Issue 29

D. Addessi et al., Frattura ed Integrità Strutturale, 29 (2014) 178-195; DOI: 10.3221/IGF-ESIS.29.16 183 cross-section on which the warping displacements are interpolated. In particular, assuming that the typical cross-section can be decomposed in a set of rectangular portions, in each of these a regular distribution of m w points is adopted (Fig. 4). Hence, 2D Lagrange polynomials M w , j ( y,z ) are defined, each of which corresponding to an additional warping displacement DOF for the element. Figure 4 : Warping interpolation points and Lagrange shape functions along beam axis (left) and on the cross-section (right). The warping displacement field u w ( x , y , z ) can thus be written firstly considering the interpolation at the section level:       , , , , =1 , , ,  , w m w i i w j w ij w w i j u x y z M y z u y z    M u (16) where   , , , w i i u x y z is the warping displacement field over the section i , u w , i is the vector collecting all the warping DOFs u w , ij of the points located on the section i and M w (y,z) is a matrix containing all the shape functions M w , j ( y,z ) defined for the generic cross-section. Then, the interpolation along the beam axis is considered:         , , , , =1 =1 , , ,      M u w w l l w w i w i w i w w i i i u x y z N x u N x y z (17) The vector p w , i , collecting the warping forces p w , ij at the points located on the section i , can be defined, which is work conjugated with u w , i . As the displacement field u w ( x , y , z ) does not contain the cross-section rigid body motions, the interpolation functions M w , j ( y,z ) need to be modify to ensure the elimination of these rigid body motions. This elimination follows a specific procedure based on the definition of a projection matrix P r and of another matrix k  , used to modify the warping stiffness matrix, as shown in Element variational formulation . Element variational formulation The equations governing the element state determination are derived on the basis of the Hu-Wahizu variational principle. The following functional, depending on the four independent fields u ( x ), d ( x ),  ( x,y,z ) and u w ( x , y , z ), is introduced:   0 Π( , , , )  dV W( , )dV  dx L T T T w w ext V V u             u d d u d d U P u p      (18) where W( d ,  w ) is the internal potential energy and p contains the loads distributed along the element axis. Enforcing the stationarity of the functional  with respect to the four independent fields, the following governing equations are derived: