Geometric Design of Retaining Wall by Bat Algorithm...Geometric Design of Retaining Wall by Bat Algorithm . G S Bath. 1, J S Dhillon. 2 . and B. S. Walia. 3. 1. Department of Civil

Geometric Design of Retaining Wall by Bat Algorithm

G S Bath1, J S Dhillon2 and B. S. Walia3

1Department of Civil Engineering

Giani Zail Singh Campus College of Engineering and Technology, Bathinda-151001, INDIA 2Department of Electrical and Instrumentation Engineering

Sant Longowal Institute of Engineering and Technology, Longowal-148106, INDIA 3Department of Civil Engineering

Maharishi Markandeshwar Engineering College Mullana (Ambala) - 133207 ( India)

Abstract: The retaining structures like retaining walls, sheet pile walls and RE walls used for the construction

of civil engineering projects. Cantilever retaining wall is most important retaining structure used for the retaining

of earth for railways, highways and basements. Economical and safe design of cantilever retaining wall can play

important role in construction industry. The trial and error method being used for dimensioning of retaining wall,

may or may not be economical one. In this paper, Bat algorithm (BA) method being a global search technique is

proposed to design the cantilever retaining wall. The stability and geometry aspects of cantilever retaining wall

are considered for the economical design of retaining wall. The algorithm exploits echolocation searching

behavior of the bats. The bat algorithm is known as a frequency based algorithm to make available a sensible

arrangement of exploration and exploitation. The weight and cost of the retaining wall are minimized subject to

stability and geometry constraints. Sensitivity of the design of retaining wall with respect to height of wall, base

soil friction, angle of shearing resistance and backfill slope have been investigated. The validity of proposed

method is authenticated by considering three design proposals under diverse conditions. The obtained design of

retaining wall gives better design than the traditional design by limit state method.

Key words: Retaining wall, Optimized design, Bat algorithm (BA) method, Multi-parameter optimization. Meta-

heuristic optimization algorithm

1. Introduction

Retaining wall is provided at a place, where soil can not attain a stable slope due the shortage of space. These

are generally provided when the structures like flyover are constructed above the natural surface and basement

is constructed below natural surface. Retaining wall is also constructed to provide stability to the unstable soil

mass of infinite natural slope. The design of this important structure should be economical and safe against

geotechnical and structural requirements. Presently, the trial and error approach is followed by professionals for

retaining wall design in accordance with the relevant codes. For the initiation of the design, the sizes of the

components of retaining wall are fixed by the geotechnical requirements and then the structural design is

completed. In case the design fails structurally, then dimensions are re-fixed as a next trial to achieve the

successful design that may or may not be the economical design. The load carrying capacity of the soil, earth

pressure, base friction are the geotechnical parameters considered for designing the retaining wall. So the design

of cantilever retaining wall is a complex engineering problem.

Complex problems in several areas of engineering and technology need solution through optimization

techniques. Sometimes optimization problems become difficult due to tangible and complex nature of the

objective function and constraints. Optimization methods involving derivative-based techniques are outlined in

(Yang, 2010). These strong techniques are robust and very effective in solving many complex optimization

problems. These optimization methods are sometimes stuck in local optima, become complex from

computational view point, and may not be applied to some types of objective functions. To overcome these

shortcomings, new optimization techniques are required to be developed. Heuristic optimization techniques are

emerging methods that can solve the drawbacks of derivative-based techniques. Meta-heuristic techniques are

computer-based methods, in which predetermined rules are used for solution of optimization problems. Nature

inspired meta-heuristics are bio-inspired (Boussaïd et al., 2012; Simon, 2008; Simon et al., 2011; Xiong et al.,

2013-2014) or ant colony (Sisworahardjo and El-Keib, 2002) techniques etc. To summarize, a quality global

solution procedure able to solve the design of retaining wall with better computational efficiency is a wide-open

research field. It is better to use a solution procedure that is able to generate single solution as a final solution,

VOLUME 24 Science and Engineering Journal ISSUE 12 2020

ISSN NO: 0103-944X Page no : 1

which meets the design aspirations. Further, to support such procedure a search algorithm having balanced

exploration and exploitation capability is needed, which ensures useful diversity in the population.

