Reinforced concrete beam failing in shear with bond-slip and CDPM2 in plane stress

Bond-slip is important for modelling the bending stiffness and crack spacing in reinforced concrete beams. Here, a reinforced beam failing in shear is modelled with a bond-slip law between reinforcement and concrete.

This example on how to model bond-slip in 2D is an extension of a previous example of a reinforced beam failing in shear modelled with perfect bond see the previous post. The same beam, mesh and concrete model are used. It is Leonhardt and Walther (1962) beam no. 5 without shear reinforcement. Symmetry at midspan is used so that only half of the beam is modelled. Concrete is modelled with the plane stress CDPM2 version with a crack band model. The input parameters are determined from the following concrete properties: fc = 28.5 MPa, ft = 2.25 MPa, G_F = 133 N/m, E = 30.5 GPa. Element size is 20 mm in the refined region. The reinforcement is modelled as truss elements. The 2 bars of 26 mm in the experiments are lumped into one truss cross-section positioned at 50 mm above the bottom face. The reinforcement is assumed to be elastic with E = 200 GPa, because it is known from the experiments that the beam fails in shear before the reinforcement reaches a stress level that could result in yielding. The main difference to the previous post is that perfect bond is replaced by bond-slip between truss elements and concrete.

Bond-slip modelling

The reinforcement is meshed in T3D as a free line, independent of the concrete background mesh. Next, each bar node is duplicated. The duplicated nodes are hanging nodes whose displacements are interpolated from the concrete elements in which they lie. In OOFEM, the concrete elements are automatically identified if the node is declared to be a hanging node. Each truss node is connected to the duplicated node by a link element, called bondlink2dtruss. In the link element, the relative displacements of the two nodes (truss node and hanging node) are split into slip along the bar and a lateral component across the bar. The component along the bar is used in a bond-slip law and the lateral component is constrained by a penalty stiffness. The bond area of each link is the length of bar assigned to the node (half of each adjacent truss element) times the perimeter. The duplicate nodes and links are written by the duplicate tool (tools/duplicate in the OOFEM fork), which has modes for 2D and 3D. For this coarse mesh, there are 64 bond links, 1262 nodes and 2247 triangular elements.

Bond-slip law

The bond-slip law follows the Model Code 2010 assuming good bond conditions for a characteristic compressive strength of 20.5 MPa. The bond strength is t0 = 2.5 sqrt(fck) = 11.3 MPa. The three slip thresholds are s1 = 1, s2 = 2 and s3 = 8.5 mm (assumed clear rib spacing). The shape factor is alpha = 0.4 and the residual bond stress is tf = 0.4 t0 = 4.5 MPa. The maximum slip in the analysis is only about 0.1 mm, which is still on the ascending branch of the bond law (s1 = 1 mm).

Load control

The load control is indirect displacement control. The displacement components to control the analysis are determined iteratively. Initially an analysis with direct displacement control is applied which shows at a certain load level the load drops suddenly. The step after this drop is not converged. The deformed shapes of the beam before and after the drop are used to choose the DOFs which should control the analysis. The deformed shapes reveal that the drop occurs because a crack forms in the horizontal direction along the reinforcement. In metastep 1 (244 steps), the load-point displacement is used with 10 µm per step up to the peak. In metastep 2 (100 steps), the vertical relative displacement of two nodes on either side of the horizontal crack is used at 0.1 µm per step. The run time of the analysis is about 8 minutes on a laptop. All steps converge.

Results

The results are shown in form of a load displacement curve and contour plots of the damage variable on a deformed mesh magnified by a factor of 20. The load displacement curve is compared to experimental results and the curve obtained from the perfect bond example see the previous post.

In the pre-peak, the bond-slip curve is close to the experimental points at 30, 40 and 50 kN. In comparison the perfect bond is stiffer after flexural cracking. The peak force of 58.1 kN at 2.54 mm midspan deflection is below the perfect bond case of 62.3 kN at 2.36 mm, and also below the experimental value of 60 kN at 2.57 mm. During snapback (both load and deflection decrease) a crack along the reinforcement starts to develop. This is the mechanism that was identified in the initial run and was used here to control the displacements so that the snapback can be traced.

Reproduce

git clone https://github.com/githubgrasp/oofem-examples.git
cd oofem-examples/cdpm2-bondslip-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-bondslip-2d; issues and questions go on the issue tracker.

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