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Concrete In Australia : June 2014
Concrete in Australia Vol 40 No 2 31 sliding failure. To determine the inclination of the critical diagonal crack and the shear capacity, Zhang (1994) combined a cracking criterion with a crack sliding criterion. According to the model, diagonal cracking is immediately followed by a crack sliding failure when the following condition fulfilled is: () () cr u Vx Vx = (1) Here, V cr (x) is the load required to develop a diagonal crack with horizontal projection x (ie the cracking load), and Vu(x) is the load required to cause a sliding failure in the same crack (ie the shear resistance of the crack). The diagonal crack is assumed to end at the point of loading. As indicated in Figure 1, V cr (x) may be calculated by considering a rotational mechanism where part I rotates around the loading point relative to part II. The shear capacity, V u (x), related to sliding failure in a diagonal crack is determined from a translation mechanism where the relative displacement, u, in the yield line is vertically directed. In both calculations, concrete is treated as a rigid plastic Modified Coulomb material. The cracking load increases with an increase in x while the shear resistance decreases with an increase in x. Details on how to calculate Vcr(x) and Vu(x) for various problems may be found in Ref. 11 . Expressions valid for prestressed hollow core slabs are provided below. The procedure to determine the shear capacity is illustrated in Figure 2. The shear capacity is found as the point of intersection between the two curves representing Vcr(x) and Vu(x), respectively. The model is also able to take into account the delaying effect of a compressive axial force, N, on the development of cracks. As a consequence, the cracking load curve is lifted upward as shown in Figure 2, which then results in a larger shear capacity (as also observed in tests). In prestressed elements (pretensioning as well as post tensioning) the prestressing force delays crack development in a similar way as a normal force. Therefore, when calculating the cracking load in the model, a prestressing force is treated as an external normal force acting at the level of the prestressing strands. Figure 2: Graphical determination of load carrying capacity. 3.0 CRACK SLIDING MODEL APPLIED TO PRESTRESSED HOLLOW CORE SLABS The Crack Sliding Model has been extended to deal with shear strength of prestressed hollow core slabs. The work was initiated by Hoang (1997). A refinement of the original work is presented in this paper. The refinement lies in a more detailed inclusion of the slip length of strands and the width of the support. As a simplification, a hollow core slab may be considered as an assembly of I-beams; see Figure 3. The web thickness bw of the idealised cross section is taken as the thinner part between two voids in the actual hollow core cross section. The flange thicknesses to and tu are taken as the thinner parts above and below the voids respectively. Figure 3: Idealised cross section of hollow core slabs. When applying the Crack Sliding Model to hollow core slabs, it is necessary to take into account the fact that the prestressing force has to be developed by bond over a certain transfer length, lt . In addition, it is also necessary to consider the initial bond slip, l slip , of the strands at the ends of the element. For pre-tensioning strands, the following simple expressions for the transfer length (Engström, 1999) and the slip length (FIP, 1988) may be adopted: 5 for slow release of strands 60 for rapid release of strands tl (2) 5 for slow release of strands 10 for rapid release of strands slip l (3) where φ is the strand diameter. A linear development of the prestressing force within the transfer length is assumed in the following. For hollow core slabs which are saw-cut from a longer extruded strip, the transfer of the prestressing force into the concrete will take place gradually when each wire in the strand is cut. For this reason, it is more reasonable to assume slow release of strands rather than rapid release. Slow release is assumed in the following. When taking lt as well as lslip into account, the variation of the prestressing force along the beam axis, x’, may be expressed as follows (see also Figure 4): 0, ' ' (') , ' ,' slip slip se slip t slip t se t slip xl xl FxF l xll l Fl x l (4) where Fse denotes the effective prestressing force outside the transfer length. Losses due to elastic shortening, relaxation, shrinkage and creep are covered by Fse . In hollow core slabs, the magnitude of Fse is typically in the order of 65% of the ultimate strength of the strands. Figure 4 also shows the schematic variation of the crack sliding resistance Vu(x) and the cracking load Vcr(x). As illustrated, the prestressing force changes the cracking load curve in a similar way as an external normal force does (Figure 2). Here, however, the building up of the prestressing force within the transfer length has to be taken into account. Therefore, in addition to a shift upwards, the shape of the cracking load curve also changes as compared CIA.indb 31 CIA.indb 31 20/05/14 12:40 PM 20/05/14 12:40 PM