1. Introduction
Cantilever soldier piles (CSP) are frequently used structures to support the generated lateral thrust of soil mass through excavation works. CSPs can be penetrated into a wide range of soil strata, including layered stratification, in addition to easing the construction works by using special equipment that ensures reducing labor. The simple design process of laterally loaded supporting piles can be generally defined as the same as other classical-retaining-structures’ design process, which is starting with the calculation of activated lateral pressures through the wall surfaces to check the stability conditions based on the foreseen dimensions. This kind of process, referred to as proportioning, ensures the application of an iterative design process takes place where the sections are adjusted at the end of the calculations if needed [
1]. Then, the calculations are followed by the implementation of structural necessities to assess structural design limits. Several studies have been performed to design CSP retaining systems depending on the determination of the lateral earth pressures along the pile length, which depend upon the soil’s geotechnical properties and the environmental conditions, analyzed using different kinds of techniques such as theoretical or experimental and so on. The single-soldier-pile walls are sometimes analyzed like sheet pile walls, although the structural components of both systems are different. In that condition, the selected type of the retaining system has to satisfy the force and moment equilibrium simultaneously [
2]. Moreover, the free embedded cantilever walls are basically used to support relatively low heights of cohesionless soils. Traditionally, these kinds of walls are designed based on limit equilibrium analysis depending on the stress distribution along the wall. Sisson et al. [
3] shaped a research study about the design of CSP retaining walls embedded in stiff clays and clay stones by taking into consideration the long-term loading conditions, the strength of the soil, and the method of analysis for design.
Wen-Ai and De-Ling [
4] applied a different design process for CSP based on the principle of minimum potential energy. An analytic determination of the pile head’s maximum displacement was produced. The relationship between the design and the excavation depth was also investigated. Gajan [
2] studied both sheet pile and soldier pile walls penetrated into cohesionless soils and used Rankine earth pressure theory to determine the lateral pressures activated for active and passive states. Relationships were developed to calculate the depth of embedment in terms of nondimensional parameters that can be used for various soil conditions, pile dimensions, and depths of excavation. Lee et al. [
5] analyzed cantilever double-soldier-pile walls with a proposed model, including an equivalent single-soldier-pile system, and centrifuge model tests were conducted in the sand to measure the performance of the suggested study.
The reinforced concrete material is assumed to be the economical and safe choice for retaining the system design in most of the projects. This situation necessitates determining the reinforcing bars’ number and diameters to calculate the volume of the concrete that is needed to complete the whole design process. This necessity shows that the said sequential geotechnical and structural design process also creates the need to provide economic control. However, it can be seen from the actual literature sources that the conceptions of the design of the CSP and the economy of the structural system do not take into consideration an integrated view. Moreover, beginning from the 20th century, the huge increase in the amount of reinforced concrete throughout the world has begun, due to the unforeseen growth of sheltering and industrialization needs. Therefore, it has become significant to design structural systems in an eco-friendly manner to protect human life from the harmful effects of global warming. In this respect, construction materials made of both steel and concrete have caused carbon dioxide emissions, which increase global warming and have destructive effects on the environment.
The mentioned mixed relationship leads designers to use many discrete design variables for the solution of related cases. For this reason, generally, the practicability of optimization-based techniques means they are preferred to be used by designers to model structural design problems to ensure enough safety, cost efficiency, and eco-friendly solutions.
In structural design, optimization-based techniques usually use the power of computer science technologies to obtain ideal design values by generating random designs. Thuswise, the structure’s type is validated and automatically calculated, and afterward, the redefinition process of the design by conducting the process of the envisaged algorithm, which controls the process of iteration numbers through the investigation of the optimum, can be carried out [
6]. Specifically, the advantage of metaheuristic algorithms, such as a genetic algorithm [
7,
8], harmony search algorithm [
9,
10], particle swarm optimization [
11,
12], and firefly algorithm [
13,
14], are preferred to be used to maintain the design analysis of retaining structures. Some of the conducted studies are focused on the optimal design of reinforced-concrete retaining structures depending on the minimization of only the cost. Sasidhar et al. [
7] performed an optimization analysis for the design of cantilever retaining walls with different heights using a GA to compare the effectiveness of the method with traditional design predictions. The single-objective algorithms optimize scalar variables; in that case, the fitness function returns scalar values, and the algorithm finds the optimum solution to minimize a scalar variable.
In multiobjective optimizations, Pareto optimality exhibits an opportunity to synchronize the optimality considering multiple objectives. In this sense, there are various studies on the civil engineering field by using multiobjective meta-heuristic algorithms. Especially in the energy modeling of a building, multiobjective algorithms have important practical outcomes. Wang et al. [
15] applied the GA multiobjective optimization model to support green building designing. Another study about green building objects to minimizes energy consumption on the other hand thermal comfort is maximized [
16]. In some studies such as [
17], multiobjective GA is used along with building heat simulation software. In the study, optimum solutions were searched to satisfy minimum energy consumption with minimum cooling equipment.
