Issue 29

D. Addessi et al., Frattura ed Integrità Strutturale, 29 (2014) 178-195; DOI: 10.3221/IGF-ESIS.29.16 186 1 ( , )  w A y y z dA z             V M According to the equilibrated beam formulation, the flexibility matrix F can be evaluated, by manipulating Eq. (29-32), as the sum of two symmetric matrices as: 1 1 1 0 0  ( ) L L T T S w w ww w dx dx          F F F b C b b C K K B (34) where S F is the classical flexibility matrix, while w F represents the contribution due to the warping effects. The matrices w K , ww K and B w are derived by substituting in the warping equilibrium Eq. (32) the expression of d obtained by Eq. (30). Hence, in compact form, Eq. (32) results as: w w ww w      P B Q K U (35) where w  U and w  P are the increments of the two vectors collecting the warping displacements of all points at all sections used for the interpolation and the warping forces at the same points, respectively, which are defined as:     ,1 ,2 , , ,1 ,2 , , w w T w w w w i w l T w w w w i w l       U u u u u  P p p p p (36) their size being w w l m  . Moreover:       ,1 ,1,1 ,1, ,1 ,2 , , , ,1 , , ˆ ˆ ˆ w w w w w w w w w l w w w w l ww w w l w l w l l                                   C C B K C C C K B B C C     (37)   , , , , , , , , , 0 0 ˆ ˆ ˆ ˆ ˆ L L w i x yz w i w i a w i w w i w ww,i,a w i w a w i N N dx  dx x          C C C C C C CC B C C b , , , , , , , , 0 ˆ L w i w n w i w n x xy yz ww,i,a ww w n w i ww w i w n ww N N N N N N N N dx x x x x                    C C C C P ROPOSED SOLUTION ALGORITHM he proposed FE is implemented in the numerical code FEAP [10], which is used to perform all the numerical analyses. Based on a step-by-step time discretization, the nonlinear problem in each time step is solved by adopting a classical iterative Newton-Raphson algorithm. At each iteration FEAP requires at the element level the computation of the element stiffness matrix +1 i K and of the force vector +1 i P , that are used to assembly the global stiffness matrix and the residual vector. In this section, the proposed algorithm for the element state determination is described. It is a generalization of the one presented in [7] and can be summarized as follows: 1. At the global Newton-Raphson current iteration ' +1' i , the nodal displacement vector +1 i U and its increment +1 i  U with respect to the previous iteration ' ' i are given; 2. The basic element deformation increment +1 i  D with respect to the previous iteration is computed, removing from +1 i  U the element rigid body motions through the matrix T : T