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

Plan the Solution
Begin by calculating the radial distances from the center of curvature, O, to the centroid
of the cross section and to the neutral-axis location. Then, use Equation (8.28) to calculate
the flexural stresses on the inner surface of the bar at A and on the outer surface at B.
Repeat this calculation for radial values from r = 50 mm to r = 120 mm to obtain the
flexural stress distribution. Next, determine the section modulus for the rectangle, and
calculate the bending stresses at A and B using σ x = M/S. Finally, calculate percent errors
for the stresses at the inner and outer surfaces.
SolutioN
The bar has a width of 35 mm and a depth of 70 mm; therefore, its cross-sectional area is
A = 2,450 mm 2 . The radial distance from the center of curvature, O, to the centroid of the
cross section is
75 mm
rc = 50 mm + = 85 mm
2
location of the Neutral Axis
Next, we will determine the location of the neutral axis r n from Equation (8.27):
The denominator can be integrated to yield
r
n
dA
∫ =
r
=
∫
A 50
∫
120
A
dA
r
A
(35mm) dr
r
⎛
(35mm) ln 120 ⎞
=
30.641406 mm
⎝
⎜
50 ⎠
⎟ =
Accordingly, the radial distance from the center of curvature, O, to the neutral axis of the
curved bar is
r
n
2
A 2,450 mm
= = = 79.957167 mm
dA 30.641406 mm
∫A
r
Note that it is important to carry additional significant digits with the value of r n . In
calculating bending stresses, the term r c − r n will appear in the denominator of the bending
stress equation. Since r c and r n are close in value to one another, the term r c − r n will
be relatively small and the extra significant digits for r n are needed to avoid substantial
rounding error.
(a) Bending Stresses
The curved bar is subjected to a bending moment M = 4,500 N ⋅ m. The moment is positive,
since it creates compressive normal stress on the inner surface (i.e., at radius r i ) of the curved
bar. (Note that the moment reduces the radius of curvature of the bar, making the curve
tighter.)
The bending stress is determined from Equation (8.28),
Mr ( n − r)
σ x =−
rA( r − r )
where r is the distance from the center of curvature to the location where the stress is to
be determined.
c
n
311