Civil Engineering Reference
In-Depth Information
The
principle of the stationary value of the
TPE may therefore be stated as:
The TPE of an elastic system has a stationary value for all small displacements when
the system is in equilibrium; further, the equilibrium is stable if the stationary value is a
minimum.
Potential energy can often be used in the approximate analysis of structures in cases
where an exact analysis does not exist. We shall illustrate such an application for a
simple beam in Ex. 15.11 below and in Chapter 21 in the case of a buckled column;
in both cases we shall suppose that the deflected form is unknown and has to be
initially assumed (this approach is called the
Rayleigh-Ritz method
). For a linearly
elastic system, of course, the methods of complementary energy and potential energy
are identical.
E
XAMPLE
15.11
Determine the deflection of the mid-span point of the linearly
elastic, simply supported beam ABC shown in Fig. 15.23(a).
W
A
B
C
EI
x
L
/2
L
/2
(a)
W
A
C
F
IGURE
15.23
Approximate value for
beam deflection using
TPE
υ
B
B
(b)
We shall suppose that the deflected shape of the beam is unknown. Initially, there-
fore, we shall assume a deflected shape that satisfies the boundary conditions for the
beam. Generally, trigonometric or polynomial functions have been found to be the
most convenient where the simpler the function the less accurate the solution. Let us
suppose that the displaced shape of the beam is given by
υ
B
sin
π
x
L
υ
=
(i)
in which
υ
B
is the deflection at the mid-span point. From Eq. (i) we see that
when
x
=
0 and
x
=
L
,
υ
=
0 and that when
x
=
L
/
2,
υ
=
υ
B
. Furthermore, d
υ/
d
x
=
(
π/
L
)
υ
B
cos (
π
x
/
L
) which is zero when
x
=
L
/
2. Thus the displacement function
satisfies the boundary conditions of the beam.
