Basic Hypotheses of Fracturing Truss Model
Consider the sheared beam in Fig. 10.1.3 (which shows only the left-end portion of the beam). For the sake of simplicity, the beam is considered to have a rectangular cross section (a generalization to flanged cross sections, however, would not be difficult). The analysis of the size effect performed by Bazant (1996b) rests on the two following hypotheses:
Hypothesis I: The failure modes at maximum load of beams of different sizes are geometrically similar.
This means that, for example, the shear span s and the length c of the material failure zone at maximum load are geometrically similar (Fig. 10.3.1). In other words, the ratios sj D and c/D are assumed to be constant. The hypothesis is applicable only within a certain range of sizes. However, experience from testing as well as finite element analysis indicates that this range covers the size range of practical interest.
Hypothesis II: The maximum load is determined by the compression failure in the inclined compression struts.
The compression failure must be interpreted as a temporary incremental strain-softening in compression (or progressive crushing) of concrete in the strut, characterized by a negative slope of the stress-strain diagram. Hypothesis 11 means that the concrete in the compression strut is suffering splitting cracks in the direction of compression only during a certain, possibly short, portion of the loading history during which the applied load is reaching its maximum. It does not mean that the concrete will get crushed completely once-the load will be reduced to zero (such complete crushing is seen only in T-beams; Leonhardt 1977). During the postpeak softening, the splitting cracks may interconnect to produce what looks like compression shear cracks in the horizontal or vertical direction (Fig. 10.3.2) (however, if the failure were assumed to be caused by propagation of a horizontal or vertical shear crack across the strut, the calculation results would be the same). Thus, after the failure is completed, the failure might not look as crushed concrete but as a diagonal crack and a shear crack. The lack of complete crushing may be caused by the failure process taking place under a decreasing load, after the maximum load. The concrete in the strut may have been partially damaged by compression splitting but need not have disintegrated.
Denying the existence of progressive failure of the compression strut at maximum load would be tantamount to denying the validity of the truss model (strut-and-tic model). If this model is valid, then (1) diagonal tensile cracks must form before the maximum load, (2) the tensile and shear stresses (crackbridging or cohesive stresses) transmitted across these cracks must be negligible compared to compression stresses in the struts, and (3) (he compression struts between these cracks must be aligned in the direction of the compressive principal stress in concrete. Only under these conditions, the concrete, stirrups, and
Figure 10.3.2 Splitting crack interconnection to form horizontal or vertical compression-shear cracks: (a, b) beams without stirrups; (c – e) beams with stirrups.
longitudinal bars may be treated as a truss. Assuming that the stirrups and longitudinal bars are designed strong enough, the truss can fail only in concrete. Because the concrete is in uniaxial compression, the failure must be compression failure.
The stresses transmitted across the diagonal cracks are, of course, nonzero, because the cracks are not open widely enough at maximum load. But the important point, which justifies the truss model, is that these stresses are much smaller in magnitude than the compression stresses in the struts.
The energy release due to fracture propagation can be calculated in two ways: (1) from the change of the strain energy of the structure-load system at constant displacement, or (2) from the change of the complementary energy of the structure at constant load (see Chapter 3). We will examine both approaches in a simplified manner and show that they give approximately the same results.