Page 136 - Civil Engineering Formulas

BEAM FORMULAS 77

TABLE 2.7 Polar Moment of Intertia and Maximum Torsional Shear
Polar moment of Maximum shear*
inertia J v max

1 4
πr 2T
2 3
2r πr
at periphery

R = 2T
208a 3
a 0.141a 4
at midpoint
of each side

T(3a – 1.8b)
b 1 b b 4 a b
2 2
ab 3 – 0.21 1 – 4
3 a 12a at midpoint
a of longer sides
20T
a 0.0214a 4 a 3
at midpoint
of each side

2TR
1 4 4
4
4
π(R – r ) π(R – r )
2r 2R 2
at outer
periphery
*T twisting moment, or torque.

Strain Energy in Torsion
For a member subjected to torsion
2
T L
U (2.40)
2JG
JG 2
U (2.41)
2L