6th World Congresses of Structural and Multidisciplinary Optimization

Rio de Janeiro, 30 May - 03 June 2005, Brazil

Multi-objective Optimization of Reinforced Concrete Frames

Matj Lep
Department of Structural Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague,
Thkurova 7, 166 28, Prague 6, Czech Republic, e-mail: leps@cml.fsv.cvut.cz
1. Abstract
This paper presents a discrete optimization of reinforced concrete structures based on an efficient combination of deterministic and
stochastic optimization strategies. The deterministic optimization algorithm is used for the detailing of a reinforced concrete crosssection for a given combination of internal forces. The multi-objective stochastic optimization algorithm is then applied to the
optimization of a whole structure in terms of basic structural characteristics like types of materials, dimensions of elements or
profiles of steel bars
2. Keywords: Reinforced concrete frames, design, multi-objective optimization, evolutionary algorithms
3. Introduction
An attempt to create an effective design procedure for a reinforced concrete structures design goes through the history of Civil
Engineering. We limit our attention to frame structures, which are the major part in this field as one of the basic building blocks of
various construction systems.
It would be highly desirable to solve the whole design problem as one optimization task but the number of all possible
solutions is too high for realistic frame structures. Therefore, it appears to be inevitable to split the process of structural design into
two parts the detailing of a reinforced concrete cross-section and the optimization of a whole structure in terms of basic structural
characteristics like types of materials, dimensions of elements or profiles of steel bars.
The main goal of the first part is to fit an interaction diagram of a RC cross-section to a given combination of load cases. Efficient
procedures for fast evaluation of internal forces for a general cross-section and stress-strain relationship were proposed in [8]. This
task, for a given reinforcing bar diameter, thus reduces to a mere checking of admissible combinations of reinforcements.
The second part of a frame design focuses on the proportioning of building blocks. The goal is to find the best combination of
discrete inputs that is, in an appropriate sense, optimal from the point of view of the total cost of the structure as well as maximum
deflection of structural members.
For the single objective case, our experience [5] shows that a modified version of the genetic algorithm based procedure called
Augmented Simulated Annealing method is capable of solving this combinatorial task. In this contribution, the multi-objective
approach is introduced to tackle several conflicting objectives. The Strength Pareto Approach algorithm [9] is then used for the
determination of trade-off surfaces for selected criteria. For this purpose, two important objectives are selected and defined - the total
price of a resulting structure and the maximum deflection of structural members. As results of the presented research, Pareto-optimal
solutions can be plotted to demonstrate the non-linearity of this design problem and to show the applicability of this approach in Civil
Engineering practice.
4. Design parameterization
As already mentioned above, we search for a frame structure simultaneously considering price of the structure and maximum
deflection as the objectives of optimization. For simplicity, we limit our attention to 2D problems and elements with rectangular
cross-sections. Hence, we consider frame structures located in the xz plane and our interest is restricted to internal forces acting in
this plane: the bending moment My, the normal force Nx and the shear force Vz.
From the construction point of view as well as optimization itself it appears to be advantageous to decompose the whole structure
into nd design elements (see Fig. 1). These user-defined blocks are parts of a structure which a-priori possess identical optimized
parameters like dimensions of the cross-section, the area and the diameter of the bending reinforcement etc. In addition, we assume
that the structure is discretized into ne finite elements, used for the determination of an internal forces distribution. In the sequel, we
will denote a quantity X related to the i-th finite element as X[i] while a quantity related to the i-th design element as X(i), i.e., values
related to finite elements are indexed by square brackets, while quantities connected with design elements are denoted by round ones.
Further, e[i] and e(i) are used for the i-th finite and design elements, respectively. The symbol E(i) is reserved for the set of finite
elements, related to the i-th design element, i.e.,

E (i ) = e[ j ] : e[ j ] e( i ) , j = 1,..., ne .

