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the effects of compressive stresses, respectively (Walraven, 2002).It can be noted that by neglecting compressive stresses effects (which means by considering αc = 1),an high scatter of results occurs in terms of ψ versus nominal shear strength curves (beingψ = ρswfyw/fc1),
while a better fitting is observed if the values of αc coefficient are evaluated according to the new formulation proposed by Eurocode 2 (EN 1992-1-1: 2004), Eq. (7):Several authors have taken into account the influence of the prestressing by the αc coefficient that multiply
the coefficient ν described above. Nielsen (Nielsen 1990) states that prestressing has a
positive influence on the shear capacity of beams with shear reinforcement. He proposed to evaluate αcwithEq. (9) :αc = 1 + 2σcp/fc with σcp/fc < 0.5 (9)A comparison with 93 tests results shows, however,that this expression is not sufficiently conservative to serve as a safe lower bond over the whole region of test results. Another proposal is given by Foure (Foure
2000), Eq. (10):– for low compressive levels (0 < σcp < 0.4fcd ):αc = (1 − 0.67σcp/fctm) (10)
– for high compression levels (0.4fcd < σcp < fcd ):αc =1.2(1 − σcp/fcd )(1 + σcp/fcd )0.5 (11)
In Fig. 5 the different formulation proposed for αc are compared to each other.
3 PARC MODEL FOR NON LINEAR FINITE
ELEMENT ANALYSES
Non linear finite element analyses have been carried out with ABAQUS Code. The PARC constitutive
matrix (Belletti, 2001), implemented in the users’s subroutine UMAT for describes the material inelasticity. Total strains are evaluated by considering as kinematic variables the crack opening w, the crack slip ν and the concrete strain εc2, Fig. 6. PARC model has been modified in this study by adopting Belarbi and Hsu (Belarbi and Hsu, 1991) relation for concrete struts in compression. The Saenz’s stress-strain relation is reduced to consider the deterioration in compression resistance due to cracking.Belarbi and Hsu (Belarbi and Hsu, 1991) suggested to
reduce both the peak strength and strain by means of a softening coefficient for strain, νε, and a softening coefficient for stresses, νσ, Fig. 7. These softening coefficients depend on the orientation of the cracks to the reinforcement (angle β):νε = 1 √1 + Kε • ε1(12)νσ = 0.9 √1 + Kσ • ε1(13)Kε is a value linearly varying from 0 to 160 for β varying from 90◦ to 45◦, respectively, while Kσ isvalue linearly varying from 250 to 400 for β varying from 90◦ to 45◦, respectively.
4 CASE STUDY
Experimental tests (Levi, Marro, 1992) have been carried out on twelve reinforced and prestressed concrete beams (RC and PC beams) differing to each other bydiameter of longitudinal reinforcement and amount of prestressing. All the beams are supported at the two extremities and loaded with two concentrated loads.In Fig. 8 the geometry of the PC beams is shown. The beams are prestressed by means of 0.6 strands, 6 to 12 per beam, to an average prestressing value at the centre of gravity of approximately fc/12. The tendons are tensioned before casting and, following the transfer, are provided with anchorages to prevent early slip induced failure. The evolution of the test carried out on thePCbeams reveals the appearance of cracks at an inclination of about 30◦, which crosses web and which are close to one other, with a distance of about 15 cm, as shown in Fig. 9. These cracks remains virtually unchanged up to failure. The failure is due to complete yielding of theshear strength Vmax. The geometrical and mechanical features of the beams analyzed in this paper are summarized inTable 2. In Fig. 10 the comparison between the experimental
tests and the variable inclination method curve are represented considering the different contributions of shear capacity of the beam tested. The shear capacity has been plotted considering the steel contribution and the concrete contribution taking into account both