https://doi.org/10.1007/s00158-019-02470-w
RESEARCH PAPER
A regression-based approach for estimating preliminary
dimensioning of reinforced concrete cantilever retaining walls
Ugur Dagdeviren1 · Burak Kaymak1
Received: 29 March 2019 / Revised: 28 October 2019 / Accepted: 3 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract
The reinforced concrete cantilever retaining walls (RCCRWs) are among the most commonly used type of structures to support the soil in civil engineering applications. In the conventional trial and error design of RCCRWs, which are based on engineering experiences and literature reviews, the preliminary dimensions of the wall are selected by considering the wall height only. However, it is known that the properties of backfill soil and surcharge loads also affect the dimensions of the wall. Therefore, in order to take into account the effects of the backfill soil properties and surcharge loads in addition to the height of the wall, a new regression-based approach is developed for predicting the preliminary dimensions of T-shaped RCCRWs. For this aim, a total of 375 optimization analyses are carried out for the optimum design of RCCRWs resting on soil with high bearing capacity by using the artificial bee colony (ABC) algorithm. Based on these calculated optimum solutions, the regression equations are developed for preliminary dimensioning of the T-shaped RCCRWs by using multiple regression analyses. Moreover, a set of 15 random problems are generated to assess prediction ability of the proposed regression equations, and their optimum dimensions are calculated by ABC algorithm and then these calculated dimensions are compared with the preliminary dimensions estimated by the proposed regression equations. From this comparison, it is observed that the maximum difference between the calculated and the estimated wall dimensions is only 6.2%. This means that the proposed preliminary dimensioning regression equations are capable of predicting dimensions that are close enough to the optimum dimensions. Therefore, for the most economical design of the T-shaped RCCRWs resting on soil with high bearing capacity, the predicted dimensions, which are supplied by the proposed regression equations, can be used as a good starting point when an optimization technique or a conventional trial and error method is employed.
Keywords Artificial bee colony (ABC)· Multiple regression model · Optimization · Preliminary dimensioning ·
Retaining walls
1 Introduction
Reinforced concrete cantilever retaining walls (RCCRWs) are important retaining structures used to support the earth in civil engineering projects such as buildings, highways,
Responsible Editor: Mehmet Polat Saka
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00158-019-02470-w) con-tains supplementary material, which is available to authorized users.
Ugur Dagdeviren
ugur.dagdeviren@dpu.edu.tr
1 Department of Civil Engineering, Kutahya Dumlupinar
University, Kutahya, 43100, Turkey
railways, embankments, and slopes. Many different failure modes must be considered in the design of RCCRWs, which are external stability requirements such as overturning, sliding, no tension condition, and bearing capacity of the soil under the foundation of the wall and internal stability requirements such as bending and shear capacity controls for the critical sections of the stem, toe, and heel of the wall. In the classical design procedure of RCCRWs, the cross-sectional dimensions, which are stem thickness at the top and bottom, widths of the toe and heel, and thickness of the foundation, are firstly selected based on previous design experiences and literature reviews, then, they are checked in terms of external and internal design criteria. In order to minimize the cost of the wall, the trial and error method is applied so that the cross-sectional dimensions and reinforcement ratios are reselected and the internal and external stability analyses are repeated until the most
14 January 2020 / Published online:
economical design is obtained. In order to come closer to the optimum solution, the selection of the preliminary cross-sectional dimensions is a very important step at the beginning of the design calculations. However, it is too laborious to reach the most economical solution with the conventional design approach. Therefore, in this study, the design of RCCRWs is considered as an optimization problem where the cost of the wall is the objective function and the constraints are various failure modes and design criteria.
In recent years, mathematical and meta-heuristic opti-mization algorithms have been widely used to solve com-plex engineering problems. Nowadays, the optimization problems are generally solved by meta-heuristic methods rather than mathematical programming methods, because of ease of use and convergence speed in finding the opti-mum value (Sun et al. 2018; Greco et al. 2019; Kaymak 2019; Vargas et al.2019; Yılmaz and Temel Gencer 2019; Wang et al.2019a; Wang et al. 2019b; Zhang et al.2019). The meta-heuristic optimization techniques have been suc-cessfully employed in the optimum design of reinforced concrete retaining walls. Some of these techniques used to optimize the reinforced concrete retaining walls are particle swarm optimization (Ahmadi-Nedushan and Varaee2009), ant colony optimization (Ghazavi and Bonab2011), bacte-rial foraging optimization algorithm (Ghazavi and Salavati 2011), harmony search algorithm (Kaveh and Abadi2011), genetic algorithm (Pei and Xia 2012), charged system search algorithm (Kaveh and Behnam2013), firefly algo-rithm (Sheikholeslami et al.2016). The artificial bee colony (ABC) algorithm is one of the meta-heuristic optimiza-tion algorithms, which was developed by Karaboga based on a particular intelligent behavior of honeybee swarms (Karaboga 2005; Karaboga and Basturk 2007; Karaboga and Akay2011). The ABC algorithm has many advantages over the other meta-heuristic methods, such as simplic-ity, flexibilsimplic-ity, stabilsimplic-ity, fewer algorithm parameters needed, easy hybridization with other optimization algorithms, and high performance of the algorithm (Karaboga and Bas-turk 2008; Bolaji et al. 2013). The ABC algorithm has been successfully applied in many different problems such as multidimensional and multimodal numerical problems (Bolaji et al.2013), sustainable groundwater management strategies (Boddula and Eldho2018), to invert the surface wave dispersion data (Song et al. 2015), design of space trusses (Sonmez 2018), design of steel frames (Aydo˘gdu et al.2016), detection of multiple complex flaw clusters in elastic solids (Ma et al.2017), and the problem of locating additional drill holes (Jafrasteh and Fathianpour2017).
In order to determine the most economical design dimensions by conventional trial and error method, the preliminary dimensions should be chosen as close as
possible to those economical dimensions. However, it is observed that the preliminary dimensions, which are given in the literature are insufficient to satisfy this requirement. The main reason for this is the fact that the preliminary wall cross-sectional dimensions are defined by considering the wall height only (Das 2016; Bowles 1997; Budhu 2008). Although the most important parameter on the cost of the wall is its height, the researchers also showed that the surcharge load and the properties of backfill soil also have significant effects on the cost of the wall (Camp and Akin 2011; Gandomi et al. 2017; Da˘gdeviren and Kaymak 2018). The aim of this study is to develop a multiple linear regression model that takes into account all these parameters in selecting the preliminary cross-sectional dimensions of T-shaped RCCRWs. For this purpose, 375 different sample retaining wall problems, which have different wall heights, surcharge loads, internal friction angles, and unit weights of backfill soil, are solved by the ABC algorithm. It is also assumed that for the sample problems, the base soil has high bearing capacity to obtain the minimum cost design of RCCRW. In this study, structural design calculations are based on ACI 318/2005, which is the building code requirements for structural concrete and commentary (American Concrete Institute 2005). The critical section dimensions of the stem, toe, heel, and foundation of the wall, which are calculated by using the ABC algorithm, are all investigated and the regression equations for the prediction of the dimensions of T-shaped RCCRWs are developed from the multiple linear regression model. Moreover, in order to assess the prediction ability of regression equations, a total of 15 test problems are randomly generated and solved by the ABC algorithm. When the calculated wall dimensions are compared with the wall dimensions which are predicted by using the proposed regression equations, it is observed that they are rather close to each other. This means that the proposed regression equations are capable of estimating the preliminary design dimensions close enough to the calculated optimal dimensions. Which means that the proposed regression-based approach is better than the conventional approaches (Das 2016; Bowles1997; Budhu 2008) in estimating the preliminary design dimensions for the T-shaped RCCRWs. In order to calculate the most economical design dimensions for the T-shaped RCCRWs, which are resting on soil with high bearing capacity, the preliminary design dimensions which are given by the proposed regression equations can be used as a good starting point when an optimization technique or a conventional trial and error method is employed.
In the following sections, a new approach for estimating the preliminary design dimensions for the T-shaped RCCRWs is presented such that: In Section2, some of the
definitions used in the artificial bee colony algorithm are presented. In Section3, some of the design details of the reinforced concrete retaining walls are given. The optimum design problem of reinforced concrete retaining walls is described and the performance of the ABC algorithm is discussed in Section 4. The multiple linear regression technique, which is used in developing the proposed regression equations for estimating the preliminary design dimensions for the T-shaped RCCRWs, is presented in Section 5. In Section 6, the multiple regression models are developed and the preliminary design dimensions predicted by the proposed regression equations and by the conventional approaches are compared for 15 test problems. This comparison shows that the proposed regression equations predict preliminary design dimensions much better than the conventional approaches. The last section is devoted to the conclusions and the findings of the present study.
2 Artificial bee colony algorithm
The artificial bee colony algorithm is a meta-heuristic optimization method based on the foraging behavior of honeybee colonies, which is proposed by Karaboga (2005). The algorithm is modified many times, in order to increase the convergence speed (Karaboga and Akay2011; Akay and Karaboga 2012). The ABC algorithm is more advantageous than other meta-heuristics algorithms because of its simplicity, flexibility, robustness, and having fewer control parameters. The details of the ABC algorithm could be found in several papers (Karaboga 2005; Karaboga and Basturk 2007; Karaboga and Akay 2011; Akay and Karaboga2012). The flowchart of the ABC algorithm which is used in this study is given in Fig.1. The main phase of the algorithm can be summarized as follows:
Initialization The ABC algorithm generates a randomized
initial population in the solution space, within the lower (xmin,j) and upper (xmax,j) bounds of each design variable
(xj) as follows:
xij = xmin,j+ rand(0, 1)(xmax,j − xmin,j) (1) Employed bees phase The employed bees identify new
resources in the neighborhood of the existing food source by using (2). In the modified ABC algorithm proposed by Karaboga and Akay (2011), the comparison of the quality between the new food source (vij) and the existing food source (xij) in the memories of the employed bees, and the
Fig. 1 The flowchart of ABC algorithm
selection process of best food source is made according to Deb’s rule (2000).
vij =
xij + Φij(xij− xkj) , Rj < MR
xij , Rj ≥ MR (2)
where k ∈ {1, 2, . . . , sn} is randomly chosen index that has to be different from i. sn is the number of employed bees in the population. Rjis random real number in the range [0, 1] and Φij is random real number in the range of [−1, 1].
xij is component j of position of ith bee. MR is a control parameter in the range of [0, 1].
Calculate probabilities for onlookers Once all employed
bees have completed the search process, they share the information about the quality and location of the food sources with the onlooker bees. Onlooker bees make the probabilistic selection to search in the vicinity of the known
resources. The selection process is made according to the probability values calculated by the roulette-wheel selection depending on the fitness values and the violation values. The probability of selecting the ith employed bee is defined in (3). Pi= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0.5+ fitnessi sn j=1fitnessj × 0.5 , for feasible ⎛ ⎜ ⎝1 − violationi sn j=1 violationj ⎞ ⎟ ⎠ × 0.5 , for infeasible (3)
The definition of fitness is given in (4), where fi is the objective function of the optimization problem.
fitnessi=
1
1+fi , fi≥ 0
1+ |fi| , fi<0 (4)
The general definition of the constraint violation is given in (5), where gj(x) are normalized inequality constraint functions and hj(x) are normalized equality constraint functions. q is the number of inequality constraints, and m is the number of equality constraints.
violationi= q j=1 max{0, gj(x)} + m j=1 |hj(x)| (5)
In (5), while the constraint violation value is calculated, when the normalized inequality constraint function gj(x)
has a negative value then, this gj(x)function value is taken as zero in the calculations. In this study, there are twelve
inequality constraints (q = 12) and there is no equality constraint (m = 0) in the optimization problems which are presented in Section 4. Therefore, the value of the hj(x)
function will be zero in this problem.
Onlooker bees phase Onlooker bees look for new food
sources in the vicinity of the existing food sources, just like the employed bees. They compare the new solutions with the old solutions and keep better solutions in their memories. If a better solution is not available, then the error value is increased by 1. If the error value exceeds the limit value, which is one of the parameters of the algorithm, then it is concluded that a better solution cannot be found in the vicinity of the food source and the related bees become the scout bees.
Scout bees phase The last stage of the ABC algorithm is the
scout bees phase. The scout bees look for random solutions in the search space by using (1) just like initialization phase. The steps which are defined above are repeated until the defined maximum number of cycles is reached and the best solution obtained is reported and the algorithm is terminated.
3 Design of the RCCRW
The typical vertical and horizontal forces acting on the T-shaped RCCRWs handled in this study are shown in
Fig.2. In the model, it is assumed that the backfill is non-cohesive soil, the ground surface on the front and back of the wall is horizontal, and there is no friction between the stem of the wall and backfill soil. The active and passive earth pressures are calculated according to Rankine’s earth pressure theory, where active earth pressure coefficients (Ka) are determined by using the angle of internal friction of the backfill soil (φ1) as given in (6). The active earth
pressure due to backfill soil and external loading tends to deflect the wall away from the backfill. The lateral earth forces depend on highly the wall and soil properties such as the height of wall (H ), the surcharge load (q), the angle of internal friction (φ1), and the unit weight (γ1) of the backfill
soil as shown in Fig.2a. The designed RCCRW must satisfy two requirements for stability of the wall, which are external stability (the geotechnical design) and internal stability (the structural design).
Ka= tan2(45−
φ1
2) (6)
For external stability, the wall as a whole has to be safe against overturning, sliding, bearing capacity, and tension stress below the foundation. The weight of natural soil on the toe, surcharge load above the heel, and passive earth pressure are generally neglected in external stability analyses. The RCCRW must have adequate resistance against overturning of the wall. The factor of safety against overturning, which is the ratio of the sum of the moments to resist against overturning around point O shown in Fig.2a to the sum of the overturning moments, can be defined as in (7). The factor of safety against overturning must be at least 1.5 for granular backfill soils (Venkatramaiah2016).
F Soverturning=
Σ MR
Σ Mo
(7)
The factor of safety against sliding of the wall foundation (F Ssliding), which is the ratio of the sum of the horizontal
resisting forces to the sum of the horizontal sliding forces (ΣFS = Pa + Pq), can be defined as in (8), and the minimum factor of safety against sliding must be greater than 1.5 for backfills such as sand, gravel, and rock (Venkatramaiah2016).
F Ssliding=
ΣFR
ΣFS
(8) The horizontal resisting forces are calculated from (9).
