Keywords: reinforced concrete, biaxial shear, stirrup design, finite element modelling, numerical analysis, parametric modelling.

Stress distributions, bearing capacities and failure of reinforced concrete beams subject to biaxial loadings are not satisfactorily researched. Few test data are available in the literature. Thus, it is suitable to investigate the complex bearing mechanisms using finite element methods. The computed shear resistances are used to verify shear design formulas [1], which are prepared for shear loads acting inclined to the principle axes of the cross sections. They base on spatial strut and tie models and the design principles of Eurocode 2 [2]. Moreover, numerical simulations open the view to the behaviour of the girders to properly understand cracking processes, strut and tie mechanisms and failure modes.

A finite element model of a three point-bending-test with inclined load applications is developed and verified to experimental data. It is variable in its properties of material, geometry and mesh generation and allows for a comfortable model generation by a parametric input file. Strength values, aspect ratios, stirrup distances as well as bar diameters and distributions of the longitudinal reinforcement are chosen in advance. 8- or 20-node solids discretise the concrete body. Bar and stirrup reinforcement - discretely modelled by hundreds of spatial truss elements - lie embedded. So, nodal displacements of the trusses are coupled to the solid host elements ("embedded modelling").

An elasto-plastic damage model idealises the material behaviour of concrete. It assumes isotropic damage [3,4] and uses a yield surface of combined Drucker-Prager and Rankine type to characterise the states of cracking, crushing or failure. Plastic flow is governed by a non-associated flow rule to achieve sufficient dilatancy. Closing or reopening of cracks as well as damage transference is taken into account by stiffness recovery or loss effects. To ensure a mesh independent formulation, crushing and fracture energies are introduced and combined with internal length parameters [5]. A uniaxial, elasto-plastic material behaviour is assumed for the reinforcement steel.

The main characteristics of biaxial shear and bending are extracted from the numerical results. Complex three-dimensional distributions of the compressive concrete stresses arise. They start from the compressive zone, branch to stiffen the stirrup's corner and finally progress diagonally to reach the most stressed reinforcement point in the tensile zone. Spatial strut and tie models idealise such distributions and lump concrete stresses in compressive struts. Tensile struts of stirrups and longitudinal bars complete the models.

Usually, deflections and shear force orientations do not comply for biaxial loadings. This misalignment even increases, if cracking occurs. Consequently, second order torsional moments arise and load redistributions take place. They significantly influence local stress concentrations. However, the overall bearing capacities are hardly affected.

1

P. Mark, "Design of reinforced concrete beams with rectangular cross sections against biaxial shear forces", Beton- Stahlbetonbau, 100(5), 370-375, 2005. doi:10.1002/best.200590092

2

ENV 1992-1-1, Eurocode2, "Design of concrete structures - part 1 general rules and rules for buildings", 1992.

3

J. Lubliner, J. Oliver, S. Oller, E. Onate, "A plastic-damage model for concrete", Int. J. Solids Structures, 25(3), 299-326, 1989. doi:10.1016/0020-7683(89)90050-4

4

J. Lee, G.L. Fenves, "Plastic-damage model for cyclic loading of concrete structures", J. Eng. Mechanics, 124(8), 892-900, 1998. doi:10.1061/(ASCE)0733-9399(1998)124:8(892)

5

W.B. Krätzig, R. Pölling, "An elasto-plastic damage model for reinforced concrete with minimum number of material parameters", Computers and Structures, 82, 1201-1215, 2004. doi:10.1016/j.compstruc.2004.03.002

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