Civil Engineering Reference
In-Depth Information
where
-½
II
ª
II
II
º
-½
II
- ½
II
t
t
t
t
u
°°
°°
° °
II
II
I
^`
1
11
12
^`
10
^`
1
II
t
;
K
;
t
;
u
«
»
®¾
®¾
® ¾
c
II
II
II
co
II
c
II
t
«
t
t
»
t
u
°°
°°
° °
¯¿
¬
¼
¯¿
¯ ¿
2
21
22
20
2
(11.22)
-½
II
ª
II
II
º
-½
II
u
u
u
u
°°
°°
II
II
^`
3
II
31
32
^`
30
x
;
A
;
x
«
»
®¾
®¾
c
co
II
II
II
II
u
«
u
u
»
u
°°
°°
¯¿
¬
¼
¯¿
4
41
42
40
The equations of compatibility or preservation of heat at the interface can be written
as
I
II
c
I
c
II
c
-
½
-
½
-
½
-
½
-
½
t
t
u
u
u
®
¾
®
¾
®
¾
®
¾
®
¾
1
c
2
1
2
2
0
;
(11.23)
I
II
c
I
II
c
¯
¿
¯
¿
¯
¿
¯
¿
¯
¿
t
t
u
u
u
2
c
1
2
c
1
3
Substituting (11.16) and (11.22) into (11.23) we obtain
^` ^`
0
c
(11.24)
K
u
t
where
ª
I
II
I
II
º
-½
-½
I
t
t
t
t
u
t
°°
°°
^`
^`
11
22
12
21
2
10
(11.25)
K
«
»
;
u
;
t
®¾
®¾
c
I
II
I
II
I
«
t
t
t
t
»
u
t
°°
°°
¬
¼
¯¿
¯¿
22
11
21
12
3
20
This system can be solved for the interface unknowns. The calculation of the other
unknowns is done separately for each region. For region I we have
I
I
I
I
(11.26)
^`
^`
^`
^`
I
I
t
K
u
;
x
A
u
c
c
f
c
Whereas for region II
II
II
II
II
II
II
^` ^`
^`
^` ^`
^`
II
II
(11.27)
t
t
K
u
;
x
x
A
u
c
c
0
c
f
f
0
c
If we consider the equivalent elasticity problem of a cantilever beam, we see (Figure
11.7) then for region
II
the problem where the interface displacements are fixed gives
the tractions at the interface corresponding to a shortened cantilever beam. If
u
x
=1 is
applied only a rigid body motion results and therefore no resulting tractions at the
interface occur. The application of
u
y
=1 however will result in shear tractions at the
interface.
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