PSI - Issue 59
Andrii Pavluk et al. / Procedia Structural Integrity 59 (2024) 566–574 Andrii Pavluk et al./ Structural Integrity Procedia 00 (2019) 000 – 000
570
5
z
z
z
2
2
2 1
1 2 2
( ) N fudA fu fbdz ( ) ( )
(
( )
) (
) a z a dz 1 2 с
E z k с с
z
1
1
2
с
2
c
z
z
z
1
1
1
3 2
z
z
2 1
1
1
1
1
2
2 3
2 2 a z dz E a )
с
( E a z
( )
( )
E z a k
a z k
1
2 2
1
2
2
1 3
с
с с с
с
с
с
z
1
2
2 2 2
4 2 4
3 2 3
3
z
z
z
z
z
1
1
1
1
1
2
2
1
1
с
с
с
с
с
( )
( )
E a
k
a
k
a
E a
E a
2
2
1
2
2
1
2
с
с
с
2
3
4
3
z
z
1
1
2
2
1
1
с
с
( )
( )
,
k
a
k
a
2
1
2
2
4
3
(11) Substituting formulas (10) and (11) into equation (9), we obtain:
2
3
3
2
4
1
1
1
1
1
z
z
z
z
z
2
2
( N b E
( )
)
( )
k
E a
E a
k
a
1
1
2
2
2
с
с
с
с
с
2
1
2
2
1
c
с
с
с
2
3
3
4 2
4
z
3
3
2
3
1
1
1
1
1
z
z
z
z
2
2
2
( )
( )
( )
.
k
a
1 E a E a с
k
1 a k с
a
2
1
1
с
с
с
2
2
1
2
2
1
2
2
с
с
3
3
2
4
3
(12) The tensile force in the normal cross-section of the bending element:
1 t t N N N t
,
2
(13)
z
z
z
2
1 z N fudA fubdz E xbdz E b 1 1 1 2 2 1 1 ( ) ( ) t t t t
,
1
t
2
0
0
0
(14)
z
z
z
z
1
1
2
2
2
2
1 z z
2
( ) N fudA fu fbdz E z az adz ( ) ( ) ( )
(
E a z
2
2
2
1
2
1
t
t
t
t
t
t
3 E a E a 1 1 1 t z
z
z
z
1
1
1
3
2
2
1
1
1
z
z
) E a z dz E a
.
E a
2
2
1
t
t
t
2
1
2
1
2
t
t
t
t
t
t
3
2
3
2
(15) By substituting formulas (14) and (15) into equation (13), we obtain:
3
2
3
2
1
1
1
1
1
z
z
z
z
.
2 N E bz E a E a t
1 E a E a t
2
1
t
t
1
1
2
1
2
t
t
t
t
t
t
t
3
2
3
2
(16) The bending moment from the neutral axis for the compressed zone in the normal cross-section is equal to:
2 ,
1 c c M M M c
(17)
Made with FlippingBook - Online Brochure Maker