Circular sections

Non-circular sections

Relationship between Torque, Rate of Twist and Shear Flow

Thin walled open sections

Thin walled closed sections

Multi cell sections
 

Torsion in  Circular Shafts

Shear Strain
Applied Torque
Where J is the polar moment of inertia equal to
For a solid circular section of radius R,
For a hollow circular shaft
Rate of twist 
Shear Stress

Torsion of Rectangular Sections

a and b are the wider and narrower sides of the cross-section, repectively.

Table of coefficients c₁ and c₂ (ref. Beer and Johnston)
 

Coefficients c₁ and c₂ (ref. Beer and Johnston)

a/b	c1	c2
1.0 1.2 1.5 2.0 2.5 3.0 4.0 5.0 10.0 infinity	0.208 0.219 0.231 0.246 0.258 0.267 0.282 0.291 0.312 0.333	0.1406 0.1661 0.1958 0.229 0.249 0.263 0.281 0.291 0.312 0.333

Torque and Shear Flow

Hence    i.e.

Therefore, 
for constant q,

Rate of Twist and Shear Flow

Where 
By equating work done with stored energy it can be found that rate of twist
and also noting that   (for constant q) this may be written as

 dq/dz is rate of twist or twist per unit length

t=shear stress. G = shear Modulus. t = thickness

therefore 

GJ is torsional stiffness. J is torsion constant given by

shear stress is assumed constant across the thickness
Shear flow 'q' is defined as shear load per unit width
Sehar Stress:

Shear flow 'q' is constant over the whole section
Applied Torque:
where A is the enclosed area of the cross section.
Twist per unit length:
where   is the closed integral of  along the perimeter of the cross section
Therefore,
J is the Torsion Constant given by

example for solid, thin walled open and thin walled closed shafts

N= the number of cells
N+1 is the number of unknowns [q₁, q₂....q_N, and rate of twist dq/dz]
note that the rate of twist is the same for all cells

for the first N equations 

and the final equation is given by 

hence N+1 unknowns can be solved with N+1equations