Reinforced concrete beam failing in shear with the reinforcement as frame elements

Adding beam elements to a solid mesh is one way of modelling reinforcement in concrete. Here, the workflow is demonstrated using the Leonhardt and Walther shear beam no. 5.

Model

In this example the reinforcement in a concrete beam is modelled with beam elements which are connected to the concrete by node-to-node link elements using a bond-slip law. The same beam, mesh, concrete model and bond-slip law are used as in the previous post: Leonhardt and Walther (1962) beam no. 5, half beam, plane stress CDPM2 with crack band, element size 20 mm. The difference to the previous post is that the reinforcement is modelled with frame elements (lattice2d) instead of truss elements. This element is a Timoshenko beam element with a single integration point. The bar has axial, shear and bending stiffness and can carry dowel forces across cracks. The sectional input parameters for the beam element are given as area and moment of inertia. As there are two rebars in the beam, the area is the area of the two 26 mm bars (1061.9 mm2, radius 18.4 mm). The moment of inertia is set to that of two separate 26 mm bars (2 π 134/4 = 44863 mm4). The material response of the reinforcement is set to be elastic with a Young's modulus of E = 200 GPa.

Bond

Each bar node is duplicated. The duplicate is a hangingnode whose displacements follow the concrete triangle it lies in. Each bar node and its duplicate are joined by a link element called bondlink2dtruss, which was already used in the post in which the rebar was modelled with truss elements. In this link element, slip along the bar versus bond stress follows the Model Code 2010 bond-slip law with the same parameters as in the previous post. The lateral jump across the bar is modelled with a high stiffness as in the previous post. Rotations of the beam nodes are kept free and are indirectly restrained by the lateral nodal displacements. The bond area of each link is the length of bar assigned to the node (half of each adjacent element) times the perimeter of the two bars. Duplicate nodes and links are written by the duplicate tool of the OOFEM fork. The mesh consists of 64 links, 1262 nodes (64 of them hanging) and 2247 elements in total (concrete, steel plates, frame elements and links).

Load control

The load is controlled with indirect displacement control using the load-point displacement with 10 µm per step. 195 steps converge, at which point the load is already past the peak. Run time about 5 minutes on a laptop.

Results

The results are shown as a load-deflection curve and contour plots of the damage variable.

In the pre-peak, beam and truss reinforcement model follow the same curve up to about 1.9 mm (50 kN). At the peak, the beam model reaches 50.8 kN at 1.99 mm midspan deflection versus 58.1 kN at 2.54 mm for the truss model discussed in the previous post. The last experimental point is 60 kN at 2.57 mm. In the post-peak regime, the load decreases to 49.7 kN at 2.03 mm (step 195). The failure pattern at the peak shows that damage grows at the tip of the diagonal crack and in a crack along the top of bar, which is a similar failure mode to the model with truss elements as reinforcement.

Reproduce

To reproduce the results, the docker setup can be used.

git clone https://github.com/githubgrasp/oofem-examples.git
cd oofem-examples/cdpm2-reinforced-beam-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-beam-2d

Issues and questions can be submitted to the issue tracker.

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