Civil Engineering Reference
In-Depth Information
THEOREM OF RECIPROCAL DISPLACEMENTS
In a linearly elastic body a load,
P
1
, applied at a point 1 will produce a displacement,
1
, at the point and in its own line of action given by
1
=
a
11
P
1
in which
a
11
is a
flexibility coefficient
which is defined as the displacement at the point
1 in the direction of
P
1
produced by a unit load at the point 1 in the direction of
P
1
.It
follows that if the elastic body is subjected to a series of loads,
P
1
,
P
2
,
...
,
P
k
,
...
,
P
r
,
each of the loads will contribute to the displacement of point 1. Thus the corresponding
displacement,
1
, at the point 1 (i.e. the total displacement in the direction of
P
1
produced by all the loads) is then
1
=
a
11
P
1
+
a
12
P
2
+···+
a
1
k
P
k
+···+
a
1
r
P
r
in which
a
12
is the displacement at the point 1 in the direction of
P
1
produced by a
unit load at 2 in the direction of
P
2
, and so on. The corresponding displacements at
the points of application of the loads are then
!
1
=
a
11
P
1
+
a
12
P
2
+···+
a
1
k
P
k
+···+
a
1
r
P
r
2
=
a
21
P
1
+
a
22
P
2
+···+
a
2
k
P
k
+···+
a
2
r
P
r
.
(15.49)
"
k
=
a
k
1
P
1
+
a
k
2
P
2
+···+
a
kk
P
k
+···+
a
kr
P
r
.
r
=
a
r
1
P
1
+
a
r
2
P
2
+···+
a
rk
P
k
+···+
a
rr
P
r
or, in matrix form
.
/
!
.
/
!
1
2
.
k
.
r
a
11
a
12
...
a
1
k
...
a
1
r
P
1
P
2
.
P
k
.
P
r
a
21
a
22
...
a
2
k
...
a
2
r
.
=
(15.50)
0
"
0
"
a
k
1
a
k
2
...
a
kk
...
a
kr
.
a
r
1
a
r
2
...
a
rk
...
a
rr
which may be written in matrix shorthand notation as
{
}=
[
A
]
{
P
}
Suppose now that a linearly elastic body is subjected to a gradually applied load,
P
1
,
at a point 1 and then, while
P
1
remains in position, a load
P
2
is gradually applied at
another point 2. The total strain energy,
U
1
, of the body is equal to the external work
done by the loads; thus
P
1
2
(
a
11
P
1
)
P
2
2
(
a
22
P
2
)
U
1
=
+
+
P
1
(
a
12
P
2
)
(15.51)