(1)

Furthermore, the analyzed structure is supposed to be loaded by nc user-supplied load cases. A quantity X related to the i-th design
element and the c-th load case is denoted as cX(i).

Figure 1. An example of a frame structure

Figure 2. An example of a design element

As described in the previous paragraph, the design element is used for the definition of basic optimization parameters (see Fig. 2). In
our context, the design optimization parameters are the cross-section dimensions b and h, the diameter of bending reinforcement b,
the number of reinforcing bars located at the upper and the bottom surfaces of the design element denoted by ns1 and ns2 and,
alternatively, the diameter of shear reinforcement w and the spacing of stirrups sw. We assume that the cross-sectional dimensions
and stirrup spacing vary with a given discrete difference (e.g., 0.025 m), while b and w are selected from a given list of available
dimensions.
5. Ultimate limit state
Generally speaking, the structural requirements imposed by a chosen design standard (e.g., EC2 [1] considered in this work) can be
divided into two basic categories: load-bearing capacity and serviceability requirements. In the present work, the load-bearing
capacity requirements, discussed in the present section, are directly incorporated into the reinforcement design. The serviceability
requirements, on the other hand, are taken into account as the second optimization objective, see Section 6.2.

(a)

(b)

Figure 3. The cross-section scheme: (a) a plane of deformation, (b) an interaction diagram
In our previous works [4] the optimization of cross-section reinforcement was carried out simultaneously with the

determination of geometrical parameters of the structure. This approach, however, does not seem to be feasible for larger structures
because it would result in a huge amount of optimized variables, rendering the whole problem unmanageable. Thus, we employ
a conceptually simple procedure aimed at the reduction of the problem size based on powerful algorithms for fast evaluation of
internal forces, that were developed in a work by R. Vondrek [7].
First of all, we briefly list the basic ideas of the procedure of the evaluation of internal forces employed in this work and refer
an interested reader to R. Vondreks works [7], [8] for more detailed discussion. To that end, we assume that a given polygonal
cross-section is subjected to a given linear distribution of the x strain given by

x ( z ) = 0 + z ,

(2)

where 0 is the strain at the coordinate system origin and is the curvature in the z direction (see Fig. 3a). Further, the response of
a material is governed by a constitutive equation

x = ( x ).

(3)

The internal forces Nx and My are then provided by the well-known relations

N x = x dA , M y = x z dA .
A

(4)

Converting the area integrals (4) into boundary integrals by the Gauss-Green formula together with the fact, that the cross-section is
polygonal, yield after some manipulations

Nx =
1
My = 3

n p 1

i =0

1
2

n p 1

i =0

1
( ss ( (i +1) ) ss ( (i ) )),
ki

( i+1)
1
[( 0 ) ss ( ) 2 ss ( ) 2 sss ( )] ( i ) ,
ki

(5)

(6)

where np is the number of polygon segments, ki is the tangent of i-th polygon segment, (i) is the value of the strain at the i-th polygon
vertex and values ss( . ) and sss( . ) follow from recursions

sss( ) = ss( ) d = [ s ( ) d ] d = [ [ ( ) d ] d ] d .

(7)

For detailed derivation and discussion of these relations together with the treatment of degenerate cases (i.e., 0 or ki 0)
we again refer to the original works [7], [8].
Once we are able to evaluate internal forces for a given plane of deformation determined by and 0, the boundary of the
interaction diagram I (see Fig. 3b) for a given cross-section can be simply constructed by evaluating the values of the bending
moment My and the normal force Nx for a given set of extremal deformation planes. Then, the cross-section can sustain the given
normal force NSd and the bending moment MSd iff

( N Sd , M Sd ) I .

(8)

In the design procedure, we assume that we are provided with the dimensions of a cross-section b and h and the diameter of the
longitudinal reinforcing bars b. Next, the Codes of Practice provide us with the minimum and maximum values of reinforcement
areas As1 + As2, which can be easily converted to a minimum/maximum number of reinforcing bars ns,min and ns,max. Then, one can
find the minimum reinforcement area such that the condition (8) holds for all elements and load cases, i.e.
[ j] c
[ j]
(c N Sd
, M Sd
) I , j E (i ) , i = 1,..., nd , c = 1,..., nc .

