» Thin-Walled Torsion Member
In Chapter 3, we introduced the TORSION MEMBER at right. If the torque, cross-section and material are constant over the length, then the following relationships hold:

g =

qR L

;   t =

T 2pR2t

;   t = Gg

Combining the above we get an expression for Angle of Twist:

q = TL 2pR3tG = TL JG

• J is called the Polar Moment of Inertia

» Solid and Thick-walled Torsion Members
SOLID and THICK-WALLED SHAFTS can be envisioned to be made up of many thin-walled cylinders of thickness, dr, one inside the other. The Shear Strain and Shear Stress are a function of the radial position, r:

g(r) =

rq L

=

r R

gmax

and

t(r) =

Tr J

=

r R

tmax

Combining we again get:

q = TL JG

• Note that the max. shear stress is t_max = TR/J
• Recall J depends on the shaft's cross-section.

» Discretely and Continually Varying Shafts
To solve problems involving DISCRETELY VARYING SHAFTS, break up the Shaft into lengths over which all the values of Force, Area and Modulus are constant. The total angle of rotation can then be obtained by treating each section of bar as a Uniform Torsion Member:

qtotal = qi = TiLi JiGi

Similarly, for problems involving CONTINUALLY VARYING SHAFTS, the Angle of Twist can be determined by integrating over the length of the member:

qtotal = L dq = L T(x) dx J(x) G(x)

» Power Transmission
POWER is transmitted through rotating shafts. The power transmitted by a shaft is given by the torque multiplied by the shaft's angular velocity:

P = T w S.I. Units:   P [Watts] = T [N-m] · w [rad/sec] English Units:   P [h.p.] = 2p · T [ft-lb] · N [r.p.m.] 33,000

• h.p. = Power in HorsePower (1 h.p. = 550 ft-lb/sec)
• r.p.m. = Revalutions Per Minute