ch01-03 stress & strain & properties

01 Solutions 46060 5/6/10 2:43 PM Page 47
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1–67. The prismatic bar has a cross-sectional area A. If it
is subjected to a distributed axial loading that increases
linearly from w = 0 at x = 0 to w = w 0 at x = a, and then
decreases linearly to w = 0 at x = 2a, determine the average
normal stress in the bar as a function of x for 0 … x 6 a.
x
a
w 0
a
Equation of Equilibrium:
: + ©F x = 0; -N + 1 2 a w 0
a x + w 0b(a - x) + 1 2 w 0 a = 0
N = w 0
2a A2a2 - x 2 B
Average Normal Stress:
s = N w 0
A = 2a (2a2 - x 2 )
= w 0
A 2aA A2a2 - x 2 B
Ans.
*1–68. The prismatic bar has a cross-sectional area A. If it is
subjected to a distributed axial loading that increases linearly
from w = 0 at x = 0 to w = w 0 at x = a, and then decreases
linearly to w = 0 at x = 2a, determine the average normal
stress in the bar as a function of x for a 6 x … 2a.
x
a
w 0
a
Equation of Equilibrium:
: + ©F x = 0; -N + 1 2 c w 0
a
(2a - x) d(2a - x) = 0
N = w 0
(2a - x)2
2a
Average Normal Stress:
s = N w 0
A = 2a
(2a - x)2
= w 0
(2a - x)2
A 2aA
Ans.
•1–69. The tapered rod has a radius of r = (2 - x>6) in.
and is subjected to the distributed loading of
w = (60 + 40x) lb>in. Determine the average normal stress
at the center of the rod, B.
r
w (60 40x) lb/ in.
x
r = (2 — 6
) in.
A = pa2 - 3 6 b 2
= 7.069 in 2
B
x
6
© F x = 0; N - (60 + 40x) dx = 0; N = 720 lb
L
3
3 in. 3 in.
s = 720 = 102 psi
7.069
Ans.
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