(9)

Although the proposed procedure is extremely simple, it performs satisfactorily thanks to the very efficient implementation of
internal forces evaluation. Furthermore, it effectively eliminates infeasible solutions and thus substantially decreases the
dimensionality of the problem.
6. Objective functions
Having defined (and appropriately reduced) the domain of all admissible structures, the most suitable solution from this set is to be
selected. For this purpose we need to measure the quality of each structure. As mentioned previously, we have selected both the total
price of a structure and the maximum deflection as the objectives to be optimized. Note that some deflection limit usually serves as
constraint during an optimization process while here is an objective.

6.1 Design economy

The total price of the structure follows from the expression

f ( X) = Vc Pc + Ws Ps + Ac PAc ,

(10)

where X stands for the vector of design variables, Vc is the volume of concrete, Ws is the weight of steel and Ac is the area of concrete
connected with form-work; Pc, Ps are the prices of concrete per unit volume and steel per kilogram, PAc is the price of form-work per
square meter, which is added to simulate construction costs1.

6.2 Serviceability limit state

As the second objective of the optimization, the maximum deflection of the analyzed structure will be considered. In the current
implementation, the maximum sagging of the i-th design element due to the c-th load case is determined on the basis of a simple
numerical integration algorithm. To this end, suppose for simplicity that the internal forces distribution for the given load case and
design element is known from the elastic analysis. For the given values of the bending moment My and the normal force Nx2, the
parameters of the deformation plane 0 and , recall Fig. 3a, can be efficiently found by the Newton-Raphson algorithm [7]. Then,
under the assumptions of small deformations and small initial curvature, the deflection curve follows from the familiar relation,

d 2 w( x)
= ( M y ( x)),
dx 2

(11)

which yields, after integrating Eq. (11) twice,

x

w( x) = [ ( M y ( )) d ] d + C1 x + C2

(12)

with integration constants C1 and C2 determined from boundary conditions for a given design element. Also the analyzed element can
be split into several equidistant parts with the length x and Eqs. (11) and (12) can be replaced by their discretized counterparts. The
maximum deflection of the design element is then straightforwardly determined as the extremal value found for all load cases.
7. Multi-objective optimization algorithm
The Strength Pareto Evolutionary Algorithm (SPEA), firstly introduced by Zitzler and Thiele [10] in 1999, was selected as the multiobjective optimizer in the present study. The key ideas of this algorithm can be summarized as [10]: storing non-dominated solutions
externally in a second, continuously updated population, fitness assignment with respect to the number of external non-dominated
points that dominate it, preserving population diversity using the Pareto dominance relationship and incorporating a clustering
procedure for the reduction of the non-dominated set. Moreover, all these features are actually independent of the form of crossover
and mutation operators. Therefore, it is possible to use operators developed for the single-objective optimization problem [4] without
any changes. Last, but certainly not least, advantage of this algorithm is its conceptual simplicity and freely available C++ source
code. An interested reader is referred to the article [10] and the Ph.D. thesis [9] for more detailed description of the algorithm as well
as extensive numerical investigation of its performance.
8. Examples and results
We demonstrate the aforementioned design procedure on the benchmark problems, already considered in a conference paper [2]. In
particular, two different statically determined structures are examined.

