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Concrete In Australia : September 2011
Concrete in Australia Vol 37 No 3 53 7.0 NUMERICAL EXAMPLES 7.1 Analysis of a reinforced concrete beam with clamped ends e RC beam tested by Sasani and Kropelnicki (2007) is analysed in this example. e geometry, details of reinforcing and loading are shown in Figure 7a. e material properties are: fy = 525 MPa, Es = 1.96×105 MPa, fcp = 42 MPa, Ec = 4.0×104 MPa and ft = 2.5 MPa. e strain corresponding with the maximum stress for compressive concrete is taken as 0 c ε = 0.0025 and ultimate strain of concrete cu ε = 0.008. One half of the sub-assembly was modelled using three elements and a composite Simpson s integration scheme with 25 integration points through the section depth and 21 integration points along the element axis is employed in this example. e resulting objective load-mid span displacement for the beam obtained from the developed formulation is compared with the experimental and with other analytical results in Figure 7b. It is observed that the value of the ultimate load predicted by the nonlocal approach of P= 74.9 kN correlates well with the experimental result of P= 70.0 kN and analytical results of the ANSYS continuum model (Sasani and Kropelnicki, 2007) P= 66.7 kN. e local flexibility 3 9.5 Add 3 9.5 Add FE Model P/2 2 9.5 Con. 600 mm 600 mm W1.4@68 mm 2 9.5 Con. C.L. 172 mm 2000 mm 305 mm Middle Column Support Column Element-3 Element-2 Element-1 S 3 9.5 Add 2 9.5 Con. W1.4@68 mm 2 9.5 Con. 191 mm Section S 191 mm Figure 7. (a) Geometry, reinforcing details and idealised FE model. (b) Load versus mid-span displacement for the beam tested by Sasani and Kropelnicki (2007). (a) (b) formulation, however, significantly underestimates the ultimate loading capacity giving P= 52.1 kN. Furthermore, the post peak response obtained from the presented nonlocal formulation correlates well with that of the experiment and captures the catenary action of the member quite accurately. e local formulation, on the other hand fails to capture these effects properly. 7.2 Cyclic analysis of a beam-column Low and Moehle (1987) studied the behaviour of a reinforced concrete cantilever beam-column subjected to a constant axial load, P= 44.54 kN, and a variable lateral load (displacement) applied at its tip. e geometry of the beam-column and cross section details are shown in Figure 8a. e material properties are adopted as fy = 450 MPa, Es = 2.0×105 MPa, fcp = 35 MPa, ft = 2.3 MPa and Ec = 2.6×104 MPa. e strain corresponding with maximum stress for compressive concrete is0 c ε = 0.002 and ultimate strain of concrete is cu ε = 0.01. e effect of confinement for concrete within the core is taken into account using Mander et al (1988) model. Unloading/reloading happens along a line with a slope equal to the initial elasticity modulus. e Menegotto and Pinto (1973) model is used for modelling the cyclic behaviour of steel bars. e entire beam is modelled by a single element and a composite Simpson s integration scheme with 15 points through the section depth is used and number of integration points along the element axis is variable. Figure 8b shows the lateral load versus tip displacement of the free end obtained from total secant formulation without and with geometrical nonlinearity included, respectively. As shown in Figure 8b the tangent approach may suffer of load step size related instability for the first few steps after load reversal. With regard to Figure 8b, it is observed that the formulation with geometrical nonlinearities captures the element response more accurately. e difference between the two formulations (i.e. with and without geometrical nonlinearity) becomes more pronounced at ultimate stages of loading. 7.3 Beam subjected to air blast pressure In this example, beam B140-D2 of Magnusson and Hallgren (2004) subjected to uniform air blast pressure is