Civil Engineering Reference
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S
y
tQ
()
F
x
UPQ
(, )
x
UPQ
(, )
dS
dS
x
tQ
()
Q
Q
Q
F
x
UPQ
(, )
x
UPQ
(, )
Q
P
P
P1
x
Load case 1
Load case 2
Figure 13.2
Application of Betti´s theorem including the effect of internal concentrated forces
The work done by the loads of load case 1, times the displacements of load case 2,
W
12
is computed by
³
Wt
(
QUP Qt
,
QUP QdS Q
,
)
(
)
12
x
xx
y
xy
S
(13.20)
FU
(, )
P Q
F QU
()
(, )
P Q
x x
y
y
The work done by the displacements of load case 1 times the forces of load case 2,
W
21
is the same as explained in Chapter 5. Using Betti's theorem the following integral
equation is obtained, which includes the effect of the concentrated laod:
³
³
u
P
U
P Q
,
t
Q dS
ȉ
P Q
,
u
Q dS
U
P Q
,
F
Q
(13.21)
S
S
where
§
·
F
()
()
Q
x
F
¨
¨
¸
(13.22)
¸
F
Q
©
¹
y
The discretised form can be written as
E
N
E
N
¦¦
¦¦
11
ee
ee
cu
P
'
T
u
'
U
t
FU
(,)
P Q
(13.23)
i
ni
ni
i
n
n
en
en
11
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