Pei, Y. and Xia, Y., (2012), designed reinforced cantilever retaining walls by using heuristic optimization

algorithms. Gravitational search algorithm for the optimization of retaining structures is used by Khajehzadeh

and Eslami (2012). CO2 optimization, ant colony optimization, harmony search, charged system search

algorithm, hybrid firefly algorithm Big Bang Crunch were applied by Yepes et al. (1997), Villalba et al. (2010), ,

Kaveh and Abadi (2011), Kaveh and Behnam (2013), Sheikholeslami et al. (2014), respectively for the optimum

design of concrete retaining wall. Poursha et al. (2011) employed harmony search based algorithms for the

optimum design of reinforced concrete cantilever retaining walls. Amin Manouchehrian et al. (2014), Malkawi et

al. (2001), Gandomi et al. (2014), Venanzio R. Greco, (1996) and Khajehzadeh et al. (2011) have applied

optimization techniques for slope stability in the field of geotechnical engineering. Hasançebi, O., and Carbas, S.

(2014), have applied bat inspired algorithm for discrete size optimization of steel Frames.

The intent of this paper is to focus on the applications of Bat algorithms to design the retaining wall that

minimizes the weight and cost subject to feasible stability and geometry constraints. Capacity constraints of the

design of retaining wall are out of scope of this paper. Three design cases are considered to check the efficiency

of the proposed algorithm. Moreover, sensitivity of the proposed algorithm to design retaining with respect to

height of wall, base soil friction, angle shearing resistance and backfill slope has been investigated. In this study,

continuous variables are undertaken for wall geometry used for optimal design of retaining wall.

2. Geometric design of cantilever retaining wall

The geometric design of the retaining wall includes the dimensioning of various components of the wall and

the design should not fail due to overturning, sliding and bearing capacity.

Proportioning of Retaining Wall

Dimensions of toe slab, heel slab, stem and key are decided in the first step for geotechnical requirements

and space requirements for proper placement of concrete and reinforcement. In general, the top of stem varies

from 150mm to 300mm for proper placement of concrete. The thickness of stem at the bottom may

approximately be ten percent of the height. The base slab thickness should be 300mm. The base of retaining wall

should be 0.5 – 0.7 times the height. Donkanda and Menon (2012) have proposed the following equations for the

calculation of the optimal length of the heel slab.

Length of heel, , length of the toe,, length of the base, height of retained soil, of the retaining wall and

computed from following equations.

where is the height in meters.

Coefficient active earth pressure, is defined mathematically as below:

where o is the slope of the retained soil. o is the shearing resistance parameter of soil.

The passive earth pressure coefficient, is stated below

The height of retained soil above heal, in m is computed by using equation 4.6

The horizontal active earth pressures force per meter length of wall, , is written below

where and is the bulk unit weight in kN/m3 of the retained soil.



https://www.researchgate.net/profile/Amir_Gandomi

The vertical active earth pressures force per meter length, , is calculated by using following equation

Figure 1: Geometry of retaining wall relabeled with algorithmic variables

Stability of Retaining Wall against Overturning

The active earth pressure in horizontal direction tries to overturn but the weight of backfill soil and self-

weight of the wall, prevents the overturning. The overturning safety factor, is given below:

where is the resisting moment (kNm) and is the overturning moment (kNm).

Mathematically, the overturning moment, (kNm)stated as:

The resisting moment, is the resisting moment (kNm) and is defined as:



Figure 2: Forces acting on retaining wall relabeled with algorithmic variables

where is the sum of all the weights (kN) and M is sum of moments (kNm).

