Issue 29

D. Addessi et al., Frattura ed Integrità Strutturale, 29 (2014) 178-195; DOI: 10.3221/IGF-ESIS.29.16 187 +1 +1 i i    D T U 3. The element basic forces are updated, using the element basic stiffness matrix at the previous iteration:   +1 1 +1 +1 +1 i i i i i i         Q F D Q Q Q 4. To enforce the satisfaction of both the element equilibrium and compatibility conditions, a nested iterative procedure is performed (the superscript ' ' j and ' +1' j denote the previous and current internal iteration, respectively): a. By means of Eq. (35), the increment of the warping displacement vector at each section, where it is not constrained, is computed:   1 +1  +1  +1  +1 j j j j j j j w ww w w w w          U K B Q U U U and the increment of the warping force vector at the sections, where the corresponding displacement is constrained, is evaluated: +1  +1  +1  +1 j j j j j j w w w w w        P B Q P P P b. At each section, the increment of the deformation vector is evaluated:   1  +1  +1  +1  +1  +1 , , 1 ˆ w l j j j j j j j j w n w n n                     d C b Q C u d d d c. At each point over the cross-section, the strains are computed and the resulting stress vector and material stiffness matrix are determined through the constitutive law: +1  +1 ,  +1  +1  +1 , , +1  +1 =1 ˆ ˆ w j j l w n j j x yz j w w n w w n j j n N N x                       d M M u c c       d. The stress vector  +1 j  and the material stiffness matrix are integrated over the cross-section to obtain the generalized section stress vector and the section stiffness matrix:  +1  +1  +1  +1           and             q  C c j T j j T j A A dA  dA     e. The warping matrices +1 j w K , 1 j+ ww K and +1 j w B are computed using Eq. (37); f. The section deformation vector is updated by adding to it the residual that arises from the difference between the balanced and the constitutive generalized section stress vectors:    +1 -1  +1  +1  +1 +1  +1  +1  +1  +1 j j j j i j j j j p           q d C bQ q q d d d       g. The compatible basic element deformation vector, including the above residual, and the basic element stiffness matrix are computed:        +1  +1 0 -1 -1 1 -1  +1  +1  +1 +1  +1 0 L j T j L j T j j j j w ww w  dx dx                    D b d F b C b K K B h. The basic element force vector is computed as follows:     1  +1 1 +1  +1  +1  +1  +1 j j i j j j j        Q F T U D Q Q Q  