Civil Engineering Reference
In-Depth Information
The strain energy due to bending of the beam is given by Eq. (9.21), i.e.
L
M
2
2
EI
d
x
U
=
(ii)
0
Also, from Eq. (13.3)
EI
d
2
υ
d
x
2
M
=
(iii)
Substituting in Eq. (iii) for
υ
from Eq. (i), and for
M
in Eq. (ii) from Eq. (iii), we have
L
υ
B
π
4
L
4
EI
2
sin
2
π
x
U
=
L
d
x
0
which gives
π
4
EI
υ
B
4
L
3
U
=
The TPE of the beam is then given by
π
4
EI
υ
B
4
L
3
TPE
=
U
+
V
=
−
W
υ
B
Hence, from the principle of the stationary value of the TPE
π
4
EI
υ
B
2
L
3
∂
(
U
+
V
)
=
−
W
=
0
∂υ
B
whence
2
WL
3
π
4
EI
=
0
.
02053
WL
3
EI
υ
B
=
(iv)
The exact expression for the deflection at the mid-span point was found in Ex. 13.5
and is
WL
3
48
EI
=
0
.
02083
WL
3
EI
υ
B
=
(v)
Comparing the exact and approximate results we see that the difference is less than
2%. Furthermore, the approximate deflection is less than the exact deflection because,
by assuming a deflected shape, we have, in effect, forced the beam into that shape by
imposing restraints; the beam is therefore stiffer.
15.4 R
ECIPROCAL
T
HEOREMS
There are two reciprocal theorems: one, attributed to Maxwell, is the
theorem of
reciprocal displacements
(often referred to as Maxwell's reciprocal theorem) and the
other, derived by Betti and Rayleigh, is the
theorem of reciprocal work.
We shall see, in
fact, that the former is a special case of the latter. We shall also see that their proofs
rely on the principle of superposition (Section 3.7) so that their application is limited
to linearly elastic structures.