The sum of weights, is computed as

The sum of all the moments, M (kNm) is computed as.

where and are the ith moment and weight respectively.

Stability of Retaining Wall against Sliding

Sliding of retaining wall is triggered by the active earth pressure. Whereas the base friction, weight of

retained soil and of retaining wall are utilized for its stability.

The safety factor, against sliding of retaining wall is given below:

where is the resisting force (kN) and is the sliding force (kN).

The resisting force, (kN) is given below:

where is the friction coefficient and is passive pressure force / m length of retaining wall in kN

The sliding force, (kN) is stated below:

Stability Against Soil Bearing Pressure

Excessive settlement of soil below the foundation is responsible for the bearing capacity failure which can

be prevented by using safety factor.

The factor of safety, against the settlement of soil is defined as:



where qa is the net allowable load carrying capacity of soil in kN/m2, and is maximum pressure in kN/m2.

The maximum, and minimum, pressure (kN/m2) acting at the base of retaining wall are stated below:

Mathematical, the parameter, is expressed below

Eccentricity, e is computed as

Passive earth pressure is shown below

where is the depth with respect to the NSL.

3. Formulation of problem

To formulate an optimization problem for the geometrical design of retaining wall by using bat algorithm,

objective function(s) and constraints are required to be described in the ensuing sub-sections.

Objective functions

In this section cost per meter length, and weight per meter length, of retaining wall are considered as two

objective functions. The first objective function is to minimize the cost per meter length, of retaining wall and is

defined below:

where is the cost of one cubic meter of concrete, are the volumes of stem, heel, toe and shear key, respectively

in m3.

The second objective function is to minimize the weight per meter length, of retaining wall and is defined below:

where is the unit weight in kN/m3 of concrete

The variable, is searched using bat algorithm to minimize the above mention objectives of optimization problem.

Constraints

The design of retaining wall should be stable, safe and economical. This design is required to satisfy the

conditions associated with capacity, stability and geometry of retaining wall.

Stability Constraints

Stability against overturning is undertaken by satisfying the following equation

Stability against sliding is limited by satisfying the following equation

Stability against settlement of the soil is handled by fulfilling requirements of the following equation

Dimension Constraints

Bound on design variables is tackled by fulfilling the following equations.



Continuous Variables

The optimal dimensions of the components of the retaining wall depend on the parameters of soil. For fixing

dimensions of wall X1 to XNv, continuous variables are used. These variables can vary between the following

limits

Aggregating the above equations constrained problem is optimized in the following equations

Minimize

Minimize

Subject to

Stability against overturning given by Equation (26)

Stability against sliding given by Equation (27)

Stability against bearing capacity given by Equation (28)

Design variable Constraints are given by Equations (29) to (30).

Bound on design variables given by Equation (31)

Above optimization problem is redefined to club both the objectives having the same nature as below

Minimize

Subject to:

Stability against overturning given by Equation (26) and represented as G1

Stability against sliding given by Equation (27) and represented as G2

Stability against bearing capacity of soil given by Equation (28) and represented as G3

Dimension constraint is given by equation (30) is represented as G4

Equality constraints are given by Equations (29)

where is conversion factor having units (₹`/Kg)

The constrained scalar objective multivariable optimization problem is converted to an unconstrained scalar

objective multivariable problem utilizing exterior penalty function and is demonstrated below

where

The penalty parameter has a large value. The aim of optimization problem is to find X so that is minimized

while bound on variables are taken care of during applying the bat algorithm.

4. BAT ALGORITHM

The bat algorithm was introduced by Yang X. S., 2010. The algorithm exploits echolocation searching

behavior of the bats. Sonar and echo techniques are used by bats to discover and avoid hindrances. Sound beats

are transmuted into a frequency which returns from hindrances. The movement of bats is guided by time gap

between release and reception of these sound beats. The short and loud sound impulses are discharged by bats.

