Torsion

Torque: A moment that twists a member around the longitudinal axis. A force that rotates
about an objects axis.


Torsion Formula: For circular shafts that are homogeneous and linear-elastic...

= Tρ/J

where (tao) = shear stress in circular shaft.

• T = the resulting internal torque which acts on the cross section of the member.
• ρ(rho) = a distance from the center of shaft where shear stress is to be determined.
• J = (pi/2) r^4 -- r is the radius for a solid shaft.
• J = (pi/2) (ro^4 - ri^4) -- ro is the outer radius and ri is the inner radius for a tubular shaft.
  

Shear stress will be zero along the shafts longitudinal axis. -- = 0
Shear stress will be a maximum at the surface of the shaft. -- = Tr/J


These shafts and tubes can be used to transmit power.
p = Tω

• p is the power, measured in watts, foot-pounds per second, or horsepower.
• T is the applied torque measured in newton-meters (N-m).
• ω is the angular velocity measured in radians per second and 2 (pi) f, where f is the frequency. Frequency can be expressed as Hertz or cycles per second.

Angle of Twist ϕ

ϕ = ∑(TL)/(JG) -- for constant torque and constant cross section throughout the shaft. The
summation come from the sum of each segment for which T, J, and G are
constant/continuous. Method of sections typically used.

• ϕ is the angle of twist of one end of the shaft with respect to the other end of the shaft,
• measured in radians.
• T is the internal torque.
• J is the shafts polar moment of inertia.
• G is the shear modulus of elasticity for the type of material that make up the shaft.

ϕ = ∫ (T(x)dx) / (J(x)G) -- for a circular cross section that varies. Integrated from zero to L.
Where L is the length of the shaft.

• ϕ is the angle of twist of one end of the shaft with respect to the other end of the shaft,
• measured in radians.
• T is the internal torque at a position x.
• J is the shafts polar moment of inertia as a function of x.
• G is the shear modulus of elasticity for the type of material that make up the shaft.

What if the shaft is not circular? (Ellipse, Square, and Equilateral triangle)

A shaft that has torque applied to it and is not axisymmetric will develop a complex warping in the cross sections. For these 3 shapes the maximum shear stress takes place on the surface of the cross section closest to the center of the shaft.

Ellipse: semi-major axis length a max = 2T / (pi)ab^2
semi-minor axis length b ϕ = (a^2 + b^2)TL / (pi)a^3 b^3 G

Square: sides length a. -- max = 4.81T / a^3
ϕ = 7.1TL / a^4 G

Equilateral Triangle: sides length a. -- max = 20T / a^3
ϕ = 46TL / a^4 G

A circular shaft is the most efficient because it'll have a smaller maximum shear stress(max ) and angle of twist.


Closed Cross Sections of Thin-Walled Tubes
Shear Flow: q = avg t

• avg is the average longitudinal shear stress.
• t is the thickness.

Average shear stress acting over the thickness: avg = T / 2t A

• T is the Resultant internal torque.
• t is the thickness.
• A is the area measured from the center of tube to the middle of the tube's thickness.

Statically Indeterminate: equations of equilibrium for moments is not enough to find unknown
torques acting on the shaft. (1 equation with multiple unknowns)
Therefore, we will say the angle of twist for one end of the shaft with respect to the other end to be zero. ϕ1/2 = 0



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