Fracturing Truss Model: Size Effect in Shear Failure of Reinforced Concrete

Abstract

The classical truss model (or strut-and-tie model) for shear failure of reinforced concrete beams is modified to describe fracture phenomena during failure. The failure is assumed to be caused by propagation of a compression fracture across the concrete strut during the portion of the loading history in which the maximum load is reached. The compression fracture may consist of a band of splitting cracks that later interconnect to form a shear crack or a shear fracture band inclined to the strut. The width of the fracture band is assumed to occupy only a portion of the strut length and to represent a fixed material property independent of the beam depth. The energy release from the truss is calculated using two alternative approximate methods: (1) using the potential energy change deduced from the concept of stress relief zones; and (2) using the complementary energy change due to stress redistribution caused by propagation of the fracture band across the compressed concrete strut. Both approaches show that a size effect on the nominal strength of shear failure must exist and that it should approximately follow the size effect law proposed by BaZant in 1984. The physical mechanism of the size effect is also explained in a clear and simple intuitive manner. Finally, it is shown that the applied nominal shear stress that causes large initial diagonal cracks to form also exhibits a size effect.

Scaling of Structural Failure

“…in which 9 is a dimensionless function expressed in terms of functions I and r and their derivatives (Baiant 1996). For fracture situations of positive geometry (increasing g), which is the usual case, the plot of function 9 at constant relative notch length ao looks roughly as shown in Fig 8. This function has the meaning of the dimensionless energy release rate modified according to the R-curve.…”

Can Stirrups Suppress Size Effect on Shear Strength of RC Beams?

“…But this can be true only as long as the stress in the imagined "compression strut" is much less than the compressive strength of concrete, f 0 c . If the deep beam fails by compression crushing of concrete, the compressive strength of the "strut" exhibits a strong size effect (Bažant 1997). This size effect, well evidenced by tests (Walraven and Lehwalter 1994;Tan and Cheng 2006), is yet to be incorporated into design codes, as well as into the usage of strut-and-tie model.…”

“…Compared with the deep beams, the failure mechanism of slender beam is more complicated and less understood (e.g., ACI-ASCE 1998). Although fracture mechanics of a modified classical strut-and-tie model can explain the size effect in concrete beams with stirrups if the energy release caused by the growth of a compression crushing band in the compression strut is taken into account (Bažant 1997), the inclination and the effective cross section of the strut cannot be determined easily, which takes away the usefulness for prediction. The experimental, as well as computational, evidence (Frosch 2000;Angelakos et al 2001;Tompos and Frosch 2002;Bažant and Yu 2005b) shows that the compressive resultant in the concrete ligament between the diagonal crack tip and the top face in slender beams has a steeper slope than what is assumed in the simple, classical version of the strut-and-tie model (Lampert and Thürlimann 1969).…”

Scaling of structural failure

“…For the case of failure after a large stable crack growth, the matching of the large size and small size asymptotic expansions for the fractal fracture yields, instead of Eq. (ll), the result: The hypothesis that the fracture propagation is fractal has been made and the consequences have been deduced (Bajant, 1997). Now, by judging the consequences we may decide whether the hypothesis was correct.…”