Normally pulse rate of bats is taken between 10 to 20 times in a second. The position of prey is guessed on

striking and reflecting. The bats convert their own beat into valuable information to predict the position of the

prey. The bats use wavelengths between 0.7 mm to 17 mm and frequencies between 20 kHz - 500 kHz. The

algorithm selects the pulse frequency and the rate, initially. The pulse rate is between 0 to 1. At 0 the pulse

emitting capacity is minimum and at 1 the pulse emitting capacity is maximum (Gandomi et al., 2013; Tsai et al.,

2012). The behavior of a bat is used to modify the BA. Basically, the bats estimate the distance with respect to

prey by using echolocation. Bats differentiate between the food or prey and also the related barriers [Tsai et al.,

2012]. To search prey, bats move arbitrarily with different wavelength λ and loudness A0. The parameter of the

bat at position can be velocity vi at frequency fmin, Bats spontaneously amend the wavelength of released pulses

and amend their rate of pulse r ϵ [0, 1], based on the proximity of the object. Loudness differs from a maximum

to a minimum value .



The frequency of emitted pulses [20 kHz, 500 kHz] corresponds to wavelengths between 0.7 mm to 17 mm

[Tsai et al. 2012]. The frequency ‘f’ or wavelength ‘λ’ for dissimilar applications are dependent on the comfort of

implementation and some other factors. Velocity ‘vi’ and position. ‘’ is associated with each bat, for any iteration,

't’ in a search space of dimension .

where is a chaotic sequence generated by Gauss map within 0 and 1. are maximum and minimum range of

frequency, respectively. is the ith best global position of a bat. is the ith position of the kth bat.

The homogeneously scattered Random number between 0 and 1 are used to generate the position of bats within

the search space for random initialization. Exactly, it is denoted as below:

where is random number for ith position (generator) and kth bat, is population size of bats.

The position of the bat is updated within the prescribed limits using the following equation:

Every bat is arbitrarily allotted a frequency taken from the range of to. That's why, the bat algorithm is known as

a frequency based algorithm to make available a sensible arrangement of exploration and exploitation. The

spontaneous control and auto-zooming into the area with a favorable solution is dependent on the loudness and

pulse rate. (Basu and Chowdhury, 2013).

Based upon the pulse rate change, the bat position is locally updated and taken as

where and Ri is the uniform random number varying between 0 and 1.

The position of the bat is adjusted within the limits of search space, as stated below:

where is the ith position of the kth bat. Uniform random number R ϵ [0, 1] is generated using the chaotic sequence.

To control the exploration and exploitation and switch to the exploitation stage when necessary, the pulse

emission rate ri, and loudness Ai are changed in the iterations. When a bat detects its prey, the loudness decreases

and pulse emission rate increases. The loudness is selected between amin and amax. Zero value of amin means bat has

just found the prey and temporarily stop emitting any sound. With these assumptions loudness, and emission

rate, are updated with the following equations (Yang X. S., 2010)

where α, β and γ are constants. ‘α’ and ‘β’ have values less than 1 such that their sum is 1. Preferably, ‘α’ is taken

as a golden number . ‘γ’ is taken near to 100.

The new position of the kth bat is determined from the previous position of (k - 1)th bat.

Local Random Search: In minor dimensional problems, the conventional heuristic search approach

accomplish better results with fast convergence. However, in huge scale and complex optimization problems, the

convergence may have the probabilities of achieving local optima because of the degradation of diversity of

population. A local random search approach is implanted to avoid from early convergence. Local random search

approach is proficient in intensifying the algorithm’s manipulation capacity in the search space.

The proposed methodology uses bat algorithm to exploit global search and then implement the local random

search approach to find near the best solution created so far to trace the global optima.

. are the random numbers whose values are varied from 0 to 1.



Target vector and subsequent trial vector are compared every time. The vector with better objective value is

utilized for the next generation.

The selection process is stated as:

The local search is upgraded until improvement in the objective function is accomplished.

