INTERNATIONAL JOURNAL OF CIVIL ENGINEERING … OPTIMIZATION OF... · International Journal of Civil Engineering and Technology ... RCC chimney in MATLAB. ... International Journal

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308

(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME

402

COST OPTIMIZATION OF REINFORCED CONCRETE CHIMNEY

Prof.Wakchaure M.R.1, Sapate S.V

2, Kuwar B.B.

3, Kulkarni P.S.

4

1(Assistant Professor, Civil Engineering Department, Amrutvahini college of Engineering,

Sangamner, Pune university, India) 2(M.E.Structures, Civil Engineering Department, Amrutvahini college of Engineering,

Sangamner, Pune university, India) 3(M.E.Structures, Civil Engineering Department, K.K.Wagh college of Engineering, Nasik,

Pune university, India) 4(M.E.Structures, Civil Engineering Department, K.K.Wagh college of Engineering, Nasik,

Pune university, India)

ABSTRACT

The design of reinforced concrete chimney structure almost always involves decision

making with a choice of set of choices along with their associated uncertainties and

outcomes. While designing such a structures, a designer may propose a large number of

feasible designs; however, only the most optimal one, with the least cost be chosen for

construction. For delivering an acceptable design, computer based programmes may help

today’s design practitioner. A program is developed for analysis and designing a low cost

RCC chimney in MATLAB. The optimtool module is used to find out the structure having

minimum cost with appropriate safety and stability. Illustrative case of chimney structure is

presented and discussed by using Interior point method from optimtool. The comparison

between conventional and optimal design is made and further results are presented. In final

result, percentages saving in overall cost of construction are presented in this paper.

Keywords: RCC chimney, Cost optimization, Interior point method, MATLAB, optimtool.

1. INTRODUCTION

During the past few years industrial chimneys have undergone considerable

developments, not only in the structural conception, modeling and method of analysis, but

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403

also in the materials employed and the methods of construction. Illustrative case of chimney

structure is presented and discussed by using Interior point Method from optimtool in

MATLAB. Interior point method and sequential quadratic programming methods are the two

alternative approaches for handling the inequality constraints.

Interior point method provides an alternative to active set method for the treatment of

inequality constraints. Interior point method have been a remerging field in optimization

since the mid of 1980s. At each iteration, an interior point algorithm computes a direction in

which to proceed, and then must decide how long of a step to take. The traditional approach

to choose a step length is to use a merit function which balances the goals of improving the

objective function and satisfying the constraints. Sequential quadratic programming (SQP)

ideas are used to efficiently handle nonlinearities in the constraints. Sequential quadratic

programming (SQP) methods find an approximate solution of a sequence of quadratic

programming (QP) sub problems in which a quadratic model of objective function is

minimized subject to the linearized constraints. Both interior method and SQP method have

an inner or outer iteration structure, with the work for an inner iteration being dominated by

cost of solving a large sparse system of symmetric indefinite linear equation, SQP method

provide a reliable certificate of infeasibility and they have potential of being able to capitalize

on a good initial starting point.

In this paper, cost optimization is done for 66 m industrial RCC Chimney (Figure1)

which is having constant outer diameter of 4m and thickness is varying from top to bottom in

three steps. Thickness of top segment (24m) shell is 200mm, and that of middle (24m) and

bottom segment (18m) it is 300mm and 400mm respectively.

Fig.1 Reinforced concrete chimney



404

2. OBJECTIVE FUNCTION

The objective function is a function of design variables the value of which provides

the basis for choice between alternate acceptable designs. Here the objective function is cost

minimization. The cost function f (cost) is:

f (cost) = Cs*Wst + Cc*Vc +Cb*Vb

Where, Cs, Cc and Cb= Unit cost of steel, concrete and brick lining respectively.

Wst is the weight of steel.

Vc and Vb= Volume of concrete, and brick lining respectively.

Cost calculation for concrete, steel and brick lining are inclusive of centering,

shuttering and cutting.

3. FORMULATION OF OPTIMIZATION PROBLEM.

The general three phases considered in the optimum design of any structure are

1) Structural modeling.

2) Optimum design modeling.

