DESIGN OPTIMIZATION OF REINFORCED CONCRETE SLABS …The two types of reinforced concrete slabs...

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International Journal of Civil Engineering and Technology (IJCIET)

Volume 9, Issue 4, April 2018, pp. 1370–1386, Article ID: IJCIET_09_04_153

Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=9&IType=4

ISSN Print: 0976-6308 and ISSN Online: 0976-6316

© IAEME Publication Scopus Indexed

DESIGN OPTIMIZATION OF REINFORCED

CONCRETE SLABS USING GENETIC

ALGORITHMS

Shaik Bepari Fayaz Basha

Assistant Professor, Department of Civil Engineering,

G. Pulla Reddy Engineering college, Kurnool, Andhra Pradesh, India

K. Madhavi Latha

Assistant Professor, Department of Civil Engineering,

G. Pulla Reddy Engineering college, Kurnool, Andhra Pradesh, India

ABSTRACT

Design Optimization is the process of finding best design parameters that satisfy

the project requirements both in terms of strength and serviceability criteria. In the

present paper this optimization is carried out for Reinforced Concrete (RC) slabs

using Genetic Algorithms (GA), an iterative procedure which is based on theory of

natural selection and evolutionary biology. The two types of reinforced concrete slabs

considered in particular for the design are: simply supported one way slab and

cantilever slab. The design of the slabs is based on IS: 456-2000 code specifications.

The objective is to include the cost of concrete, the cost of reinforcement and the cost

of formwork respectively with their volume of materials.

This genetic algorithm iterative values are developed using MATLAB and the

codes are used for optimization. The results obtained are compared with the known

published results.

Further, the changes in the optimum values are studied by the variation of several

constraints such as the number of generations, population size and mutation rate.

Also by changing the fixed parameters i.e., the characteristic cube strength of

concrete and characteristic yield strength of steel, the effect on the optimum cost for a

particular loading condition and also for various loading conditions is studied. By

changing the span length of the slab, effect on the optimum cost and optimum

reinforcement ratio is studied for different loading conditions.

Key words: Cantilever slab, Design optimization, Genetic Algorithm, One way slab,

Reinforced Concrete

Cite this Article: Shaik Bepari Fayaz Basha and K. Madhavi Latha, Design

Optimization of Reinforced Concrete Slabs Using Genetic Algorithms, International

Journal of Civil Engineering and Technology, 9(4), 2018, pp. 1370–1386.

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=9&IType=4

Shaik Bepari Fayaz Basha and K. Madhavi Latha


1. INTRODUCTION

In the construction stream of civil engineering, in particular the construction of reinforced

concrete structures, optimization is playing a vital role in terms of economy as well as safety

considerations. The overall cost of the reinforced concrete structure including the cost of

concrete, reinforcement steel and cost of formwork are considered for optimization[1].

The

optimum solution obtained by using conventional methods face some difficulties while

solving problems related to executing same moment of resistance exhibited by slab

measurements and percent of reinforcement steel. To avoid the difficulty, the Genetic

Algorithm (GA) method of optimization of RC simply supported one way slab and cantilever

slab according to IS 456:2000[2]

is studied in the present paper.

1.1. Optimum Design of Structure

Optimum design indicates the most economical design that is practically possible which

should also satisfy the safety requirements as per IS code. Optimum structural design

indicates maximum utilisation of the existing material sources.

1.2. Objective of the Study

The main objective of the study is minimisation of total cost of the RC simply supported slab

and cantilever slab using Genetic Algorithm optimization program[3]

in MATLAB software

with IS 456:2000 code as a constraint.

1.3. Approach of the Study

After the detailed inspection of the preceding related works, genetic algorithm optimization

programs are established and the efficiency of the optimization programs are verified by

applying illustrative problems on RC simply supported one way slab and cantilever slab. The

design results obtained from MATLAB are compared with the previous works. Also the

known parameters are changed and the variation of optimum values as influenced by the

variation of the constraint is studied

2. METHODOLOGY OF THE STUDY

Design of any member indicates the determination of dimensions of the structure which fulfil

the requirements of the code and should be most economical and safe. The present study is

conducted on slabs which are the reinforced concrete elements transferring the transverse

loads which creates the bending moments and shear forces[4].

