Cost optimization of reinforced concrete chimney

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
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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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*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 % 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
276.45 213.442 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
290.09 223.152 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
TotalCostinRs
0
15000
30000
45000
60000
75000
90000
105000
120000
135000
150000
0
steelinmm2
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
CostinRs
0
15000
30000
45000
60000
75000
90000
105000
120000
135000
150000
0
steelinmm2
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
TotalCostinRs
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
steelinmm2
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
2000000
TotalCostinRs
150
170
190
210
230
Volumeof
concreteinm3
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
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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.
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Laxmi Publication (P) Ltd. New Delhi-110002.
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