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

or, since a + b = L,
U
Pab 2 2 2
= Ans.
6LEI
This example demonstrates that the strain energy for a beam can be computed with
any suitable x coordinate. For this beam, the bending moment equation for segment BC is
much easier to derive and integrate if we consider a free-body diagram taken at the far end
of the beam (around roller C).
pRoBLEmS
p17.1 Determine the modulus of resilience for aluminum alloys
with the following properties:
(a) 7075-T651
(b) 5082-H112
(c) 6262-T651
E = 71.7 GPa, σ Y = 503 MPa
E = 70.3 GPa, σ Y = 190 MPa
E = 69.0 GPa, σ Y = 241 MPa
p17.2 Determine the modulus of resilience for each of the following
metals:
(a) Red Brass UNS C23000
(b) Titanium Ti-6Al-4V
(Grade 5) Annealed
(c) 304 Stainless Steel
E = 115 GPa, σ Y = 125 MPa
E = 114 GPa, σ Y = 830 MPa
E = 193 GPa, σ Y = 215 MPa
p17.3 The compound solid steel rod shown in Figure P17.3/4 is
subjected to a tensile force P. Assume that E = 29,000 ksi, d 1 = 0.50 in.,
L 1 =18 in., d 2 = 0.875 in., L 2 = 27 in., and P = 5.5 kips. Determine
(a) the elastic strain energy in rod ABC.
(b) the strain-energy density in segments (1) and (2) of the rod.
L 2
L 1
C
B
A
P
FIGURE p17.3/4
(1)
(2)
d 1
d 2
p17.4 The compound solid aluminum rod shown in Figure
P17.3/4 is subjected to a tensile force P. Assume that E = 69 GPa,
d 1 = 16 mm, L 1 = 600 mm, d 2 = 25 mm, L 2 = 900 mm, and σ Y =
276 MPa. Calculate the largest amount of strain energy that can be
stored in the rod without causing any yielding.
p17.5 A solid 2.5 m long stainless steel rod has a yield strength
of 276 MPa and an elastic modulus of 193 GPa. A strain energy
U = 13 N ⋅ m must be stored in the rod when a tensile load P is
applied to rod. Calculate
(a) the maximum strain-energy density that can be stored in the
solid rod if a factor of safety of 4.0 with respect to yielding is
specified.
(b) the minimum diameter d required for the solid rod.
p17.6 A solid stepped shaft made of AISI 1020 cold-rolled steel
(G = 11,600 ksi) is shown in Figure P17.6/7/8. The diameters of
segments (1) and (2) are d 1 = 2.25 in. and d 2 = 1.00 in., respectively.
The segment lengths are L 1 = 36 in. and L 2 = 27 in. Determine the
elastic strain energy U stored in the shaft if the torque T C produces
a rotation angle of 4° at C.
p17.7 A solid stepped shaft made of AISI 1020 cold-rolled steel
(G = 80 GPa) is shown in Figure P17.6/7/8. The diameters of segments
(1) and (2) are d 1 = 30 mm and d 2 = 15 mm, respectively. The
segment lengths are L 1 = 320 mm and L 2 = 250 mm. Determine the
maximum torque T C that can be applied to the shaft if the elastic
strain energy must be limited to U = 5.0 J.
A
(1)
L 1
B
T B
FIGURE p17.6/7/8
(2)
L 2
C
T C
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