ΣFR= Pp+ ΣV tan(k1φ2)+ B(k2c2) (9)
ΣV is the total vertical load transferred from the wall to the soil and B is the width of the foundation of the wall, which is equal to the sum of design variables x2, x3, and x4. Pp
is the passive earth force, which is generally neglected for calculation of sliding resistance. φ2and c2are shear strength
parameters of the foundation soil and k1and k2are reduction
coefficients in the range of 1/2 to 2/3.
The total base stress below the foundation of the retaining wall must be checked against the ultimate bearing capacity of the foundation soil (qult). The maximum base
stress (qmax) must not exceed the bearing capacity of the
foundation as in (10). In general, the minimum value of the factor of safety against bearing capacity failure (F Sb,min)
should not be less than 3. Furthermore, the minimum base stress below the foundation of the RCCRW must be greater than or equal to zero. This failure mode can be expressed as in (11) depending on the eccentricity (e).
F Sbearing=
qult
qmax
(10)
Table 1 Design variables and their limits
Design variables Description Lower limits Upper limits
x1 Stem thickness at the top 0.3 m 4.0 m
x2 Width of the toe 0.0 m 4.0 m
x3 Stem thickness at the bottom 0.3 m 4.0 m
x4 Width of the heel 0.2 m 4.0 m
x5 Thickness of the foundation 0.2 m 4.0 m
x6 Main reinforcement ratio of the toe ρmin ρmax
x7 Main reinforcement ratio of the heel ρmin ρmax
x8 Main reinforcement ratio of the stem ρmin ρmax
F Stension=
B/6
e (11)
For the internal stability, all sections of the wall must be safe against shear and bending failures. The internal stability analyses for RCCRWs are generally applied to at least three critical sections, which are the stem of the wall, toe, and heel of the foundation. On the critical sections (1–1, 2–2, and 3–3), which are shown in Fig.2a, the distribution of the stresses is shown in Fig. 3. The behavior of the stem, toe, and heel is the same as that at the support of the cantilever slabs. All sections must satisfy both flexure and shear requirements. The moment and shear capacity of these sections must be greater than the design moment (Md) and design shear force (Vd). In ACI 318/2005, the moment capacity (Mn) and shear capacity (Vn) for the reinforced concrete members are given in relations (12) and (13), respectively. Mn= φMAsfy d−a 2 ≥ Md (12) Vn= φV0.17 fcbd≥ Vd (13)
where the strength reduction factors are φM = 0.9 and
φV = 0.75 according to ACI 318/2005. The reinforcement
Table 2 Input parameters and used data for the analysis sets Input parameters Used data values Height of wall (H ) 4, 5, 6, 7, and 8 m Surcharge load (q) 0, 10, 20, 30, and 40 kPa Internal friction
Angle of backfill (φ1) 28, 30, 32, 34, and 36◦
Unit weight of backfill (γ1) 16, 17, and 18 kN/m3
Depth of soil in front of wall (Df) 1.0 m
Internal friction
Angle of base soil (φ2) 30◦
Cohesion of base soil (c2) 200 kPa
Compression strength of concrete 30 MPa Yield strength of steel 420 MPa
ratio in each section of the RCCRW must be between the minimum (ρmin) and maximum (ρmax) reinforcement ratios.
4 Definition of the optimization problem
In this study, the design variables of the RCCRW optimization problem are divided into two groups: (1) geometric dimensions of the cross sections of the wall and (2) main reinforcement ratios in the critical sections. There are five geometric design variables and three reinforcement ratio design variables in this optimum design model. The design variables are shown in Fig.2b and defined in Table1. The properties of the backfill and foundation soil, height of the wall, external loads, desired minimum factors of safety, and concrete and steel specifications are considered as the input parameters in the optimization analyses. The values of the input parameters are given in Table2. In the analyses, the unit weights of base soil and concrete are taken as 18 kN/m3 and 23.5 kN/m3, respectively. The minimum factors of safety against overturning, sliding, and bearing capacity failure are chosen as 1.5, 1.5, and 3.0, respectively. The height of retaining wall (H ), the surcharge load (q), the angle of internal friction of the backfill (φ1), and the unit
weight of the backfill (γ1) are selected as variables. The
knowledge obtained from the previous design experiences shows that the cost of the walls is not affected from the shear strength parameters of base soil with high bearing capacity. Therefore, in this study, all of the analyses are carried out for the base soil with high bearing capacity. In order to reflect the strong soil conditions, the shear strength parameters of the base soil are selected as c2= 200 kPa and φ2= 30◦.
In this study for the RCCRW optimization problems, all the constraints are inequality constraints and there are no equality constraints. These inequality constraints are called design constraints and they are classified into three categories: failure modes for external stability, g1−4(x);
failure modes for internal stability, g5−10(x); and additional
Table 3 Normalized design constraints
Design constraints Description
g1(x) = F SF Soverturningo,min − 1 ≤ 0 Overturning failure mode g2(x) = F SF Sslidings,min − 1 ≤ 0 Sliding failure mode g3(x) = F SF Sbearingb,min − 1 ≤ 0 Bearing failure mode g4(x) = F Stension1 − 1 ≤ 0 No tension failure mode
g5−7(x) = Md,i
Mn,i − 1 ≤ 0 Bending failure mode for toe, heel, and stem
g8−10(x) = Vd,i
Vn,i − 1 ≤ 0 Shear failure mode for toe, heel, and stem
g11(x) = 50(xH−x3−x51)− 1 ≤ 0 Geometric constraint
g12(x) = xH5 − 1 ≤ 0 Geometric constraint
constraints for the optimization problems are given in Table3.
The objective function for the optimum design of RCCRW is to minimize the cost of the wall. When the cost of excavation, formwork, and backfill compaction is neglected, the objective function is formulated as;
f (x)= CstWst+ CcVc,net (14)
In (14), Cst is unit cost of steel and accepted as
$0.40/kg (Sarıbas¸ and Erbatur 2003), Cc is unit cost of
concrete and accepted as $40/m3 (Sarıbas¸ and Erbatur 2003), Wstis the weight of steel per unit length of the wall,
and Vc,net is the net volume of concrete per unit length of
the wall.
4.1 The performance of the algorithm
The aim of the present study is to determine how the cross-sectional dimensions of RCCRW, which is resting on soil with high bearing capacity, are affected by the design variables such as the angle of internal friction of the backfill (φ1), height of retaining wall (H ), surcharge
load (q), and unit weight of backfill (γ1). In order to find
out the answer to this question, 375 different optimization problems are solved by using the ABC algorithm and the optimum cross-sectional dimensions are calculated. In these optimization calculations, combinations of the following design variables are used: angles of internal friction of the backfill (28, 30, 32, 34, and 36◦), wall heights (4, 5, 6, 7, and 8 m), surcharge loads (0, 10, 20, 30, and 40 kPa), and unit weights of backfill (16, 17, and 18 kN/m3) which are all realistic design variables. In order to assure high levels of stability, performance, and repeatability of the ABC algorithm, the parameters of the algorithm are also tested for different combinations and the appropriate parameter values are determined and used in the calculations (number of colonies, CS = 100; modification rate, MR = 0.8; limit = 400; number of maximum cycles, MCN = 1000). The ABC algorithm is implemented by C++ programming language. The analyses are run on a computer with Intel CoreTM
i7-2760QM CPU @ 2.40 GHz× 8 and 8-GB RAM memory.
