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Concrete In Australia : March 2014
32 Concrete in Australia Vol 40 No 1 COVER STORY: SHEAR DESIGN used to predict the shear stress distribution over the depth of the beam and the complete load-deformation response of concrete sections subjected to shear, flexure and axial load; see Figure 10. If only the shear strength of a beam cross-section is required then the web of the beam can be approximated by just one biaxial element located at mid-depth and the shear stress on the element can be assumed to be V/(bwd v ) where bw is the web width and dv is the flexural lever arm which can be taken as 0.9d. The longitudinal strain, εx , at mid-depth of the beam can be found from the calculated strain in the longitudinal flexural reinforcement and the assumption that plane sections remain plane. For a given value of εx the failure shear stress can then be calculated from the MCFT as the sum of two terms, V c and Vs , see Figure 10. This simplified MCFT 24, 25 sectional design model for shear is the method used in the current Canadian Standards Association (CSA) document “Design of Concrete Structures” A23.3 -044. Based on the MCFT, the CSA shear provisions provide a set of equations that depend on geometric terms, material properties, and a measure of the average longitudinal strain at mid-depth of the beam at shear failure. The reliable shear strength (Vr ) of a beam with partial safety factors ( fc =0.65, fs = 0.85) can be taken as: cot 25 . 0 v y v s v w c c r v w c c s c r d f s A d b f V d b f V V V (1) where, b w is the web width (mm), d v is the shear depth taken as 0.9d (mm), f c ’ is the concrete cylinder strength (MPa), A v is the area of shear reinforcement (mm2) at a spacing of s (mm) and a yield strength of fy (MPa). For higher strength concretes the square root of fc ’ in the Vc term should not be taken as greater than 8.0 MPa. Two parameters are needed to solve this equation, the first, β, is a measure of the ability of the cracked reinforced concrete to transmit shear stresses across cracks. The second, θ , indicates the direction of principal compression and governs the contribution of the stirrups. Derived from the MCFT equations, the following equation is included in the code to determine β: xe x s 1000 1300 1500 1 40 . 0 (2) where εx is a measure of the average longitudinal strain at mid- depth of the section under the combined loading, see below. The term sxe defines the effective crack spacing of the member and is taken as 300 mm for members with at least minimum stirrups or dv = 0.9d for other members with normal strength concrete and 19 mm coarse aggregate size. In general, s xe is given as: v ge v xe d a d s 85 . 0 16 35 (3) Figure 7: The Modified Compression Field Theory (MCFT). Figure 8: Shell element tester with 60 computer controlled actuators. strength of the elements in these zones making it probable that the shear failure will occur outside of these zones. For beams with short shear spans the zones with significant clamping stresses will overlap and the shear strength of the beam will be considerably increased. It is important to recognise that in these “disturbed regions” the shear stress distribution over the depth of the beam is influenced by the distribution of the clamping stresses and near the loads and reactions, plane sections do not remain plane. Outside of the disturbed regions it is appropriate to assume that plane sections remain plane and that the clamping stresses are negligible. With these two assumptions a beam cross-section can be modeled as a stack of biaxially stressed elements with the response of each element being predicted by the MCFT. This is the basis of program Response-2000 22, 23 which can be 28-38 - Cover story.indd 32 28-38 - Cover story.indd 32 28/01/14 3:09 PM 28/01/14 3:09 PM