Keywords: continuum damage mechanics, Mazars damage model, hybrid-mixed stress models, finite elements, nonlinear dynamics.

This paper presents a hybrid mixed stress model for the physically nonlinear dynamic analysis of reinforced concrete frame structures. The model is called hybrid because both the stresses in the domain and the displacements on the static are simultaneously approximated. It is termed mixed because both the stresses and the displacements in the domain are directly approximated. All the fundamental conditions are imposed in a weighted residual form designed to ensure that the numerical discrete model embodies all the relevant properties presented by the continuum it represents. It is considered a stress model due to the type of enforcement followed for the equilibrium and compatibility conditions in the domain and because the connection between elements is ensured by averaging the equilibrium conditions on the static boundary.

The model presented in this communication is based on the use of orthonormal Legendre polynomials as approximation functions. The properties of these functions allow the definition of analytical closed form solutions for the computation of all structural operators in linear analysis. Numerical integration schemes are thus completely avoided. The numerical stability associated with the use of Legendre polynomial bases enables the use of macro-element meshes where the definition of highly effective p-adaptive refinement procedures is simplified.

To model the concrete physically non linear behaviour the damage model introduced by Mazars is adopted. To validate and demonstrate the potential of the model, several numerical examples are presented and discussed, in which displacement conventional finite elements are compared with hybrid-mixed stress models.

For the time integration procedure a variant of the mixed formulation presented in [1] is adopted. The displacements, the velocity and the acceleration fields are independently approximated in time using hierarchical orthonormal Legendre polynomial bases. This time integration algorithm consists essentially in a modal decomposition technique implemented in the time domain, as opposed to the classical method of modal decomposition in the space domain. This modal decomposition in the time domain develops from the basic simplification of separating the space and time fields of the solution. The numerical efficiency of the resulting time integration scheme depends on the possibility of uncoupling the equations of motion. This is achieved by solving an eigenvalue problem in the time domain that depends only on the time approximation basis being considered [1].

The accuracy and the efficiency of this technique are assessed by comparing the results obtained with those provided by the classical Newmark method in nonlinear dynamic analysis.

1

J.A.T. Freitas, "Mixed Finite Element Solution of Time-Dependent Problems", Computer Methods in Applied Mechanics and Engineering, 197(45-48), 3657-3678, 2008. doi:10.1016/j.cma.2008.02.014

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