Modelling Aggregate Interlock to Determine
Aggregate interlock is assumed to carry all shear forces for members without stirrups, and a potentially significant proportion of the total shear force for members with stirrups. The ability of a crack to resist these stresses is predicted by the MCFT to decrease with decreasing concrete strength; decreasing maximum specified coarse aggregate size, representing crack roughness; and increasing absolute crack width. The suggestion that wider cracks are less able to resist sliding shear stresses is hopefully intuitive. To estimate a crack width, w, the following relationship can be used:
w = є ■ s, (2)
where s is the crack spacing and є is the average strain perpendicular to the crack. As wider cracks will be associated with lower aggregate interlock strength, anything that increases the value of w can be expected to result in decreased shear capacity. Thus if the crack spacing increases due to, say, the construction of a larger member, it can be expected that the shear strength will decrease. This is called the size effect and is an important part of the behavior of members without stirrups. Secondly, if the average strain in the concrete increases due to, say, applied tension then the shear strength is also predicted to drop. This is called the strain effect in shear and is less well known than the size effect, though it is of comparable importance.
Fig. 5. Crack pattern in member without stirrups. ^ is spacing at mid-depth.
Overall, then, a size effect and a strain effect are predicted to be important aspects of the concrete component of shear strength of members. Experiments show that these are indeed the two most important aspects influencing shear stress at failure and should be included in any state of the art shear provisions.
The full derivation of the proposed method is given elsewhere (Bentz, 2006; Bentz et al., 2006), but the important concepts are explained here. Determining shear strength will depend on the terms s and є in Equation (2). The value of the crack spacing will depend largely on the size of the member. The crack spacing in the longitudinal direction, sz, is taken as sz = jd = 0.9d if no stirrups are provided. If the member is constructed with an aggregate size, ag, different from 20 mm, the aggregate interlock capacity will be affected, and this is accounted for by using an effective crack spacing, sze, given by:
sze = —— > 0.85dv. (3)
15 + ag
For high strength concrete, the aggregate fractures and does not contribute to crack roughness. To account for this, take ag = 0 for f > 70 MPa. To avoid a discontinuity, linearly interpolate ag from the specified value at f = 60 MPa down to zero at f = 70 MPa. For members with stirrups, the stirrups will control the crack spacing and the term sze may be simply taken as 300 mm.
Figure 5 shows the test of a 300 mm wide strip taken from a 1500 mm (5 foot) thick slab. It can be seen that the spacing of the cracks at the mid-depth of the member is much greater than the spacing of the cracks at the flexural tension face. The parameter sz refers to the longitudinal spacing of the cracks at mid-depth of the member where the shear stress is generally critical.
The value of є is slightly more complex to determine compared to the effective crack spacing as it depends on the currently applied load level, amount of prestress, material properties of the flexural reinforcement, etc. Consider that a given amount of applied load will be associated with a given strain in the longitudinal reinforcement based on a free body diagram such as that in Figure 4. This value of strain will be present in the reinforcement, whereas Equation (2) requires the strain at 90° to the diagonal crack. As such, the equations of the MCFT are employed to derive a relationship between the width of a diagonal crack, for a given crack spacing, at shear failure given that the longitudinal strain is a known quantity. This involves the simultaneous solution of 15 nonlinear equations and is described elsewhere (Bentz, 2006).
Figure 6 shows the results of the calculation of diagonal crack width for various longitudinal strains by the MCFT. As the longitudinal strain increases, the critical crack width also increases. The analysis results are nonlinear, but a simplified equation is also shown that conservatively approximates the nonlinear behaviour. This equation is intentionally selected to provide a good match to the MCFT for mid-depth strains that are expected for a member reinforced with 400 MPa flexural
reinforcement. If a member is to be subjected to larger longitudinal strains, say with FRP reinforcement, these provisions should be conservative as the crack width will be overestimated by the simplified equation.
When the simplified equation in the figure is substituted into the MCFT equation for aggregate interlock, and a size effect terms is also added (Bentz, 2006), the following equation is obtained for the value of в which defines the concrete contribution based on the MCFT:
The first term in this equation accounts for the strain effect whereby members with smaller longitudinal strains are stronger in shear. The second term accounts for the size effect which cancels out for members with stirrups (sze = 300 mm). The estimation of the longitudinal strain in the member, ex, will be discussed below.