Civil Engineering Reference
In-Depth Information
The third term on the right-hand side of Eq. (15.51) results from the additional work
done by
P
1
as it is displaced through a further distance
a
12
P
2
by the action of
P
2
.If
we now remove the loads and then apply
P
2
followed by
P
1
, the strain energy,
U
2
,is
given by
P
2
2
(
a
22
P
2
)
P
1
2
(
a
11
P
1
)
U
2
=
+
+
P
2
(
a
21
P
1
)
(15.52)
By the principle of superposition the strain energy of the body is independent of the
order in which the loads are applied. Hence
U
1
=
U
2
so that
a
12
=
a
21
(15.53)
Thus, in its simplest form, the theorem of reciprocal displacements states that:
The displacement at a point 1 in a given direction due to a unit load at a point 2 in a second
direction is equal to the displacement at the point 2 in the second direction due to a unit
load at the point 1 in the given direction.
The theorem of reciprocal displacements may also be expressed in terms of moments
and rotations. Thus:
The rotation at a point 1 due to a unit moment at a point 2 is equal to the rotation at the
point 2 produced by a unit moment at the point 1.
Finally we have:
The rotation in radians at a point 1 due to a unit load at a point 2 is numerically equal to
the displacement at the point 2 in the direction of the unit load due to a unit moment at
the point 1.
E
XAMPLE
15.12
A cantilever 800 mm long with a prop 500mm from its built-in
end deflects in accordance with the following observations when a concentrated load
of 40 kN is applied at its free end:
Distance from
0
100
200
300
400
500
600
700
800
fixed end (mm)
Deflection (mm)
0
0.3
1.4
2.5
1.9
0
−
2.3
−
4.8
−
10.6
What will be the angular rotation of the beam at the prop due to a 30 kN load applied
200mm from the built-in end together with a 10 kN load applied 350mm from the
built-in end?
