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

p17.54 Determine the minimum moment of inertia I required
for the beam shown in Figure P17.54 if the maximum beam deflection
must not exceed 35 mm. Assume that E = 200 GPa.
p17.55 Determine the minimum moment of inertia I required
for the beam shown in Figure P17.55 if the maximum beam deflection
must not exceed 0.5 in. Assume that E = 29,000 ksi.
125 kN
125 kN
1.5 kips/ft
75 kip·ft
A
B
4 m 4 m 4 m
C
D
A
15 ft
B
FIGURE p17.54
FIGURE p17.55
17.11 castigliano’s Second Theorem
(a)
Strain-energy techniques are frequently used to analyze the deflections of beams and structures.
Of the many available methods, the application of Castigliano’s second theorem, to
be developed here, is one of the most widely used. The theorem was presented in 1873 by
the Italian engineer Alberto Castigliano (1847–1884). Although we will derive it by considering
the strain energy stored in beams, it is applicable to any structure for which the
force–deformation relations are linear. 1 The method incorporates strain-energy principles
developed earlier. Further, it is remarkably similar to the virtual-work method
developed previously.
If the beam shown in Figure 17.25a is slowly and simultaneously loaded by two
forces P 1 and P 2 , with resulting deflections D 1 and D 2 , the strain energy U of the beam
is equal to the work done by the forces. Therefore,
(b)
FIGURE 17.25 Beam subjected to
an incremental load.
1 1
U = P∆ + P D
2 2
1 1 2 2
Recall that the factor ½ in each term is required because the loads build up from
zero to their final magnitude. [See Equation (17.1).]
Let the force P 1 be increased by a small amount dP 1 while force P 2 remains
constant, as shown in Figure 17.25b. The changes in deflection due to this incremental
load will be denoted dD 1 and dD 2 . The strain energy in the beam increases by the
amount ½dP 1 dD 1 as the incremental force dP 1 deflects through the distance dD 1 .
However, forces P 1 and P 2 , which remain present on the beam, also perform work as
the beam deflects. Altogether, the increase in the strain energy due to the application
of dP 1 is
1
dU = Pd 1 D 1 + Pd 2 D 2 + dP1d
D 1
(a)
2
1 Castigliano’s first theorem, which can be used to establish equations of equilibrium, will not be discussed in
this text. That theorem is a powerful method for solving problems for statically indeterminate structures and has
application in many computer-based analytical methods, such as finite-element analysis.
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