ch01-03 stress & strain & properties

02 Solutions 46060 5/6/10 1:45 PM Page 18
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*2–32. The bar is originally 300 mm long when it is flat. If it
is subjected to a shear strain defined by g xy = 0.02x, where
x is in meters, determine the displacement ¢y at the end of
its bottom edge. It is distorted into the shape shown, where
no elongation of the bar occurs in the x direction.
y
y
300 mm
x
Shear Strain:
dy
dx = tan g xy ; dy
dx
= tan (0.02 x)
L0
¢y
300 mm
dy = tan (0.02 x)dx
L
0
300 mm
¢y = -50[ln cos (0.02x)]| 0
= 2.03 mm
Ans.
•2–33. The fiber AB has a length L and orientation u. If its
ends A and B undergo very small displacements u A and v B ,
respectively, determine the normal strain in the fiber when
it is in position A¿B¿.
y
B¿
v B
B
Geometry:
L A¿B¿ = 2(L cos u - u A ) 2 + (L sin u + y B ) 2
A
u A A¿
L
u
x
= 2L 3 + u A 2 + y B 2 + 2L(y B sin u - u A cos u)
Average Normal Strain:
e AB = L A¿B¿ - L
L
= 1 + u A 2 + y 2 B
A L 2 + 2(y B sin u - u A cos u)
- 1
L
Neglecting higher terms
u A
2
and
y 2 B
e AB = B1 + 2(y B sin u - u A cos u)
L
Using the binomial theorem:
1
2
R - 1
e AB = 1 + 1 2 ¢ 2y B sin u
L
- 2u A cos u ≤ + . . . - 1
L
= y B sin u
L
- u A cos u
L
Ans.
18