Issue 29

D. Addessi et al., Frattura ed Integrità Strutturale, 29 (2014) 178-195; DOI: 10.3221/IGF-ESIS.29.16 184 Stationarity with respect to Governing equation ( ) u x T rp   T Q P P (19) ( ) x d ˆ ( , , ) [ ( )]  x y z x, y,z    (20) ( ) x  0 ( ) ( )  L T x x dx   D b d (21) ( ) w u x,y,z 0 0 L L w,i x yz w,i w w,i w N dx N dx x       p q  q (22) Eq. (19) is the element equilibrium equation, rp P being the element nodal forces that arise from the distributed loads, Eq. (20) represents the constitutive law characterizing the response at the generic point of the cross-section, while Eq. (21) is the element compatibility equation, here enforced in weak form. Note that these are the classical equations derived from a standard three-field mixed FE formulation. Eq. (22), instead, represents the section equilibrium condition related to the warping effects, i.e., it requires that the warping loads p w , i at each cross-section are equal to the integral of the stresses x w q and yz w q due to the section warping, with:   ( , ) ( , , )  ( ) ( ) ( , ) ( , , )  T x w x w A w yz T w yz w A y z x y z dA x x x y z x y z dA                                     M q q q M   (23)   , x w y z M ,   , yz w y z M being matrices with dimensions 3  m w , composed as follows:     ( , ) ( , ) ,               and             , ( , )                                   0 M M M 0 M 0 M yz w x w w w w y z y z y z y z y y z z (24) To determine the solution of the nonlinear structural problems, the governing Eq. (19-22) need to be linearized, resulting as:             and          T       T Q P q b Q (25) [ ] ˆ        c       (26) 0 L T dx     D b d (27) , , , 0 0 L L w i x yz w i w w i w N dx N dx x          p q q (28) After some manipulations, the following set of equations is obtained: 1             and            ( )           K U P q b Q b F D (29)