Simplified Cohesive Crack Models
In the literature, there are two slightly different approaches that use the cohesive crack model in a simplified form that avoids finite element computations. One model was put forward by the research group at Aalborg University (Ulfkjmr, Brincker and Krenk 1990; Ulfkjter et al. 1994) and the other by the group at the University of New Mexico (Gerstle et al. 1992).
The basic idea is to describe the fictitious crack as a smeared crack of width h as shown in Fig. 10.4.6a. It is further assumed, based on the hypothesis of plane cross sections remaining plane, that the strain distribution along the beam depth is linear (Fig. 10.4.6b). Then, assuming a linear softening curve, the stress distribution can be computed as sketched in Fig. 10.4.6c. The stress-strain curve in tension is completely determined by the tensile strength and the critical crack opening wc, as shown in Fig. 10.4.6d. The stress-strain curve in compression is assumed to be linear all the way to complete fracture. The essential difference between the two approaches is that Ulfkjter ct al. consider a smeared band width proportional to the size, h — yD, while Gerstle et al. consider a width twice the instantaneous cohesive crack length (h — 2y, see Fig. 10.4.6).
Although the analytical approaches vary, the essential steps are the same: (l)wrile the strain distribution as a function of the curvature к and the position of the neutral axis x (2) use the stress-strain curves for concrete and steel to express the stress distribution and the steel force as a function of к and x (this includes the determination of y) (3) write the equation of equilibrium of forces and solve for ж as a function of re; (4) write the equation of equilibrium of moments; (5) finally, from the three equations deduced in (2)-(4), solve for y, x, and M for any given к. Obviously, the system of equations depends on the load level because of the discontinuity in the derivatives of the stress-strain curves for concrete and steel.
As pointed out before, the smearing band width assumed in these models is different. Ulfkjaer et al. assume h = yD, with у = 0.5. This value of у was based on comparisons of the load-displacement (moment-curvature) diagrams for unrei nforced beams predicted by the approximate model and by accurate finite element computations. Gerstle etal. assume apriori that h = 2y and verify that, foranunreinforced
Figure 10.4.6 Simplifi&l cohesive crack models, (a) Smearing band, (b) Strain distribution, (c) Stress distribution, (d) Approximate stress-strain curve, (e) Experimental and theoretical nominal stress-rotation curves computed by Ulfkjseret al. (1994). (f) Theoretical nominal stress-cohesive crack length curves computed according to the model by Gerstle et al. (1992).
beam, the moment vs. cohesive crack length curve predicted by their model is reasonably close to the curve obtained by finite elements. However, this verification has been done only for the ascending part of the curve and only for one size. Further validation is necessary.
Secondary differences between these models are that, in the formulation ofUlfkjteret al., the steel cover is arbitrary and the steel is allowed to yield in a perfectly plastic manner, while Gerstle et al. consider only a vanishing cover thickness to obtain simpler expressions and assume the steel to be always elastic, as is manifested by the increasing load value at the tail of the curves in Fig 10.4.6f.