The optimization analyses are repeated 30 times for each problem starting with independent random populations. For 30 independent analysis results, the best, the worst and
Table 4 Selected example results in 375 RCCRW optimization problems
Objective function ($/m)
No. H(m) φ1(◦) q(kPa) γ1(kN/m3) Best Mean Worst COV (%) Time (s)
A1 4 30 10 16 97.30 97.60 97.85 0.14 2.48 A2 4 32 20 18 108.33 108.56 108.78 0.11 2.32 A3 5 28 0 16 126.78 127.25 128.28 0.23 2.55 A4 5 32 40 18 201.61 202.07 202.65 0.13 2.57 A5 6 34 30 16 239.50 240.29 241.21 0.14 2.62 A6 6 36 20 18 207.72 208.70 209.44 0.18 2.45 A7 7 28 40 17 411.19 414.48 422.95 0.71 2.83 A8 7 36 30 17 306.15 307.53 312.67 0.39 2.73 A9 8 30 0 16 326.03 327.48 329.36 0.28 2.61 A10 8 34 10 18 350.57 352.97 355.83 0.29 2.45
Fig. 4 The distribution of the coefficients of variation with wall height
average objective function values, coefficient of variation (COV), and the time taken for the analyses are given in Table 4 for the selected 10 sample problems. In order to test the repeatability of the algorithm, the distribution of the objective functions, which are obtained from 30 independent analyses, are examined and it is seen in Table4 that the maximum coefficient of variation is equal to 0.71% for the selected examples. Similar evaluations for coefficients of variation are seen in Fig. 4 for a total of 375 optimization problems handled in the study. Figure4 shows that the coefficient of variation increases with wall height, because of the narrowing of the feasible region with increasing wall height. In 90% of the 375 optimization problems, it is seen that the coefficients of variation are below 0.39% and in 95% of the 375 optimization problems,
Fig. 5 The convergence rate for ten sample problems
the coefficients of variation are below 0.54%. Almost for all optimization problems, the coefficients of variation are less than 1%. It is possible to reduce the variability of the optimum values obtained from the 30 independent analyses by changing the parameters of the algorithm, but the repeatability of the algorithm is found to be sufficient for the purposes of this study. Also, it is seen that the time taken to perform 30 independent analyses is less than 3 s for all optimization problems.
In order to evaluate the performance of the algorithm, the convergence rate, normalized design constraints, and the values of the design variables are examined for the ten sample problems. The variations of the cost of the wall with the number of iterations for the ten sample problems are given in Fig.5and from this figure, it is observed that the convergence rates are quite high. Additionally, Table 5is prepared to show that the algorithm is capable of calculating the optima of the RCCRWs where all the normalized constraints remain in between 0.000 and −1.000 for 375 optimization problems. If the normalized constraint values are equal to 0.000, it means that the normalized constraints are equal to the given limits. If the normalized constraint values are close to −1.000, it means that the normalized constraints are far from the given limits. Active constraints are shown in bold in Table 5. Because it is assumed that the base soil is high bearing capacity in the definition of the optimization problem, bearing capacity design constraint,
g3(x), is not the active constraints. However, the no tension
failure mode constraints, g4(x), are the active constraint
for all external stability analyses. Bending failure mode constraints (g5(x), g6(x), g7(x)) and shear failure mode
constraints (g8(x), g9(x), g10(x)) in the critical sections of
the wall should be considered together in the evaluations. The g5(x), g6(x), g8(x), and g9(x) constraints relate
with the foundation of the wall and these constraints are the parameters that to determine the cross-sectional dimensions (x2, x4, x5) and reinforcement ratios (x6, x7)
of the foundation. The g5(x)and/or g6(x)constraints are
active constraints in almost all of the analyses. Also, it is understood that the g8(x) and g9(x) constraints are also
involved in determining dimensions in some problems. At least one of the g7(x)constraint or g10(x)constraint or the
wall face slope constraint, g11(x), which is determinative in
determining the design variables x3and x8, is active in all
problems. Therefore, it is clearly shown that the ability of finding the effective solution of the algorithm is high in the stem dimensions just as on the foundation of the wall. The
g12(x) geometric constraint, which keeps the foundation
thickness within the structural rules, is not active at the RCCRWs resting on soil with high bearing capacity.
The values of the design variables at the optima of the ten sample problems are shown in Table6. The bold text shows the values which are near the design variable limit.
Table 5 Normalized design constraints for the best solutions No. g1(x) g2(x) g3(x) g4(x) g5(x) g6(x) g7(x) g8(x) g9(x) g10(x) g11(x) g12(x) A1 −0.269 −0.766 −0.967 −0.004 −0.014 −0.008 −0.076 −0.079 −0.257 −0.635 −0.001 −0.945 A2 −0.255 −0.737 −0.965 −0.002 −0.006 −0.001 −0.005 −0.117 −0.143 −0.588 −0.254 −0.936 A3 −0.313 −0.733 −0.954 −0.001 −0.015 −0.025 −0.002 −0.112 −0.327 −0.565 −0.039 −0.948 A4 −0.245 −0.656 −0.959 −0.003 −0.007 −0.008 −0.003 −0.148 −0.032 −0.509 −0.685 −0.917 A5 −0.262 −0.670 −0.953 −0.003 −0.003 −0.005 −0.005 −0.104 −0.084 −0.516 −0.642 −0.927 A6 −0.275 −0.675 −0.945 −0.000 −0.034 −0.037 −0.496 −0.002 −0.004 −0.009 −0.501 −0.937 A7 −0.256 −0.551 −0.945 −0.001 −0.005 −0.005 −0.000 −0.054 −0.001 −0.418 −0.791 −0.908 A8 −0.253 −0.633 −0.944 −0.003 −0.009 −0.024 −0.003 −0.000 −0.034 −0.455 −0.640 −0.929 A9 −0.281 −0.615 −0.934 −0.006 −0.117 −0.011 −0.001 −0.004 −0.273 −0.423 −0.573 −0.936 A10 −0.264 −0.595 −0.930 −0.001 −0.000 −0.174 −0.431 −0.157 −0.001 −0.002 −0.636 −0.930
The stem thickness at the top is equal to the lower limit value of the design variable x1 for all problems. Other
geometric design variables are not close to the lower/upper limit values. For the concrete and steel properties used in this study, reinforcement ratios must remain between 0.0033 and 0.0304 according to ACI 318/2005. When the values of the reinforcement ratios (x6, x7, and x8) are examined in
Table6, then, it is seen that most of the reinforcement ratio values are very close to the lower limit value and some of them are seen to be equal to the lower limit value 0.0033. It proves that the ABC algorithm for the RCCRW problem works effectively.
When all these evaluations for RCCRWs, which are resting on soil with high bearing capacity, are taken into consideration, it is seen that the ABC algorithm successfully calculates the optimum solutions. The purpose of this study is not to discuss the applicability of the ABC algorithm in solving the optimum RCCRW problems but to use the optimum solutions in generating the regression model equations.
