Issue 53

I. Monetto et alii, Frattura ed Integrità Strutturale, 53 (2020) 372-393; DOI: 10.3221/IGF-ESIS.53.29 376 The general solution of the differential equation can be written in three different forms: 1 1 2 2 3 1 4 2 1 2 3 4 1 3 4 2 3 4 3 3 4 4 3 4 ˆ exp( ) exp( ) exp( ) exp( ) for 0 ˆ ( )exp( ) ( )exp( ) for 0 exp( )cos( ) exp( )sin( ) ˆ for 0 exp( )cos( ) exp( )sin( ) n C x C C x C x d p C C x x C C x x d C x x C x x d C x x C x x x                                           (9) where 2 ˆ ˆ ˆ4 d a e   , ˆ 2 a   (10a) 1 ˆ ˆ 2 a d    , 2 ˆ ˆ 2 a d    , 3 1 ˆ ˆ 2 2 e a    and 4 1 ˆ ˆ 2 2 e a    (10b) with ˆ a and ˆ e given by Eqns. (6) and (8). When the normal tractions along the interface between the beams have been obtained, closed form expressions for the shear tractions and for the internal forces and displacements for each beam arm can be derived, as detailed in what follows. Firstly, integration of the second of Eqn. (5) gives the shear tractions along the interface between the two beams: 1 1 5 ˆ ˆ d t n p p g k c x C       (11) where for brevity, hereinafter, d x   indicates integration of · once with respect to x ; whereas C 5 is an additional arbitrary constant. Secondly, integration of Eqn. (2) gives, respectively, the axial forces ' 5 ( 1) d i i x i i t i N E h u p x C       (12a) the shear forces   ' 7 ( 1) d i i s xz i i i n i T G h w p x C          (12b) and the bending moments in the beams ( i =1, 2) 3 ' 9 1 d 12 2 i i x i i i t i E h h M T p x C              (12c) which contain six more arbitrary constants C 5+ i , C 7+ i and C 9+ i ( i =1,2). Thirdly, integration of the internal forces gives the rotations 11 3 12 d i i i x i M x C E h      (13a) the axial displacements