The basic design begins by selecting the cross section dimensions which should withstand

the bending moment and also selecting the necessity of steel reinforcement. Then the slab

dimensions should be checked for shear and deflection.

2.1. Design Preview for Slabs

The programs developed in the thesis for the optimization of simply supported and cantilever

slab follows certain provisions of IS 456:2000, which include the clear cover to the steel, trial

sections for depth, maximum moment coefficients of slabs, requirements for reinforcing steel,

shear reinforcement and deflection control.

2.1.1. Simply Supported One Way Slab

The loading diagram and the geometry of simply supported one way slab are shown in fig.1

and fig.2 respectively.

Design Optimization of Reinforced Concrete Slabs Using Genetic Algorithms


Figure 1 Typical simply supported RC one way slab with distributed load[5]

Figure 2 Geometry of a RC one way slab[6]

2.1.2. Constant Parameters

The constant values considered for the reinforced concrete one way slab model are Span

length of the slab, Span to depth ratio, The characteristic compressive cube strength of the

concrete, The characteristic strength of the steel, The uniformly distributed super imposed

loads, The cost of the quantity of concrete, The cost of required steel reinforcement, The cost

of the required formwork.

2.1.3. Design Variables

The following variables are treated in the simply supported one way slab problem are: d =

effective depth of slab (x1), pt = percentage of steel (x2), sv = spacing of reinforcement

bars(x3).

2.1.4. Design Constraints

The design constraints used in the study include the span to depth ratio, minimum depth

constraint, constraints to be considered in flexural design, minimum and maximum spacing

between reinforcement bars, deflection constraints and the constraints for shear design

according to IS 456:2000

2.1.5. Objective Function

The objective function in this RC one way slab in which the total cost including the cost of

concrete, the cost of reinforcement steel and the cost of formwork is defined as follows:



'

'

'

(x) C [ ( (1) d ) breadth {(x(2) 0.01 width x(1) x(3) ( 1) breadth)(2)

(0.0012 width (x(1) d ))}] [(x(2) 0.01 width x(1) x(3) ( 1) breadth) (0.0012(2)

width (x(1) d ))] [breadth

c

s

f

widthf width x

x

widthC

x

C

'(x(1) d )](1)

Where,

x(1) = effective depth of the slab, d

x(2) = percentage reinforcement ratio of steel, pt

2

2

100 4.5981 (1 )

2 8

ck ut

y ck

f w lp

f bd f

x(3) = spacing of reinforcement, sv

278.5398v

t

sb d p

Cc = cost of concrete including labour charges(Rs./m3)

Cs = cost of steel including bending of bars (Rs./m3)

Cf = cost of formwork(Rs./m2)

d’ = effective nominal cover to the reinforcement (mm)

fck = characteristic compressive strength of the concrete in N/mm2

fy = characteristic yield strength of the steel in N/mm2

l = effective span of the slab in metre.

Mu = bending moment due to super imposed load and self-weight in kN-m

wu = design load in kN/m2 = 1.5(dead load +live load)

b = width of the slab = 1000 mm

Ø = diameter of the reinforcing bar (mm)

2.1.6. Cantilever Slab

The loading diagram and the geometry of cantilever slab are shown in fig.3 and fig.4

respectively.

Figure 3 Typical RC cantilever slab with distributed load



Figure 4 Geometry of a RC cantilever slab

The constant parameters, design variables are same as that of simply supported one way

slab. The design constraints same as that of the simply supported one way slab corresponding

to the cantilever slab are used according to IS 456:2000.

2.1.7. Objective Function

The objective function is same as that of simply supported one way slab given by (1)

Except the percentage reinforcement ratio of steel is given by:

2

2

100 4.5981 (1 )

2 2

ck ut

y ck

f w lp

f bd f

2.1.8. Values of Cost

The value of Cc, Cs, Cf as per USSR (Unified schedule of rates and specifications) are given

below[7]:

Cost of concrete:

For M20 grade, cost of concrete, Cc = 6854.7(Rs. /m3)

For M25 grade, cost of concrete, Cc = 7091.05(Rs. /m3)

Cost of steel:

For Fe415 grade, cost of steel, Cs = 353250(Rs. /m3)

For Fe500 grade, cost of steel, Cs = 392500(Rs. /m3)

Cost of formwork, Cf = 401.65(Rs. /m2)

3. GENETIC ALGORITHM

Genetic algorithms are the search algorithms that have been evolved from th eevolution

observed in nature, namely the proess of natural selection, genetisc and survival of the fittest.

The main operators include:selection, fitness function, reproduction, crossover, mutation[8]

etc.,



3.1. Implementation of Genetic Algorithm in Matlab

Figure 5 Flow chart for the developed GA optimization model[9]

MATLAB, an acronym for MAT-rix LAB-oratory, is a very effective technical language

for mathematical programming. It offers a broad form of options that are useful to a designer

who utilises GA and to those who want to experiment with optimization using genetic

algorithms to learn about possible applications. The step-by-step procedure followed in

Genetic Algorithm is shown by the flow chart in fig.5

4. RESULTS

Reinforced concrete simply supported one way slab and cantilever slab are studied by making

valid generated GA programs, the results obtained are compared with the previous work.

4.1. Reinforced Concrete One Way Slab Problem

The loading diagram considered for the slab is as shown in fig. 6



Figure 6 The RC one way slab numerical example

4.1.1. The Fixed Parameters

Span of the slab is 3.96 m

Uniformly distributed load of 1.34 kN/m2

Characteristic cube strength of the concrete fck= 20.68 MPa

Cost of concrete, Cc = 610 (Rs. /m3)

Characteristic strength for the steel fy = 275.8 MPa

Cost of steel bars, Cs =95.2809 (Rs. /kg).

Solution

The above problem is solved using Genetic algorithm coding and the results obtained are as

follows: Cost = 1650.34 Rs. /m

d = 167.80 mm pt = 0.3738% sv = 262.9 mm This study is compared with one of the previous works

Cost = 1770.36 Rs. /m

d = 158.75 mm

pt = 0.42%

sv = 220 mm

The optimal cost obtained by genetic algorithm coding is compared with the optimal cost

obtained by one of the previous works and it is observed that former showed a reduction of

6.78% in total cost.

4.2. Reinforced Concrete Cantilever Slab Problem

The loading diagram considered for the slab is as shown in fig. 7

Figure 7 The RC cantilever slab numerical example



4.2.1. The Fixed Parameters

Span of the slab is 3.96 m

Uniformly distributed load of 1.34 kN/m2

Characteristic cube strength of the concrete fck= 20.68 MPa

Characteristic strength for the steel fy = 275.8 MPa

Cost of concrete, Cc = 610 (Rs. /m3)

Cost of steel bars, Cs =95.2809 (Rs. /kg).

Solution

The above problem is solved using Genetic algorithm coding and the results obtained are as

follows: Cost = 3204.05 Rs. /m d = 343.90 mm

pt = 0.3907% sv = 191.8 mm This study is compared with one of the previous works

[10]

Cost = 3951.83 Rs. /m d = 317.50 mm

pt = 0.54% sv = 127.0 mm

The optimal cost obtained by genetic algorithm coding is compared with the optimal cost

obtained by one of the previous works and it is observed that former showed a reduction of

18.92% in total cost.

4.3. Effect of Change of Mutation, Generation and Population

Input data for design problem

Span of the slab is 4 m

Uniformly distributed live load of 3 kN/m2

The characteristic cube strength for concrete fck= 20 MPa

The characteristic yield strength of steel fy = 415 MPa.

Output: For 100 generation, 100 population size, and 0.01mutation.

The optimum cost value for one way slab is shown in fig: 8

Figure 8 Showing the optimum cost value for one way slab



The optimum cost value for cantilever slab is shown in fig: 9

Figure 9 Showing the optimum cost value for cantilever slab

4.3.1. The Number of Generations

A study is carried out by changing the number of generations while making the other values

as unchanged as follows:

The number of population size = 100, and the mutation rate = 0.01, the method of

selection is selected as Roulette Wheel function.

fig.10 and fig.11 shows the results of changing the number of generations for both the

slabs.

