Issue 38

P. Lonetti et alii, Frattura ed Integrità Strutturale, 38 (2016) 359-376; DOI: 10.3221/IGF-ESIS.38.46 364 Cable elements The nonlinear behavior of the cables is defined according to finite deformation theory in which large displacement effects and inelastic behavior due to plastic deformations are considered. In order to reproduce kinematic nonlinearities arising from the sag effect, a multiple truss element formulation is considered, in which each cable or part of cable is discretized by using multiple truss elements [35-37]. The strain measure is described by Green-Lagrange tensor, which is defined, consistently to Green-Nadghi approach [38, 39]. Therefore, the description of the stress variable coincides with the Second Piola-Kirchhoff stress, which can be expressed as:   2 2 2 1 1 1, 0 1 1, 1, 2, 3, 1 1 1 1 1 with 2 C C C C C C C p X X X X S C E E S E U U U U              (8) where 1 C E is the total axial strain, 1, C p E is the axial plastic strain and 0 C S is the initial stress. A scalar valued variable, i.e. C  , is considered to describe isotropic hardening evolution law, by means of a classical linear relationship. Moreover, it is postulated the existence of a convex, differentiable yield function expressed in the stress space by means of the following expression: 1 1, 1 0 ( , , ) C C C C C C p y f f S E S S K       (9) where 0 C y S and K are the initial hardening threshold and the evolution parameter, respectively. Consistently with classical plasticity approach, the evolution of the plastic variables is based on a standard formulation, whose incremental relationship is defined according to the normality evolution rule, i.e. * 1, 1 C C p E f S     . Finally loading-unloading relationships concerning the consistency of the incremental constitutive equations can be defined by the following expressions: * * * 0, 0 and 0 f f f         (10) F INITE ELEMENT IMPLEMENTATION he governing equations reported in the previous section introduce a nonlinear differential system, whose analytical solution is quite complex to be extracted. As a consequence, a numerical approach based on the finite element formulation is utilized. In particular, the weak forms for the i -th finite element related to the girder ( G ), pylon ( P ) and the cable system ( C ), respectively, are defined by the following expressions: Girder and Pylon           2 ( ) ( ) ( ) ( ) 1, 1 1 1 1 1, 1 1 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2, 2, 1 2 1 3 3 2 3 3, 1 1 1 1 1 1 2 ( ) ( ) ( ) ( ) ( ) 3, 3, 1 3 1 2 2 3 2, 1 1 1 1 1 1 0 0 G P G P G P G P X j j X i j l e G P G P G P G P G P G P G P X X X j j j j X i j j l e G P G P G P G P G P X X X j j j X i j l e N U w dX N U M w N U w dX T U M M w N U w dX T U M                        2 ( ) ( ) 2 1 2 ( ) ( ) ( ) 4, 1 1 1 1 1 1 0 0 G P G P j j G P G P G P X j j i j l e M w dX M           (11) T