5.4 Some remarks concerning the evaluation of the θ angle Because of PARC constitutive model is a fixed crack model, the stresses can be referred to the orthotropicaxes which remain fixed after cracking, Fig. 16(a). Stresses in concrete struts are due to axial compressive
stresses along 2 direction, axial tensile stresses along 1 direction and tangential stresses due to aggregate interlock phenomena. In the perspective of the variable inclination angle, the inclination θ of concrete struts coming from Eurocode 2 formulations, has been compared with the inclination of principal direction of stresses (σI and σII) of cracked concrete, Fig. 16(b).
In Fig. 17 the elements used to calculate θ value are represented. In Table 6 the values of strut inclination obtained with experimental tests (Levi and Marro, 1992), θexp, with Eurocode 2 formulations (EN 1992- 1-1: 2004), θEC2, and with NLFEA results, θfem are reported.
5.5 Some remarks concerning the evaluation of the coefficient αc For PC60B1 the coefficient αc,fem has been calculated from the NLFEA analyses as: αc,fem = Vconcrete/VR,c (14)In Table 7 the values of this coefficient evaluated with Eurocode 2 (EN 1992-1-1: 2004) prescriptions, αc, and NLFEA results, αfem, for PC30A1, PC60A2 and PC60B1 beams are reported. The value of compressive stresses due to prestressing are quite low for these beams, so a parametric study on PC60B1 beam is carried out by varying the applied prestressing force. This parametric study helps to evaluate the effects of high compressive stress values on shear capacity of beams. In Table 8 (Boiardi, 2008) the values of prestressign force considered are reported; it should be noted that in order to distinguish the applied prestressing force, the name of each beams contains a digit which
represents the ratio of the applied prestressing force to the prestressing force of PC60B1. For example beam PC60B1(8/4) is a beam with a prestressing force which is the double of the prestressing force applied to PC60B1. The αfem values obtained with NLFE analyses by applying different values of prestressing force are reported in Table 8 and compared in Fig. 18 with
the different formulations proposed for αc coefficient. It should be noted that the assumed prestressing forces values and consequently the assumed compressive stresses and σcp/fc ratios are those usual for reinforced concrete beams and correspond to the first branch of the three linear relation proposed by EC2 (EN 1992-1-1: 2004).
6 CONCLUSIONS
In this paper the shear capacity of reinforced concrete prestressed beams is analyzed. To this aim the prestressed beams tested by Levi and Marro are analyzed and experimental measurements are compared with NLFEA results and Code prescriptions. Particular attention has been paid on the values of αc and ν parameters proposed by Eurocode 2 to evaluate the shear resistance provided by inclined concrete struts forming an angle θ with respect longitudinal beam axis. The comparison between experimental measurements and NLFEA results demonstrated that the effect of prestressing forces on shear capacity is properly considered by Eurocode 2 prescriptions for low values of compressive stresses. The case of high values of compressive stresses are not analyzed in this paper and will be deeply investigated in further works. The case of high value of compressive stresses is important because is typical of reinforced concrete columns. Consequently, the right evaluation of the dependency of the shear capacity on the compressive stress level is really important also for the design of frames subjected to seismic actions. Indeed, in the perspective of the capacity design approach the evaluation of the shear capacity becomes an essential task for the prediction of the earthquake resistance of the entire building.REFERENCES ACI Committee 318, 2005, ‘‘Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (318R-05)’’, American Concrete Institute, Farmington Hills, Michigan. Belarbi A., Hsu T.T.C, 1991, ‘‘Constitutive lawsof reinforced concrete in biaxial tension-compression Research Report UHCEE 91-2, University of Houston, Houston, Texas.Belletti B., Cerioni, R., Iori, I., 2001, A Physical Approach for Reinforced Concrete (PARC) membrane elements’’, Journal of Structural Engineering, ASCE, 127(12), pp. 1412–426. Boiardi A., 2008, ‘‘Studio sulla resistenza a taglio  di travi in cement armatopre compresso’MAScthesis, Department of Civil Engineering, University of Parma, Italy. Cladera, A., Mari, A., 2007, ‘‘Shear strength in the new Eurocode 2. A step forward?’’ Print ISSN: 1464–177, Volume: 8, Issue: 2, Thomas Telford and fib, pp. 57–66 CSA Commettee A23.3, 2004, ‘‘Design of concrete structures’’, Canadian Standards Association, Mississauga, Ontario, Canada. Dei Poli S.D., Gambarova P.G., Karakoc C., 1987, ‘‘Aggregate interlock role in R.C. thin- webbed beams in shear’’, Journal of the Structural Division, ASCE V. 113, No. 1, pp. 1–9. Eurocode 2,’’Design of concrete structures - Part 1-1: General rules and rules for buildings’’, ENV 1992-1-1: 1991. Eurocode 2, 2005. Eurocode 2,’’Design of concrete structures—art 1-1: General rules and rules for buildings’’,EN 1992-1-1: 2004. Foure B., 2000, ‘‘Proposal for rewriting item 6.2.4 in section 6.2 (Ultimate Limit State-Shear’’, Note to Project Team for EC-2, May. Gupta P.R., 1995, ‘‘Shear behavior of reinforced concrete beams subjected to high axial compression’’, PhD thesis, University of Toronto, Canada. Joint ACI-ASCE Committee 326, 1962, ‘‘Shear and diagonal tension’’, ACI Journal, Proceedings V. 59, No. 1-3, pp. 1- 30, 277–44, 352–96.Haddadin M., Hong S.T., Mattok A.H., 1971, ‘‘ Stirrup effectiveness in reinforced concrete beams with axial force’’, Journal of the structural pision, ASCE, V. 97, No. ST9, pp. 2277–297. Levi F., Marro P., 1993, ‘‘Shear tests on HSC prestressed beams - proposals of new interpretative models’’, Conference on High Strength Concrete, Lillehammer 20–4 June, Proceedings, pp. 293–05. Nielsen M.P., 1990, ‘‘Commentaries on Shear and Torsion’’, Eurocode 2 editorial group, 1st draft, October. Vecchio F.J., Collins M.P.,1988, ‘‘Predicting the response of reinforced concrete beams subjected to shear using modified compression field theory’’, ACI Structural Journal, V. 85, No. 3, pp. 258–68. Vecchio F., Collins M.P., 1986, ‘‘ The modified compression field theory for reinforced concrete elements subjected to shear’’, ACI Journal, Proceedings, V. 83 No. 2, pp. 219–31. Watanabe F., Kabeyasawa T., 1998, ‘‘Shear Strength of RC Members with High-Strength Concrete’’, High Strength Concrete in Seismic Regions, SP-176, C.W.W. French and M.E. Kreder, eds., American Concrete Institute, Farmington Hills, Mich., pp. 379–96. Walraven J.C., 2002, ‘‘Background document for prENV 1992-1-1:2001, 6.2 SHEAR’’, Delft University of Technology, The Netherlands. Yoshida Y., 2000, ‘‘Shear reinforcement for large, lightly, 混凝土结构模型英文文献和中文翻译(5):http://www.751com.cn/fanyi/lunwen_22283.html