Pseudo code used for the local random search algorithm is presented in the following Algorithm -1. The detailed

bat algorithm considering local search is presented in Algorithm 2.

Algorithm-1: Local Random Search Algorithm

Enter

DO

Generate random number

DO

randomly with in variable limits

Compute using Eq. (43) using Eq.(24)

ENDDO

Compute

IF THEN

ENDIF

ENDDO

RETURN



Algorithm-2: Bat Algorithm (BA) using local search

Objective function,

Initialize the bat population using Eq. (4.102)

Initialize bat velocity, randomly

Compute fitness function using Eq. (24Define pulse frequency

Initialize pulse rate and loudness

t=0

WHILE ) DO

t=t+1

FOR

Generate new solution by adjusting frequency using Eq. (38)

Update velocity, using Eq. (4.101), position, by using Eq. (40)

Compute the fitness function using Eq. (24) IF THEN

Update solution, locally using Eq. (41)

Adjust the limits of design variables using Eq. (42)

ENDIF

IF THEN

Accept the new solution and update fitness function

Increase and reduce using Eq. (44) and (43), respectively

ENDIF

ENDFOR

Rank the bats based on their fitness function and find the current best

Apply Algorithm-4.2 to update current positions of Bat

Select global best solution

ENDDO

STOP



5. DESIGN DATA

To validate the optimal design using bat algorithm for the (i) geometrical design of retaining wall by using

properties of retained soil, (ii) geometrical design of retaining wall by using properties of retained soil and soil

below its base and (iii) structural design of retaining wall by using properties of retained soil and soil below its

base. Three different design cases are undertaken for 4 m, 5m and 6m high. Input data required is elaborated as

below.Soil is taken as dry and cohesion-less. Properties of soil below the base of the retaining wall are not

considered, and fixed load carrying capacity of the soil is taken. The parameters of soil and concrete are

Bulk unit weight of retained soil, = 16 kN/m3.

Unit weight of concrete, = 25 kN/m3.

The angle of shearing resistance, = 30o.

Depth of foundation below natural ground surface, = 1.25 m.

The load carrying capacity of soil = 160 kN/m3.

Inclination of backfill surface with horizontal, = 15o

Coefficient of friction between soil and concrete, = 0.5

Cohesion intercept, c’ = 0.0 kN/m2

6. Lower and Upper Limits on Continuous Variables

The upper and lower values on the continuous variables of retaining wall for a height of 4.0 m, 5.0 m and 6.0 m

are given in Tables 4.3, 4.4 and 4.5 for Formulation-I, Formulation-II and Formulation-III, respectively.

Table 1: Lower and upper limits on continuous variables

Variable, Case-1 Case-2 Case-3

Width of base, 1.25 3.50 1.25 3.50 1.25 4.0

Width of toe slab, 0.50 1.25 0.50 1.25 0.50 1.75

Thickness of stem at base, 0.30 0.45 0.30 0.45 0.30 0.45

Thickness of stem at top, 0.15 0.20 0.15 0.20 0.15 0.20

Thickness of the base slab, 0.30 0.50 0.30 0.50 0.30 0.50

Length of the heel slab, 1.00 2.50 1.00 2.50 1.00 2.50

Depth of key, 0.25 0.50 0.25 0.50 0.25 0.50

Width of key, 0.25 0.50 0.25 0.50 0.25 0.50



7. Results For Geometric Design By Bat Algorithm

Bat algorithm has been applied for the design of cantilever retaining wall. Cost, weight and other design

parameters have been optimized for three design cases of retaining wall of different heights viz. 4.0 m, 5.0 m

and 6.0 m. To implement bat algorithm, bat population size, NP is taken as 100 and maximum generations Tmax is

chosen as 1000. The minimum and maximum frequencies, fmin and fmax are set to 0 and 2, respectively. The

minimum and maximum values for loudness, amin and amax are set to 0 and 1, respectively. Other parameters viz. α

is set to 0.618 and is set to 100.The best, mean and worst values and also standard deviation of cost and weight

per unit length by applying bat algorithm for three design cases of retaining wall are presented in Table 4.6. The

result of the traditional limit state design of the retaining wall has been compared with the optimized design by

bat algorithm for three different heights; 4.0 m, 5.0 m and 6.0 m of retaining wall. These values are obtained by

performing 100 independent trial runs.

