Reinforced concrete beam failing in shear: CDPM2 in plane stress with embedded reinforcement

In OOFEM, the reinforcement does not have to follow the concrete mesh. Here, the bars are truss elements whose nodes hang inside the plane stress CDPM2 concrete elements. Applied to a beam failing in shear, the analysis follows the snap-back after the peak.

An example of CDPM2 applied to reinforced concrete

One simple way of modelling reinforced concrete in OOFEM is to put truss elements across the mesh that models the concrete. Here we demonstrate it for beam no. 5 of the shear test by Leonhardt and Walther (1962). This is a four-point bending test without shear reinforcement, which was designed to fail in shear. The beam is 320 mm deep and 190 mm thick with a shear span of 810 mm from support to load. The longitudinal reinforcement consists of two 26 mm bars placed 50 mm above the bottom face (bar axis), which gives a concrete cover of 37 mm. We modelled this beam previously in Xenos and Grassl (2016) with a plane strain version of CDPM2. Here, we use the plane stress implementation of CDPM2 with a crack band approach that was presented in the previous post. The material input used for CDPM2 is the same as in the previous study. The main concrete parameters are fc = 28.5 MPa, ft = 2.25 MPa, G_F = 133 N/m and E = 30.5 GPa. Here, E, ft and G_F are determined from Model Code 2010. The element size in the refined region is 20 mm. Only one mesh is considered. Mesh size dependence was studied in detail in the 2016 paper.

Meshing the reinforcement

In T3D, the mesher used here, the reinforcement is placed as a line across the concrete domain. The line is not attached to any of the outline curves and not linked to the concrete patch. When T3D meshes the domain, it divides the rebar into line elements independently from the division of the domain into triangles. Line elements are converted into truss elements in the OOFEM input file. At the start of the OOFEM analysis, nodes of the truss elements are made hanging nodes in the triangular elements in which they are located. The displacements of each hanging reinforcement node are interpolated from the displacements of the nodes of the element in which the hanging node is located. The advantage of this approach is that the concrete element does not have to follow the reinforcement arrangement, so that complex reinforcement arrangements (bends, loops, stirrups) can be modelled.

Loading control

The analysis uses indirect displacement control (arc length with selected DOFs) with two metasteps. In metastep 1, the load-point displacement increases by 10 µm per step up to just before the peak. After the peak, the load drops while the load-point and midspan displacements decrease (snap-back). With load-point control alone, the snap-back cannot be modelled. Therefore, in metastep 2, a weighted sum of the vertical load-point displacement and the relative displacement across the region in which the crack opens is used. With this control, the analysis converges well and runs in about 4 minutes on a laptop.

Results

The results are shown in the form of load versus midspan deflection. The peak load at the load point is 62.3 kN at a midspan deflection of 2.36 mm. In comparison, the last recorded experimental point is 60 kN at 2.57 mm. The analysis stops at about 48 kN during softening.

Reproduce

git clone https://github.com/githubgrasp/oofem-examples.git
cd oofem-examples/cdpm2-reinforced-2d
docker run --rm -v "$PWD":/work ghcr.io/githubgrasp/oofem-public:latest bash run.sh

The example folder is at github.com/githubgrasp/oofem-examples/tree/main/cdpm2-reinforced-2d; issues and questions go on the issue tracker.

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