the evaluation process of emission couple 1. The clusters of the Pareto data are also shown as grey crosses in Figure 24. The distribution of the Pareto sets is proportional to the excavation depth and the density of the Pareto data is not changed with depth. The distribution range of the data enlarges with the increased excavation depth. This condition increases the relative difference between the obtained upper and lower limits of the results. In that situation, the coefficient-of-determination value that is calculated for the average conditions makes it possible to obtain an applicable mathematical expression for the design of CSP walls.

_{CO2}according to conducted regression analyses.

_{2}emission are differentiated from the solution results. Moreover, the distribution of Pareto data is also similar for Figure 24 and Figure 27 wherein the amount of the unit emission value of the concrete is the same. Therefore, it may be stated that the unit emission value of the concrete material forms the general shape of the distribution of the Pareto data for the design problem of CSP. This may be related to the volume content of the construction materials. The concrete material plays a dominant role in terms of volume rendering during the construction of CSP wall systems. This situation has to be also controlled with the change in design dimensions.

## 5. Conclusions

_{2}emission amount minimization. Additionally, the optimum results were obtained and the whole data were stored in the spreadsheet file. All combinations and their outcomes were recorded. The graphs were plotted that represent various relationships between the total cost and the ultimate CO

_{2}emission; the total length of the pile and the total cost; the diameter of the pile and the ultimate CO

_{2}emission; the total length of the pile and the diameter of the pile; the total cost and the diameter of the pile; and the total cost and the total length of the pile, which were evaluated depending on all the foreseen parameters of the whole design process by the usage of HS. The Pareto fronts of trade-offs between total costs and CO

_{2}emissions were categorized based on the generated combinations and evaluated by performing regression analyses, and proper design expressions were suggested depending upon the Pareto graphs and charts. All the conducting regression analyses led to attaining a coefficient-of-determination number bigger than 0.85 which seems to be a satisfactory amount for an expression to be applicable for the design problem of cantilever soldier piles. Consequently, the analyses performed within this study ensure the development of more exhaustive mathematical expressions; hence, it allows the process of seeking the design of a cantilever soldier pile wall to be automated. Two practical comparative-design objectives were considered with relevant applicable results to involve the most proper design application of the pile, and this system could be extended to the design of other soil-supporting structures and other metaheuristic algorithms with the use of more objective functions.

## 6. Recommendations for Further Studies

_{2}emission minimization. Therefore, it aimed to be a guide to designers and researchers to focus on the design problem of CSPs with the multiobjective approach. In this context, within this study, it was utilized from the logic of Pareto optimality, and the HS algorithm was used as the tool of solution. Although the factors such as the excavation depth, the shear strength parameters of foundation soil strata, and the unit costs and unit emission amounts of structural materials were considered during the analysis, it is a clear fact that factors such as the existence of groundwater, the change in the loading type, environmental effects, etc., affect the design, performance, and cost of the structural system. On the other hand, the applicability of other more powerful methods can be controlled and the statistical optimum estimation techniques can also be used to justify the number of numerical tests required to provide a solution. In addition, the control parameters’ sensitivity analysis can be conducted to clarify the quality of the final solution. Further studies in which these above-mentioned factors are also considered are ongoing.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The classical geometrical section of a single CSP (

