Simplified models/procedures for estimation of ... - ELSA - Europa

24concrete struts can form with the member axis an angle θ that can vary up to a value of 45°. Themethod recognises that due to shear mechanisms different from those considered by the Ritter–Mörsch truss model, the compressive stresses in the member web may have an inclination lowerthan 45°. According to this model the shear resistance of a member may be reached either foryielding of the stirrups:AvV = ⋅dv⋅ fvy⋅cotθ (8.4)sor for crushing of the concrete web struts:ν ′ ( θ θ)V = b⋅d ⋅ ⋅ f / cot + tan (8.5)vcwhere ν is the strength reduction factor for concrete cracked in shear and is computed as afunction of the compressive stress f’ c derived from cylinder tests:⎛ f ′ ⎞ν = 0.6⋅⎜1−c⎟ (f ’ c in MPa) (8.6)⎝ 250 ⎠Equation (8.6) accounts for the lower compression resistance of the concrete forming the strutswith respect to the compression resistance derived from standard cylinder tests. This reduction inresistance is due to the high tensile strains that exist in the struts normal to the direction of thestruts and to the effects of the stirrups crossing the struts.For a given amount of transverse reinforcement the maximum shear resistance is attained whenthe yielding of the shear reinforcement and the crushing of the web concrete struts are reachedsimultaneously. This assumption leads to the following condition for angle θ:1−ωvcot θ = (8.7)ωvWith the mechanical percentage of web reinforcement ω v equal to:Av⋅ fvyωv=s⋅b⋅ν⋅ f ′cConsidering the values of θ given by Equation (8.7), the shear strength may be expressed as afunction of the amount of shear reinforcement, and may be plotted in a dimensionless format asshown in Fig. 2.1, indicating that larger shear capacities are obtained with respect to the Ritter –Mörsch truss model, which assumes a strut inclination of θ equal to 45°.The norm prEN 1992-1 [CEN, 2003] suggests to use as shear resistance of a member the lowestvalue resulting from Equations (8.4) and (8.5) for a given amount of shear reinforcement and fora given value of θ. For members subjected to axial compressive forces, the same code suggests tomultiply the value given by Equation (8.5) by a factor α c ranging between 0 and 1.25 (see Fig.2.2), which depends on the mean compressive stress σ p acting on the section of the member.prEN 1992-1 [CEN, 2003] does not consider any distinction between the loads inducing thecompressive stresses, which may be either external (i.e., gravity loads) or due to prestressing orpostensioning. It should be noted that prEN 1992-1 [CEN, 2003] does not give any indication onthe effects on shear resistance associated to loads inducing tensile stresses on the section.Both of the approaches presented in the previous paragraphs (e.g., concrete contribution methodas given by Equation (8.1) and the variable angle truss model) are insensitive to the magnitudeand consequences of the member deformations. Mechanisms such as tension stiffening,(8.8)