Notched unreinforced concrete beam under 3-point bending

The three finite element meshes described earlier are used to show the influence of mesh refinement on the load-displacement response of the concrete beam.

Since there is no reinforcement in this problem, the postfailure behavior is specified in terms of the stress-displacement response to minimize mesh sensitivity. We can also specify the postfailure behavior directly in terms of the fracture energy, $G_f^I$. The fracture energy method assumes a linear loss of strength after cracking. Thus, if we specify the tension softening behavior in terms of stress versus cracking displacement and assume a linear curve (${\sigma }_{tu}^I$, 0), (0, $2G_f^I$/${\sigma }_{tu}^I$) as shown in Figure 3, the above two methods will give the same results. Tensile damage is specified in terms of the tension damage variable, $d_t$, versus the cracking displacement. A linear dependence—(0, 0), (0.9, $2G_f^I/{\sigma }_{tu}^I$)—is assumed for this study, as shown in Figure 4. For the constitutive calculations, Abaqus automatically converts the cracking displacement values to “plastic” displacement values using the relationship

where the specimen length, $l_0$, is assumed to be one unit; (i.e., $l_0=1$). Care must be taken in specifying the tension damage to ensure that the calculated plastic strain (or displacement) is positive and monotonically increasing with increasing cracking strain (or displacement).

The load-displacement response of the notched beam obtained for the three meshes with Abaqus/Standard is shown in Figure 5 for the three-dimensional models and in Figure 6 for the plane stress models. The load-displacement response obtained with Abaqus/Explicit is shown in Figure 7 for the three-dimensional models and in Figure 8 for the plane stress models. These figures show that the three-dimensional and plane stress models in Abaqus/Standard are in close agreement. Minor differences are observed in the results obtained with Abaqus/Explicit; these can be attributed primarily to dynamic effects. Three-dimensional models have a relatively higher level of mesh sensitivity due to the effect of possible cracking in the out-of-plane direction. For the two-dimensional models, although a small amount of mesh sensitivity remains between the coarse mesh and the other two meshes, the medium and fine meshes give similar results. Based on these observations, all subsequent studies are done using the plane stress medium mesh. All the curves shown are smoothed. Displaced shapes obtained with Abaqus/Standard for the three plane stress meshes are shown in Figure 9. The three-dimensional meshes and the Abaqus/Explicit meshes show essentially the same deformation. The expected Mode I fracture pattern is observed consistently in all meshes.

The results described above are obtained using linear tension softening. Such a choice of softening leads to a response that is too stiff compared with the experimental observations of Petersson. In this study we use three different evolutions of the stress as a function of the cracking displacement. We compare the linear variation used previously to two tension softening functions where the cracking stress is reduced more rapidly as the crack initiates. These functions are shown in Figure 10: one consists of a two-segment representation of softening, and the other is a four-segment representation. The area under the softening curve is the same in all cases so that the value of the Mode I fracture energy of the material is preserved. Different linear tension damage curves are used for each tension softening model in this study to ensure that the plastic displacement is positive and monotonically increasing with increasing cracking displacement for all three tension softening curves.

The load-displacement responses obtained with Abaqus/Standard for the three tension softening representations are shown in Figure 11 for the plane stress medium mesh. For Abaqus/Explicit the responses are shown in Figure 12 for the plane stress medium mesh. It is clear that more rapid reductions of the cracking stress after initial cracking lead to less stiff responses. The modeling of tension softening is a key determinant of the peak/failure response. The two-segment and four-segment softening models provide peak/failure responses that agree well with the experimental observations of Petersson. The initial linear responses of the calculated results are slightly softer than the experimental results. This small difference is because a relatively blunt notch is used in this study, while a much sharper cast notch was used in Petersson (1981). All the curves shown have been smoothed.

The quasi-static solutions obtained in the previous Abaqus/Explicit studies still show some oscillations due to inertia effects, albeit somewhat hidden by the fact that curve smoothing is used. This additional exercise is intended to show the difference between the unsmoothed and smoothed responses obtained at the loading speed used thus far (0.06 m/s) and an analysis where the loading is applied at a much lower speed (0.005 m/s).

Figure 13 shows the results obtained for the plane stress medium mesh with four-segment tension softening. Smoothing of the faster load-displacement response (19635 analysis increments) is shown to match reasonably well the load-displacement response obtained at the slower speed (235830 analysis increments). Since the slower response does not provide much more useful information, we conclude that we are justified to run at the faster speed and to use smoothing to present the quasi-static response.