PSI - Issue 47

Daniela Scorza et al. / Procedia Structural Integrity 47 (2023) 30–36 Scorza et al. / Structural Integrity Procedia 00 (2023) 000–000

32

3

being 1 v and 1 M the transversal displacement and the moment related to the left part of the beam, whereas 2 v and 2 M those related to the right part, E the elastic modulus, I the moment of inertia, and c L the material internal characteristic length. ( ) ( ) n  stays for the n th -order derivative with respect to x . The constitutive boundary conditions are: ( ) ( ) (3) (2) 1 1 1 0 0 0 c v v L − = (3) ( ) ( ) (3) (2) 2 2 1 0 c v L v L L + = (4)

and the constitutive continuity conditions are:

1

2

( ) 1 v L v L + ( ) 1 (3) (2) 1 1

( ) L

χ

=

(5)

12 1 ,

L

L

c

c

1

2

( ) 1 v L v L − ( ) 1 (3) (2) 2 2

( ) L

χ

=−

(6)

21 1 ,

L

L

c

c

being 12 , χ and 21 , χ defined by means of two integrals on the beam segments (Scorza et al. (2023)). The kinematic boundary conditions are given by: ( ) 1 0 0 v =

(7)

( ) ( ) 1 1 0 0 v =

(8)

the static boundary conditions for x L = are given by:

1

( ) c v L v L L − ( ) (4) (2) 2 2 2

0

=

(9)

1

1

F

( ) v L v L L − ( ) (5) (3) 2 2 2

=

(10)

2 c L EI

c

and the kinematic and static continuity conditions are given by: ( ) ( ) 1 1 2 1 v L v L =

(11)

( ) 1 1

EI M L

( ) 1

( ) 1

( ) 1

( ) 1

(1) v L v L v L L v L  − = − ⋅ (1) (2) 2 (4)

 

=

(12)



2

1

1

1

c

k

k