According to the documented methods, cantilever soldier piles are not optimized for multiple objectives. In this study, the total cost and ultimate CO2 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 CO2 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.
3. Parametrical Analysis
Multivariate parametrical analyses are conducted in this comparative study for exemplifying the optimal design process of CSP systems. It is assumed that the foundation soil is purely homogeneous frictional. Therefore, the design variants that concern the soil remain only the shear strength angle (ϕ) and the unit weight of the soil (γ). ϕ is assumed to be 20°, 25°, 30°, 35°, and 40°, respectively, but γ is taken as a constant 18 kN/m
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.
The chosen values of the CO
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.
In total, in the context of the study, 4500 case analyses were fictionalized and 250,000 different solutions were searched depending on the huge variety in the design parameters. However, to reduce the informational convergence that can occur during the interpretation process of the outcomes of the analyses, some special cases were selected to explain the main theme of the study.
Figure 4 shows a pseudocode for generating the different cases to generate optimum design(s) regarding the related case. There are four different encapsulated loops that generate variations in conditions. The first loop generates different excavation depths. The second loop generates concrete and steel cost couples.
Cc and
Cs 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.
Figure 5 shows a Pareto front for the optimum designs for 7 m excavation depths. The other values are given at the right side of the graph. Each data point represents cost and CO
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
For each iteration, a Pareto front of optimum solutions is obtained. For instance,
Figure 6a,b include six different Pareto fronts regarding different excavation depths. The related condition is given on the upper-left side of the graphs. There is no possible optimum design for excavation depths larger than 9 m.
The output file is inserted into a spreadsheet file. By using a pivot table, whole data are filtered and can be directly viewed. After the graphs are generated, a curve-fitting tool is used for obtaining the optimum relationship between the output variables. Some special cases are envisaged to show the applicability of Pareto HS analyses have been used to discuss the assessment of the sustainable design of CSP via multiobjective Pareto front optimal analysis. The first type of analysis is conducted to compare the effects of selected amounts of the design variants. The relationships between the total cost and the ultimate CO
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.
Regression analyses are conducted with the use of Microsoft front analysis for the CSP design problem. The abbreviations a and b are used to define the arranged lower- and upper-limit conditions of the fictionalized cases. The lower- and upper-limit definitions can be described by the selected parameter specifications. The lower limit of the parameters is the smallest value of the design parameters. In such a case that if the application of the lower limits is admissible, the shear strength angle of the foundation soil is 20°, the unit costs of the concrete and steel are USD 50 and USD 700, respectively, and couple 3 is chosen as the unit emission values of the materials. However, on the contrary, if the application of the upper limits is admissible, the shear strength angle of the foundation soil is 35°, the unit costs of the concrete and steel are USD 150 and USD 1100, respectively, and couple 4 is chosen as the unit emission values of the materials. In
Figure 6a,b, the relationship between ΣC and ΣE
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.
In
Figure 7, the relationship between ΣL and ΣE
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.
In
Figure 8, the relationship between D and ΣE
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.
In
Figure 8, the relationship between ΣL and D 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 0.98 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.
In
Figure 10, the relationship between ΣC and D 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 0.84 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.
In
Figure 11, the relationship between ΣC and ΣL is investigated against the change in the excavation depth for both the lower and upper limits of the design variants. The coefficient of determination, denoted R
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.
In addition to these evaluations, the effect of the change in the shear strength angle of the foundation soil was also investigated individually. The upper and lower boundaries of the design variants were also used, respectively, for the interpretation of the change in different CO
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.
Figure 13a,b represent the analyses conducted for the lower limits of the costs and the emission couple 2. The same logic is also used to specify the average Pareto points in
Figure 13 as above mentioned. The coefficient-of-determination values (minimum 0.9999) show that the prediction rates for the foreseen conditions have been determined almost exactly.
Figure 14a,b represent the analyses conducted for the lower limits of the costs and the emission couple 3. The coefficient-of-determination values (minimum 0.92) show that the prediction rates for the foreseen conditions have been determined almost exactly, too. It has to be noted that emission couple 3 consists of the minimum CO
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.
Figure 15a,b represent the analyses conducted for the lower limits of the costs and the emission couple 4. It has to be noted that the emission couple 4 consists of the maximum CO
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.
The same Pareto-optimal analyses were repeated for the upper limits of the material costs. The unit cost of the concrete and steel was assumed to be USD 150 and USD 1100 for the continuing Pareto-optimal analyses. In
Figure 16,
Figure 17,
Figure 18 and
Figure 19, the results of the analyses are shown for the emission couples 1, 2, 3, and 4, respectively. The huge increase in the total cost of the construction is the remarkable point of the analyses.
