Keywords: reinforced concrete, circular sections, shear design, parameterised finite element modelling, continuum damage model, strut and tie model.

The shear bearing capacities of reinforced concrete (RC) beams with circular cross sections are often evaluated on the basis of formulas valid for rectangular sections. In doing so, an equivalent width is defined and the resistance of the stirrups is calculated from the notion of a uniform distribution of the shear force in two halves on both stirrup legs. Effects from the specific circular shape of the stirrups and the axisymmetric distribution of the longitudinal reinforcement are neglected. However, such effects dominate the inner load bearing behaviour, as deviation forces perpendicular to the stirrups and a non-uniform shear flow arise.

In the paper, the specific inner load bearing behaviour of RC beams is investigated, theoretically and numerically using nonlinear finite element methods, and elaborated to gain general formulas for a shear resistant design.

First, theoretical methods are applied. The inner shear flow, resolved into radial and tangential components, is derived from the equilibrium conditions of slender spatial beams and the coupling of bending moments and shear forces assuming the validity of the Bernoulli-hypothesis. It is supplemented by radial deviation forces that act inside the section plane, ensure equilibrium and interact with the inclined compressive shear struts. Integrations over the tensile zone lead to the maximum stirrup stresses that occur at the neutral axis. Stresses exceed those in rectangular sections by up to about 35%. Similarly, the stresses exceed the concrete stresses in the inclined shear struts. Thus, the resistances of the tensile and of the compressive shear strut evidently decrease. An efficiency factor is introduced to account for such decreases [1]. It is applicable to many international design codes, like it is presented for the "Standard shear design method" of Eurocode 2.

In a second step, the inner load bearing behaviour governed by the nonlinearity imposed by cracking and stress redistributions into strut and tie mechanisms is investigated by numerical methods. Hence, a finite element model of a three-point-bending test of RC beams with circular cross sections is developed. It consists of three-dimensional solid elements of the concrete body and three-dimensional truss elements discretely incorporating stirrups and longitudinal bars. The model is variable in its material properties and its basic geometric parameters, like section and bar diameters, stirrup spacing or amounts of longitudinal bars. A plasticity based continuum damage model describes the material behaviour of concrete [2,3,4]. It assumes isotropic stiffness degradations due to cracking under tensile or compressive loadings and includes stiffness recovery or stiffness loss effects if cracks close or reopen. A yielding condition of combined Drucker-Prager and Rankine type models the failure surface under multiaxial loading conditions assuming non-associated flow. The energy criteria on tensile and compressive sides and internal length parameters minimise mesh sensitivity. An elasto-plastic material law with a gradually rising plastic branch is adopted for the reinforcement.

The investigations focus on two representative beams. One beam exhibits a brittle failure, the other one a ductile behaviour with a pronounced yielding plateau. The figures of concrete and stirrup stresses show that the development of shear struts and localised yielding zones, where compressive struts and cracks cross. Distributions of compressive concrete stresses correspond to crack patterns taken from comparable experimental data. Vector plots in section plane illustrate the interaction of shear flow, deviation forces and tensile stresses in the stirrups. They agree well with the theoretically derived force flow in the circular section. Moreover, the numerically determined ultimate shear forces are used to verify the design formulas including the efficiency factor.

1

Bender, M., Mark, P., "Shear Design of Girders with circular Sections. Part 2: Girders with shear Reinforcement", Beton- und Stahlbetonbau 101(5), 322-329, 2006. doi:10.1002/best.200600477

2

Lee, J., Fenves, G.L., "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)

3

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

4

ABAQUS Theory Manual, Version 6.4, ABAQUS Inc., USA, 2003.

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