3.2.12 Mixed-mode failure of a notched unreinforced concrete beam

Product: ABAQUS/Explicit  

ABAQUS/Explicit provides a cracking constitutive model (Cracking model for concrete, Section 18.5.2 of the ABAQUS Analysis User's Manual) suitable for brittle materials such as concrete. The model is intended for unreinforced as well as reinforced concrete structures, and this manual includes examples of both types of applications. The problem described here illustrates the use of this model for the analysis of an unreinforced notched concrete beam subject to loading that causes mixed-mode cracking. This problem has been chosen because it has been studied extensively both experimentally by Arrea and Ingraffea (1982) and analytically by Rots et al. (1984, 1985, 1987, 1989, 1991, 1992), de Borst (1986, 1987), and Meyer et al. (1994), among others. The behavior in this problem is a combination of Mode I and Mode II cracking. It, therefore, provides verification of the model for general mixed-mode loading. We also have the advantage that this beam experiment has been repeated by a number of different researchers, and there is good material information about important parameters such as the Mode I fracture energy, . We investigate the sensitivity of the numerical results to the finite element discretization as well as the choice of cracking material properties.

Problem description

Loading and solution control

Since ABAQUS/Explicit is a dynamic analysis program, and in this case we are interested in static solutions, care must be taken that the beam is loaded slowly enough to eliminate any significant inertia effects. For problems involving brittle failure, this is especially important since the sudden drops in load carrying capacity that normally accompany brittle behavior generally lead to increases in the kinetic energy content of the response.

The beam is loaded by applying a velocity that increases linearly from zero to 0.75 mm/second over a period of 0.38 seconds. The velocity is applied at point C and transmitted to the notched beam through the rigid beam AB. The beam itself is not modeled since its kinematic motion can easily be modeled using the *EQUATION option. The load transmitted at points D and B is distributed over a 30 mm length to avoid hourglassing of the elements in the vicinity of these points where the highest loads are transmitted. The velocity chosen ensures that a quasi-static solution is obtained. The kinetic energy in the beam is small until the crack has propagated across the entire depth of the beam. Nevertheless, oscillations in the load-displacement response caused by inertia effects are still visible, mainly after the concrete has cracked significantly.

Results and discussion

Input files

References

Table

Table 3.2.12–1 Concrete material properties.

Young's modulus:24800 N/mm2 (3.60 × 106 lb/in2)
Poisson's ratio:0.18
Cracking failure stress:2.8 N/mm2 (406.09 lb/in2)
Mode I fracture energy :0.055 N/mm (0.314 lb/in)
Density:2.4 × 10–6 kg/mm3 (0.225 × 10–3 lb s2/in4)


Figures

Figure 3.2.12–1 Notched, mixed-mode beam: geometry and dimensions.

Figure 3.2.12–2 Finite element meshes used for notched, mixed-mode concrete beam.

Figure 3.2.12–3 Tension softening model used for mesh refinement study.

Figure 3.2.12–4 Shear retention model used for mesh refinement study.

Figure 3.2.12–5 Mesh refinement study: load-CMSD responses.

Figure 3.2.12–6 Displaced shapes obtained in mesh refinement study (magnification factor 200).

Figure 3.2.12–7 Crack patterns obtained in mesh refinement study (detail of the concrete beam around its notch).

Figure 3.2.12–8 Tension softening models.

Figure 3.2.12–9 Tension softening study; fine mesh.

Figure 3.2.12–10 Shear retention models.

Figure 3.2.12–11 Shear retention study; fine mesh with two-segment tension softening.

Figure 3.2.12–12 Shear retention study; fine mesh with four-segment tension softening.

Figure 3.2.12–13 Element removal: tension softening study for plane stress fine mesh.

Figure 3.2.12–14 Element removal: plane stress fine mesh with a two-segment tension softening curve.