A Modern Method of Shear Design
A considerable step forward in shear design methods was the development of a general method of shear design based on the MCFT. Design methods based on the MCFT have a firm theoretical base and are not derived by empirical curve fits to experimental data. As such, MCFT-based shear provisions are able to predict the behaviour of reinforced concrete elements in shear where no experimental data is available. Particular strengths of MCFT-based shear design methods include the ability to accurately predict the size and aggregate effects.
The recently developed SMCFT (Bentz et al., 2005) is based on the methods in the AASHTO-LRFD and the 1994 CSA Standards, but has been considerably simplified. Simple expressions have been developed for P, the crack angle, 0 and the longitudinal strain in the web, sx, thereby eliminating the need to iterate to solve for these values.
The SMCFT employs the following relationship to determine the shear resistance of a concrete section:
V = Vc + Vs =pfbwdv + – A~f^dv cot0 (4)
The term P in Equation (4) is a parameter that models the ability of cracked concrete to transfer shear. It is a function of 1) the longitudinal strain at the mid-depth of the web, sx, 2) the crack spacing at the middepth of the web and 3) the maximum coarse aggregate size, ag. It is calculated using an expression that consists of a strain effect term and a size effect term:
The longitudinal strain at the mid-depth of a beam web is conservatively assumed to be equal to one-half the strain in the longitudinal tensile reinforcing steel. For sections that are neither prestressed nor subjected to axial loads, sx is calculated by:
The effect of the crack spacing at the beam mid-depth is accounted for by use of a crack spacing parameter, sz. This crack spacing parameter is equal to the smaller of either the flexural lever arm (dv=0.9d or 0.72h, whichever is smaller) or the maximum distance between layers of longitudinal crack control steel distributed along the height of the web. To be effective, the area of the crack control steel in a particular layer must be greater than 0.003bwsz.
The term sze is referred to as an “equivalent crack spacing factor” and has been developed to model the effects of different maximum aggregate size on the shear strength of concrete sections by modifying the crack spacing parameter. For concrete sections with less than the minimum quantity of transverse reinforcement and constructed with a maximum aggregate size of 20mm, sze is equal to sz. For concrete with a maximum aggregate size other than 20mm, sze is calculated as follows:
sze * 0.85sz (7)
15 + ag
To account for aggregate fracturing at high concrete strengths, an effective maximum aggregate size is calculated by linearly reducing ag to zero as f’c increases from 60 to 70MPa. The term ag is equal to zero if f’c is greater than 70MPa. The square root of the concrete strength is limited to a maximum of 8MPa.
Since specimens with transverse reinforcement do not exhibit a size effect, sze is set equal to 300mm for specimens with at least the minimum quantity of stirrups as per Equation (8). This has the effect of reducing the size effect term to 1.
The angle of inclination of the cracks at the beam mid-depth, 0, is calculated by the following equation:
0 = (29° + 7000sx)(0.88 + sze/2500) < 75° (9)