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
571
6
1 0 z
1 0 z
1 0 z
3
1
1
1
z
2 ( ) M fuzdA fuzbdz b E z k ( ) (
2 3 ( ) )
( z dz b E
1
с
1
1
1
2
c
с
с
с
с
с
с
3
4 2 1 z
1 ( )
),
k
с
2
4
(18)
z
z
z
1
1
2
2
2
2 E z k (
2 3 ( ) ) (
( ) M fuzdA fuz fbdz ( ) ( )
)
1 z a z a dz z 2 4 1 с с
2
1
1
2
c
с
с
с
с
z
z
z
1
1
1
z
1
1
1
1
2
3 E a z E z a k 2 (
2
4 a z k
2
3 a z dz E a )
( )
( )
2
с
1
2
2
1
2
2
1
с
с
с
с
с
с
с
4
z
1
3
5
4
4
3
1
1
1
1
1
z
z
z
z
z
2
2
( )
( )
a
E a
k
a
k
a
E a
E
2
2
2
1
1
с
с
с
с
с
2
2
1
2
2
1
2
с
с
с
3
3
5
4
4
5
4
1
1
z
z
2
2
( )
( )
.
k
1 a k с
a
1
с
2
1
2
2
5
4
(19) By substituting formulas (18) and (19) into equation (17), we obtain:
3
4
4
3
5
1
1
1
1
1
z
z
z
z
z
2
2
( M b E
( )
)
( )
1 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
3
4
4
3
5
z
a
4
4
3
5
4
1
1
1
1
z
z
z
z
2
2
2
( )
( )
( )
.
k
a
1 E a E a с
k
1 a k с
2
1
1
с
с
с
2
2
1
2
2
1
2
2
с
с
4
4
3
5
4
(20) The bending moment from the neutral axis for the tensile zone in the normal cross-section is equal to:
,
1 t t M M M t
2
(21)
z
z
z
3
1 z M f udA f ubzdz E zbdz E b 1 1 2 1 2 2 1 1 ( ) ( ) t t t t t
,
1
t
3
0
0
0
(22)
z
z
z
1
2
2
2
2
1 M fuzdA fuz fbdz E z az adz z z z E az E azdzEa E a 1 1 2 2 2 2 1 2 4 3 4 3 2 2 2 ( ) ( ) ( ) ( ) 1 1 1 1 1 ( ) t t t t t t z z z z t t t
3
1
z
.
1 E a E a
1
t
1
2
1
2
1
2
t
t
t
t
t
t
t
4
3
4
3
z
1
(23) By substituting formulas (22) and (23) into equation (21), we obtain:
3
4
3
4
3
1
1
1
1
1
z
z
z
z
z
.
1 M E b E a E a 2 t t
1 E a E a t
2
1
t
t
1
2
1
2
t
t
t
t
t
t
3
4
3
4
3
(24) By substituting formulas (24) and (20) into equation (7), we obtain:
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