# Modelling of Diagonal Crushing of Concrete to Determine

Members with at least a minimum quantity of well-anchored stirrups are predicted not to fail by sliding on the crack, but by yielding of the stirrups and eventual crushing of the concrete in the web. In the 1984 shear provisions which allowed the engineer to select the value of в, equations were provided to ensure that the concrete did not crush before reaching the design shear strength, which provided a lower limit on в, and additional rules were provided to ensure that the stirrups would yield at design shear failure, which provided an upper limit on в. At low applied shear forces, the range over which the value of в could be selected was large, but as the shear stress increased, the range became more restrictive. For the 2004 shear provisions, it was decided to maintain the maximum shear limits that were present in the 1994 standard, so this high shear loading would control the selection of в for all applied loading levels.

Figure 7 shows the limits on allowable angle of principal compression, в, based on the MCFT for members heavily loaded in shear and for different strains in the member at mid-depth (ex). As can be seen, the range of allowable angles to select from at this high shear loading is rather narrow. Members designed based on angles in the upper shaded region would be expected to fail in shear before yielding of the transverse steel making the use of Equation (1) unconservative. Members designed based on angles from the lower shaded region would also be unconservative as here the

 Stirrups do not yield 20-70 MPa__ before shear failue ^ ^ 70 MPa 1 і / / X* Concrete Crushes before reaching desired shear stress ^ 20 MPa CSA Equation 0 = 29 t 7000ex 1 f v//t‘» 0.25
 45 43 41 39 ‘ 37 35 33 31 29 27

 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Longitudinal strain at mid-depth of member ( e, ) (x 10 3) Fig. 7. Selection of equation for 9.

member is predicted to fail by crushing of the concrete in diagonal compression before achieving the design shear strength. Only within the unshaded region would a member as heavily loaded as this be predicted to be able to resist the applied shear force. Shown in Figure 7 is a simple equation that lies within the allowable range as:

9 = 29° + 7000ex. (5)