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

8 in.
0.859 in.
z
53.6°
H
2.859 in.
Compressive
bending stress
K
y
4 in.
Tensile
bending stress
9,000 lb·in.
and the bending stress at point K is
0.50 in.
(typ)
Neutral
axis
⎛ Iz z − Iyz
y⎞
Iyy
Iyzz
σ K = ⎜ M y
M
2 2
⎝ II − I ⎠
⎟ + ⎛ − + ⎞
⎜
⎝ II − I
⎟
⎠
y z
yz
y z
yz
⎡
4 4
(6.7393 in. )( 2.859 in.) (9.1304 in. )(0.859 in.)
= 0 + − − + ⎤
⎢
( 9,000 lb in.)
4 4 4 2 ⎥ − ⋅
⎣ (6.7393 in. )(38.4893 in. ) − (9.1304 in. ) ⎦
=− 1,386 psi = 1,386 psi(C)
orientation of the Neutral Axis
The orientation of the neutral axis can be calculated from Equation (8.23):
tan β =
MI
MI
+ MI
y z z yz
+ MI
z y y yz
z
4
( 9,000 lb in.)(9.1304 in. )
= 0+ − ⋅
= 1.3548
4
( −9,000 lb⋅ in.)(6.7393 in. ) + 0
∴ β = 53.6°
Positive β angles are rotated clockwise from the z axis; therefore, the neutral axis
is oriented as shown in the sketch, which has been shaded to indicate the tensile
and compressive normal stress regions of the cross section.
pRoBLEmS
p8.60 A wooden beam with a rectangular cross section is subjected
to bending moment magnitudes M z = 1,250 lb ⋅ ft and M y =
460 lb ⋅ ft, acting in the directions shown in Figure P8.60. The crosssectional
dimensions are b = 4 in. and d = 7 in. Determine
(a) the maximum magnitude of the bending stress in the beam.
(b) the angle β that the neutral axis makes with the +z axis. Note
that positive β angles rotate clockwise from the +z axis.
z
FIGURE p8.60
p8.61 A hollow-core concrete plank is subjected to bending
moment magnitudes M z = 50 kip ⋅ ft and M y = 20 kip ⋅ ft, acting in the
directions shown in Figure P8.61. The cross-sectional dimensions
are b = 24 in., h = 12 in., d = 7 in., and a = 5.5 in. Determine
(a) the bending stress at B.
(b) the bending stress at C.
(c) the angle β that the neutral axis makes with the +z axis. Note
that positive β angles rotate clockwise from the +z axis.
y
M z
A B
D
M y
C
b
d
z
h
2
h
2
A
D
FIGURE p8.61
p8.62 The moment M acting on the cross section of a certain tee
beam is oriented at an angle of θ = 55° as shown in Figure P8.62.
The dimensions of the cross section are b f = 180 mm, t f = 16 mm,
d = 200 mm, and t w = 10 mm. The allowable bending stress is
165 MPa. What is the largest bending moment M that can be
applied as shown to this cross section?
M
A
z
M z
θ
D
b f
y
C
FIGURE p8.62
b
t w
M y
a
(typ)
B
d
t f
d
B
C
h
300