Keywords: reinforced concrete beam, eight-node solid element, rigid arm, hanging node, slave degree of freedom, dynamic relaxation.

In this paper the use of the microplane model in an analysis of reinforced concrete beams is discussed. The microplane model [1], is generally a constitutive, fully three-dimensional model capable of describing concrete in its very complex behaviour. Thanks to its qualities, the microplane model is convenient for the simulation of experimental results and a comprehensive analysis of various concrete structures. In particular, the authors have focused on reinforced concrete (RC) frames and have proposed an adaptive analysis of RC frames with microplane joints. It is based on a geometrical model of frame compound of one-dimensional beams and three-dimensional elements (bricks). The microplane model is applied only on bricks, that will be used for segments of the frame with significant nonlinear strains. The rest of the frame will be modelled using beam elements with fibered cross-sections, which partly allows simulation of nonlinear behaviour of material.

To obtain an appropriate model of the three-dimensional segment of a RC beam the authors implemented into a finite element code a linear eight-node solid element with rotational degrees of freedom [2]. This element has an advantage over a common linear eight-node solid element in that it has six degrees of freedom per node. So it can be combined with both truss elements (three DOFs per node) and beam elements (six DOFs per node). Because the solution time of a commonly used nonlinear static analysis is rather large the authors also employed explicit dynamic relaxation analysis.

The adaptive analysis employes d-refinement which combines a one-dimensional and three-dimensional geometrical model of RC beams. This approach leads to a complicated finite element mesh consisting of one-dimensional (beam and truss) and three-dimensional (brick) elements and brings two problems - generation of the mesh and connection of all finite elements into one compact unit. To satisfy the first one it was necessary to develop a special preprocessor capable to generate such a composite finite element mesh. In order to achieve a true response it was necessary to provide a suitable connection of one-dimensional and three-dimensional elements. There are two types of this connection. The first connection is between one-dimensional and three-dimensional models of the RC beam (segment). The second one is realized inside a three-dimensional segment between one-dimensional elements (representing bars of reinforcement) and three-dimensional elements (representing concrete). To provide these connections the rigid arm node and hanging nodes was implemented in the finite element code. Both nodes were based on a so-called slave degree of freedom.

All above mentioned elements and computational tools and schemes was tested for a fixed-ended beam loaded with a mid-span point load.

1

Z. Bazant, F. Caner, I. Carol, M. Adley, and S. Akers, "Microplane model M4 for concrete. Part I: Formulation with work-conjugate deviatoric stress", Journal of Engineering Mechanics, 126, 944-953, September 2000. doi:10.1061/(ASCE)0733-9399(2000)126:9(944)

2

C. Choi, K. Chung, and N. Lee, "Three dimensional non-conforming 8-node solid elements with rotational degrees of freedom", Structural Engineering And Mechanics, 4, 569-586, 1996.

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