Figure 10 No. of generations vs. Optimum cost for one way slab

6336.35

6326.63

6319.93 6319.93

6318

6320

6322

6324

6326

6328

6330

6332

6334

6336

6338

20 40 60 80 100

Op

tim

um

cost

(R

s.)

Generations

fck = 20 N/mm², fy = 415 N/mm²



Figure 11 No. of generations vs. Optimum cost for cantilever slab

4.3.2. The Population Size

A study is carried out by changing the population size while making the other values as

unchanged as follows:

The number of generations = 100, and the mutation rate = 0.01, the method of selection is

selected as Roulette Wheel function.

fig.12 and fig.13 showing the outcomes by changing the number of population size for

both the slabs.

Figure 12 Population size vs. Optimum cost for one way slab

10525.8

10518.6

10515.4 10513.52

10513

10514

10515

10516

10517

10518

10519

10520

10521

10522

10523

10524

10525

10526

20 40 60 80 100

Op

tim

um

cost

(R

s.)

Generations

fck = 20 N/mm², fy = 415 N/mm²

6369.24

6337.81

6321.41 6319.93

63186322632663306334633863426346635063546358636263666370

20 40 60 80 100

Op

tim

um

cost

(R

s.)

Population size

fck = 20 N/mm², fy = 415…



Figure 13 Population size vs. Optimum cost for cantilever slab

4.3.3. Mutation Rate

A study is carried out by changing the mutation rate while making the other values as

unchanged as follows:

The number of generations = 100, and the number of population size = 100, the method of

selection is selected as Roulette Wheel function.

fig.14 and fig.15 showing the outcomes by changing the mutation rate for both the slabs

Figure 14 Mutation rate vs. Optimum cost for one way slab

10550.5

10526.9

10520.6

10513.52

10512

10516

10520

10524

10528

10532

10536

10540

10544

10548

10552

20 40 60 80 100

Op

tim

um

cost

(R

s.)

Population size

fck = 20 N/mm², fy = 415 N/mm²

6336.35

6319.93

6320.57 6321.34

6322.17

6318

6320

6322

6324

6326

6328

6330

6332

6334

6336

6338

0 0.02 0.04 0.06 0.08 0.1Op

tim

um

cost

(R

s.)

Mutation Rate

fck = 20 N/mm², fy = 415 N/mm²



Figure 15 Mutation rate vs. Optimum cost for cantilever slab

4.4. Effect on Optimum Cost by Changing the Constant Parameters

The effect on the optimum cost value by changing the constant parameters is observed using

illustrated problems of both the RC one way and the cantilever slab.

4.4.1. Effect of Variation in Grade of Concrete and Steel

By changing the value of the compressive cube strength of the concrete and for the steel, there

will be correspondingly change in the unit cost of the concrete and steel. In this region genetic

algorithm programs are applied for different values of the compressive cube strength of the

concrete and for the steel and the respective obtained optimum values of the solution is

studied. The changing values of concrete are 20 and 25 MPa and for steel the changing values

are 415 and 500 MPa for the RC one way slab and cantilever examples.

The results of the illustrated problems shown in fig.16 and fig.18 subsequently for a

particular load is studied and by changing loading values the effect of variation in grade of

concrete and steel is studied in fig.17 and fig.19 respectively for both the simply supported

RC one way and cantilever slabs.

Input Data for Design Problem

Span of the slab is 4 m

Uniformly distributed live load of 3,5 and 7 kN/m2

The characteristic cube strength for concrete fck= 20 and 25 MPa

The characteristic yield strength of steel fy = 415 and 500 MPa.

10537.6

10513.52

10515.3 10517

10527

1051210514105161051810520105221052410526105281053010532105341053610538

0 0.02 0.04 0.06 0.08 0.1Op

tim

um

cost

(R

s.)