The coefficient-of-determination values are bigger than 0.99 for all the conducted analyses and this condition can convince the designers to use the envisaged mathematical expressions to use at the design stage of the CSP systems.
It is a significant point that the inclination of the achieved curve-fitting line and the accuracy rate of the representative linear mathematical expressions are increased depending on the rise in the material costs except for the condition of the usage of the maximum emission values (
Figure 19). Moreover, the usage of emission couple 3 remarkably decreases the amount of ultimate CO
2 emission as expected (
Figure 18b).
These above-mentioned expressions only belong to the used values of the design parameters and all different parameter usage processes lead to creating new correlations. In this context, the excavation depth and shear strength angle are also effective parameters from the first step of the analysis, but the derivation of an integrated relationship is possible with the use of different values of the specified parameters as seen from the given graph. The solution of the objective function is directly proportional to the amounts of costs and emissions; therefore, there arises a new requirement to generate design graphs or charts to estimate the outcomes of CSP design directly. In this connection, design charts are also given in
Table 5 and
Table 6 to ease the sustainable design process of the CSP walls.
Table 5 represents the design chart for the lower limit of the material costs, and
Table 6 includes the same analyses for the upper limit of the material costs considering different excavation depths and shear strength angles.
The linear proportional interpolation can be applicable for unavailable intermediate values.
Table 5 represents the design chart for the upper limit of the material costs and the emission couple 1 and 2, and
Table 6 includes the same analyses for couple 3 and couple 4 for different excavation depths and shear strength angles. The linear proportional interpolation is also applicable for unavailable intermediate values.
The Pareto-optimal analyses were evaluated with a different perspective by also the consideration of some special cases. In
Figure 20, the Pareto data are investigated with the change in ΣC and ΣE
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.
In
Figure 21, the rise in the unit cost of concrete is also investigated with the assumption of a constant unit cost of steel value. In these analyses, the unit cost of the steel material is assumed to be at its upper limit of USD 1100. The comparison of
Figure 20 and
Figure 21 shows that the increase in the steel costs only affects the total cost of the CSP but does not lead to a change in the design of the CSP system. Moreover, the prediction rate of the mathematical expressions obtained from regression analyses for the upper, average, and lower limits of the Pareto data results is high.
The effects of the change in the steel costs are also investigated with the assumption of a constant concrete cost. The concrete cost is selected as USD 50 and USD 150, respectively, and the results of the Pareto-optimal solution are given in
Figure 22 and
Figure 23.
The comparison of
Figure 22 with
Figure 19 shows that the change in the unit cost of the steel is not an important factor to change the whole design of the CSP if the unit cost of the concrete remains at its envisaged lower limit. However, the increase in the unit cost of the concrete causes a narrowing in the range of Pareto sets. In
Figure 23, the average values of the ultimate CO
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.
Figure 25 represents this mentioned relationship within the evaluation process of emission couple 2. The distribution of the Pareto sets is differentiated from the data set that is obtained in
Figure 24. The distribution of the data is highly regular and linear. This condition increases the prediction rate of the relationship between ΣC and ΣE
CO2 according to conducted regression analyses.
Figure 26 represents this mentioned relationship within the evaluation process of emission couple 3 and
Figure 27 represents this mentioned relationship within the evaluation process of emission couple 4. It should be remembered that couple 3 represents the minimum limit of the emission values and couple 4 represents the maximum of the emission values. The distribution of the Pareto data is similar for
Figure 25 and
Figure 26 which have the same amount of emission values for the concrete material. However, the amounts of the total cost and ultimate CO
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.
To evaluate the dimensional variance in the CSP designs, integrated graphs were drawn considering different unit costs, emission values, and excavation depths together.
Figure 28a,b represent the mentioned kind of relationship of ΣL and D within the limits of emission couple 1 and emission couple 2, respectively. In addition,
Figure 29a,b represent the mentioned kind of relationship of ΣL and D within the limits of emission couple 3 and emission couple 4, respectively.
Figure 28 and
Figure 29 show the relationships between the diameter and the length of CSP structures that are obtained from Pareto front designs. Additionally, there is a similar linear relationship which appears between the length and the diameter. The calculated coefficient-of-determination values are bigger than 0.98 for all the fictionalized cases. Therefore, this situation may lead to the thought that these graphs can be used for the predesign of CSP systems and can be adapted for different shear-strength-angle values. The envisaged mathematical expressions can be associated based on the similarity of the coefficients and constraints values in the equation. Hence,
Figure 30 is produced to query the expressibility situation of the integrated relationship with the use of a simple ordinary expression.
Figure 30 shows the general dimensional change in the CSP walls depending on the change in unit cost, emission, and depth values. The integrated relation can ease the design process by producing charts or graphs for only different shear strength values.