5 Multiple linear regression
In many engineering and science problems, it is observed that the values taken by the design variables are also related to the other design variables. There are various methods for determining the relationship between the dependent and independent variables, and the most commonly used statistical technique is the regression analysis. The regression analysis is used in many fields such as in engineering, science, and social sciences problems. The purpose of the regression analysis is to determine whether there is a significant relationship between the variables, to find an appropriate regression equation and to calculate the confidence interval of the predictions using this equation. In the regression analysis, many different types of mathematical functions (linear and nonlinear) are used to model a response which is a function of one or more independent variables. If a linear relationship between more than two variables is accepted then, these analyses are defined as multiple linear regression analysis. Multiple
Table 6 The design variables and objective functions for the best solutions
No. x1(m) x2(m) x3(m) x4(m) x5(m) x6 x7 x8 fbest($/m) A1 0.3000 0.9079 0.3757 1.1083 0.2184 0.0055 0.0054 0.0033 97.30 A2 0.3003 1.0131 0.4008 1.1798 0.2543 0.0048 0.0053 0.0034 108.33 A3 0.3000 0.7496 0.3987 1.4416 0.2607 0.0033 0.0055 0.0038 126.78 A4 0.3000 1.5143 0.5915 1.5811 0.4126 0.0038 0.0044 0.0033 201.61 A5 0.3001 1.4506 0.6110 1.7517 0.4384 0.0035 0.0043 0.0037 239.50 A6 0.3000 1.0891 0.5256 1.6789 0.3780 0.0033 0.0053 0.0045 207.72 A7 0.3000 1.9896 0.9075 2.2662 0.6419 0.0033 0.0039 0.0033 411.19 A8 0.3014 1.5780 0.6622 1.8473 0.5004 0.0037 0.0042 0.0044 306.15 A9 0.3002 1.3222 0.6506 1.9954 0.5112 0.0033 0.0034 0.0047 326.03 A10 0.3000 1.3886 0.7086 1.9471 0.5602 0.0033 0.0033 0.0044 350.57
linear regression analysis is widely used in various fields of geosciences and geotechnical engineering problems such as to predict the factor of safety of the slopes (Chakraborty and Goswami 2017), compressibility parameters of clays (Yoon et al.2005), and compressive strength of artificially structured soils (Sharma and Singh2018).
When a dependent variable (y) is considered to be affected by more than one independent variables (x1,
x2, . . ., xn) and assuming that the independent variables
have a linear relationships with the dependent variable, the multiple linear regression model with n independent variables is described as:
yi= β0+ β1xi,1+ β2xi,2+ . . . + βnxi,n+ εi (15) where yiis the observation value of the dependent variable,
iis the number of observation, n is the number of predictor variables, the parameter β0is the intercept of the regression
plane, and parameters β1, β2, . . . , βn are called regression coefficients for the predictor variables. The actual y value to any observation (yi) may not be equal to a predicted value (ˆyi) then a random error (εi) occurs. εi is called the ith residual, which is the difference between the observed and the predicted values in the ith observation. If it is assumed that the expected error value is zero, then the regression equation can be written as
E(Y )= ˆy = b0+ b1x1+ b2x2+ . . . + bnxn (16) where, b0, b1, . . . , bnare estimated regression coefficients of the proposed model. This model describes a hyperplane in the n-dimensional space of the independent variables. For the calculation of the regression coefficients in this function, the sum of squares of error (SSE), which is the sum of squares of deviations of the observed value (yi) from the estimated values (ˆyi), will be minimized by using the least squares method as defined in (17). The minimum SSE can be achieved when the partial derivatives of SSE with respect to regression coefficients are equal to zero.
min SSE= min n i=1 yi− ˆyi 2 (17) Before using the proposed regression models, the models must be also verified with different methods. There are many methods available to check the validity of the regression models. The statistical measure parameters, hypothesis tests, and using the verification data set are only a few of them. In this study, the performance of the proposed regression models is checked by the additional test results and by using the correlation coefficient (R), the root mean squared error (RMSE) and the mean absolute relative error (MARE) values, which are defined in (18), (19), and (20) respectively. In these definitions, yi and ˆyi are calculated and predicted output values, respectively. n is the number of analyses presented in the database. If the coefficient
of determination (R2), which is the square of correlation coefficient (R), is equal to one, then this implies that the dependent variable is estimated without error. When the RMSE or MARE value approaches zero, then this implies that the regression model provides accurate predictions.
R= n i=1 (yi− yi)(ˆyi− ˆyi) n i=1 (yi− yi)2 n i=1 (ˆyi− ˆyi)2 (18) RMSE= n i=1 (yi− ˆyi)2 n (19) MARE= n i=1 yi− ˆyi yi n (20)
6 Regression models for dimensioning
the RCCRW
6.1 Selection of the predictor variables in regression models
One of the key points in multivariate regression analysis is deciding the number of independent variables to be used in the model. When the number of input parameters used for estimating the dependent variable is few, then the model becomes easier to use. While deciding on the variables that will be used in the analyses, some clues can be obtained from the correlations between the dependent and independent variables. Thus, a correlation matrix must be constructed for the input variables (φ1, H , q, γ1) and for
the cross-sectional dimensions of the wall (x1, x2, x3, x4,
x5), which are obtained as a result of the 375 optimization
analyses conducted. When the correlation matrix given in Table 7 is examined, it is clearly observed that the input parameters do not have significant interdependence.
When the correlation matrix is studied, it is observed that the highest binary correlation between dimension x1 and
input parameter (γ1) is−0.107 and statistically this means
that x1has no significant relationship with the independent
variables. When the correlation between dimension x2with
the input parameters is studied, then the highest correlations are with surcharge load (q), wall height (H ), and angle of internal friction of the backfill (φ1), respectively. However,
the correlation between x2and unit weight of backfill (γ1)
Table 7 Correlation matrix for dependent and independent variables
Input parameters Output parameters
φ1 H q γ1 x1 x2 x3 x4 x5 φ1 1.000 H 0.000 1.000 q 0.000 0.000 1.000 γ1 0.000 0.000 0.000 1.000 x1 −0.069 −0.047 0.024 −0.107 1.000 x2 −0.207 0.619 0.741 −0.052 0.002 1.000 x3 −0.183 0.832 0.456 0.022 −0.061 0.892 1.000 x4 −0.271 0.880 0.350 0.000 0.000 0.846 0.929 1.000 x5 −0.145 0.888 0.412 0.048 −0.045 0.889 0.969 0.963 1.000
coefficients of the remaining three dependent variables (x3, x4, x5) with the independent variables is aligned as
wall height (H ), surcharge load (q), and angle of internal friction of the backfill (φ1), respectively, but the unit
weight of backfill (γ1) does not have statistically significant
correlation with these dependent variables.
Interpreting the existence of a linear relationship between the dependent variables and each of the independent vari-ables by solely considering the correlation coefficient may be statistically inadequate. Thus, a series of statistical tests must be carried out to determine the existence of a possible relationship between the dependent and indepen-dent variables. In this study, the null hypothesis (Ho :
ρX,Y = 0) is tested against the alternative hypothesis (H1 : ρX,Y = 0) at the significance level of 0.01. When
the joint distribution of dependent and independent vari-ables is assumed to be a normal distribution, the t-statistic values are obtained which are shown in Table 8. After the hypothesis testing, the test results which are marked by (*) indicate that the null hypothesis is accepted, which means that the dependent variable does not have a statis-tically significant relationship with the related independent variable.
As a result of the hypothesis testing, which is aimed at discovering whether unit weight of backfill will be taken as an input parameter in regression analyses, Table8 shows that no statistically significant relationship exists between
the wall cross-sectional dimensions and unit weight of backfill (γ1).
