Issue 38

P. Lonetti et alii, Frattura ed Integrità Strutturale, 38 (2016) 359-376; DOI: 10.3221/IGF-ESIS.38.46 362 where S N is the number of stays, H N is the number of hangers and the superscripts S , H and M refer to the stays ( S ), hangers ( H ) and main cable ( M ), respectively. The displacement conditions utilized to achieve zero displacements at the cable anchorages are expressed as follows: 1 1 1 1 [( , , ), ] 0 [( , , ), ] 0 [( ), ] 0 S S S S S S S S N N H H H H H H H H N N M M M M L S S S S U L S S S S U L S S U                     (2) where S L  , H L  and M L  are the constraint operators referred to the stays, hangers and main cable variables, respectively. In Eqs. (2), S U  is the vector containing horizontal displacements of the left and right top pylon cross-sections, which are identified as 1 L P U and 1 R P U , respectively, and vertical displacements of the stays at the girder connections: 3, (1) 1 1 3, (N ) [ , , , , ] T P P S G G L R S S S U U U U U        (3) Moreover, H U  is the vector containing vertical displacements of the hangers at the girder connections, i.e. 3, (1) 3, ( ) [ , , ] TH G G H H H N U U U        . Finally, M U is the vertical displacement of the main cable at the midspan cross- section. It is worth noting that, in Eqs. (2), the total initial stress is expressed as a combination of a constant quantity , , ( ) S H M i S and an incremental contribution , , ( ) S H M i S  . The former is a set of trial initial post-tensioning cable forces, which are estimated by means of simple design rules commonly adopted and verified in the context of long-span bridges [1, 29- 32], whereas the latter is defined by an additional stress vector, whose components correspond to unknown quantities to be identified. Figure 2 : Displacement and control variables for initial configuration. Girder and pylons The kinematic model is consistent with a geometric nonlinear Euler-Bernoulli theory, in which moderately large rotations are considered. The torsional behavior owing to eccentric loading is described by means of classical De Saint Venant theory. In addition, since girder and pylons are mainly subjected to axial forces and bending moments, the nonlinear material behavior can be taken into account by a gradual yielding theory based on the combination of the Column Research Council (CRC) tangent modulus concept and a plastic hinge model [33]. The former is suitable to take into account for gradual yielding along the length of an axially loaded element between plastic hinges, whereas the latter is used to represent the partial plasticization effect associated to bending mechanisms. Starting from the status obtained in the initial configuration, in which only dead and permanent loads are considered, the internal stresses are defined by the following relationships between generalized strain and stress variables as follows: