Issue 29

D. Addessi et al., Frattura ed Integrità Strutturale, 29 (2014) 178-195; DOI: 10.3221/IGF-ESIS.29.16 191 where i Σ is the stress tensor containing only the positive or the negative part of the principal stresses, and Λ is the secant stiffness matrix, depending on the damage D. The parameter  is a shear factor assumed equal to 1.06. The adopted stress-strain law is presented in Fig. 9.a in the case of uniaxial tension and compression, by adopting the material parameters reported in Fig. 11. In Fig. 9.b the damage evolution law is presented. (a) (b) Figure 9 : Damage constitutive model for concrete: (a) Uniaxial stress-strain law; (b) Damage evolution law. Non local formulation As known, in presence of a strain-softening constitutive law, localization problems and related mesh-dependency of the FE solution arise, thus requiring the adoption of a regularization technique. In the proposed FE, the localization of the generalized deformations at the integration points along the beam axis can appear, as typical in the force–based element [11]. To overcome this problem, a nonlocal integral regularization technique is introduced. It is based on the nonlocal definition of the generalized section deformation ˆ d , computed in each cross-section as:         2 L  w              with      w w  ˆ G c x x S S x x dx x e x dx             d d (44) where G x x  represents the distance between the quadrature point located at G x and all the others, and the parameter c L influences the extension of the averaging process. Then, at each point of the cross-section the total strains are evaluated as: , , , =1 ˆ ˆ w l w n x yz w w n w w n n N x            d M N M u   (45) and on the basis of these the damage parameters used for the constitutive law in the step 4.c are computed. Plain concrete beams subjected to torsional loads The applications presented in this section were introduced by Karayannis [14], based on experimental tests, and were also analyzed by Mazars [15]. It is a beam, whose total length is 160 cm , divided into three parts: two end parts, properly reinforced, so as to remain elastic, and a plain concrete middle part, characterized by a cracking behavior (Fig. 10). A torque x M is applied at both the ends with a particular system that allows all the cross-sections of the beam to undergo warping displacements. Two shapes are considered for the beam cross-section: a rectangular one and a T-shaped one. For the rectangular beam, the adopted mesh is made from three FEs, one for each part of the specimen. Only one warping interpolation point is located along the axis, 1 w l  , while 4 4 16 w m    warping points are used over the cross- section. In fact, in this case the warping distribution along the element axis is uniform and has a cubic distribution in the principal directions over the section. Four Gauss-Lobatto integration points are used in each FE and a grid of 8 16  fibers is adopted. The material parameters are given in Fig. 11. The modulus E is not specified in [14], thus it is taken so as to reproduce the experimental initial stiffness.