Torsional stiffness

[EcoMan]

Hello,
The chapter on torsion in my Advanced Mechanics of Materials textbook from university says that, for a solid section with a given cross-sectional area A, stiffness is greatest when the centroidal polar moment of inertia K is least. What confuses me is that twist (= TL / GK) is inversely proportional to K. Wouldn't lower K mean more twist and, therefore, less stiffness?

 

[JAE]

Are you sure they didn't say that "STRESS" was greatest with the least polar moment of inertia?

Torsional stress, [τ] = Tc / K

Torsional twist, [θ] = TL/GK

 

[EcoMan]

I'm sure. Roark (1954) says the same thing.

 

[BAretired]

Unless the shape is circular, polar moment of inertia is the wrong term to use. The correct term is J or torsional constant. The expression JG is the torsional stiffness and [θ] = M_t/JG where M_t is the torsional moment and [θ] is the rotation in radians per unit length of member.

For a rectangular cross section, the maximum torsional stress occurs at the middle of the long side and is equal to M_t/J .

BA

 

[EcoMan]

You're right. I shouldn't have used the same variable (K) to denote both the polar moment of inertia and the torsional constant unless the cross section is circular. They are different for, as an example, an elliptical cross section. Consequently, twist of noncircular sections is not related to polar moment of inertia.