Another test is carried out by checking the coefficient of determination (R2). For this test, the input variables are taken as 3 (φ1, H , q) and as 4 (φ1, H , q, γ1) in the
regression model equations developed in this study and then the coefficients of determination (R2) for these two models are calculated and then compared with each other. It is observed that almost the same coefficients of determination (R2) are calculated for both regression model equations such that the difference between the (R2) coefficients is as follows: 0.002 for dimension (x2), 0.002 for dimension
(x5), zero for dimension (x3), and zero for dimension (x4).
This means that the unit weight of the backfill (γ1) is
not an important parameter that affects the cross-sectional dimensions (x2, x3, x4, x5) of the wall and need not be
included in the regression model equations.
When the correlations between the input parameters and wall cross- sectional dimensions are evaluated together with the hypothesis tests shown in Table 8, it is observed that the dependent variable x1 does not have a statistically
significant relationship with any of the independent variables. The dimension x1remains within a very narrow
range (0.300 ∼ 0.304 m) in the 375 analyses carried out, and therefore it is deduced that x1 is not influenced by
any of the input parameters and it takes a value very close to 0.3 m which is the constructively essential lower
Table 8 The ρX,Y = 0 hypothesis test for dependent and independent variables
t-statistic for φ1 t-statistic for H t-statistic for q t-statistic for γ1
x1 −1.342∗ −0.917∗ 0.472∗ −2.079∗
x2 −4.088 15.217 21.315 −0.996∗
x3 −3.593 28.971 9.893 0.429∗
x4 −5.446 35.812 7.211 −0.003∗
Table 9 Regression models for the estimation of dependent variable x2and their statistical evaluations
Models Variables Coefficients St. error t stat P value Multiple R Sign. F
Model 1 Intercept 0.3438 0.0677 5.080 <0.001 0.619 5.1E−41
H 0.1670 0.0110 15.217 <0.001
Model 2 Intercept −0.0563 0.0236 −2.382 0.0177 0.965 4.4E−218
H 0.1670 0.0036 45.833 <0.001
q 0.0200 0.0004 54.879 <0.001
Model 3 Intercept 0.8381 0.0382 21.939 <0.001 0.987 6.4E−297
H 0.1670 0.0022 75.485 <0.001
q 0.0200 0.0002 90.383 <0.001
φ1 −0.0280 0.0011 −25.259 <0.001
design limit. Therefore, the regression equation estimating the preliminary design dimension x1is a constant which is
equal to the lower design limit, 0.3 m. 6.2 Regression models
In this study, three separate regression models for each dependent variable are constructed to estimate the prelimi-nary design dimensions of the RCCRW. In the selection of these models, the parameters which have the highest corre-lation between the dependent and independent variables are added to the regression equations. However, although the x2
variable has the highest binary correlation with surcharge load, to preserve the general form of all the models, in the first model, the first variable is taken as H instead of q. Dur-ing the regression analyses for model 1, only the H-based estimation model is used. For model 2, the q variable is added to variable H . In model 3, the φ1variable is added to
variables H and q. In this study, the multiple linear regres-sion model form shown in (21) is used. In order to preserve this general form, some of the regression coefficients of the variables which are not used in models 1 and 2 are taken as zero. Similarly, since x1dimension is not influenced by any
of the independent parameters, then in all the models, the regression coefficients (b1,1, b2,1, b3,1) will be zero.
ˆxi= b0,i+ b1,iH+ b2,iq+ b3,iφ1 (21)
The regression coefficients and the criteria for perfor-mance evaluation for each regression model developed for estimating the width of toe (x2) are presented in Table9.
In model 1, x2dimension is estimated as (x2 = 0.3438 +
0.1670H ). As seen in Table9, for the one-parameter model based on wall height only (model 1), the correlation coeffi-cient is equal to 0.619, which is a very low value indicating a poor estimate of x2dimension. In the literature, it is also
common to see the preliminary dimensioning approaches, which are based on wall height only. For instance, the pro-portional x2dimension is suggested as (0.1H ) in reference
(Das2016) or as (0.133H ∼ 0.233H) in reference (Bowles 1997) or as (0.12H ∼ 0.21H) in reference (Budhu2008). Also as seen in Table9, for model 2, where surcharge load is added as the second parameter, the correlation coeffi-cient value increases to R = 0.965 and for model 3, where the angle of internal friction of the backfill is added as the third parameter, the correlation coefficient increases to R = 0.987. This increase in the correlation coefficient value (R) is an indication of improved estimates of x2dimension.
Therefore, model 3 is more effective in estimating x2wall
dimension than model 1 and model 2. Moreover, it is also clearly observed from Table9that the standard errors of the regression coefficients and significant F-value determined for model 3 are very low, which means that the regression coefficients and the model are statistically significant.
Similar regression models and statistical evaluations are also made for the other design variables (x3, x4, and x5)
and the results are presented in Tables 10, 11, and 12, respectively. In one-parameter models, x3dimension and x5
dimension are determined as (x3 = 0.0077 + 0.0957H)
and (x5 = −0.1088 + 0.0888H), respectively. When
the regression coefficients in regression models with one parameter based only on wall height are investigated, it is determined that the values for x3 and x5 dimensions are
quite close to the wall height ratios (0.08H ∼ 0.1H) given in the literature (Das 2016; Bowles 1997; Budhu 2008). However, the correlation coefficients are equal to R = 0.832 and 0.888, which are low values estimating of x3 and x5
dimensions. It is seen that in Table 10, the correlation coefficient value increases to R = 0.949 for model 2 and
R = 0.966 for model 3. Similarly, for x5 dimension, the
correlation coefficient value is equal to R = 0.990 for model 3 as shown in Table12. It is clear that these three-parameter models are statistically significant.
For model 1, the regression coefficient is calculated to be
b1,4 = 0.2356 for dimension x4, whereas the proportional
preliminary dimension x4is defined as (0.3H ∼ 0.5H) in
Table 10 Regression models for the estimation of dependent variable x3and their statistical evaluations
Models Variables Coefficients St. error t stat P value Multiple R Sign. F
Model 1 Intercept 0.0077 0.0204 0.380 0.7043 0.832 1.7E−97
H 0.0957 0.0033 28.971 <0.001
Model 2 Intercept −0.0972 0.0122 −7.956 <0.001 0.949 7.0E−187
H 0.0957 0.0019 50.797 <0.001
q 0.0052 0.0002 27.834 <0.001
Model 3 Intercept 0.2395 0.0266 9.018 <0.001 0.966 4.3E−218
H 0.0957 0.0015 62.217 <0.001
q 0.0052 0.0002 34.091 <0.001
φ1 −0.0105 0.0008 −13.677 <0.001
(Bowles1997) or as (0.18H ∼ 0.41H) in reference (Budhu 2008). However, the correlation coefficient in the one-parameter model is equal to 0.880, which is the indication of the poor estimates. Also as seen in Table11, the correlation coefficients are R = 0.947 and 0.985 for model 2 and model 3, respectively. Model 3 is very effective in estimating x4
wall dimension.
