Issue 53

I. Monetto et alii, Frattura ed Integrità Strutturale, 53 (2020) 372-393; DOI: 10.3221/IGF-ESIS.53.29 377 13 1 d i i i x i u N x C E h     (13b) and the deflections of the beams ( i =1,2) 15 d i i i i s xz i G h T w x C                (13c) with six more arbitrary constants C 11+ i , C 13+ i and C 15+ i ( i =1,2). From Eqn. (13a) the relative rotation between the two beam arms is given by 2 1       (14a) whereas the relative deflection between the two beam arms follows directly from Eqn. (3) 2 1 / n w p k w w     (14b) The general solution derived above contains 17 arbitrary constants, which are evaluated using suitable boundary conditions. Seven conditions are obtained by imposing that the expressions given by Eqn. (13) for displacements and rotations satisfy the bond conditions Eqn. (4) of perfect connection in the longitudinal x -direction and Eqn. (3) of elastic connection in the transverse z -direction for all points of the interface. The prescription of free end boundary at infinity ( x →∞ ) gives five further conditions. These twelve conditions result in the following relations: 1 2 5 6 7 8 9 10 11 0 C C C C C C C C C          , 13 12 C C  (15a) 17 16 C C  and   15 14 12 1 2 2 ( ) / 2 C C C h h    (15b) Then, imposing that the static quantities equate the applied forces at the end x =0 yields C 3 and C 4 , which enter the expressions given by Eqn. (14) for the relative rotation and deflection between the two beam arms. The remaining arbitrary constants C 12 , C 14 and C 16 remain indeterminate but are not of interest here because they describe a rigid motion of the whole self-equilibrated two-layered system. Finally, substituting such results into the expressions given by Eqn. (14), evaluated at x =0, gives the end relative deflection and rotation, say  w 0 and  0 respectively, between the two beams in terms of the end forces ( V 0 , M 0 and N 0 ). The compliance coefficients describing the rotations and displacements follow directly from these relationships, as detailed below. R OOT ROTATION AND DISPLACEMENT COMPLIANCE COEFFICIENTS he three elementary schematics of the bi-layered system shown in Fig. 2 are considered. The self-equilibrated end loading conditions shown in Fig. 1b are reproduced by the superposition of these schematics. The relative rotation at x =0 given in Eqn. (14) defines the root rotation and is related to the end forces by 0 0 0 0 0 0 0 1 1 1 V M N x M d V d d N E h h            (16) and the relative deflection at x =0 defines the root displacement and is given by 0 0 0 0 0 0 0 1 1 V M N x w M c V c c N E h           (17) T