Table 2: Optimal geometrical design of retaining wall using bat algorithm

Design Cases Case-1 Case-2 Case-3

Height (m) 4.0 5.0 6.0

Best Cost, (₹/m) 10518.40 13118.07 16021.42

Best Weight, (Kg/m) 4382.66 5465.86 6675.59

Mean Cost, (₹/m) 10633.99 14010.78 17070.79

Mean Weight, (Kg/m) 4430.83 5837.82 7112.83

Worst Cost, (₹/m) 11082.60 14395.34 19492.55

Worst Weight, (Kg/m) 4617.75 5998.06 8121.89

SD, Cost (₹/m) 138.46 426.72 1006.22

SD, Weight (Kg/m) 0.577 1.77 4.19

Cost (₹/m), (LSD Method) 12495.0 18045.0 23715.0

Weight(Kg/m), (LSD Method) 5206.25 7518.75 9881.25

Percentage Saving in cost 15.81% 27.30% 32.04%

Table 3: Comparison between design parameters by bat algorithm and LSD method: Case-1

Parameter Limit State Design Method Optimized design with Bat Algorithm

Best Design Worst Design

Cost (₹/m) 12495.0 10518.40 11082.60

Weight (Kg/m) 5206.25 4382.66 4617.75

(m) 2.550 2.4885 2.6590

(m) 0.850 1.0369 1.1670

(m) 0.400 0.3378 0.3729

(m) 0.150 0.1787 0.1817

(m) 0.346 0.3119 0.3117

(m) 1.664 1.4436 1.4931

(m) 0.300 0.3519 0.2767

(m) 0.300 0.3756 0.4014



Table 4: Comparison between design parameters by bat algorithm and LSD method:Case-2

Parameter Limit State Design Method Optimized design with BAT


Cost (₹/m) 18045.0 12479.83 13118.07

Weight (Kg/m) 7518.75 5199.92 5465.86

(m) 3.150 3.1885 3.1415

(m) 1.050 1.1775 1.1844

(m) 0.450 0.3725 0.3402

(m) 0.150 0.1670 0.1853

(m) 0.432 0.3792 0.3053

(m) 2.045 1.9154 1.8107

(m) 0.300 0.3267 0.3692

(m) 0.300 0.3697 0.3843

Table 5: Comparison between design parameters by bat algorithm and LSD method: Case-3

Parameter Limit State Design Method Optimized design with BAT algorithm


Cost (₹/m) 23715.0 16021.42 19492.55

Weight (Kg/m) 9881.25 6675.59 8121.89

(m) 3.750 3.8382 3.9061

(m) 1.250 1.6474 1.6387

(m) 0.450 0.3280 0.3047

(m) 0.150 0.1518 0.1817

(m) 0.528 0.3371 0.4374

(m) 2.426 2.1819 2.2553

(m) 0.300 0.3636 0.4788

(m) 0.300 0.3601 0.6797

Tables 4.7, Table 4.8, and Table 4.9 show the parameters of retaining wall obtained by traditional limit

state method along with best and worst design parameters obtained by Bat optimization technique. These results

are also graphically depicted in Figures 4.3 and 4.4.



Figure 3: Variation of cost with height of retaining wall using LSD method and bat algorithm

Figure 4: Variation of weight with height of retaining wall using LSD method and bat algorithm

It is observed from the results that optimized design obtained from Bat algorithm technique is better than the

design by limit state method. Even the worst values of weight per meter length and cost per meter length

obtained after performing independent 50 trial runs are better than the traditional design by limit state method. A

saving of 15.81%, 27.30% and 32.04% in cost and weight per meter length is obtained in optimal design for 4.0

m, 5.0 m, and 6.0 m high retaining wall.