It is also observed in Tables 9, 10, 11, and 12 that as the number of the input parameters is increased, the correlation coefficients are also significantly increased to the levels so that improved wall dimensions are estimated which are close to the optimum dimensions. For the three-parameter models (model 3), the regression coefficients and the regression equations are highly significant and the standard errors are smaller than 0.065 for all design variables. It is also observed that the P values in the three-parameter models are smaller than 0.05. This means that regression models are significant for all independent variables. Similarly, since the F-significant values are much less than 0.05, it is understood that the dependent variables can be significantly predicted by the independent variables. Therefore, three-parameter regression models (model 3),
where wall heights, surcharge loads, and the angles of internal friction of the backfills are used as the predictor variables, can be accepted as strong models giving more realistic preliminary dimension estimates for the most economical design of T-shaped RCCRWs.
Table 13 presents the error evaluations concerning the proposed regression models for estimating the preliminary design dimensions. For all three models, when dimension x1
is taken as 0.3 m, which is the design lower limit value, the error values are found to be 0.001, which are close to zero, and this is adequate for estimating the preliminary design dimension as x1 = 0.3. But in estimating the other design
dimensions (x2, x3, x4, x5), it is observed from Table13that
for the one-parameter model (model 1), the error values are rather high. Therefore, it can be asserted that the estimation capacity of the one-parameter models presented in this study is poor. As seen from Table 13, in estimating the design dimensions x2, x3, x4, x5, the error values for the
two-parameter model (model 2) are significantly less than those for the model 1. But they are still high, and therefore the estimation capacity of the two-parameter models presented in this study is also rather poor. When the three-parameter
Table 11 Regression models for the estimation of dependent variable x4and their statistical evaluations
Models Variables Coefficients St. error t stat P value Multiple R Sign. F
Model 1 Intercept 0.2477 0.0406 6.108 <0.001 0.880 9.2E−123
H 0.2356 0.0066 35.812 <0.001
Model 2 Intercept 0.0604 0.0289 2.094 0.0369 0.947 2.2E−184
H 0.2356 0.0045 52.909 <0.001
q 0.0094 0.0004 21.027 <0.001
Model 3 Intercept 1.2229 0.0411 29.783 <0.001 0.985 5.7E−284
H 0.2356 0.0024 99.059 <0.001
q 0.0094 0.0002 39.368 <0.001
Table 12 Regression models for the estimation of dependent variable x5and their statistical evaluations
Models Variables Coefficients St. Error t Stat P-value Multiple R Sign. F
Model 1 Intercept -0.1088 0.0146 -7.429 < 0.001 0.888 3.4E−128
H 0.0888 0.0024 37.380 < 0.001
Model 2 Intercept -0.1912 0.0068 -28.168 < 0.001 0.979 4.6E−259
H 0.0888 0.0010 84.789 < 0.001 q 0.0041 0.0001 39.334 < 0.001 Model 3 Intercept 0.0404 0.0126 3.205 0.0015 0.990 0 H 0.0888 0.0007 121.423 < 0.001 q 0.0041 0.0001 56.329 < 0.001 φ1 -0.0072 0.0004 -19.797 < 0.001
model (model 3) is used, it is seen from Table13that the error values are not very high and the highest RMSE and MARE are found to be 0.065 m and 5.81%, respectively. These results indicate that the estimation capacity of the three-parameter model presented in this study is very good. The multiple linear regression equations, which are proposed in this study for estimating the preliminary design dimensions for the most economical design of T-shaped RCCRWs, are summarized in (22). In case the values obtained using these regression equations are smaller than the design lower limits, then the preliminary dimension for related wall dimension must be selected as the design lower limit value. In the multiple linear regression equations,
H is defined in meters, q is defined in kilopascal, φ1 is
defined in degrees, and the corresponding wall dimensions are calculated in meters.
ˆx1= 0.3000 ˆx2= 0.8381 + 0.1670H + 0.0200q − 0.0279φ1 ˆx3= 0.2395 + 0.0957H + 0.0052q − 0.0105φ1 ˆx4= 1.2229 + 0.2356H + 0.0094q − 0.0363φ1 ˆx5= 0.0404 + 0.0888H + 0.0041q − 0.0072φ1 (22)
6.3 Testing of the proposed models
In order to test the preliminary dimension estimates of the proposed regression models (model 1, model 2, and model 3), 15 different data sets, which are randomly generated as given in Table 14, are handled. These data sets are used and the minimum-cost wall designs are obtained by using the ABC algorithm and the optimum wall dimensions are calculated which satisfy the design requirements. Afterward, using the data sets given in Table 14, the preliminary dimension values are estimated using the proposed regression models summarized in (22).
Figure 6 shows a comparison between the calculated optimum wall dimensions and the estimated wall dimen-sions obtained from the regression models. In view of the graphics shown in Fig.6; it is seen that the scatter between the estimated wall dimensions and the calculated optimum wall dimensions is quite high for one-parameter models (model 1). For model 1, the highest error between the calculated and the estimated values reaches to 41.7%. As shown in Fig. 6, estimating the wall dimensions from the
Table 13 Errors of the
proposed regression models Design variable Model RMSE (m) MARE (%)
x1 Model 1 0.001 0.06 x2 Model 1 0.300 21.53 Model 2 0.099 6.13 Model 3 0.060 4.00 x3 Model 1 0.090 12.08 Model 2 0.051 7.17 Model 3 0.042 5.81 x4 Model 1 0.180 9.55 Model 2 0.121 6.21 Model 3 0.065 3.28 x5 Model 1 0.065 13.81 Model 2 0.025 5.13 Model 3 0.018 3.22
Table 14 The random data sets used to test the proposed models
Test No. H(m) q(kPa) φ1(◦) γ1(kN/m3)
#1 6.6 38 36 17.0 #2 4.7 27 32 16.8 #3 5.9 10 29 17.0 #4 5.5 20 32 16.9 #5 6.9 24 34 16.6 #6 5.2 7 31 16.0 #7 5.8 5 29 17.4 #8 6.3 31 35 17.6 #9 5.6 30 30 16.4 #10 5.7 16 28 16.8 #11 7.5 36 33 17.8 #12 4.2 13 33 17.8 #13 8.0 23 34 16.9 #14 6.7 34 35 17.9 #15 7.1 0 36 16.1
two-parameter models (model 2) is closer to the calculated optimum wall dimensions. The scatter is reduced and the maximum error is decreased to 14.7% for the two-parameter models. When the three-parameter models (model 3) are examined, the maximum error in the distribution of esti-mated and calculated wall dimensions is insignificant, and thus the estimation ability of the proposed three-parameter regression models is proved to be very high.
When the distribution between calculated and estimated dimensions for three-parameter models is examined, it is seen that the correlation coefficients for model 3 are found to be: R = 0.995 for x2, R = 0.991 for x3, R = 0.987 for x4,
and R = 0.994 for x5. These correlation coefficient values
correspond to the perfect level for a multivariate problem. When the errors for the 15 test samples are studied, the maximum difference between the estimated and calculated wall dimensions is 2 mm, 62 mm, 34 mm, 77 mm, and 24 mm for x1, x2, x3, x4, and x5, respectively, where wall
heights range from 4 to 8 m. The maximum difference is only 6.2% for the 15 test problems, and this confirms that the proposed regression models (model 3) are very effective. The 15 different data sets are also used to compare the preliminary dimension estimates of the conventional approaches and the proposed three-parameter regression model (model 3). Using the data sets given in Table 14, the preliminary dimension values that are estimated from the conventional approaches (Das 2016; Bowles 1997; Budhu2008) and also from the proposed three-parameter regression models (model 3) are shown in Fig.7. If some ranges are defined for the related geometric dimensions in the conventional approaches, then the mean geometric
dimension values are marked on the graphs and this is indicated by the expression “mean” on the legend.
