Cost Optimization of Reinforced Concrete Chimney-libre

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.V2, 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 todays 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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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


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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.


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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

X3


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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*103 Characteristic strength of concrete kN/m2 fck 25*103 Unit wt of concrete kN/m3 dc 25 Density of steel kg/m3 ds 7894.09 % minimum steel for vertical steel % min 0.3 % maximum steel for vertical steel % max 4 % minimum steel for horizontal steel % hmin 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


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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 E(Print), ISSN 0976 6316(Onlin

Graph1: CASE (I)

Graph2: CASE (I)

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(I) comparison of optimum and conventional cost

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 steelconventional steel

Height in m

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el.


Graph3: CASE (II)

Graph4: CASE (I

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) comparison of optimum and of conventional cos

II) comparison optimum and conventional steel

12 24 36 48 60 72

Optimal Costconventional Cost

Height in m

12 24 36 48 60 72


Height in m

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ost


Table 6: CASE (II) 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

TabSr. 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)

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) Conventional values by taking six (6) segmentsX2

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

ble 7: CASE (II) Cost comparison.

Co (Rs)

Ct (Rs)

Total Optimum Cost(Rs)

Total Conventiona

cost(Rs)

192016 237198 192016 237198231931 237198 423947 474396291723 374438 715670 848834346096 374438 1061766 1223272295728 363579 1357494 1586851332517 363579 1690011 1950431

) comparison of optimum and of conventional co

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

Optimal Costconventional Cost

Height in m

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ts. Volume

of concrete

m3

223.15

nal %

saving

98 19.05 96 10.63 34 15.69 72 13.20 51 14.45 31 13.35

cost


Graph6: CASE (II

Table 8: CASE (III)

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

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ste

el

in

mm

2


411

II) comparison optimum and conventional steel

II) Optimal values by taking eleven (11) segments

X1 mm

X2 mm2

Total X2

mm2

X3 mm2

WeightedAvg.

Thicknesmm

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

0000000000000000000000000000000

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


Height in m

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ed

ess

Volume of

concrete m3

187.90


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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 OPTIMU

Weighted thickness in ecalculated and is compared wiconcrete saving in each case.

Graph7

Graph8: Compa

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1200000

1400000

1600000

1800000

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150

170

190

210

230

Vo

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f

co

ncre

te

in

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MUM CONCRETE AND CONVENTIONAL CON

each case is calculated, from which volume owith conventional one. Following graph show

h7: Numbers of segment Vs Total cost

parison of optimum and conventional concrete

00

00

00

00

00

00

0 3 6 9 12 15 18 21 24

Optimal Cost

Number of segments

3 6 11 22

Optimal Concreteconventional concrete

No of segments

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ONCRETE

of concrete is ws amount of


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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-Gonzlez and Jos Mara 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 Reinforced Concrete Columns by the Transfer-Matrix Method, International Journal of Civil Engineering & Technology (IJCIET), Volume 3, Issue 1, 2012, pp. 111 - 127, ISSN Print: 0976 6308, ISSN Online: 0976 6316.

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Cost Optimization of Reinforced Concrete Chimney-libre