Issue 38

P. Lonetti et alii, Frattura ed Integrità Strutturale, 38 (2016) 359-376; DOI: 10.3221/IGF-ESIS.38.46

Cable system

2







C

C

C





1 1, 1 1, N U w dX b w dX N U    1 1 1 1 1 1 j 1 X X



0

j

1



i e

i e

j

1

l

l

2

1 2, N w dX T U  1 2 2 C C j C X 

i l e 



(12)

0

j

1



j

1

2



1 3, 3 3 1 N w dX b w dX T U   3 3 C C j C X 1







0

j

1



i e

i e

j

1

l

l

where with k = C,G,P represent the internal forces applied at the end nodes of the generic cable, girder or pylon element. Finite element expressions are written starting from the weak forms previously reported, introducing Hermit cubic interpolation functions for the girder and pylon flexures in the 1 2 X X and 2 3 X X deformation planes and Lagrange linear interpolation functions for the cable system variables and the remaining variables of the girder and the pylons:               , , , , , C C C G G G P P P U r t N r q t U r t N q t U r t N q t                 (13) where , , C G P q q q    are the vectors collecting the nodal degrees of freedom of the cable, girder and pylon respectively, , , C G P N N N    are the matrixes containing the displacement interpolation functions for cable element ( C ), girder ( G ) and pylons ( P ), r  is the local coordinate vector of the i -th finite element. The discrete equations in the local reference system of the i -th element are derived substituting Eq. (13) into Eqs. (11)-(12), leading to the following equations in matrix notation: with , , i i i i K U P R i G P C        (14) where i K  is the stiffness matrix, i P  is the load vector produced by the dead and live loading, i R  is the unknown force vector collecting point sources. In order to reproduce the bridge kinematic correctly, additional relationships to define the connections between girder, pylon and cable system should be introduced. In particular, the cable system displacements should be equal to those of the girder and the pylons at the corresponding intersection points. Hence, the bridge kinematic is restricted by means of the following constrain conditions:             3 1 3 1 3 1 , , , , , , , 2 2 G G G G C G G C C C C C C C i i i i i i b b U X t X t U X t U X t X t U X t             (15)             1 1 2 2 3 3 , , , , , , , , P C P C P C P P P P P P U X t U X t U X t U X t U X t U X t          (16) where i C X  and P X  represent the vectors containing the intersection positions of the i -th cable element and the pylon top cross section, respectively, 1 3 ( , ) G G U U and 1 3 ( , ) G G   are the displacement and rotation fields of the centroid axis of the girder with respect to the global reference system, respectively. It is worth nothing that, Eqs. (15) are constraint equations imposed between the off-set nodes of the girder and those associated to the cable elements. Finally, starting from Eqs.(14) taking into account of Eqs. (15)-(16) as well as the balance of secondary variables at the interelement boundaries, the resulting equations of the finite element model are: KQ T     (17) where Q  with C G P Q U U U        is the generalized coordinate vector containing the kinematic variables associated with the girder, the pylons and the cable system, K  is the global stiffness matrix and T  is the loading vector. Since the structural behavior of each element depends on the deformation state of the members, the governing equations defined by Eq. (17) will change continuously as the structure deforms. 1 2 3 ( , , , j j j j N T T M M M 1 2 3 , , ) k j j

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