3) Optimization algorithm.

In structural modeling, the problem is formulated as the determination of a set of

design variables for which the objective of the design is achieved without violating the design

constraints. For the optimum design modeling, Study the problem parameter in depth, so as to

decide on design parameter, design variables, constraints, and the objective function. In the

search for finding optimum design starts from a design or from a set of designs to proceed

towards optimum.

3.1 Structural Modeling

In cost optimization of RCC chimney the aim is to minimize the overall construction cost

under constraints. This optimization problem can be expressed as follows:

Minimize f(X)

Subject to the constraints

gi (X) ≤ 0 i=1, 2, . . . . p

hj (X) = 0 j=1, 2,. . . . m

Where, f(X) is the objective function and

gi(X), hj(X) are inequality and equality constraints respectively.

3.2 Optimum Design Model

3.2.1 Design Variables

In optimization process, we required decision variables, design constraints, and

objective function. Decision variables are defined by a set of quantities some of which are

viewed as variables during the design process. The design variables cannot be chosen

arbitrarily, rather they have to satisfy certain specified functional and other requirements.

Figure 2 shows the design variables considered for RCC chimney. h = Height of chimney

structure, X1=Thickness of segment, X2=Vertical reinforcement, X3=Horizontal

reinforcement, X4 = Thickness of brick lining.



405

Fig.2: Mathematical model used for optimization of R.C.C. Chimney.

3.2.2 Design Constraints The restrictions that must be satisfied to produce an acceptable design are collectively called

as design constraint. The following design constraints are imposed on the variables.

1. Actual eccentricity (E) should be less than allowable eccentricity (Ea).

2. Maximum compressive stress should be less than allowable compressive stress.

3. Maximum Tensile stress should be less than allowable tensile stress (0.85Mpa).

4. Restriction on maximum and minimum vertical reinforcement percentage as per

CICIND Model code for concrete chimney shell.

5. Restriction on horizontal reinforcement percentage as CICIND Model code for

concrete chimney shell.

6. Stresses due to temperature gradient should be less than permissible stresses.

7. Bearing capacity criterion.

In design of RCC chimney structure, the objective function is taken for minimizing

the overall cost of construction. Structurally, a chimney is designed for its own weight, wind

pressure or seismic forces and the temperature stresses. Its own weight cause direct

compression in the section which increases towards the base. The wind pressure tends to

bend the chimney as a cantilever about its base, causing compression on leeward side and

tension on windward side. These stresses should not exceed the permissible values for

different grades of concrete and steel. So in this particular optimization, constraint is given

for stresses in leeward and windward side. The temperature stresses are developed in

chimney due to difference of temperature on its outside and inside surfaces. So the constraint

is given so that stresses induced due to temperature should be within permissible limit. Other

constraints are for maximum and minimum reinforcement percentage, Eccentricity which

must satisfy the standard code requirement. Bearing capacity criterion includes maximum

reaction pressure on footing should be less than safe bearing capacity of soil.

h

X2

X1

111

X3



406

4. RESULTS OF OPTIMIZATION

The programs developed were applied to obtained optimal solution for 66 m height

RCC chimney. Optimal values are obtained for three cases which include segments of

different heights as mentioned below and compared with conventional values. CASE (I) 3

segments of 24m, 24m, and 18m. CASE (II) 6 segments of 12m, 12m, 12m, 12m, 9m, and

9m. CASE (III) 11 segments of 6m each. The design parameters considered in above cases

that are related to wind pressure on chimney, code specifications, unit cost and other

characteristics of construction materials. Optimal solution changes with the variation of these

parameters which is an important issue as far as practical design is concerned. This is

constrained nonlinear programming problem for the numerical solution of the RCC chimney

structure using MATLAB, optimtool. A constrained equation and objective function has been

prepared for various height segments. Following are the input parameters of chimney which

is used in the optimtool for making constrained equations.

