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

pRoBLEmS
p8.1 During the fabrication of a laminated-timber arch, one of
the 10 in. wide by 1 in. thick Douglas fir [E = 1,900 ksi] planks is
bent to a radius of curvature of 40 ft. Determine the maximum
bending stress developed in the plank.
p8.2 A copper wire of diameter d = 2 mm is coiled around a
spool of radius r. The elastic modulus of the copper is E = 117 GPa
and its yield strength is 310 MPa. Determine the minimum spool
radius r that may be used if the bending stress in the wire is not to
exceed the yield strength.
p8.3 The boards for a concrete form are to be bent into a circular
shape having an inside radius of 10 m. What maximum thickness can
be used for the boards if the normal stress is not to exceed 7 MPa?
Assume that the modulus of elasticity of the wood is 12 GPa.
p8.4 A beam is subjected to equal bending moments of M z =
3,200 N ⋅ m, as shown in Figure P8.4a. The cross-sectional dimensions
(Figure P8.4b) are b = 150 mm, c = 30 mm, d = 70 mm, and
t = 6 mm. Determine
(a) the centroid location, the moment of inertia about the z axis,
and the controlling section modulus about the z axis.
(b) the bending stress at point H. State whether the normal stress
at H is tension or compression.
(c) the bending stress at point K. State whether the normal stress
at K is tension or compression.
(d) the maximum bending stress produced in the cross section.
State whether the stress is tension or compression.
M z
z
FIGURE p8.5a
p8.6 A beam is subjected to equal bending moments of M z =
240 N ⋅ m, as shown in Figure P8.6a. The cross-sectional dimensions
(Figure P8.6b) are a = 20 mm, b = 40 mm, d = 80 mm, and r =
12 mm. Determine
(a) the centroid location, the moment of inertia about the z axis,
and the controlling section modulus about the z axis.
(b) the bending stress at point H. State whether the normal stress
at H is tension or compression.
(c) the bending stress at point K. State whether the normal stress
at K is tension or compression.
(d) the maximum bending stress produced in the cross section.
State whether the stress is tension or compression.
y
y
M z
x
z
H
b 2
K
b 1
y
b 3
FIGURE p8.5b
b
d 1
d 2
d 3
M z
z
FIGURE p8.4a
y
M z
x
t
(typ)
FIGURE p8.4b
p8.5 A beam is subjected to equal bending moments of M z =
45 kip ⋅ ft, as shown in Figure P8.5a. The cross-sectional dimensions
(Figure P8.5b) are b 1 = 7.5 in., d 1 = 1.5 in., b 2 = 0.75 in., d 2 =
6.0 in., b 3 = 3.0 in., and d 3 = 2.0 in. Determine
(a) the centroid location, the moment of inertia about the z axis,
and the controlling section modulus about the z axis.
(b) the bending stress at point H. State whether the normal stress
at H is tension or compression.
(c) the bending stress at point K. State whether the normal stress
at K is tension or compression.
(d) the maximum bending stress produced in the cross section.
State whether the stress is tension or compression.
c
z
y
b
c
H
K
d
M z
z
FIGURE p8.6a
M z
p8.7 The dimensions of the beam
cross section shown in Figure P8.7
are a = 7 mm and b = 45 mm. The
internal bending moment about the z
centroidal axis is M z = 325 N ⋅ m.
What is the magnitude of the maximum
bending stress in the beam?
x
FIGURE p8.6b
p8.8 The dimensions of the double-box beam cross section
shown in Figure P8.8 are b = 150 mm, d = 50 mm, and t = 4 mm.
If the maximum allowable bending stress is σ b = 17 MPa, determine
the magnitude of the maximum internal bending moment M z that
can be applied to the beam.
a
2a
2a
a
z
y
K
r
z
FIGURE p8.7
H
a
a
y
b
d
2a
252