Mutation rate

fck = 20 N/mm², fy = 415 N/mm²



Figure 16 Effect of variation of grade of concrete and steel (RC one way slab)

Figure 17 Effect of variation of grade of concrete and steel for different loading values (RC one way slab)

Figure 18 Effect of variation of grade of concrete and steel (RC cantilever slab)

6319.93

6238.82

6370.27

6306.34

6000

6050

6100

6150

6200

6250

6300

6350

6400

For load 3 kN/mm²

Op

tim

um

cost

(R

s.)

fck = 20 N/mm², fy = 415 N/mm²

fck = 20 N/mm², fy = 500 N/mm²

6000

6100

6200

6300

6400

6500

6600

6700

6800

6900

7000

3 5 7

Op

tim

um

cost

(R

s.)

Load in kN/mm² fck = 20 N/mm², fy = 415 N/mm² fck = 20 N/mm², fy = 500 N/mm²

fck = 25 N/mm², fy = 415 N/mm² fck = 25 N/mm², fy = 500 N/mm²

10513.52

10416.86

10687.46

10593.84

10250

10300

10350

10400

10450

10500

10550

10600

10650

10700

10750

For load 3 kN/mm²

Op

tim

um

cost

(R

s.)





Figure 19 Effect of variation of grade of concrete and steel for different loading values (RC cantilever slab)

4.4.2. Cost Optimization Results for Different Span Lengths

Two problems of the reinforced concrete slabs are studied by changing the support conditions.

Problem 1 is a reinforced concrete simply supported slab. Problem 2 is a reinforced concrete

cantilever slab. The present study investigates the effect of slab span length on optimal design

variables[12],

the cost components and optimal reinforcement ratio.

Total cost obtained for two examples for different span lengths for different loading

values are presented in fig.20, fig.22 and fig.24. Steel reinforcement ratio obtained for two

examples for different span lengths for different loading values are presented in fig.21, fig.23

and fig.25

It should be noted that cantilever slabs are rarely used in high span lengths and for most

practical span lengths (3-4 m). [11]

Input Data for Design Problem

Span of the slab is 3,3.5,4,4.5,5 and 5.5,

Uniformly distributed live load of 3,5 and 7 kN/m2,

The characteristic cube strength for concrete fck= 20 MPa,

The characteristic yield strength of steel fy = 415 MPa.

Figure 20 Optimum cost for one way and cantilever slabs for different span lengths (For load 3kN/mm2)

9800

10000

10200

10400

10600

10800

11000

11200

11400

11600

3 5 7

Op

tim

um

cost

(R

s.)

Load in kN/mm² fck = 20 N/mm², fy = 415 N/mm²

fck = 20 N/mm², fy = 500 N/mm²

1500

4000

6500

9000

11500

14000

16500

19000

21500

24000

3 3.5 4 4.5 5 5.5

Op

tim

um

cost

(R

s.)

Span length(m)

one way slab cantilever



Figure 21 Optimum reinforcement ratio for one way and cantilever slabs for different span lengths

(For load 3kN/mm2)

Figure 22 Optimum cost for one way and cantilever slabs for different span lengths (For load 5kN/mm2)

Figure 23 Optimum reinforcement ratio for one way and cantilever slabs for different span lengths (For load 5kN/mm2)

0.25

0.26

0.27

0.28

0.29

0.30

0.31

3 3.5 4 4.5 5 5.5

Op

tim

um

rei

nfo

rcem

ent

rati

o

(%)

Span length(m)


1500

4000

6500

9000

11500

14000

16500

19000

21500

24000

3 3.5 4 4.5 5 5.5

Op

tim

um

cost

(R

s.)

Span length(m)


0.28

0.29

0.30

0.31

0.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

3 3.5 4 4.5 5 5.5

Op

tim

um

rei

nfo

rcem

ent

rati

o

(%)

Span length(m)




Figure 24 Optimal cost for one way and cantilever slabs for different span lengths (For load 7kN/mm2)

Figure 25 Optimum reinforcement ratio for one way and cantilever slabs for different span lengths (For load

7kN/mm2)

5. CONCLUSIONS

It was observed that when the no. of generations and population size is smaller the obtained

value of the GA codes for the both the slabs show far away from the optimum value of the

solution and the best optimum value is obtained by changing the generation value to 100 after

this value also there is no change in this best solution.