Table 1: Input parameters

Input parameter Unit Symbol Design Value

Height m h 66

Yield strength of steel kN/m2 fy 500*10

3

Characteristic strength of concrete kN/m2 fck 25*10

3

Unit wt of concrete kN/m3 dc 25

Density of steel kg/m3 ds 7894.09

% minimum steel for vertical steel % ρmin 0.3

% maximum steel for vertical steel % ρmax 4

% minimum steel for horizontal steel % ρhmin 0.2

spacing for horizontal steel mm s 250

Cost of steel Rs/kg Cs 60

Cost of concrete Rs/m3 Cc 8000

Cost of concrete Rs/m3 Cb 2500

S.B.C. kN/m2 b 180

Table 2: CASE (I) Optimal values for three (3) segments

Sr.

No

h

m

Seg-

ment

Segment

Length

m

X1

mm

X2

mm2

X3

mm2

Total

X2

mm2

Weighted

Avg.

Thickness

mm

Volume of

concrete

m3

1 24 0-24 24 196 23462 393 23462

276.45 213.44 2 48 24-48 24 289 33666 578 57128

3 66 48-66 18 367 41888 734 99016



407

Table 3: CASE(I) Conventional values for three (3) segments

Sr.

No

h

m

Seg-

ment

X1

mm

X2

mm2

Total

X2

mm2

X3

mm2

Weighted

Avg.

Thickness

mm

Volume of

concrete

m3

1 24 0-24 200 24127 24127 400

290.09 223.15 2 48 24-48 300 37699 61826 600

3 66 48-66 400 58904 120730 800

Table 4: CASE (I) Cost comparison.

Sr.

No

h

m

Seg-

ment

Segment

Length

(m)

Co

(Rs)

Ct

(Rs)

Total

Optimum

Cost (Rs)

Total

Conventional

cost (Rs)

%

saving

1 24 0-24 24 466063 474396 466063 474396 1.76

2 48 24-48 24 696700 748876 1162763 1223272 4.95

3 66 48-66 18 670284 727159 1833047 1950431 6.02

Table 5: CASE (II) Optimal values by taking six (6) segments.

Sr.

No

h

m

Seg-

ment

Segment

Length

m

X1

mm

X2

mm2

Total

X2

mm2

X3

mm2

Weighted

Avg.

Thickness

mm

Volume of

concrete

m3

1 12 0-12 12 160 9667 9667 321

252.68 196.33

2 24 12-24 12 195 11676 21343 391

3 36 24-36 12 237 13985 35328 473

4 48 36-48 12 287 16720 52048 573

5 57 48-57 9 317 18323 70371 633

6 66 57-66 9 364 20770 91141 727

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976

(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March

Graph1: CASE (I) comparison of

Graph2: CASE (I) comparison of

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

1800000

2000000

0

Tota

lC

ost

inR

s

0

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

0

stee

l i

n m

m2

rnational Journal of Civil Engineering and Technology (IJCIET), ISSN 0976

6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME

408

CASE (I) comparison of optimum and conventional cost

CASE (I) comparison of optimum and conventional steel.

0 24 48 72

Optimal Cost

conventional Cost

Height in m

12 24 36 48 60 72

Optimal steel

conventional steel

Height in m

rnational Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308

April (2013), © IAEME

optimum and conventional steel.



Graph3: CASE (II) comparison of optimum and

Graph4: CASE (I

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

1800000

2000000

0

Co

st in

Rs

0

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

0

stee

l i

n m

m2



409

) comparison of optimum and of conventional cost

II) comparison optimum and conventional steel

12 24 36 48 60 72

Optimal Cost

conventional Cost

Height in m

12 24 36 48 60 72

Optimal steel

conventional steel

Height in m



conventional cost



Table 6: CASE (II) Conventional

Sr.

No

h

m

Seg-

ment

X1

mm

1 12 0-12 200

2 24 12-24 200

3 36 24-36 300

4 48 36-48 300

5 60 48-57 400

6 66 57-66 400

Table 7

Sr.

No

h

m

Seg-

ment

Seg-

ment

Length

m

1 12 0-12 12

2 24 12-24 12

3 36 24-36 12

4 48 36-48 12

5 57 48-57 9

6 66 57-66 9

Graph5: CASE (III) comparison of optimum and

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

1800000

2000000

Tota

l C

ost

in

Rs



410

) Conventional values by taking six (6) segments

X2

mm2

Total

X2

mm2

X3

mm2

Weighted

Avg.