Figure 4. First example - a cantilever beam

8.1 A cantilever beam

Firstly, a cantilever beam, see Fig. 4, with the 4.0 meter span was studied. A concrete model with cylindrical ultimate strength equal
to 20 MPa (Class C 16/20) was considered with steel model with the 410 MPa yield stress (Class V 10 425). The cantilever was
loaded with two loading cases: ( 1N1 =1800 kN, 1N2 =100 kN) and (2N1=300 kN, 2N2 =100 kN). The theoretical cover of steel
reinforcement was set to 0.05 m and the supposed diameter of shear reinforcement was 0.06 m. In the design procedure, the beam
width was restricted to b {0.3, 0.35, 0.4, and 0.45} m while the heights h {0.4, 0.5, and 0.6} m were considered. The
longitudinal reinforcement profiles were selected from the list b {10, 12, 14, 16, 18, 20, 22, 25, 28, 32, and 36} mm. The
individual unit prices appearing in Eq. (33) were considered Pc = 2,500 CZK/m3, Ps = 25 CZK/kg and PAc = 1,250 CZK/m2,

1
2

See the paper [6] for more than seventy references dealing with cost optimization of reinforced concrete structures.
Note that for the notational simplicity, indices c and i are omitted in the present section.

The symbol CZK stands for Czech Crowns.

Deflection

(c)

Price

d [m]
0

10

20

30

40

50

60

70

80

90

0.010

0.015

0.020

0.025

0.030

0.035

0.040

(d)

Price

(b)

Price

Figure 5. Results for the cantilever beam example: (a) Pareto-front and Pareto sets - (b) steel profiles d, (c) the width b and the height h, (d)
the number of steel bars n and the amount of steel ws

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

(a)

Price

2
3
8
2
8
2
1
8
4
1
2
4
5
08 65 06 5 68 586 6 48 678 7 37 7 82 825 8 88 948 0 14
14 14 15
1
1
1
1
1
1
1
1
1
2

0.00600

0.00800

0.01000

0.01200

0.01400

0.01600

0.01800

b,h [m]

n,ws [kg]

0.02000

respectively3. Finally, the integration step x = 0.25 m was considered for the deflection analysis.

Results are shown in Fig. 5 and Fig. 6 by the visualization methodology presented in the authors thesis [3]. The principle is
that all Pareto-optimal solutions can be sorted in the terms of individual functions and also their x values can be sorted/drawn in this
order. Such picture can give us sensitivity information on the variables influence on the objective function and, therefore, to
significantly help the designer to choose the proper solution.
It can be seen that there are 39 non-dominated solutions, which are characterized by the maximal value of the height of the
beam h and by non-monotonously increasing amount of steel, see Fig. 5(d). It is also important, that solutions are not created by the
small steel profiles which are probably not able to sustain applied internal forces.

ws
n
d

Deflection

b
h

0.02000

1.00

0.01800

0.80
0.60

0.01600
19967

0.01400

18882

[% ]
0.40
0.20
0.00

0.01200

18034

0.01000

17067
Price

0.00800

16482

0.00600

15839

0.00400

15053

0.00200
0.00000

14082

Figure 6. Results for the cantilever beam example depicted in 3D

Figure 7. Second example - a simply supported beam

8.2 A simply supported beam

The second example studied was a simply supported beam, see Fig. 7. The span was considered 6 m. The concrete and the steel
model were the same as in the previous example, as well as a reinforcement cover, a shear reinforcement profile, geometrical
parameters b, h and b . The beam was loaded with three loading cases: (1 p1 = 62.5 kN/m, 1N1 = -240 kN), (2p1 = 62.5 kN/m, 2N1 = 1440 kN) and (3p1 = 62.5 kN/m, 3N1 = 480 kN).
At this example, we simulated the scenario of a growing price of steel. The question placed here is: What will happen if
a price of steel grows for 20%?. Therefore, the Case 1 is characterized by unit prices Pc = 2,500 CZK/m3, Ps = 25 CZK/kg and
PAc = 1,250 CZK/m2 and the Case 2 by the same values for Pc and PAc, but the value of Ps is set to 30 CZK/kg.

