Related Variational and Energy Principles - Mechanics of Structures: Variational and Computational Methods - page 142

The variations P k are arbitrary, except that they must satisfy the equilibrium conditions,

so that each term in the parentheses must vanish. Then

δ

U i
∂

∂

P k =

V k (3.10)

In vector notation,

U i
∂

∂

P =

V (3.11)

This is Castigliano's theorem, part II. This theorem asserts that:

If the complementary energy of a body is expressed in terms of the forces, then the first
partial derivative of the complementary energy, with respect to any one of the forces, is
equal to the corresponding displacement at the point where the force is located.

The subscript k is arbitrary in that it can represent any force from 1 to n . To utilize this

theorem, it is necessary to express U i in terms of the forces P .
The theorem implies that any temperature distribution to which the structure is subjected
must remain constant. This can be indicated by writing Eq. (3.11) as U i /∂

.
This theorem applies for linear and nonlinear elastic bodies. If large displacements are to
be taken into account, it is necessary to appropriately redefine U i

(∂ P ) T = constant = V

Sometimes the theorem
is referred to as Engesser's 1 first theorem and if the structure is linearly elastic, so that U i

.

=

U i

and

∂ U i

∂ P =

V (3.12)

then the theorem is almost always called Castigliano's second theorem.
“Forces” as used here include both forces and moments. Thus, the theorem may also take
the form

U i ∂

M k = θ k

(3.13)

∂

In summary, Castigliano's theorem, part II can be used to compute displacements and
slopes of surface displacements at the locations of concentrated forces or moments.

EXAMPLE 3.6 Deflection of a Cantilevered Beam
Find the deflection and slope at the free end of the linearly elastic cantilevered beam shown
in Fig. 3.3a.
For a linearly elastic structure, U i =

U i . Then Castigliano's theorem, part II in the form

∂ θ 0 are the deflection and slope at

the free end. As shown in Chapter 2, Example 2.1, the complementary strain energy in a

U i /∂ P = w 0 and ∂ U i /∂ M 0 = θ 0 applies, where w 0 and

1 Friedrich Engesser (1848-1931) was a German engineer who made many significant contributions to the analysis
of statically indeterminate systems. He began his career as a railroad design engineer specializing in bridge design.
He worked extensively in developing theories for the buckling of members, in particular, lateral instability. He
presented the general form of Castigliano's theorem, part II, in 1889 in “ Uber Statisch Unbestimmte Trager bei
beliebigem Formanderungs-Gesetz,” Z. Arch. Ing. Ver. Hannover , 35, 733-744, 1889.