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αc = 1 and αc = 1.
5 DISCUSSION OF RESULTS
5.1 Modeling
The behavior of the beams tested by Levi and Marro (Levi and Marro, 1992) has been studied by NLFE analyses carried out with ABAQUS Code. Since the structure is symmetric along y axis in term of geometry, boundary and loading conditions, half structure has been analyzed in order to reduce the time for the implementation. A mesh of four node membrane elements has been adopted. The average element size is 50 mm; non linear analyses have been carried out in displacement control. Different materials have been used to distinguish elements having different thicknesses and reinforcement ratios (smeared reinforcement is assumed inPARC model). In Figure 11 the mesh and the boundary conditions are shown; each grey shade represents a different material.A Gauss integration technique in the element plane has been adopted. Rebar layers, embedded in the
‘‘host’’ memebrane elements, have been used in order to model strands. By specifying the cross-sectional area, the material and the isoparametric direction of each rebar, the strands can be modeled in a quite appropriate way. The prestressing action is defined in the rebars as a given initial condition which can change during an equilibrate static analysis step. Finally a selfequilibrating stress state established itself in the structure. In this way the prestressing force does not remain constant during all the analysis steps but changes following the development of the crack pattern up to failure.
5.2 NLFEA results
In Table 3 the mechanical characteristics of material adopted for the NLFEA of the PC30A1, PC60A2 and PC60B1 beams are shown. In Fig. 12 the shear- deflection curves obtained with NLFE analyses are shown. It can be noted that the shear resistances obtained with NLFE analyses are very close to the experimental ones. As the prestressing increases, the ultimate shear capacity of the beam
increases and as the shear reinforcement increases the stiffness of the structure in the cracked phase increases. The cracks develop most of all in the zones between the load and the support, where the shear force is constant and maximum, while in the middle zone the cracks are limited and mostly due to bending moment. In Fig. 13(a), (b) and (c) are reported, for PC60B1 beam, the values of crack opening, strain of stirrups and strain of concrete, respectively. 586Fig. 14 demonstrates that for low values of prestressing force (PC30A1 beam) the stirrups start to
work earlier than for high values of prestressing force (PC60 beams). Furthermore, for PC60 beams the activation of stirrups is quite contemporary but the slope of shear versus stirrups stresses curves and the yielding points are different since the stirrup spacing is Different. In Table 4 the experimental shear capacity of the beams tested by Levi and Marro (Levi and Marro, 1992) is compared with the shear resistance obtained with different codes and with the shear resistance
obtained with NLFE analyses.
5.3 Some remarks regarding the evaluation of the internal lever arm Initially the internal lever arm was fixed, as a common simplified assumption, equal to z = 0.9d. Thisapproximation doesnt take into account the effects of prestressing forces and that the occurrence of cracking
changes the application point of the resultant of the internal forces, so afterward the lever arm (zfem) value has been calculated in a rigorous way by consideringthe arm of the couple produced by the internal forces coming from NLFE analyses. In Fig. 15 the results obtained from the analyses
carried out for PC60B1 beam are shown and compared with the shear capacity calculated with Eurocode 2, by adopting z = 0.9d = 897 mm and z = zfem =873 mm. It can be noted that considering zfem the shear capacity calculated with Eurocode 2 (EN 1992-1-1:2004) are closer to NLFEA results.In Table 5 the values of the internal lever arm evaluated with NLFEA results, zfem, and by assuming z =
0.9d are reported. It can be noted that the difference between the z values increases as the prestressing force increases.