Effect of back slope angle on cost and weight

Table 4.10 shows the variation of cost per meter length with respect to backfill slope, θo, which is observed by

varying the value from 0° to 25° in steps of 5° for the three-design cases of retaining wall. It is perceived that the

cost increases with the rise of backfill slope. Figure 4.5 represents the results graphically also for three design

cases undertaken in study. Variation of weight with respect to backfill slope θo is shown in Table 4.11, which is

observed by varying the value from 0° to 25° with an increment of 5° for all the three-design cases of retaining

wall. It is observed that the weight of material increases with the increase of backfill slope. Bar charts in Figure

4.6 show the effect of angle of backfill slope on weight/m length

Figure 5: Variation of cost with the angle of backfill slope using bat algorithm

Table 6: Variation of cost with the angle of backfill slope using bat algorithm

Angle of backfill slope,

θo

Cost (₹/m)

Case-1 Case-2 Case-3

0o 10274.84 13192.56 14535.80

5 o 10086.53 13310.91 14382.76

10 o 10113.35 13562.79 15874.53

15 o 10518.40 13118.07 16021.42

20 o 10669.73 13315.74 16261.74

25 o 11796.05 14647.32 18050.53

Variation % 14.80% 11.03% 24.17%



Table 7: Variation of weight with the angle of backfill slope using bat algorithm

Angle of backfill slope,

θo

Weight (kg/m)


0 o 4281.18 5496.89 6056.58

5 o 4202.72 5546.21 5992.28

10 o 4213.96 5651.16 6614.38

15 o 4382.66 5465.86 6675.59

20 o 4445.72 5546.97 6775.72

25 o 4915.02 6105.30 7615.46

Variation % 14.80% 11.06% 25.73%

Figure 6: Variation of weight with the angle of backfill slope using bat algorithm

Effect of angle of shearing resistance on cost and weight

Cost / m length and weight / m length of the retaining wall is also affected by the angle of shearing resistance.

Table 4.12 and 4.13 show the variation of cost and weight with respect to the angle of shearing resistance, .

These results are also depicted graphically in Figure 4.7 and 4.8.

Table 8: Variation of cost with the angle of shearing resistance using bat algorithm

Angle of Shearing

Resistance

Cost (₹/m)


26 o 11507.56 14122.04 17579.65

28 o 10686.08 13198.17 16277.76

30 o 10518.40 13118.07 16021.42

32 o 10567.04 13076.97 16149.38

34 o 9697.54 13310.91 14373.80

36 o 9489.36 12550.43 14191.93

38 o 9288.11 12171.02 13293.60

Variation % 23.89% 16.03% 32.24%



Figure 7: Variation of cost with the angle of shearing resistance using bat algorithm

Table 9: Variation of weight with angle of shearing resistance using bat algorithm

Angle of Shearing

Resistance

Weight (kg/m)


26 o 4794.81 5884.48 7324.99

28 o 4452.53 5499.23 6782.40

30 o 4382.66 5465.86 6675.59

32 o 4402.93 5448.73 6728.90

34 o 4040.64 5546.21 5989.08

36 o 3953.90 5229.34 5913.30

38 o 3870.04 5071.25 5539.00

Variation % 23.89% 16.03% 23.24%

Figure 8: Variation of weight with the angle of shearing resistance using bat algorithm

Angle of shearing resistance in degrees

38 36 34 32 30 28 26

8000

7000

6000

5000

4000

3000

2000

1000

0

4m height 5m height 6m height

Wei

gh

t k

g/m

len

gth



This variation is observed by varying the value of shearing resistance, Φ^∘ from 26° to 38° in steps of 2° for three-design cases of retaining wall. It is observed that the cost of material and the weight of retaining wall per

meter length decreases with the increase in the angle of shearing resistance, Φ^∘.

