Some aspects on three-dimensional numerical modelling of reinforced concrete structures using the ®nite element method H.M. Gomes, A.M. Awruch * Graduate Program in Civil Engineering, Federal University of Rio Grande do Sul, Av. Osvaldo Aranha, 99, 38 andar-90035-190, Porto Alegre, RS, Brazil Received 4 November 1999; revised 18 August 2000; accepted 23 August 2000 Abstract This paper deals with some aspects related to three-dimensional numerical modelling of reinforced concrete structures using the Finite Element Method (FEM). Some subjects such as the solution technique of the non-linear equilibrium equations and the constitutive model for concrete and reinforcement steel are emphasised and commented. A robust method for the evaluation of the intersecting points of the embedded reinforcement bars into the three-dimensional ®nite element mesh is also presented. The main advantages of the Generalised Displacement Control Method with the Generalised Displacement Parameter to improve the response of the concrete and reinforced concrete analyses are highlighted. Finally, a series of numerical examples related to the above-mentioned aspects are presented. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Reinforced concrete; Structural analysis; Finite elements 1. Introduction In the last three decades, a large number of numerical models for reinforced concrete analysis have emerged. They met fairly well the needs for which they were conceived. These models deal with the behaviour of concrete structures till a certain limit given by the service load or prior to the ultimate load. Lately, much attention has been given to some model re®nements with the aim to eval- uate more accurately the post-peak behaviour. For this purpose, some progresses in material constitutive modelling and in numerical methods to solve the non-linear equili- brium equations have been obtained. With respect to the last aspect, several control methods have been employed to capture the post-peak behaviour of the structure. However, the generalised displacement control method [24] (usually applied to geometrically non-linear analysis) and the quasi-Newton methods, such as the BFGS [10] (due to Broyden±Fletcher±Goldfarb±Shanno), seem to be the best candidates for the complete evaluation of the load displacements curves for structures with geometrically and/or material non-linear behaviours. Use of the inverse mapping technique improved the steel reinforcement modelling, because using this technique, the reinforcement positions are independent of the FE mesh and one may use irregular or distorted meshes. However, problems like bond between concrete and reinforcement bars are not completely solved, especially when using the reinforcement bar embedding model. 1.1. Concrete material modelling The concrete is a material, which presents a different behaviour in tension and compression. For a uniaxial compression curve, at approximately 0.3f c , where f c is the compression peak stress, the concrete changes from a linear elastic behaviour to a non-linear behaviour, originating the ®rst microcracks at this stage. This value is usually proposed as the elastic limit. At approximately 0.75f c , the process of unstable crack propagation starts. After this level, the beha- viour is quite non-linear, with a progressive collapse and loss of the load capacity of the concrete to resist stress increments till the complete specimen rupture. For a uni- axial tension stress±elongation curve test, a brittle material behaviour is observed. The elastic limit is observed at approximately 0.8f t where f t is the tensile peak stress. As in compression, there is a progressive loss of the load capa- city, which depends on factors such as material properties and dimensions of the tested specimen. For the non-linear concrete behaviour description, an elasto-viscoplastic formulation with a smeared crack approach, which is a slight modi®cation of Chen's Model [10] is used in this work. A description of the model is Advances in Engineering Software 32 (2001) 257±277 0965-9978/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S0965-9978(00)00093-4 www.elsevier.com/locate/advengsoft * Corresponding author. Tel.: 155-51-316-3587; fax: 155-51-227-1807. E-mail address: awruch@adufrgs.ufrgs.br (A.M. Awruch).