Keywords: finite elements, hybrid-mixed stress formulations, continuum damage mechanics, Legendre polynomials, concrete three-dimensional structures.

This paper presents a hybrid-mixed stress model applied to the physically nonlinear analysis of concrete structures. In this model, both the stress and the displacement fields are simultaneously and independently approximated in the domain of each element [1]. The displacements on the static boundary are also independently modelled. None of the fundamental relations are locally enforced. The compatibility and the equilibrium conditions in the domain and the equilibrium conditions on the static boundary are enforced on an average basis. The connection between elements is ensured by imposing that adjacent elements share the same boundary displacement approximation. The special interest in using hybrid-mixed stress models is due to the possibility of obtaining locally quasi-equilibrated solutions, which are very convenient for structural design purposes. As none of the fundamental equations have to be locally satisfied, the use of such models also enables the use of a wide range of approximation functions.

As in [2], the stress approximation is replaced by an independent effective stress approach. Complete sets of orthonormal Legendre polynomials are used to define all the required approximations. The orthonormality property of these polynomials enables the construction of analytical solutions for the integration involved in a definition of the linear operators, which minimizes processing times and ensures high accuracy in the computations. With the development of damage, the governing system becomes nonlinear and its resolution leads to the adoption of an incremental and iterative process. The iterative method used is a modified version of the Newton-Raphson technique, which uses the secant matrix in the computation of the increments of all generalized variables in each stage of the iterative process.

The nonlinear behaviour of concrete is modelled using continuum damage mechanics. Two isotropic and elastic damage models with scalar variables have been adopted associated with a regularization technique known as nonlocal integral model [3].

To validate the numerical model and to assess its efficiency and accuracy, a set of three-dimensional problems with known experimental and numerical results is analysed and discussed.

1

C.M. Silva, L.M.S.S. Castro, "Hybrid-Mixed Stress Model for the Non-Linear Analysis of Concrete Strutures", Computers & Structures, 83, 2381-2394, 2005. doi:10.1016/j.compstruc.2005.03.031

2

C.M. Silva, L.M.S.S. Castro, "Hybrid-mixed stress formulation using continuum damage models", Communications in Numerical Methods in Engineering, 22, 605-617, 2006. doi:10.1002/cnm.837

3

C.M. Silva, L.M.S.S. Castro, "Continuum damage models with non-conventional finite element formulations", International Journal of Non-Linear Mechanics, 45(2), 83-99, 2010. doi:10.1016/j.ijnonlinmec.2009.09.005

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