**a**); the activated lateral forces (

**b**); beams on elastic soil assumption (

**c**,

**d**).

**Figure 6.**The relationship between the ultimate CO

_{2}emission and the total cost. (

**a**) Lower limits, (

**b**) Upper limits.

**Figure 7.**The relationship between the ultimate CO

_{2}emission and the total length of the pile. (

**a**) Lower limits, (

**b**) Upper limits.

**Figure 8.**The relationship between the ultimate CO

_{2}emission and the diameter of the pile. (

**a**) Lower limits, (

**b**) Upper limits.

**Figure 9.**The relationship between the total length of the pile and the diameter of the pile. (

**a**) Lower limits, (

**b**) Upper limits.

**Figure 10.**The relationship between the total cost and the diameter of the pile. (

**a**) Lower limits, (

**b**) Upper limits.

**Figure 11.**The relationship between the total cost and the total length of the pile. (

**a**) Lower limits, (

**b**) Upper limits.

**Figure 12.**The relationship between the ultimate CO

_{2}emission and the total cost (

**a**) classified with pile lengths (

**b**) not classified by any criteria (Lower limit of costs—Emission couple 1).

**Figure 13.**The relationship between the ultimate CO

_{2}emission and the total cost (

**a**) classified with pile lengths (

**b**) not classified by any criteria (Lower limit of costs—Emission couple 2).

**Figure 14.**The relationship between the ultimate CO

_{2}emission and the total cost (

**a**) classified with pile lengths (

**b**) not classified by any criteria (Lower limit of costs—Emission couple 3).

**Figure 15.**The relationship between the ultimate CO

_{2}emission and the total cost (

**a**) classified with pile lengths (

**b**) not classified by any criteria (Lower limit of costs—Emission couple 4).

**Figure 16.**The relationship between the ultimate CO

_{2}emission and the total cost (

**a**) classified with pile lengths (

**b**) not classified by any criteria (Upper limit of costs—Emission couple 1).

**Figure 17.**The relationship between the ultimate CO

_{2}emission and the total cost (

**a**) classified with pile lengths (

**b**) not classified by any criteria (Upper limit of costs—Emission couple 2).

**Figure 18.**The relationship between the ultimate CO

_{2}emission and the total cost (

**a**) classified with pile lengths (

**b**) not classified by any criteria (Upper limit of costs—Emission couple 3).

**Figure 19.**The relationship between the ultimate CO

_{2}emission and the total cost (

**a**) classified with pile lengths (

**b**) not classified by any criteria (Upper limit of costs—Emission couple 4).

**Figure 20.**The change in ΣC and ΣE

_{CO2}against the rise in the unit cost of the concrete (C

_{s}= USD 700).

**Figure 21.**The change in ΣC and ΣE

_{CO2}against the rise in the unit cost of the concrete (C

_{s}= USD 1100).

**Figure 22.**The change in ΣC and ΣE

_{CO2}against the rise in the unit cost of the steel (C

_{c}= USD 50).

**Figure 23.**The change in ΣC and ΣE

_{CO2}against the rise in the unit cost of the steel (C

_{c}= USD 150).

**Figure 24.**The change in ΣC and ΣE

_{CO2}against the rise in the unit cost of steel and concrete (Couple 1).

**Figure 25.**The change in ΣC and ΣE

_{CO2}against the rise in the unit cost of steel and concrete (Couple 2).

**Figure 26.**The change in ΣC and ΣE

_{CO2}against the rise in the unit cost of steel and concrete (Couple 3).

**Figure 27.**The change in ΣC and ΣE

_{CO2}against the rise in the unit cost of steel and concrete (Couple 4).

**Figure 28.**The change in ΣL and D against the rise in the unit cost of steel and concrete. (

**a**) Couple 1, (

**b**) Couple 2.

**Figure 29.**The change in ΣL and D against the rise in the unit cost of steel and concrete. (

**a**) Couple 3, (

**b**) Couple 4.

Symbol | Parameter Description |
---|---|

X1 | Diameter of the CSP (D) |

X2 | Diameter of the reinforcing bars of the CSP (ϕp) |

X3 | Number of the reinforcing bars of the CSP |

Description of the Constraints | Constraints |
---|---|

Flexural-strength capacities of critical sections (M_{d}) | g_{1}(X): M_{d} ≥ M_{u} |

Shear strength capacities of critical sections (V_{d}) | g_{2}(X): V_{d} ≥ V_{u} |

Minimum reinforcement areas of critical sections (A_{smin}) | g_{3}(X): A_{s} ≥ A_{smin} |

Maximum reinforcement areas of critical sections (A_{smax}) | g_{4}(X): A_{s} ≤ A_{smax} |

Symbol | Definition | Value | Unit |
---|---|---|---|

h | Depth of excavation | 4 to 12 | m |

f_{y} | Yield strength of steel | 420 | MPa |

f’_{c} | Compressive strength of concrete | 30 | MPa |

c_{c} | Concrete cover | 30 | mm |

E_{steel} | The elasticity modulus of steel | 200 | GPa |

E_{concrete} | Elasticity modulus of concrete | 23,5 | GPa |

γ_{steel} | Unit weight of steel | 7.85 | t/m^{3} |

γ_{concrete} | Unit weight of concrete | 25 | kN/m^{3} |

C_{c} | Cost of concrete per m^{3} | 50, 75, 100, 125, 150 | USD |

C_{s} | Cost of steel per ton | 700, 800, 900, 1000, 1100 | USD |

q | Surcharge load | 10 | kPa |

ϕ | Shear strength angle of soil | 20, 25, 30, 35, 40 | ° |

γ | Unit weight of soil | 18 | kN/m^{3} |

D | Diameter of pile | 0.3–2 | m |

ϕ_{p} | Diameter of reinforcing bars of soldier pile | 14–40 | - |

n | Number of reinforcing bars of soldier pile | 6–20 | - |

The CO_{2} Emission Couples | Concrete (C30) | Steel (S420) |
---|---|---|

Couple 1 (C1) | 376 | 352 |

Couple 2 (C2) | 143.48 | 3010 |

Couple 3 (C3) | 143.48 | 352 |

Couple 4 (C4) | 376 | 3010 |

C_{c} = USD 50 and C_{s} = USD 700 | Couple 1 | Couple 2 | |||||||
---|---|---|---|---|---|---|---|---|---|

