1 Static and dynamic physically non-linear analysis of concrete structures using a hybrid mixed finite element model Mário R.T. Arruda, Luís Manuel Santos Castro Departamento de Engenharia Civil, Arquitectura e Georrecursos Instituto Superior Técnico Universidade Técnica de Lisboa Av. Rovisco Pais, 1049-001 Lisboa e-mail: marruda@civil.ist.utl.pt, luis@civil.ist.utl.pt ABSTRACT A new hybrid-mixed stress finite element model for the static and dynamic non-linear analysis of concrete structures is presented and discussed in this paper. The main feature of this model is the simultaneous and independent approximation of the stress, the strain and the displacement fields in the domain of each element. The displacements along the static boundary, which is considered to include inter-element boundaries, are also directly approximated. To define the approximation bases in the space domain, complete sets of orthonormal Legendre polynomials are used. The adoption of these functions enables the use of analytical closed form solutions for the computation of all linear structural operators and leads to the development of very effective p-refinement procedures. To represent the material quasi-brittle behaviour, a physically non- linear model is considered by using damage mechanics. A simple isotropic damage model is adopted and to control strain localization problems a non-local integral formulation is considered. To solve the dynamic non-linear governing system, a time integration procedure based on the use of the -HHT method is used. For each time step, the solution of the non-linear governing system is achieved using an iterative algorithm based on a secant method. The model being discussed is applied to the solution of two-dimensional structures. To validate the model, to illustrate its potential and to assess its accuracy and numerical efficiency, several numerical examples are discussed and comparisons are made with solutions provided by experimental tests and with other numerical results obtained using conventional finite elements (CFE). Keywords: hybrid-mixed formulations, stress models, time integration, Legendre polynomials, damage mechanics, static and dynamic non-linear analysis. 1 Introduction It is possible to list three main classes of hybrid and mixed non-conventional finite element formulations [27], namely the hybrid-mixed, the hybrid and the hybrid-Trefftz models [27,29,30,48,49]. All formulations evolve directly from the first principles of mechanics, in particular equilibrium, compatibility and constitutive relations. What distinguishes the three types of formulation is the set of constraints enforced, a priori, on the domain approximations. Two models may be derived for each formulation, the displacement and the stress models [40]. In recent years, some of these non-conventional finite element formulations have been extended to the non-linear analysis of concrete structures using isotropic damage models [52]. In [54] the hybrid-mixed stress model based on the use of orthonormal Legendre polynomials is chosen. The stress and the displacement fields in the domain of each element and the displacements on the static boundary are independently approximated. None of the fundamental relations is locally verified and all field equations are enforced in a weighted residual form, ensuring that the discrete numerical model embodies all the relevant properties of the continuum it represents. The Mazars’ isotropic model [38] is adopted and a non-local integral formulation where the damage variable is taken as the non-local variable is considered.