Gustafsson (1985) and Gustafsson and llilierborg (1988) performed approximate analysis of diagonal shear failure of reinforced beams of various depths. In their analysis, they assumed that a single polygonal cohesive crack with linear softening was formed as depicted in Fig. 10.2.5a, while the bulk of the concrete remained linear elastic. The interaction between steel and concrete was represented by the curve of shear stress vs. bond slip displacement, which was assumed to be of the clastic-perfectly plastic type (Fig. 10.2.5b). The behavior of the reinforcing steel was assumed to be linear elastic all the time. To determine the strength of the beam, 5 possible crack paths, as shown in Fig. 10.2.5c, were analyzed using finite elements. A typical, albeit idealized, load-displacement curve is shown in Fig. 10.2.5d. There is a maximum M caused by the failure of the concrete in tension followed, eventually, by a snapback. However, since in the computation the material surrounding the crack is assumed to be linear elastic, the load starts to increase again due to progressive stressing of the reinforcement. If the material behavior were really elastic, the load would increase forever along the dashed curve, approaching an asymptote (dash-dot line) corresponding to a fully cracked concrete sewed up by the reinforcement.
Of course, this is not actually possible, and failure does occur either by yielding of reinforcement or by crushing of concrete in the compressed ligament. In the analysis of Gustafsson and Hillerborg, only the crushing of concrete is considered. To this end, at each crack growth step Gustafsson and Hillerborg make a check of the integrity of the ligament based on the criterion described below. The computation ends at a certain point C in Fig. 10.2.5d when the crushing criterion is satisfied. Gustafsson and Hillerborg found that point C lies above point M for the cracks closer to the loading cross-section (path 1) and goes down as the path deviates more and more from the vertical. Fig. 10.2.5e sketches the values of the load
Figure 10.2.5 Gustafsson-Hillerborg model, (a) Geometry of the problem with polygonal cohesive crack path, (b) Bond stress-slip relationship, (c) Crack paths considered in the calculation, (d) Idealized load-displacement curve, (e) Cracking and crushing load vs, crack path mouth position and definition of beam strength. (0 Normal and shear forces across the ligament, (g) Crushing criterion. (Adapted from Gustafsson 1985.)
corresponding to points M (circles) and C (crosses) vs. the relative position of the crack mouth x/D (see Fig. 10.2.5a). Since the strength for a given path is given by the upper branch (heavy line), Gustafsson and Hillerborg assumed that the actual strength of the beam corresponds to the path with less strength, given by point A in the figure. For this path, the loads for point M and C are identical (identical cracking and crushing strength).
A few words regarding the crushing failure criterion. At each step in the calculation, the resultant normal and shear forces(A’q, Tq) across the uncracked ligament were computed (Fig. 10.2.5f). From them and the equilibrium condition, the normal and shear forces (JV, T) across any plane at an arbitrary angle could be computed, as well as the corresponding average stresses (tf, r). Gustafsson and Petersson postulated that crushing failure occurred as soon as, for some orientation, a criterion defined by a condition F(ef, r) = 0 was met, where the criterion was graphically defined as depicted in Fig. 10.2.5g.
Using the foregoing approach, Gustafsson and Hillerborg analyzed the influence of the size (beam depth), the steel ratio p, and the shear span ratio s/D. Fig. 10.2.6 summarizes the results of their computations. Although Gustafsson and Hillerborg proposed a size effect in which vu oc D~1/14 for 0.4 < D/£ch. < 5, as suggested by other researchers, it appears that an exponent of —0.3 instead of –0.25 fits the results better (dashed lines in Fig. 10.2.6).