D (m) | ΣE_{CO2} (kg) | ΣC (USD) | ΣL (m) | D (m) | ΣE_{CO2} (kg) | ΣC (USD) | ΣL (m) | ||

h = 4 m | Φ = 20° | 0.34 | 838.90 | 357.63 | 5.86 | 0.41 | 875.00 | 241.57 | 6.28 |

Φ = 25° | 0.32 | 739.36 | 293.20 | 5.73 | 0.35 | 799.44 | 218.08 | 5.93 | |

Φ = 30° | 0.30 | 627.56 | 300.21 | 5.60 | 0.32 | 730.01 | 197.34 | 5.69 | |

Φ = 35° | 0.26 | 556.15 | 231.85 | 5.28 | 0.30 | 732.51 | 196.46 | 5.55 | |

h = 5 m | Φ = 20° | 0.47 | 1487.35 | 616.64 | 7.74 | 0.55 | 1597.32 | 440.45 | 8.31 |

Φ = 25° | 0.44 | 1330.01 | 493.05 | 7.53 | 0.50 | 1359.65 | 374.88 | 7.96 | |

Φ = 30° | 0.40 | 1140.39 | 417.10 | 7.27 | 0.45 | 1157.62 | 319.06 | 7.59 | |

Φ = 35° | 0.37 | 978.32 | 359.50 | 7.01 | 0.39 | 991.33 | 272.94 | 7.20 | |

h = 6 m | Φ = 20° | 0.68 | 2450.54 | 872.75 | 10.34 | 0.76 | 2537.52 | 699.84 | 10.96 |

Φ = 25° | 0.60 | 2100.66 | 768.23 | 9.74 | 0.64 | 2186.33 | 603.07 | 10.06 | |

Φ = 30° | 0.54 | 1810.32 | 686.31 | 9.28 | 0.59 | 1883.63 | 519.18 | 9.65 | |

Φ = 35° | 0.46 | 1535.87 | 621.53 | 8.64 | 0.53 | 1576.60 | 435.38 | 9.22 | |

h = 7 m | Φ = 20° | 0.84 | 3534.35 | 1371.71 | 12.71 | 0.93 | 3789.14 | 1045.34 | 13.44 |

Φ = 25° | 0.77 | 3040.87 | 1228.84 | 12.11 | 0.85 | 3244.72 | 896.63 | 12.78 | |

Φ = 30° | 0.67 | 2591.12 | 1064.26 | 11.29 | 0.75 | 2789.64 | 770.37 | 11.90 | |

Φ = 35° | 0.59 | 2285.70 | 867.60 | 10.64 | 0.64 | 2382.45 | 656.59 | 11.06 | |

h = 8 m | Φ = 20° | 1.07 | 5120.19 | 1817.27 | 15.72 | 1.14 | 5351.55 | 1478.54 | 16.27 |

Φ = 25° | 1.00 | 4368.82 | 1617.13 | 15.02 | 1.03 | 4596.35 | 1269.34 | 15.31 | |

Φ = 30° | 0.85 | 3741.03 | 1424.62 | 13.75 | 0.92 | 3988.92 | 1099.31 | 14.34 | |

Φ = 35° | 0.72 | 3195.20 | 1243.84 | 12.70 | 0.81 | 3382.50 | 932.23 | 13.42 | |

h = 9 m | Φ = 20° | 1.32 | 7166.50 | 2312.24 | 18.94 | 1.38 | 7309.53 | 2013.84 | 19.51 |

