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

I xy
(38.8, 32.3) y
1.654 in.
y
R = 38.5 80.8 − 59.8 = 21.0
p2
axis
p1 axis
I p2
(21.3, 0)
(59.8, 0) C
2θ p = 56.97°
(98.3, 0)
I p1
I x , I y
8 in.
28.5°
x
|−32.3|
1 in.
2.654 in.
x (80.8, −32.3)
1 in. 5 in.
6 in.
Using the coordinates of point x and center C, the radius R of the circle can be computed
from the Pythagorean theorem:
R =
The principal moments of inertia are given by
2
⎛80.8 − 59.8⎞
2
⎜ ⎟ + ( − 32.3) = 38.5
⎝ 2 ⎠
I = C + R = 59.8 + 38.5 = 98.3 and I = C − R = 59.8 − 38.5 = 21.3
p1 p2
The orientation of the principal axes is found from the angle between the radius to
point x and the horizontal axis:
−32.3
tan2θ p =
= 1.538095
80.8 − 59.8
∴ 2θ p = 56.97°
Note that the absolute value is used in the numerator because only the magnitude of 2θ p
is needed here. From inspection of Mohr’s circle, it is evident that the angle from point x
to I p1 turns in a counterclockwise sense.
Finally, the results obtained from the Mohr’s circle must be referred back to the
actual unequal-leg angle shape. Since the angles found in Mohr’s circle are doubled, the
angle from the x axis to the axis of maximum moment of inertia is θ p = 28.5°, turned in a
counterclockwise direction. The maximum moment of inertia for the unequal-leg angle
shape occurs about the p1 axis. The axis of minimum moment of inertia I p2 is perpendicular
to the p1 axis.
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