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Torsion of annular rod with longitudinal slit

I have a question regarding simple torsion of a circular shaft with a uniform cross section. Given a hollow circular shaft with inner radius R1 and outer radius R2, length L, shear modulus G, fixed-free boundary conditions and applied torque T about the central axis, the equation for the rotation angle at the end of the beam is

 phi = T*L/(J*G)

where J = pi/4*(R2^4-R1^4)

This is from an undergraduate solid mechanics textbook and very straightforward. The question now is: what if you add a slit to the cross section of the beam (see attached image), turning it into a nearly-closed "C" shape? In the limit that the thickness t of the slit goes to zero, the polar moment of area J does not change; however, intuitively the beam should now be more torsionally compliant since the inner faces of the slit can shear with respect to each other (ignore any friction between the faces). I am not sure how to model this mathematically, or if it is even possible to do so. Is there some correction factor that can be applied to the aboe equation, an "equivalent" polar moment of area for a C shape? I looked through my undergrad solid mechanics book (R.C. Hibbeler, Mechanics of Materials) and Marks' Handbook for mechanical engineers without finding anything. I would appreciate any suggestions for a resource, or just knowing if this is impossible to do analytically, if there's an empirically derived formula somewhere, etc.

Image icon torsion_rod_C.jpg566.1 KB
arash_yavari's picture

When you cut the cross section you need to worry about warping. For thin "open" cross sections there is an approximate formula (J=1/3 Lt, where L is length of t is thickness). Look at the following link:


Jayadeep U. B.'s picture

As Arash suggested, this problem needs to be analyzed using the concept of "open sections".  For that matter, the problem mentioed is a standard one, and is discussed in books of Theory of Elasticity or Advanced Mechanics of Solids (It is available in Advanced Mechanics of Solids by L S Srinath, but I am not sure whether that book is readily available outside India!).  I feel, more than warping, the main issue is that the shear stresses can not have a component perpendicular to the free surfaces (of the slit) due to the necessity of equality of cross-shears (say, Tau-xz and Tau-zx).  As you correctly guessed, the torsional compliance will be much higher for a thin shaft with a slit.  However, I don't remember, whether any ready-made formula is available.



Found the answer on page 5-53 of the "Handbook of Engineering Fundamentals" by Esbach. For a thin-walled, open-section tube of radius r and thickness t, the "equivalent" J = 2/3*pi*r*t^3 (same as the Wikipedia article). This J is much lower than pi/4(r2^4-r1^4) for the same radii. 

The table in the handbook references "Advanced Mechanics of Materials" by Seely.

 Thanks for the help,



Ben Finio
Microrobotics Lab

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