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Based in the implicit-explicit integration scheme (IM-EX) proposed by Oliver ([5]), in this paper, a modified version of the IMPL-EX scheme is presented. Likewise the standard IMPL-EX scheme (see also [4]), the proposed scheme consists of using two integration scheme in the same time step: an standard implicit (Backward-Euler) scheme furnishing "implicit" values of stresses σ n+1 (ε n+1) in terms of the current strains (more details in Sect. 4) and an explicit integration scheme furnishing other "explicit" values of the stresses σ n+1 (ε n+1) (also to be detailed in Sect. 4 below). In [4] it is argued that, since the extrapolated values of the plastic multiplier ˜ n+1 are obtained in terms of the implicit integrated values, n and n−1 , one can expect that the well-known stability properties of the implicit integration procedures are being inherited by their explicit extrapolations. Likewise, the same argument can be made in favor of the proposed scheme. Another property pointed out in [4], that is automatically inherited by the Modified IMPL-EX, is the robustness (stability) of the standard IMPL-EX integration scheme. To explain that, in [4] it is said that, the IMPL-EX "can be understood as composed of a prediction stage, at the end of time step n, followed by a linear correction, characterized by the constant and positive definite operator" C tan (σ n+1), "during time step n + 1". Such property derives in an algorithmic stiffness matrix always positive definite resulting that "no numerical instabilities should be expected at any time and the robustness of the analysis should be dramatically increased". Also, it is said in [4] that "since the structural tangent stiffness matrix is now constant and well conditioned, the Newton-Raphson procedure should converge in just one iteration per time step".

This work presents a numerical model to simulate the failure behavior of slender reinforced concrete columns subjected to eccentric compression loads. Due to the significant influence of the lateral displacements on the loading state provided by an eccentric load, geometric nonlinearity is considered. The responses of the concrete in tension and compression are described by two scalar damage variables that reduce, respectively, the positive and negative effective stress tensors, which lead to two different damage surfaces that control the dimension of the elastic domain. To describe the behavior of the reinforcements, truss finite elements with elastoplastic material model are employed. Interaction between the steel bars and concrete is modeled through the use of interface finite elements with high aspect ratio and a damage model designed to describe the bond-slip behavior. The results showed that the numerical model is able to represent the nonlinear behavior of slender concrete columns with good accuracy, taking into account: formation of cracks; steel yielding; crushing of the concrete in the compressive region; and interaction between rebars and concrete. KeywordsReinforced concrete columns: nonlinear analysis; continuum damage model; interface finite element.Failure behavior modeling of slender reinforced concrete columns subjected to eccentric load INTRODUCTIONDue to the important role played by reinforced concrete (RC) columns as structural elements, there are a number of papers available in the literature investigating their behavior. In most cases, the variables investigated are the slenderness ratio, concrete strength, steel ratio, boundary conditions, type of cross-section, concrete shrinkage and creep, and eccentricity loading (Kim and Yang, 1995;Claeson and Gylltoft, 1998;Diniz and Frangopol, 2003;Elwood, 2004;Majewski et al., 2008;Pallarés et al., 2008;Leite et al., 2013;Carmona et al., 2013;Bouchaboub and Samai, 2013 After the advent of the high strength concrete (HSC) a large number of slender RC columns has been constructed, and more researchers have given special attention to this subject (Pallarés et al., 2008;Kim and Yang, 1995). Consequently, the development of numerical tools to support the analyses has increased and different numerical models to simulate the failure mechanism have been proposed. For instance, in order to investigate the behavior of RC columns under eccentric compression load, with consideration of cracks, Majewski et al. (2008) proposed an approach based on elasto-plasticity with non-local softening model. Recently, Carmona et al. (2013) developed a fracture mechanics model to describe the buckling behavior of lightly concrete columns, wherein the structural response is a result of the propagation of one single crack at the column's central region. Using the finite difference method, Bouchaboub and Samai (2013) proposed an approach to simulate the behavior of slender high strength RC columns under combined biaxial and axial compression. For all the aforementioned...

This paper presents a novel approach to the simulation of NATM tunnel construction using the Boundary Element Method (BEM) as principal numerical method. This new approach has the advantage that only the excavation surface, the possible plastic zones and the tunnel lining have to be discretised. The whole rock mass is represented by the BEM whereas the Finite Element Method (FEM) is used to represent the tunnel lining only. Thus, a general coupling strategy for coupling three-dimensional boundary elements with shell finite elements (shotcrete) and beam finite elements (steel arches) is presented. To achieve realistic results the effect of hydration of the shotcrete and yielding of the steel arches is considered in the excavation process. Furthermore, the nonlinear rock behaviour is modelled more realistically by using a powerful hierarchical constitutive model which considers a large range of rock materials. The combination of these ideas results in higher user-friendliness and efficiency. Some verification tests and practical applications in tunnelling are presented.

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