Mutation rate plays an important role in the genetic algorithm process. It was shown that

without and with larger values of mutation rate the obtaining solution is far away from the best

obtained solution and at value of 0.01 mutation rate the optimum solution is obtained.

In this study the illustrated examples are accomplished which shows that the total cost of slab

increases when there is increase in characteristic strength of concrete, while fixing the applied

load and grade of steel.

In this study the illustrated examples are applied which shows that while there is increase in

grade of the steel, there will be decrease in the optimum cost value while fixing the applied

load and compressive strength of concrete.

1500

4000

6500

9000

11500

14000

16500

19000

21500

24000

3 3.5 4 4.5 5 5.5Op

tim

um

co

st (

Rs.

)

Span length(m)


0.300.310.320.330.340.350.360.370.380.390.400.410.420.430.440.450.46

3 3.5 4 4.5 5 5.5Op

tim

um

rei

nfo

rcem

ent

rati

o (

%)

Span length(m)




The optimum cost for the both the slab is achieved by M20 grade of concrete and Fe500 grade

of steel.

Cantilever slab has the maximum total cost among the two examples. It should be noted that

cantilever slabs are rarely used in high span lengths and for most practical span lengths (3-4

m).

As expected in all examples the optimum value of the cost at higher span lengths increases

predominantly.

The percent reduction in optimum reinforcement ratio for the slab is directly proportion to

number of span length increases.

On comparison with an earlier literature related to cost optimization of reinforced concrete

slabs, it was concluded that there was cost reduction of 6.78% and 18.92% for the RC one way

and cantilever slab respectively.

REFERENCES

[1] Chakrabarty, B.K., (1992), “A model for optimal design of reinforced concrete beams”,

Asian Journal of Civil Engineering, Vol. 118, pp. 3238-3242.

[2] IS 456-2000, “Code of Practice for Plain and Reinforced Concrete”, Bureau of Indian

Standards, New Delhi.

[3] Saini, B., Sehgal, V.K., and Gambhir, M.L., (2006), “Genetically optimized artificial

neural network based optimum design of singly and doubly reinforced concrete beams”,

Asian Journal of Civil Engineering, Building and Housing, Vol.7, pp. 82-97.

[4] Sahab, M.G., Ashaour, A.F., and Toropov, V.V., (2004), “Cost optimization of

reinforced concrete flat slab buildings”, Engineering Structures, Vol.12, pp. 124 - 256.

[5] http://www.buildinghow.com/enus/Products/Books/VolumeB/Slabs/Cantilevers-one-way-

slabs.

[6] Krishna Raju, N., and Pranesh, R.N., (2003), Reinforced Concrete Design, Second

Edition, New Age International Limited, Publishers.

[7] Dutta, B.N., (1992), Estimation and Costing in Civil Engineering, Twenty fourth Edition,

UBS Publishers and Distributors Pvt.Ltd.

[8] Whitely, D., (1994), “A genetic algorithm tutorial”, Statistics and computing, Vol.4, pp.

65-85.

[9] Alqedra, M., Arafa, M., and Ismail, M., (2011), “Optimum cost of prestressed concrete

beams using genetic algorithms”, Journal of Artificial Intelligence, Vol.4, pp.76-88.

[10] Behrouz, A.N., and Hesam, V., (2011), “Minimum cost design of concrete slabs using

Particle Swarm Optimization”, World Applied Sciences Journal, Vol. 13, pp.2484-2494.

[11] Gunaratnam, D.J., and Sivakumaran, N.S., (1978), “Optimum design of reinforced

concrete slabs ", Structural Engineering, Vol.3, pp. 61 – 67.

[12] Camp, C.V., Pezeshk, S., and Hansso, H., (2003), Flexural design of reinforced concrete

frames using a genetic algorithm”, Asian Journal of Civil Engineering, Vol.23, pp. 733 –

944.

[13] Şengül CAN and Mustafa GER Şİ L, A Literature Review On The Use of Genetic

Algorithms In Data Mining. International Journal of Computer Engineering &

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DESIGN OPTIMIZATION OF REINFORCED CONCRETE SLABS …The two types of reinforced concrete slabs...

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