Φ = 25° | 1.18 | 6164.62 | 2048.37 | 17.71 | 1.23 | 6289.53 | 1737.25 | 18.16 | |

Φ = 30° | 1.12 | 5138.84 | 1884.60 | 16.45 | 1.10 | 5393.80 | 1487.76 | 16.91 | |

Φ = 35° | 0.88 | 4420.71 | 1565.74 | 15.04 | 0.96 | 4581.94 | 1264.22 | 15.71 | |

C_{c} = USD 50 and C_{s} = USD 700 | Couple 3 | Couple 4 | |||||||

D (m) | ΣE_{CO2} (kg) | ΣC (USD) | ΣL (m) | D (m) | ΣE_{CO2} (kg) | ΣC (USD) | ΣL (m) | ||

h = 4 m | Φ = 20° | 0.37 | 391.95 | 257.43 | 6.04 | 0.34 | 839.20 | 374.05 | 5.85 |

Φ = 25° | 0.34 | 334.87 | 217.56 | 5.83 | 0.33 | 738.13 | 291.70 | 5.76 | |

Φ = 30° | 0.32 | 288.57 | 196.59 | 5.67 | 0.28 | 636.24 | 256.28 | 5.42 | |

Φ = 35° | 0.28 | 256.13 | 195.20 | 5.42 | 0.27 | 560.41 | 219.26 | 5.36 | |

h = 5 m | Φ = 20° | 0.53 | 702.71 | 466.35 | 8.18 | 0.45 | 1472.74 | 632.92 | 7.63 |

Φ = 25° | 0.50 | 605.85 | 410.05 | 7.93 | 0.44 | 1278.39 | 553.62 | 7.54 | |

Φ = 30° | 0.43 | 516.34 | 334.85 | 7.48 | 0.40 | 1127.05 | 432.68 | 7.27 | |

Φ = 35° | 0.40 | 438.39 | 281.71 | 7.21 | 0.36 | 975.94 | 363.53 | 6.95 | |

h = 6 m | Φ = 20° | 0.65 | 1115.48 | 803.96 | 10.13 | 0.65 | 2339.40 | 996.31 | 10.15 |

Φ = 25° | 0.63 | 968.07 | 655.72 | 9.96 | 0.62 | 2097.61 | 779.16 | 9.86 | |

Φ = 30° | 0.57 | 826.30 | 562.00 | 9.46 | 0.52 | 1785.22 | 697.08 | 9.10 | |

Φ = 35° | 0.50 | 704.18 | 434.77 | 8.99 | 0.47 | 1538.44 | 616.95 | 8.77 | |

h = 7 m | Φ = 20° | 0.88 | 1660.88 | 1224.45 | 13.00 | 0.83 | 3534.57 | 1372.64 | 12.63 |

Φ = 25° | 0.80 | 1439.31 | 989.41 | 12.34 | 0.74 | 3084.81 | 1163.27 | 11.87 | |

Φ = 30° | 0.73 | 1230.84 | 895.15 | 11.73 | 0.71 | 2744.30 | 966.92 | 11.56 | |

Φ = 35° | 0.61 | 1055.10 | 711.19 | 10.80 | 0.60 | 2252.50 | 940.67 | 10.71 | |

h = 8 m | Φ = 20° | 1.09 | 2349.02 | 1744.68 | 15.85 | 1.06 | 5184.75 | 1788.43 | 15.58 |

Φ = 25° | 0.97 | 2021.56 | 1511.07 | 14.75 | 0.93 | 4430.03 | 1606.75 | 14.48 | |

Φ = 30° | 0.87 | 1746.74 | 1220.93 | 13.92 | 0.80 | 3702.39 | 1446.62 | 13.31 | |

Φ = 35° | 0.79 | 1487.05 | 1045.76 | 13.24 | 0.74 | 3193.07 | 1258.60 | 12.83 | |

h = 9 m | Φ = 20° | 1.33 | 3166.57 | 2345.37 | 19.10 | 1.29 | 7189.82 | 2269.49 | 18.66 |

Φ = 25° | 1.14 | 2742.12 | 1961.46 | 17.34 | 1.16 | 5945.58 | 2145.63 | 17.44 | |

Φ = 30° | 1.00 | 2357.97 | 1764.10 | 16.02 | 1.01 | 5161.12 | 1900.92 | 16.10 | |

Φ = 35° | 0.92 | 2022.00 | 1389.70 | 15.34 | 0.90 | 4389.98 | 1602.59 | 15.16 |

C_{c} = USD 150 and C_{s} = USD 1100 | Couple 1 | Couple 2 | |||||||
---|---|---|---|---|---|---|---|---|---|

D (m) | ΣE_{CO2} (kg) | ΣC (USD) | ΣL (m) | D (m) | ΣE_{CO2} (kg) | ΣC (USD) | ΣL (m) | ||

h = 4 m | Φ = 20° | 0.36 | 854.87 | 676.40 | 5.96 | 0.41 | 875.19 | 543.91 | 6.30 |

