PSI - Issue 41
Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 103–114 Author name / Structural Integrity Procedia 00 (2019) 000 – 000
107
5
The material properties, E , 1 and 2 , vary continuously in both longitudinal and transversal directions of the beam. The variation in transversal direction is written as 1
1 h E E
h z
2
E E
,
(24)
2
h z
2 2
1
1
,
(25)
1
1
2
h
3
h z
2
2
,
(26)
2
2
3
h
where
2 2 h z h
.
(27)
In formulae (24) – (27), z is the vertical centric axis of the beam, E , 2 are the values of E , 1 and 2 at the upper surface of the beam. The values of E , 1 and 2 at the lower surface of the beam are E , 1 and 2 , respectively. The parameters, 1 , 2 and 3 , govern the variation of E , 1 and 2 in transversal direction. The variation of E , 1 , 2 , E , 1 and 2 in longitudinal direction is expressed as 1 and
4 l E E II
I
4
E E
x
,
(28)
I
I
1
1
II
5
x
,
(29)
1
1
I
5
l
I
2
2
II
6
x
,
(30)
2
2
I
6
l
7 l E E E E I II I
7
x
,
(31)
1 II II I
I
8
x
1
1
,
(32)
1
8
l
I
9
x
2
2
,
(33)
2
2
I
9
l
where
x l 0 .
(34) 1 and
E ,
I E ,
I E ,
E ,
1 I and
2 I are the values of
1 I ,
2 I ,
2 ,
1 ,
In formulae (28) – (34),
II E ,
II E ,
E ,
2 at the left-hand end of the beam,
1 II and
2 II are the values of
1 II ,
2 II ,
1 ,
5, 6, ..., 9 i are parameters governing the
E ,
2 at the right-hand end of the beam, i where
1 and
2 ,
variation of 2 in longitudinal direction of the beam. The beam is subjected to temperature, T , that varies periodically with time as depicted in Fig. 3.The period, T T , is found as i i T t t T 1 . (35) E , 1 , 2 , E , 1 and
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