In this study, x1 dimension is deduced to be almost
equal to the construction lower limit value (0.3 m). This
x1dimension is also equated to 0.3 m by Das (2016) and
Bowles (1997). But Budhu (2008) equated x1 dimension
to 0.25 m. Therefore, it can be stated that in the literature, the stem thickness at the top of the wall is also equated to the construction lower limit value. As seen in Fig. 7, the preliminary dimensions (x1) for the fifteen test samples
proposed by Das (2016) and Bowles (1997) and the proposed model 3 are all equal to the calculated value and for this reason, there is no error in dimension (x1). But the
preliminary dimensions (x1) proposed by Budhu (2008) are
below the calculated values and for this case, the error in estimating the x1dimension is found to be 16.6%.
The preliminary dimensions for the width of the toe (x2)
for the fifteen test samples proposed by Das (2016) are far from the calculated values as shown in Fig. 7. Similarly, the preliminary mean dimensions (x2) for the fifteen test
samples proposed by Bowles (1997) and Budhu (2008) are not close to the calculated optimum values by ABC algorithm. It will be very difficult to obtain the optimum design in a preliminary design started with the selection of these values. For the fifteen test samples used in Fig. 7, if the preliminary dimensions (x2) are compared with the
calculated values, then the average errors in estimating the
x2 dimensions are found to be 54.5%, 21.2%, and 26.0%
for Das (2016), Bowles (1997), and Budhu (2008) and if the maximum errors are examined, then 64.4%, 36.7%, and 41.3% are found, respectively. These error values are
Fig. 6 Comparison of calculated and estimated wall dimensions by the proposed regression models
high. It is clear that the proposed model 3 estimates the
x2dimensions very close to the calculated optimum
cross-sectional dimension. The average error in estimating the
x2 dimensions in the tests is found to be 1.9% and the
maximum error is found to be 5.7%, which are low values. As seen in Fig.7, the preliminary dimensions (x3) for the
fifteen test samples proposed by Das (2016), Bowles (1997), and Budhu (2008) are spread around the calculated values.
If the preliminary dimensions for x3 are compared with
the calculated values, then the average error in estimating the x3 dimensions is found to be 11% and the maximum
error is equal to 37.7% for the three references. Whereas the maximum error between the estimated and calculated x3
dimensions is found to be 6.1% for the proposed model 3. The width of the heel (x4) is of great importance for
Fig. 7 Estimated wall dimensions come from the proposed model (model 3) and the conventional approaches
that the mean preliminary x4 dimension proposed by Das
(2016) is above the calculated optimum values. The errors in estimating the x4 dimensions are found to be between
29.1 and 87.9% for the reference (Das 2016), which are very high. The mean preliminary dimensions (x4) proposed
by Bowles (1997) are just below the calculated values. The average error for the x4dimensions is 6.4% for the reference
(Bowles1997). For these test samples, the estimated mean
dimensions (x4) are a bit above the calculated values for the
reference (Budhu2008), where the errors are between 2.8 and 38.6%. On the other hand, if the preliminary dimensions (x4) obtained from the proposed model 3 are compared with
the calculated values, then the average error is found to be 2.2%, which reflects a successful estimation.
The preliminary dimensions for thickness of the foun-dation (x5) proposed by Das (2016), Bowles (1997), and
Budhu (2008) are all above the calculated values for the fifteen test samples. The average error between the calcu-lated and estimated x5dimensions from the three references
is found to be 37%. But the average error is 6.2% for the proposed model 3, which is rather low.
For the 15 different test problems in this study, it is seen that the proposed three-parameter regression-based models (model 3) estimate preliminary wall dimensions better than conventional approaches. It is thought that there are some main reasons for the observations. The most important of these reasons is that conventional approaches in the literature are based only on wall heights. The approaches do not take into account the effects of the other important design parameters. Essentially, the proposed one-parameter regression model (model 1) in this study is generated based on only wall height. However, each one of the 375 different optimization problems is repeated 30 times starting with independent random populations. In the implemented ABC algorithm, the number of colonies and the number of maximum cycles are selected 100 and 1000, respectively. So, the total number of objective function evaluations is 1,125,000,000 in this study. It is impossible to make so many analyses with traditional methods. The high number of analyses provides that regression analyses are performed with a data set with very low variability. In this way, even in model 1, proposed regression-based models estimate successfully the wall dimensions more than the conventional methods, as shown in Figs. 6 and 7. Considering that these conventional approaches are developed based on engineering experiences and observations, it is natural that they do not reach the optimum wall dimensions. As it is not known that the conventional approaches are valid for which base soil conditions, the wall dimensions are defined in wide ranges in conventional approaches.
7 Conclusions
This study aims to propose a regression-based approach for estimating preliminary dimensioning of the minimum cost T-shaped RCCRW which is resting on soil with high bearing capacity. To achieve this purpose, 375 different sample retain-ing wall problems, which have different heights, surcharge loads, internal friction angles, and unit weights of backfill soil, are solved by the artificial bee colony algorithm and the minimum cost dimensions of the T-shaped RCCRW are calculated. The other properties of the backfill and founda-tion soil, desired minimum factors of safety, and concrete and steel specifications are assumed constant in the opti-mization analyses. The critical section dimensions of the stem, toe, and heel of the wall are all investigated and one-parameter, two-one-parameter, and three-parameter multiple
linear regression models for the wall dimensions are obtained. The three-parameter models (model 3), where the wall height, surcharge load, and internal friction angle of backfill soil are used as the predictor variables, yield sta-tistically significant and highly correlated results than the other models. In order to assess the predictive ability of the proposed regression equations, a total of 15 test problems are randomly generated and solved by the ABC algorithm. The calculated wall dimensions are compared with the esti-mated wall dimensions by using the proposed regression equations. The maximum difference between the calculated optimum wall dimensions and estimated wall dimensions is only 6.2% for model 3 for the 15 test problems. This means that the preliminary dimensions obtained from the proposed three-parameter regression equations are very close to the optimum design dimensions calculated by the ABC algo-rithm. Moreover, the results obtained from the proposed three-parameter regression model (model 3) are compared and discussed the preliminary dimension estimates of the conventional approaches for the 15 different data sets. It is seen that the proposed three-parameter regression models estimate preliminary wall dimensions better than conven-tional approaches. The most important of these reasons is that conventional approaches in the literature are based only on wall heights, and they do not take into account the effects of the other important design parameters. Finally, the proposed regression equations may be used in the actual wall dimension calculations of the T-shaped RCCRWs. This design will be very close to the optimum design and it is up to the designer to decide whether an optimization method is needed or not needed to improve the design dimensions any further.
Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interest.
Replication of results The optimization algorithm and the design details of RCCRWs used the study are given in Sections2,3, and4. The data set used in the developed regression models for estimation of preliminary dimensions of the RCCRWs is obtained from a total of 375 optimization problem results using the artificial bee colony algorithm, and the data set is given in the Supplementary Table1. Also, 15 different data set used to test and verify the proposed regression models are given in the Supplementary Table2.
References
Ahmadi-Nedushan B, Varaee H (2009) Optimal design of reinforced concrete retaining walls using a swarm intelligence technique. In: The first international conference on soft computing technology in civil, structural and environmental engineering, pp 1–12
