Issue 52
N. Hebbar et alii, Frattura ed Integrità Strutturale, 52 (2020) 230-246; DOI: 10.3221/IGF-ESIS.52.18
2 d w
du
d
s
0 A B 11
0
B
11
11
2
dx
dx
dx
x b N M M
1
h
/2
2 d w
du
d
s
0
0
z dz B
D
D
(17b)
x
11
11
11
2
dx
dx
dx
h
/2
f z
s
2 d w
du
d
s
s
s
0
0
B
D
H
11
11
11
2
dx
dx
dx
h
h
/2
/2
g z dz Q b ' ,
g z dz
z
N
(17c)
z
xz
xz
h
h
/2
/2
The variation of the kinetic energy is expressed by:
L V q w dx
(18)
0
Furthermore, the potential energy of the distributed load is expressed by:
L h
/2
( ) u u w w z dzdx
K
(19)
0 /2 h
Substitute the expressions displacement by deformation as well as stress by deformation which are respectively defined by the Eqns. (16), (18) and (19) in Eqn. (14) and by integrating by parts while putting the coefficients u, v, w and equal to zero. As a result, the governing equations obtained are given as follows:
x N
w
0
(20a)
0 : u
I u I
J
0 0 1
1
x
x
x
2
2
2
x
0 u x
b M
u
w
0
0
0 : w
q N
I w I
J
I
(20b)
0
0 0 1
2
2
2
2
2
x
x
x
xz Q M
w
2
s
0
1 0 N K J u K J z a a
: A K
K
(20c)
a
2
2
x x
x
with:
h
h
/2
/2
( ) , A b Q z dz B b Q z dz ( )z ,
11
11
11
11
h
h
/2
/2
h
h
/2
/2
s
2
( ) D b Q z z dz B b Q z f z dz , ( ) ( ) ,
11
11
11
11
h
h
/2
/2
(21)
h
h
/2
/2
2
S
S
( ) ( ) , D b Q z z f z dz H b Q z f z dz ( ) ( )
,
11
11
11
11
h
h
/2 /2
/2
h
2
S
( ) ( ) A b Q z g z dz
55
55
h
/2
236
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