Keywords: reliability analysis, response surface, computational efficiency, RC structures.

Reinforced concrete (RC) grids are very often adopted as the building floor structure and therefore appropriate to receive the loading transferred by the slabs. These grids are characterized by exhibiting a large number of critical cross-sections, where the failure most probably occurs. The most common numerical technique used so far to carry out reliable analysis or reinforced concrete building structures or members is the finite element code accomplished with an accurate reinforced concrete model.

The evaluation of the stiffness reduction and the ultimate strain values are often carried out by means of simple models based on bending moment-curvature diagrams, as for instance, the models proposed by Galli-Favre [1] and Debernardi [2]. Better elaborated models to predict the stiffness reduction could lead to more reliable results in engineering practice. Herein, the mean compressive branch was borrowed from the CEB MC 90 [3], while the tension branch, taken from Figueiras [4], has been chosen to simulate the global tension stiffening effects. For the reinforcement bar material points, the behaviour has been assumed elastic-plastic with constant strain hardening.

Monte Carlo simulations [5] and the Response Surface Method [6] are very common procedures to evaluate the probability of failure of a single reinforced concrete member or a more complex structural system. In both situations, the computer time required for the numerical analysis may be prohibitive. Particularly for the case of reinforced concrete building floor grids, one may face the definition of a large number of nodes and consequently a large number of degrees of freedom. As the behaviour of the grid elements are strongly non-linear with very significant stiffness reductions depending on the local curvatures required by the equilibrium, the redistribution of displacements and internal forces are very important and become mandatory for the accurate reliability assessment. Moreover, to perform reliability analysis, many finite element runs will be required and therefore computing time may be inconvenient large.

In this paper, reliability analysis is carried out by using the Monte Carlo simulations and the Response Surface Method (RSM) coupled with finite elements to reduce the global computational effort and also to improve the accuracy to evaluate the reliability index. Failures are assumed when either a concrete strained fibre or a steel bar strain reach the ultimate limit given by the Eurocode. The incremental and iterative procedure is used to achieve the ultimate load, which is defined by the failure of one grid cross-sections and captured within a tolerance previously chosen.

In order to define the failure and the safe domains, the mechanical analysis has to be performed for different values of concrete strength. The adopted reliability procedure is divided into three steps. The first step consists in performing random sampling by Monte Carlo simulations, to get the ultimate loads distribution, as well as the corresponding failure modes. This Monte Carlo simulation allows us to select the most important failure modes.

The second step aims to define the global structural response surface, by non linear regression on the basis of the random sampling performed in the first step. For a given applied load, the response surface divides the design space into safe and failure domains. In the third step, the response surface is used to determine the reliability index of the structure. The probability of failure is then evaluated by approximation methods, such as First and Second Order Reliability Methods (FORM/SORM).

The coupling between Monte Carlo simulations and response surface techniques aims to reduce significantly the number of calls to the finite element model, and hence to deal with real RC grids where large number of failure components can be found. The proposed procedure is then applied to simple reinforced concrete grids in order to show the advantages and the drawbacks for real grid structures.

1

Ghali, A. Favre, R., "Concrete Structures: Stresses and Deformations", Chapman & Hall, 1th edition. 1986.

2

Debernardi, P.G., "Behaviour of Concrete Structures in Service", Journal of Structural Mechanics, 115, 32-50, 1989. doi:10.1061/(ASCE)0733-9445(1989)115:1(32)

3

Comité Euro-Internacional du Béton. CEB-FIP Model Codes. Bulletins d'Information n. 196/203/204/205, 1990.

4

Figueiras, J.A., "Ultimate Load Analysis of Anisotropic and Reinforced Concrete Plates and Shells, Ph.D. Thesis, University of Wales, 1983.

5

Frangopol, D.M., Ide, Y., Spacone, E., Iwaki, I., "A New Look at Reliability of Reinforced Concrete Columns", J Structural Safety, 18 (2/3), 123-50, 1996. doi:10.1016/0167-4730(96)00015-X

6

Mohamed, A., Lemaire, M., "Discussion on: Structural Reliability Analysis Using a Standard Deterministic Finite Element Code", Structural Safety, 20, 391-397, 1998. doi:10.1016/S0167-4730(98)00022-8

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