by clicking the arrows at the side of the page, or by using the toolbar.
by clicking anywhere on the page.
by dragging the page around when zoomed in.
by clicking anywhere on the page when zoomed in.
web sites or send emails by clicking on hyperlinks.
Email this page to a friend
Search this issue
Index - jump to page or section
Archive - view past issues
Concrete In Australia : June 2013
Concrete in Australia Vol 39 No 2 21 11.2 Minimum shear reinforcement e smallest reasonably ductile rebar manufactured in Australia is N10 with an ideal yield strength of 78.5 * 500 = 39.2 kN/vertical leg. AS3600 c22.214.171.124 requires maximum spacing of: • MIN(0.50*D, 300 mm) near, say within 2*D, of a support and • MIN(0.75*D, 450 mm) closer to mid-span. ese values seem reasonable. AS3600 also requires a maximum spacing across the width of the beam: MIN(D, 600 mm). is does not seem reasonable given the lower-bound solution of Figure 7. e horizontal and vertical components of the arch thrusts do have to be transmitted to the longitudinal rebars and also to the shear reinforcement respectively so there must be some connection between the bending rebars and the shear rebars. is author prefers that there be a vertical stirrup leg on every second main longitudinal rebar but only, say, within 2*D of a support. Perhaps this comes from NZS3101 or ACI-318. ere is also an AS3600 requirement that the ideal smeared yield strength of the shear reinforcement: rfy = Asv*fsy bw*s >0.35MPafor concrete upto32MPa. is is quite a low value, but perhaps unlikely to be critical if one stands by the maximum spacing limits discussed above. If the first dogleg hinge already requires less shear rebar than that meeting the minimum size and all of the maximum spacing limits in the paragraph immediately above, then there is no point in going any further. e lower-bound solution of Figure 7 is indeed interesting but there is no need to routinely calculate such solutions in every set of calculations. Just consider as many dogleg hinges as are required to get down to minimum bar-size (N10) at maximum spacing as one moves towards mid-span. 11.3 Extent of shear reinforcement zones Figure 7 summarises the tie-spacing and number of vertical legs in each of the three zones h1, h2 and h3. Clearly, the heavier reinforcement will need to extend slightly beyond zones h1 and h2 in order to have a whole, integer number of spaces between tie-sets. e implication is that h1 and h1 + h2 will be replaced by ROUNDUP(h1) and ROUNDUP(h1 + h2) where ROUNDUP() is a Microsoft Excel function. However, note AS3600 c126.96.36.199 which requires that: (1) shear reinforcement theoretically required at any point be extended D beyond that point so that, for detailing on drawings the functions become ROUNDUP(h1+D) and ROUNDUP(h1+h2+D), and (2) that the first tie be 50 mm from the face of support. Spacing at 100 mm, as calculated, may seem a little close and some designers may prefer to go to N12 ties perhaps just in the h1 zone so as to increase spacing. Also, there needs to be some vertical reinforcement above the brick pier so as to control shrinkage cracks. For a 350 mm pier, it would be convenient to provide one tie-set at 100 mm from the end of the beam leaving a spacing of 300 mm to the first tie-set in the clear-span. For larger piers, tie-sets over the pier at spacing ≤ 300 mm seems appropriate. If the support is a concrete column then the vertical bars from the column will need to be properly anchored and there is then no need for ties over the support. 12.0 CONCLUSION e author believes that these are "exact" yield-line solutions and that dogleg yield hinges and nested tied parabolic half- arches are the most natural way to explore stress in reinforced beams. He is particularly obliged to friends in Copenhagen who first brought this area to his attention while visiting there in 1983. NOTATION Asv = Total area of the vertical legs in one set of shear reinforcement (mm2). is note could but does not consider angled shear reinforcement because now quite rarely used. bw = Width of beam or web-width of T-beam (mm). C1, C2, ... = Bending compression force at hinge 1, hinge 2, ... D = Overall depth of beam (mm). d = Effective depth of beam measured to centroid tensile rebar (mm). db = Rebar diameter usually longitudinal rebars but maybe also shear reinforcement (mm). dM21 = Moment increment = M2 -- M1 (kNm). fc = Concrete cylinder strength (MPa). fsy = Yield strength of longitudinal rebars and of shear rebars (500 MPa in Australia). h = Horizontal projected length of dog-leg yield hinge (mm). h0 = Width of support (mm). h1, h2, .. = Horizontal projection of dogleg hinge 1, 2 .. (mm). h21 = Averaged horizontal projection for hinges 2 and 1 = lever-arm of vertical forces (mm). k = Neutral-axis depth ratio. kd = Depth from compressed face to the neutral-axis where strain = 0 (mm). Mu = Ideal (φ =1) moment strength (kNm). M* = Moment associated with factored design load (kNm). M1, M2, ... Moment strength at hinge 1, hinge 2, ... often controlled by anchorage (kNm). rfy = Smeared yield strength of the vertical shear reinforcement rfy=V bw*h=Asv*fsy*φV bw * s (MPa). R = Reaction (kN). s = Spacing of shear rebar tie-sets (mm). T = Bending tension force (kN). T1, T2, ... = Bending tension force at dogleg yield line 1, 2 .. V = Vertical shear force (kN) under load wu (kN/m). V1, V2 .. = Vertical shear force crossing dogleg yield line 1, 2 .. wB = at part of wu applied at the bottom of the beam. wT = at part of wu applied at the top of the beam. wu = wB + wT = Ideal (φ =1) uniformly distributed collapse load (kN/m). w* = Factored design load (kN/m). x,y = coordinates see Figure 2 φ = Capacity Reduction Factor aka Reliability Factor AS3600 Table 2.2.2. φ M = 0.8 0 Factor for ductile bending failure. φ V = 0.7 0 Factor for brittle shear failure.