ch01-03 stress & strain & properties
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02 Solutions 46060 5/6/10 1:45 PM Page 10<br />
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*2–16. The square deforms into the position shown by the<br />
dashed lines. Determine the average normal <strong>strain</strong> along<br />
each diagonal, AB and CD. Side D¿B¿ remains horizontal.<br />
Geometry:<br />
D¿<br />
y<br />
D<br />
B¿<br />
B<br />
3 mm<br />
AB = CD = 250 2 + 50 2 = 70.7107 mm<br />
C¿D¿ =253 2 + 58 2 - 2(53)(58) cos 91.5°<br />
= 79.5860 mm<br />
B¿D¿ =50 + 53 sin 1.5° - 3 = 48.3874 mm<br />
53 mm<br />
A<br />
91.5<br />
C<br />
50 mm<br />
C¿<br />
x<br />
AB¿ =253 2 + 48.3874 2 - 2(53)(48.3874) cos 88.5°<br />
= 70.8243 mm<br />
50 mm<br />
8 mm<br />
Average Normal Strain:<br />
e AB =<br />
AB¿ -AB<br />
AB<br />
=<br />
70.8243 - 70.7107<br />
70.7107<br />
= 1.61A10 - 3 B mm>mm<br />
Ans.<br />
e CD =<br />
C¿D¿ -CD<br />
CD<br />
=<br />
79.5860 - 70.7107<br />
70.7107<br />
= 126A10 - 3 B mm>mm<br />
Ans.<br />
•2–17. The three cords are attached to the ring at B.When<br />
a force is applied to the ring it moves it to point B¿ , such<br />
that the normal <strong>strain</strong> in AB is P AB and the normal <strong>strain</strong> in<br />
CB is P CB . Provided these <strong>strain</strong>s are small, determine the<br />
normal <strong>strain</strong> in DB. Note that AB and CB remain<br />
horizontal and vertical, respectively, due to the roller guides<br />
at A and C.<br />
A<br />
A¿<br />
L<br />
B<br />
B¿<br />
Coordinates of B (L cos u, L sin u)<br />
Coordinates of B¿ (L cos u + e AB L cos u, L sin u + e CB L sin u)<br />
L DB¿ = 2(L cos u + e AB L cos u) 2 + (L sin u + e CB L sin u) 2<br />
D<br />
u<br />
C<br />
C¿<br />
L DB¿ = L2cos 2 u(1 + 2e AB + e 2 AB) + sin 2 u(1 + 2e CB + e 2 CB)<br />
e AB<br />
e CB<br />
Since and are small,<br />
L DB¿ = L21 + (2 e AB cos 2 u + 2e CB sin 2 u)<br />
Use the binomial theorem,<br />
L DB¿ = L ( 1 + 1 2 (2 e AB cos 2 u + 2e CB sin 2 u))<br />
= L ( 1 + e AB cos 2 u + e CB sin 2 u)<br />
Thus,<br />
e DB = L( 1 + e AB cos 2 u + e CB sin 2 u) - L<br />
L<br />
e DB = e AB cos 2 u + e CB sin 2 u<br />
Ans.<br />
10