Keywords: reinforcement, concrete, plates, shells, optimization, FEM.

Traditionally, the failure conditions of the materials in the field of concrete shells have been applied to the stresses at the structure points. Even though this check methodology has been adopted by most researchers, the basic hypotheses and methods to solve the problem at present are still open to debate. As proof of this, the references in the design codes bound to the design of plates and shells are quite scarce and, in the majority of cases, they are dealt with rather superficially.

Eurocode 2-1991 (EC2) does not include any reference to shells, and only refers to plates that are loaded on their plane, while the ACI 318-99 code considers any method of design which assures sufficient strength with equilibrium to be considered applicable. In the Model Code CEB-FIP 1990 (MC90) there is more in-depth treatment, given that design hypotheses are included which are based on the use of layer models [1], resisting the external membrane forces and the internal shear force.

Nevertheless, there are design methods that use plate or shell elements as dimensioning units resisting their nodal forces. The objective is to reach the equilibrium between the external and internal forces due to the contributions of the reinforcement and the concrete. To this end, calculus algorithms are used to provide the quantity of reinforcement at the outer layers of the element in two orthogonal directions [2,3]. Some strategies have been developed to be able to obtain more rational reinforcement distributions, with less weight in the complete concrete plate or shell [4].

The development of optimization techniques has been strongly boosted by the tremendous increase in computational and graphical capacities. These techniques represent an effective means to obtain alternative reinforcement distributions, complying with the design conditions (stress constraints, construction prescriptions, etc.) in an optimum way (minimum weight, minimum stress level, etc.). They could become a standard procedure to design elements subject to membrane and flexure forces [5].

In the paper, the amount of reinforcement in any direction is optimized locally for each finite element of the mesh that models the geometry of the problem. Starting from the equilibrium between applied and internal forces, it leads to an indeterminate system of nonlinear equations. The formulation of the method includes biaxial behaviour of the concrete and different lever arms of the reinforcement, assuming ideal plastic behaviour for both materials [6].

The ANSYS finite element program has been used in order to analyse the structure and to obtain the forces in the shell elements. Furthermore, the formulation has been implemented by means of user routines within the optimization module of the program [7]. The objective function is the combination of the tensile forces in the reinforcement.

Some numerical examples are given and compared to the results provided by some authors. According to the results obtained, it may be concluded that the differences observed between the amount of reinforcement obtained, by using optimization techniques and by means of traditional methods, may be considerable. It is possible to achieve savings of over 30% in some cross sections of the structure, and savings of between 10% and 15% for the complete structure.

1

P. Marti, "Dimensioning and detailing", IABSE, 62, 411-443, 1991.

2

P.B. Lourenço, J.A. Figueiras, "Solution for the design of reinforced concrete plates and shells", Journal of Structural Engineering, ASCE, 121(5), 815-823, 1995. doi:10.1061/(ASCE)0733-9445(1995)121:5(815)

3

C.-S. Min, "Design and ultimate behavior of reinforced concrete hyperbolic paraboloid saddle shell", Proceedings of the 40th Congress of the IASS, Vol. 1, A89-A98, R. Astudillo, A.J. Madrid (eds.), CEDEX, Madrid, Spain, 1999.

4

S.M.R. Tabatabai, "Finite element-based elasto-plastic optimum reinforcement dimensioning of spatial concrete panel structures", Dissertation, ETH Zurich, Birkhäuser Verlag Basel, Zurich, Switzerland, 1996.

5

P. Martí, A. Tomás, M.S. Torrano, "Application of optimization techniques in structural concrete teaching", Proceedings of the 1st ACHE Conference in Structural Concrete Teaching, 203-209, ACHE, Madrid, Spain, 2001.

6

P.B. Lourenço, J.A. Figueiras, "Automatic design of reinforcement in concrete plates and shells", Engineering Computers, 10, 519-541, 1993. doi:10.1108/eb023923

7

"Advanced Analysis Techniques Guide", ANSYS Inc., Canonsburg, USA, 2004.

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