Thickness

mm

12063 12063 400

290.09

12063 24127 400

18849 42976 600

18849 61826 600

29452 91278 800

29452 120730 800

Table 7: CASE (II) Cost comparison.

Co

(Rs)

Ct

(Rs)

Total

Optimum

Cost(Rs)

Total

Conventional

cost(Rs)

192016 237198 192016 237198

231931 237198 423947 474396

291723 374438 715670 848834

346096 374438 1061766 1223272

295728 363579 1357494 1586851

332517 363579 1690011 1950431

) comparison of optimum and of conventional cost

0 6 12 18 24 30 36 42 48 54 60 66 72

Optimal Cost

conventional Cost

Height in m



values by taking six (6) segments.

Volume

of

concrete

m3

223.15

Conventional

%

saving

237198 19.05

474396 10.63

848834 15.69

1223272 13.20

1586851 14.45

1950431 13.35

conventional cost



Graph6: CASE (II

Table 8: CASE (III) Optimal values by taking eleven (11) segments

Sr.

No

h

m

Seg-

ment

Segment

Length

m

1 6 0-6 6

2 12 6-12 6

3 18 12-18 6

4 24 18-24 6

5 30 24-30 6

6 36 30-36 6

7 42 36-42 6

8 48 42-48 6

9 54 48-54 6

10 60 54-60 6

11 66 60-66 6

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

ste

el

in

mm

2



411

II) comparison optimum and conventional steel

(III) Optimal values by taking eleven (11) segments

X1

mm

X2

mm2

Total

X2

mm2

X3

mm2

Weighted

Avg.

Thickness

mm

141 4266 4266 281

241.09

159 4796 9062 318

175 5244 14306 349

194 5811 20117 389

207 6158 26275 413

234 6914 33189 467

260 7639 40828 520

285 8304 49132 569

307 11865 60997 614

330 12683 73680 660

360 13732 87412 720

0

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

0 6 12 18 24 30 36 42 48 54 60 66 72

Optimal steel

conventional steel

Height in m



(III) Optimal values by taking eleven (11) segments

Weighted

Thickness

Volume

of

concrete

m3

187.90



412

Table 9: CASE (III) Conventional values by taking eleven (11) segments

Sr.

No

h

m

Seg-

ment

X1

mm

X2

mm2

Total

X2

mm2

X3

mm2

Weighted

Avg.

Thickness

mm

Volume

of

concrete

m3

1 6 0-6 200 6032 6032 400

290.09 233.15

2 12 6-12 200 6032 12063 400

3 18 12-18 200 6032 18095 400

4 24 18-24 200 6032 24127 400

5 30 24-30 300 9425 33552 600

6 36 30-36 300 9425 42976 600

7 42 36-42 300 9425 52401 600

8 48 42-48 300 9425 61826 600

9 54 48-54 400 19635 81461 800

10 60 54-60 400 19635 101095 800

11 66 60-66 400 19635 120730 800

Table 10: CASE (III) Cost comparison

Sr.

No

h

m

Seg-

ment

Segment

Length

m

Co

(Rs)

Ct

(Rs)

Total

Optimum

Cost

(Rs)

Total

Conventional

cost

(Rs)

%

saving

1 6 0-6 6 84733 118599 84733 118599 28.56

2 12 6-12 6 95263 118599 179996 237198 24.12

3 18 12-18 6 104158 118599 284154 355797 20.14

4 24 18-24 6 115421 118599 399575 474396 15.77

5 30 24-30 6 129269 187219 528844 661615 20.07

6 36 30-36 6 144309 187219 673153 848834 20.70

7 42 36-42 6 158716 187219 831869 1036053 19.71

8 48 42-48 6 171936 187219 1003805 1223272 17.94

9 54 48-54 6 191891 242386 1195696 1465658 18.42

10 60 54-60 6 204178 242386 1399874 1708045 18.04

11 66 60-66 6 219957 242386 1619831 1950431 16.95

5. TOTAL COST COMPARISONS

Graph is plotted which shows total cost of chimney obtained by optimization. In each

case i.e. by taking 3, 6, and 11 segments, total cost is plotted and compare it with

conventional cost. As numbers of segment goes on increasing, more optimum values we get.