Φ = 25° | 0.31 | 734.29 | 584.98 | 5.60 | 0.35 | 799.44 | 480.88 | 5.90 | |

Φ = 30° | 0.27 | 631.03 | 533.80 | 5.37 | 0.32 | 729.83 | 429.75 | 5.73 | |

Φ = 35° | 0.24 | 544.24 | 489.00 | 5.21 | 0.31 | 734.38 | 417.35 | 5.61 | |

h = 5 m | Φ = 20° | 0.49 | 1466.65 | 1212.08 | 7.87 | 0.55 | 1573.83 | 977.23 | 8.36 |

Φ = 25° | 0.43 | 1313.93 | 1020.53 | 7.49 | 0.50 | 1359.60 | 840.91 | 7.99 | |

Φ = 30° | 0.38 | 1112.08 | 909.71 | 7.14 | 0.47 | 1157.40 | 714.24 | 7.73 | |

Φ = 35° | 0.35 | 964.44 | 782.78 | 6.91 | 0.39 | 991.59 | 612.89 | 7.20 | |

h = 6 m | Φ = 20° | 0.62 | 2299.67 | 1922.43 | 9.88 | 0.75 | 2538.04 | 1572.80 | 10.92 |

Φ = 25° | 0.57 | 2013.28 | 1677.16 | 9.47 | 0.65 | 2190.76 | 1357.81 | 10.11 | |

Φ = 30° | 0.52 | 1755.54 | 1449.43 | 9.10 | 0.58 | 1878.18 | 1155.61 | 9.56 | |

Φ = 35° | 0.46 | 1533.39 | 1211.13 | 8.68 | 0.52 | 1576.41 | 978.29 | 9.09 | |

h = 7 m | Φ = 20° | 0.79 | 3453.94 | 2810.57 | 12.27 | 0.93 | 3786.58 | 2351.82 | 13.42 |

Φ = 25° | 0.71 | 3004.59 | 2415.12 | 11.60 | 0.82 | 3261.06 | 2019.45 | 12.48 | |

Φ = 30° | 0.67 | 2653.39 | 2046.73 | 11.27 | 0.74 | 2795.82 | 1738.26 | 11.82 | |

Φ = 35° | 0.55 | 2222.12 | 1861.22 | 10.32 | 0.66 | 2382.84 | 1474.39 | 11.22 | |

h = 8 m | Φ = 20° | 1.03 | 5097.77 | 3721.04 | 15.27 | 1.13 | 5357.60 | 3321.14 | 16.17 |

Φ = 25° | 0.86 | 4227.64 | 3390.74 | 13.86 | 1.03 | 4608.59 | 2867.28 | 15.28 | |

Φ = 30° | 0.83 | 3730.58 | 2851.10 | 13.57 | 0.91 | 3985.00 | 2466.12 | 14.22 | |

Φ = 35° | 0.71 | 3188.26 | 2482.86 | 12.59 | 0.79 | 3382.17 | 2088.35 | 13.21 | |

h = 9 m | Φ = 20° | 1.27 | 7094.63 | 4847.67 | 18.53 | 1.35 | 7281.64 | 4513.61 | 19.23 |

Φ = 25° | 1.13 | 6055.69 | 4256.64 | 17.22 | 1.23 | 6283.85 | 3888.48 | 18.17 | |

Φ = 30° | 0.98 | 5088.89 | 3794.27 | 15.91 | 1.10 | 5392.48 | 3334.49 | 16.97 | |

Φ = 35° | 0.85 | 4271.36 | 3363.08 | 14.79 | 0.96 | 4586.72 | 2840.12 | 15.68 | |

C_{c} = USD 150 and C_{s} = USD 1100 | Couple 3 | Couple 4 | |||||||

D (m) | ΣE_{CO2} (kg) | ΣC (USD) | ΣL (m) | D (m) | ΣE_{CO2} (kg) | ΣC (USD) | ΣL (m) | ||

h = 4 m | Φ = 20° | 0.38 | 391.74 | 557.62 | 6.13 | 0.35 | 849.91 | 674.02 | 5.91 |

