Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

776
ENERgy METHOdS
For the general case in which there are n loads involved, Equation (f) is written as
∂U
∂P
i
=∆ ( i = 1, …, n)
(17.37)
i
Castigliano’s second theorem
applies to any elastic system at
constant temperature and on
unyielding supports and that
obeys the law of superposition.
The following is a statement of Castigliano’s second theorem:
If the strain energy of a linearly elastic structure is expressed in terms of the system of
external loads, then the partial derivative of the strain energy with respect to a concentrated
external load is the deflection of the structure at the point of application and in the direction
of that load.
By a similar development, Castigliano’s theorem can also be shown to be valid for applied
moments and the resulting rotations (or changes in slope) of the structure. Thus, for n moments,
∂U
∂M
i
= θ ( i = 1, …,
n)
(17.38)
i
If the deflection is required to be known either at a point where there is no unique
point load or in a direction that is not aligned with the applied load, a dummy load acting in
the proper direction is introduced at the desired point. We then obtain the deflection by first
differentiating the strain energy with respect to the dummy load and then taking the limit
as the dummy load approaches zero. Also, for the application of Equation (17.38), either a
unique point moment or a dummy moment must be applied at point i. The moment will be
in the direction of rotation at the point. Note that, if the loading consists of a number of
point loads, all expressed in terms of a single parameter (e.g., P, 2P, 3P, wL, or 2wL), and
if the deflection at one of the applied loads is what is to be determined, then we must either
write the moment equation with this load as a separate identifiable term or add a dummy
load at the point so that the partial derivative can be taken with respect to that load only.
17.12 Calculating Deflections of Trusses
by Castigliano’s Theorem
The strain energy in an axial member was developed in Section 17.3. For compound axial
members and structures consisting of n prismatic axial members, the total strain energy in
the member or structure can be computed with Equation (17.14),
U =
n 2
∑
i=
1
Fi
Li
2AE
To compute the deflection of a truss, the general expression for strain energy given by
Equation (17.14) can be substituted into Equation (17.37) to obtain
∆= ∂
∂P
∑
i
i
2
F L
2AE
where the subscripts i have been omitted. It is generally easier to differentiate before summing,
in which case we express the preceding equation as
∆=
∑
2
∂F
L
∂P
2AE