6. COMPARISON OF OPTIMUM CONCRETE

Weighted thickness in each case is

calculated and is compared with conventional one. Following graph shows amount of

concrete saving in each case.

Graph7

Graph8: Comparison of optimum and

1000000

1200000

1400000

1600000

1800000

2000000T

ota

l C

ost

in R

s

150

170

190

210

230

Vo

lum

eo

f

co

ncre

te

in

m3



413

TIMUM CONCRETE AND CONVENTIONAL CONCRETE

Weighted thickness in each case is calculated, from which volume of concrete is


Graph7: Numbers of segment Vs Total cost

Comparison of optimum and conventional concrete

1000000

1200000

1400000

1600000

1800000

2000000

0 3 6 9 12 15 18 21 24

Optimal

Cost

Number of segments

3 6 11 22

Optimal Concrete

conventional concrete

No of segments



CONCRETE

calculated, from which volume of concrete is




414

7. CONCLUSIONS

• Optimum values for cost, steel and concrete are then compared with the conventional

values. It is revealed from the graphs plotted for each case that the optimum values

are getting more precise as number of segments goes on increasing. Optimal design

shows total percentage cost saving of 6% in case (I), 13% in case (II) and 16% in case

(III). This shows that optimization is more cost effective as numbers of segment go on

increasing. From graph, for conventional and optimal design consideration; it shows

that overall cost of structure can be reduced by using optimization technique with

stability.

• In optimtool, interior point method is more iterative method. So the results are more

elaborated by using interior point method.

• The solver is giving optimum solution based on initial guess. If solver has been

changed that case optimum values of design problem also changed according to initial

guess. From above results, it is indicated that initial guess in solver is important for

getting more precise optimum values of respective height of chimney.

REFERENCES

1. Johannes C. Kloppers and Detlev G. Kroger, “Cost Optimization of Cooling Tower

Geometry”, Engineering Optimization, Vol.36, No.5, Year 2004, pp.575-584.

2. F.W. Yu and K.T. Chan, “Economic Benefits of Optimal Control for water-cooled Chiller

Systems Serving Hotels in a Subtropical Climate”, Energy and Buildings, Vol. 42, No.02,

Year 2010. pp. 203-209.

3. Izuru Takewaki, “Semi-explicit optimal frequency design of chimneys with geometrical

constraints”, Department of Architectural Engineering, Kyoto University, Sakyo, Kyoto

606, Japan Available online 3 May 1999.

4. Eusiel Rubio-Castro, Medardo Serna-González and José María Ponce-Ortega, “Optimal

Design of effluent-cooling Systems Using a Mathematical Programming Model”,

International Journal of Refrigeration, Vol.34, No.1, Year 2011. pp. 243-256.

5. Shravya Donkonda and Dr.Devdas Menon, “Optimal design of reinforced concrete

retaining walls”, The Indian Concrete Journal, Vol.86, No.04, pp. 9-18.

6. A Model code for concrete chimneys, Part-A-The shell (1984)-CICIND, 136 North street,

Brighton, England.

7. Geoffrey.M.Pinfold, “Reinforced concrete chimneys and Towers”, A viewpoint

Publication limited.

8. B.C.Punmia, Ashok K Jain and Arun K Jain, “Reinforced concrete structures- Vol.II”,

Laxmi Publication (P) Ltd. New Delhi-110002.

9. Mohammed S. Al-Ansari, “Flexural Safety Cost of Optimized Reinforced Concrete

Beams”, International Journal of Civil Engineering & Technology (IJCIET), Volume 4,

Issue 2, 2013, pp. 15 - 35, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316.

10. H.Taibi Zinai, A. Plumier and D. Kerdal, “Computation of Buckling Strength of

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ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316.

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INTERNATIONAL JOURNAL OF CIVIL ENGINEERING … OPTIMIZATION OF... · International Journal of Civil Engineering and Technology ... RCC chimney in MATLAB. ... International Journal