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Concrete In Australia : June 2013
22 Concrete in Australia Vol 39 No 2 REFERENCES Gurley, CR (2011c). Exact theoretical plane-stress yield-line analysis of shear in ordinary beams. Magazine of Concrete Research, Paper 900200 ICE, London. Gurley, CR (2011). Why plastic methods for structural concrete design? Concrete in Australia Vol 37 No 3, Discussion. Vol 38 No 2 (2012) and Vol 38 No 3 (2012). Sydney. Nielsen, MP, Braestrup MW, Jensen, BC and Bach, F (1978). Concrete Plasticity Beam Shear -- Shear in Joints -- Punching Shear. Danske Selskab for Byginngsstatik, Lyngby. Nielsen, MP and Hoang, (2010) LC. Limit Analysis and Concrete Plasticity. ird edition 816 pp. CRC Press. Whitney CS (1940) Plastic theory of reinforced concrete design. Proceedings ASCE. APPENDIX A A1: ALGORITHM FOR FIRST DOGLEG HINGE e first dogleg hinge initiates at the face of the support. Calculate the force T1 < Ty for the anchorage length h0 from the span face of support to the end of rebar. If T1 = Ty then, theoretically, the beam can work as an internally-tied arch and needs no shear reinforcement but one would not want to get into an argument about this. Calculate the stress-block depth for the bending compression force C1 = T1, the vertical lever-arm jd1 of the horizontal forces and the moment strength M1 at hinge 1. e vertical forces on the first segment are the reaction R to the left end and the shears and loads that equilibrate that reaction to the right. Calculate the horizontal lever-arm h10 = M1 R Sinceh10 = h1+h0 2 , calculate the horizontal projection of hinge1fromh1 =2*h10--h0. Calculate the shear force across hinge 1 from: V1 =R-wT*h1=RifwT=0,calculatethestressontheshear rebar smeared over the concrete area from: rfy1 = V1 bw *h1 Calculate the spacing of shear reinforcement from: s1 = Asv1*fsy*(φV =0.70) bw * rfy1 Determine actual size and spacing of shear reinforcement taking note of the maximum spacing requirements mentioned above. A2: ALGORITHM FOR "NEXT" DOGLEG HINGE For example, for the second dogleg hinge following the first. Later dogleg hinges initiate on the tension face directly below the rotation centre for the previous dogleg hinge. Calculate T2 < Ty -- the tension force in the bending rebar where it crosses the new hinge. is may be limited by anchorage and cannot exceed yield. Calculate the moment strength M2 across the current hinge. Calculate the incremental moment: dM21 = M2 -- M1 Calculate the averaged lever-arm h21 = dM 21 V1 Calculate the horizontal projection of the current dogleg: h2=2*h21--h1 Calculate the shear across hinge 2: V2 = V1-h1*wB-h2*wT Calculate the ideal value of the smeared yield strength of the shear reinforcement across hinge 2: rfy2 V2 bw *h2 > 0.35 MPa for design of shear reinforcement. Calculate the spacing of the shear reinforcement across hinge2from s2 = Asv2*fsy*φv =0.70 () bw *rfy2 Determine actual size and spacing of shear reinforcement taking note of the maximum spacing requirements mentioned above. A3: SHEAR IN SIMPLY-SUPPORTED REINFORCED CONCRETE BEAMS One does want to be careful to ensure that the content of this note is as simple as it can be and directly related to the conditions of simple static equilibrium. is is not a "flash-in-the-pan" note. It follows the publication of about a dozen papers overseas on plastic methods for the design of concrete structures mostly with ICE, London in the Magazine of Concrete Research, and also a 2011 paper in Concrete in Australia Vol 37 Issue 3, which was followed by further discussion in Vol 38 Issues 2 and 3 (2012). e present AS3600 empirical method for design of concrete beams in shear is still essentially that adopted as a temporary compromise by an ACI-ASCE Joint Committee in the 1950s after some collapses of major American hanger roofs. ere is no underlying theory and the empirical method is past use-by date. Consider then, the role of plastic design in the major codes for the structural design of buildings and of civil engineering structures such as bridges, wharfs and offshore oil-platforms: • Structural steel currently AS4100:1994 • Reinforced concrete currently AS3600:2009 and Eurocode 2. is author first saw Eurocode 2 on 15 May 2013. Basic plastic theory was developed at Cambridge University, UK and at Lehigh University, Pennsylvania, US and included methods that were both "upper-bound" collapse-mechanisms and "lower-bound" complete equilibrium solutions. is theory was brought to Australia by Jack Roderick when he came from Cambridge about 1950 as Head of the School of Civil Engineering at the University of Sydney. In the mid-1960s plastic theory was incorporated into the new Australian code for steel structures written by Dr Max Lay (committee of 1) at the University of Sydney. It was an excellent code. Roderick was quite clear that "upper-bound" mechanism methods were an important part of the mix and that, without them, there was no hope of "exact" solutions. Perhaps this is why AS4100 has now been able to remain largely unchanged for almost 20 years. Nevertheless, most concrete codes, including AS3600 and Eurocode 2 are still trying to do it all by solely lower-bound methods. e only exceptions known to this author are codes in