Modulus of Rupture: Size Effect due to Fracture Initiation in Boundary Layer

The modulus of rupture, which characterizes the apparent tensile strength of unreinforced concrete beams, is known to depend on the size of the beam. Since no large stable growth of a crack exists before the maximum load is reached, the size effect cannot be explained by energy release due to fracture. Rather, it is explained by the fact that distributed microcracking and slips with strain softening occur in the boundary layer of the beam before the maximum load is reached. The beam has actually failed before any macroscopic cracks are formed. A simple formula describing the size effect is derived. Asymptotic analysis of the strain softening in the boundary layer reveals that the excess of the modulus of rupture over the direct tensile strength is inversely proportional to the beam depth and proportional to the thickness of the boundary layer, which is nearly proportional to the maximum aggregate size. The proposed formula agrees with existing experimental data. The formula, further generalized, describes the effect of the gradient of normal strains near the concrete surface. Also, approximate analysis of the size effect by linear elastic fracture mechanics yields similar formulas. Determining the effect of beam size on the modulus of rupture by this short formula is simpler than would be possible with a finite element solution.