Φ = 25° | 0.33 | 336.90 | 489.67 | 5.76 | 0.32 | 746.37 | 553.08 | 5.69 | |

Φ = 30° | 0.34 | 288.03 | 423.05 | 5.80 | 0.27 | 634.81 | 520.41 | 5.40 | |

Φ = 35° | 0.26 | 252.11 | 399.82 | 5.30 | 0.23 | 542.59 | 515.98 | 5.11 | |

h = 5 m | Φ = 20° | 0.52 | 698.26 | 1049.06 | 8.13 | 0.48 | 1471.05 | 1217.76 | 7.84 |

Φ = 25° | 0.49 | 602.96 | 876.55 | 7.89 | 0.45 | 1298.00 | 1027.97 | 7.59 | |

Φ = 30° | 0.43 | 513.69 | 714.31 | 7.47 | 0.40 | 1118.10 | 911.04 | 7.22 | |

Φ = 35° | 0.39 | 438.09 | 610.33 | 7.15 | 0.36 | 974.13 | 742.98 | 6.98 | |

h = 6 m | Φ = 20° | 0.71 | 1114.66 | 1701.85 | 10.57 | 0.63 | 2334.79 | 1928.00 | 9.96 |

Φ = 25° | 0.60 | 959.96 | 1505.01 | 9.70 | 0.98 | 1917.18 | 1907.71 | 16.26 | |

Φ = 30° | 0.56 | 825.16 | 1217.10 | 9.41 | 0.57 | 1813.46 | 1367.67 | 9.45 | |

Φ = 35° | 0.51 | 705.67 | 1020.54 | 9.01 | 0.44 | 1507.62 | 1273.01 | 8.53 | |

h = 7 m | Φ = 20° | 0.86 | 1668.71 | 2542.89 | 12.83 | 0.84 | 3555.94 | 2741.11 | 12.67 |

Φ = 25° | 0.74 | 1427.16 | 2231.99 | 11.85 | 0.69 | 2967.25 | 2460.43 | 11.45 | |

Φ = 30° | 0.70 | 1234.84 | 1819.20 | 11.52 | 0.67 | 2622.00 | 2080.24 | 11.24 | |

Φ = 35° | 0.64 | 1054.47 | 1515.04 | 11.03 | 0.58 | 2262.52 | 1808.06 | 10.57 | |

h = 8 m | Φ = 20° | 1.06 | 2345.24 | 3591.19 | 15.58 | 1.00 | 5069.91 | 3714.12 | 15.07 |

Φ = 25° | 0.99 | 2020.73 | 3108.28 | 14.92 | 0.91 | 4278.56 | 3350.59 | 14.27 | |

Φ = 30° | 0.85 | 1742.08 | 2660.49 | 13.75 | 0.82 | 3766.09 | 2823.68 | 13.54 | |

Φ = 35° | 0.79 | 1497.31 | 2180.83 | 13.26 | 0.74 | 3279.98 | 2369.96 | 12.87 | |

h = 9 m | Φ = 20° | 1.29 | 3181.93 | 4747.67 | 18.68 | 1.27 | 7176.81 | 4803.42 | 18.51 |

Φ = 25° | 1.18 | 2744.23 | 4214.54 | 17.63 | 1.12 | 5956.37 | 4318.99 | 17.09 | |

Φ = 30° | 1.02 | 2365.03 | 3571.15 | 16.25 | 0.98 | 4925.80 | 3952.83 | 15.89 | |

Φ = 35° | 0.90 | 2019.77 | 3020.58 | 15.20 | 0.86 | 4320.60 | 3340.01 | 14.81 |

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## Share and Cite

**MDPI and ACS Style**

Bekdaş, G.; Arama, Z.A.; Türkakın, O.H.; Kayabekir, A.E.; Geem, Z.W.
Cantilever Soldier Pile Design: The Multiobjective Optimization of Cost and CO_{2} Emission via Pareto Front Analysis. *Sustainability* **2022**, *14*, 9416.
https://doi.org/10.3390/su14159416

**AMA Style**

Bekdaş G, Arama ZA, Türkakın OH, Kayabekir AE, Geem ZW.
Cantilever Soldier Pile Design: The Multiobjective Optimization of Cost and CO_{2} Emission via Pareto Front Analysis. *Sustainability*. 2022; 14(15):9416.
https://doi.org/10.3390/su14159416

**Chicago/Turabian Style**

Bekdaş, Gebrail, Zülal Akbay Arama, Osman Hürol Türkakın, Aylin Ece Kayabekir, and Zong Woo Geem.
2022. "Cantilever Soldier Pile Design: The Multiobjective Optimization of Cost and CO_{2} Emission via Pareto Front Analysis" *Sustainability* 14, no. 15: 9416.
https://doi.org/10.3390/su14159416