Effect of back base friction on cost and weight

The variation of cost with respect to base friction, is observed by varying the value of from 0 to 0.6 with a step

of 0.1 as shown in Table 4.14 for all the three cases of design. The results are demonstrated graphically in figure

4.9. It is observed that the cost of material per meter length of retaining wall decreases with the increase of base

friction noticeably in the range of 0 to 0.1 for wall height of 4m, from 0 to 0.2 for wall height of 5m and from 0

to 0.3 for wall height of 6 m. After these values base friction has no effect on cost.

Table 10: Variation in cost with base friction using bat algorithm

Base Friction Cost (₹/m)


0 10760.32 13923.15 17773.78

0.1 10518.40 13323.59 16799.42

0.2 10518.40 13172.11 16239.80

0.3 10518.40 13118.07 16021.42

0.4 10518.40 13118.07 16021.42

0.5 10518.40 13118.07 16021.42

0.6 10518.40 13118.07 16021.42

Variation % 2.30% 6.14% 10.93%

Figure 9: Variation of cost with base friction using bat algorithm

Similarly, a variation of weight of material with regard to base friction, is observed by varying the value

from 0 to 0.6 with a step of 0.1 is shown in Table 4.15 for all the three-design cases of retaining wall. The figure

4.10 is used to explain the results in the form of bar charts. It is observed that the weight of material decreases

with the increase in base friction. Effect of base friction on weight has similar trend as discussed for cost.



Table 11: Variation of weight with base friction using Bat algorithm

Base Friction Weight (kg/m)


0 4483.46 5802.22 7405.75

0.1 4382.66 5528.44 6999.77

0.2 4382.66 5465.60 6766.58

0.3 4382.66 5465.86 6675.59

0.4 4382.66 5465.86 6675.59

0.5 4382.66 5465.86 6675.59

0.6 4382.66 5465.86 6675.59

Variation % 2.30 6.15 10.93

Figure 10: Variation of weight with base friction using Bat algorithm

Figure 11: Variation of objective function with number of iterations using bat algorithm: Case-3



Convergence trend of objective function with respect to iterations per run by bat algorithm is shown in Figure

there is an abrupt decrease in objective function after 5th , 433th and 481th iterations. The result does not improve

after 721 iterations.

8. Conclusion

The objective functions, stability constraints, dimension constraints, continuous variables, discrete variables, and

input variables are taken for geometrical design is achieved. Local search is included to maintain a good

population of bats and to improve the exploitation. The effect of various parameters of design viz. wall height,

angle of shearing resistance, angle of retained soil, coefficient of base friction has been observed. The optimal

design of the retaining wall has been obtained for different cases of three heights by using the bat algorithm and

is compared with design obtained by the traditional limit state design method. A saving in cost of 15.81%,

27.37%, 32.04% for 4.0 m, 5.0 m and 6.0 m wall height respectively is achieved with reference to the limit state

design method. Cost and weight per meter length of retaining wall decreases as the angle of shearing resistance

increases from 26o to 38o. With the increase in the angle of backfill soil from 0o to 25o the cost and weight of wall

increases With the increase in base friction value from 0.0 to 0.6, the cost and weight of retaining wall decreases

initially and then become constant. So optimal design by Bat algorithm is better than the design by limit state

method.

Acknowledgments The corresponding author wish to thank GZS campus college of Engineering & Technology, Bathinda and I.K. Gujral, Punjab Technical University, Jalandhar (Punjab), for providing advanced research facilities during research work.

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Geometric Design of Retaining Wall by Bat Algorithm...Geometric Design of Retaining Wall by Bat Algorithm . G S Bath. 1, J S Dhillon. 2 . and B. S. Walia. 3. 1. Department of Civil