0.014
Deflection

0.012
0.010

Case 2

0.008

Case 1

0.006
0.004
20000

22000

24000

26000

28000

30000

32000

34000

Price

0.040
0.035
d [m]

0.030
d: Case 2

0.025

d: Case 1

0.020
0.015
0.010
21000

23000

25000

27000

29000

31000

33000

35000

b,h [m]

Price

0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
21000

b: Case 2
h: Case 2
b: Case 1
h: Case 1

23000

25000

27000

29000

31000

33000

35000

n,ws [kg]

Price
90
80
70
60
50
40
30
20
10
0
21000

n: Case 2
ws: Case 2
n: Case 1
ws: Case 1

23000

25000

27000

29000

31000

33000

35000

Price

Figure 8. Results for the simply supported beam example: (a) Pareto-fronts and Pareto sets - (b) steel profiles d, (c) the widths b and
the heights h, (d) the number of steel bars n and the amount of steel ws
Results are shown again in Fig. 8. The Case 1 is created by 30 non-dominated solutions and the Case 2 by 29 and both cases are
characterized by the maximal value of the height of the beam h. On the first sight, the growth of the steel price shifts the Pareto-front
of the more expensive Case 2 to the right, see Fig. 8(a). But still, there are some designs, where both cases meet each other. The next
interesting point is the decrease of the amount of steel, as can be visible in the Fig. 8(d). And finally, by inspecting both Pareto-sets it
comes that the last 15 solution are the same - they differ only in the price. Thus, such optimal designs can be seen as stable (or at
least less sensitive) with respect to perturbation of steel price and hence more robust from the practical point of view.

9. Conclusions
Nowadays engineering tasks are different types of designs and, especially, the structural design is very frequent one. And the design
of a structure is internally an optimization problem. Moreover, the real design task is always multi-objective. Unfortunately, till
nowadays it is often solved as a single-objective problem by combining different, usually conflicting, objectives into only one. This
is so inappropriate intervention into the process of finding an optimal solution that the multi-objective methodology presented in this
paper seems to be rather a necessity than a choice. As an addition, multi-objective algorithms enable solution for constrains in a more
natural way as another objectives.
As an illustrative example, the design of RC frames is introduced. Although the number of all possible solutions for this
particular design in a detail is manifold, it was shown that the computational cost can be minimized and the optimized design can be
shifted closely to a practical use. To show its applicability, typical examples are solved and the Pareto-fronts in terms of the total
price of a structure against its deflection are depicted. The results of the SPEA algorithm revealed that there are 39, 30 and 29 nondominated solutions for the cantilever and simply supported beam problems, respectively. The trade-off surfaces for both problems
appear in Fig. 5(a) and Fig. 8(a). It is clearly visible that even for these rather elementary design tasks, both Pareto-optimal fronts are
non-convex and non-smooth due to discrete nature of the optimization problem. This fact justifies the choice of the selected
optimization strategy and suggests its applicability to more complex structural design problems.
Acknowledgements
The financial support of this work by research project of the Czech Ministry of Education, MSM 6840770003, is gratefully
acknowledged.
10. References
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In Concrete Days 99, pages 272279. The Czech Concrete Society, 1999.
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1966, August 2003.
5. K. Matou, M. Lep, J. Zeman, and M. ejnoha. Applying genetic algorithms to selected topics commonly encountered in
engineering practice. Computer Methods in Applied Mechanics and Engineering, 190(1314):16291650, 2000.
6. Kamal C. Sarma and Hojjat Adeli. Cost optimization of concrete structures. Journal of Structural Engineering, 124(5):570578,
1998.
7. R. Vondrek. Numerical methods in nonlinear concrete design. Masters thesis, Czech Technical University in Prague, 2001.
8. R. Vondrek and Z. Bittnar. Area integral of a stress function over a beam cross-section. In Proceedings of The Sixth
International Conference on Computational Structures Technology, pages 1112. Civil-Comp Press, 4-6 September 2002.
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Institute of Technology (ETH), Zurich, Switzerland, November 1999.
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Approach. IEEE Transactions on Evolutionary Computation, 3(4):257271, November 1999.