Related Variational and Energy Principles - Mechanics of Structures: Variational and Computational Methods - page 144

FIGURE 3.4
Truss.

For this structure, Castigliano's theorem, part II states that the displacement due to P
follo ws from the differentiation of the total complementary strain energy with respect to the
force P Here the total complementary strain energy, which is equal to the total strain energy,

i.e., U i =

.

U i , for linear elastic structures, is the sum of the strain energies of each member.
This is a statically determinate structure in which all members are assumed to be pin-
connected bars and are subjected only to axial forces with no bending. The complementary
strain energy in bar j is that due to its axial force N and has the value (Chapter 2, Example 2.1)
N 2 L /( 2 EA ). The total complementary strain energy is then

U i =

N 2 L

/( 2 EA ) (1)

All bars

As in the case of the beam of Example 3.6, the simplest procedure is to take the derivative of
the complementary strain energy before integration (summation), rather than after the total
complementary strain energy is computed. Then

U i
∂

= ∂

EA N ∂

L N

V b

P = ∂ P

All bars

is the vertical displacement at b .

In almo st all cases, the most rapid computation technique is to take advantage of the fact
that

P in eac h bar is equal to the axial load produced in that bar by a unit load at b
in the direction of P

∂ N /∂

P
as the rate at which the internal axial force N in the j th member changes as P changes.
Numerically, this rate of change is equal to the internal axial force produced by P set equal
to unity with all actual loads on the structure removed. This calculation involves only
the conditions of equilibrium, as is to be expected for this complementary energy-related
theorem. This approach is sometimes referred to as the unit load method which is discussed
in a subsequent section.
The required calculations can be done in tabular form. The N,

. This follows from the interpretation of the partial deriva tiv e ∂ N /∂

P entries are de-
termined using the conditions of equilibrium for the truss. Assume the bars are made of

∂ N /∂