The Size Effect in Shear


At the time Equation (3) was developed, it was not understood that the failure shear stress for members constructed without web reinforcement decreases as the member depth increases in a phenomenon known as the “size effect.” Unfortunately, the average height of the specimens tested to develop Equation (3) was 340mm. As a result, the original researchers did not notice a size effect in their tests, and the ACI expression predicts a continuous and linear increase in shear capacity as the beam depth increases. Equation (3), while originally intended to be a conservative estimate, is therefore unconservative for deeper members constructed without web reinforcement because it can not predict this size effect. The failure of the warehouse beams was attributed to an unexpected tensile force in the beams, while it is far more likely that it was simply a result of the size effect (Lubell et al., 2004).

The MCFT predicts that the size effect in shear is related to the crack spacing in the web and the crack widths. As discussed by Sherwood et al. (2004): “.. .the larger crack widths that occur in larger members reduce aggregate interlock. Crack widths increase nearly linearly both with the tensile strain in the reinforcement and with the spacing between cracks. for the same reinforcement strain, doubling the depth of the beam will double the crack widths at mid-depth. To maintain beam action, a shear stress equal to about V/bwd must be transmitted across these cracks. The shear stress that can be transmitted across such cracks, however, decreases as the crack width increases and as the maximum aggregate size decreases.” Thus, the limiting stress that can be transferred across cracks due to aggregate interlock in deep members is reached at a lower shear stress than in equivalent small beams. Once the limiting stress is reached, equilibrium cannot be maintained, and failure occurs.

The use of at least the minimum quantity of stirrups will largely eliminate the size effect by allowing more closely spaced cracks to form and by preventing the loss of aggregate interlock as the cracks widen. Web reinforcement consisting of additional longitudinal steel placed in layers along the height of a beam will also reduce the size effect. In this case, the size effect is related to the vertical spacing of the layers of reinforcement, rather than the overall height of the beam.