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Concrete In Australia : June 2013
18 Concrete in Australia Vol 39 No 2 TECHNICAL NOTE also appropriate if the tensile strength of the main rebar varies along the length of the rebar because limited by anchorage considerations. One should be clear that the dogleg hinge in Figure 2 is quite definitely a bending hinge. It is the same as a normal plane- section perpendicular (Whitney, 1940) hinge except that an angular kink or "dogleg" at the rotation centre is allowed so that the hinge is no longer in a single perpendicular plane. It follows that the principal directions for each leg of the hinge, bending-compression and bending-tension, are parallel and perpendicular to the local yield-leg. is is so the bending- compression forces in the vertical upper bending-compression yield-leg are definitely horizontal and the neutral-axis depth kd is to be calculated so as to equilibrate the horizontal bending- tension force T crossing the bending-tension yield-leg at location x. ere cannot be any diagonal concrete struts across the lower bending-tension yield-leg and all of the vertical shear force must be transferred by being hung on the vertical shear reinforcement. is mechanism involves simultaneous yield of the shear reinforcement and failure in anchorage of the main longitudinal reinforcement. It is, in that sense, a "combined mechanism". 5.0 NOTATION rfy: WHAT DOES IT MEAN? notation rfy as one remembers, was first suggested by Professor MP Nielsen of Copenhagen. Denmark is, perhaps, the leading country in the plastic analysis of concrete structures and Nielsen is the leading Danish engineer in this field. e notation r = Asv bw*sistheratio ofthe areaAsvofonesetof vertical ties to the corresponding projected horizontal area bw*s of the concrete. Alternatively, the ratio of the total area of vertical tie legs crossing the hinge to the projected horizontal area bw*h1 of the hinge. is is a direct indication of how heavy the shear reinforcement needs to be regardless of rebar size. It is comparable to the ratio 1% to 4% for the area of main rebar to the gross concrete area of a column. Such ratios are useful for designers to remember because they provide an overview regardless of specific member size. fy (Denmark) = fsy (Australia) so rfy is the vertical yield-strength of the shear reinforcement smeared over the corresponding horizontal area of concrete at a dogleg hinge. Units are those for stress: MPa. 6.0 IS THE LOAD AT THE TOP OR THE BOTTOM OF THE BEAM? Figure 1 shows the load wu at the top of the beam as do, indeed, most textbooks. e solely-bending strength will not be affected by the vertical position (top, bottom or in- between) of the load. However, the mechanism of Figure 2 does suggest that the vertical position of the load will have an effect on combined collapse strength. Presumably, most experimental programs use top loads without perhaps noting or being aware that this can affect results. It seems clear that various components of the total load will be applied at various levels over the depth of the beam. For example: • A load from an unreinforced brick wall built on top of the beam may be treated as a top load. • A load from an adjacent supported slab will be applied close to the soffit (bottom) of the slab. o For a down-turn beam that will be at about mid-depth of the beam. • If the beam is upturned from the slab then the load from the slab will be close to the bottom of the beam. • e self-weight of the beam will be smeared over its own height. A detailed consideration of these differences would make this analysis more protracted, complicated and difficult. Let us simplify this by limiting ourselves to loads that are at the top or at the bottom of the beam but not in-between. is author s previous papers treated the load as 100% top load, as do most textbooks. inking about Figure 2 suggests that bottom loads will lead to somewhat weaker beams in combined mechanisms, although there is no such suggestion in the usual textbooks or in AS3600. inking further about this and the structural principle of superposition suggests that the difference between a load wu =wT =100%loadatthetopandanotherloadwu =wB = 100% load at the bottom is that the bottom load will add an extra uniform vertical tensile stress over the full span which is, in the case of Figure 1; Part rfy = wu bw = 156 350 = 0.446 MPa. is is significant but not major. In due course, one would expect AS3600 to provide guidance on this matter. In the meantime, it seems safe and appropriate to treat part-loads as bottom-applied loads wB except only for part-loads that are quite definitely applied at the top. e extra Figure 2. Danish 1978 combined-collapse mechanism with symmetric dogleg hinges.