Torsion: Shear Stress

1
Torsion Shear Stress Twist (3.1-3.5)

• MAE 314 Solid Mechanics
• Yun Jing

2
Torsion of Circular Shafts

• In this chapter, we will examine uniaxial bars
  subject to torque.
• Where does this occur?

Transmission Shaft
Force Couples
3
Torsion of Circular Shafts

• We assume
• Bar is in pure torsion
• Small rotations (the length and radius will not
  change)
• How does the bar deform?
• Cross-section of the bar remains the same shape,
  bar is simply rotating.
• Cross-section remains perpendicular to axis of
  cylinder (cylinder does not warp).

Not true for most non-circular bars
4
Angle of Twist

• Deformation of a circular shaft subjected to pure
  torsion
• Fix left end of shaft
• A moves to A
• ? angle of twist (in radians)
• What are the boundary conditions on ??
• ?(x) 0 at x 0
• ?(x) ? at x L
• For pure torsion, ? is linear.

x
5
Shearing Strain

• Calculate the surface shear strain in
  thecylinder.
• For pure torsion ?(x) ?x / L, so

6
Shearing Strain
Maximum shear strain on surface

• The maximum shear strain on the surface ofthe
  cylinder occurs when ?c.
• We can express the shearing strain at
  anydistance from the axis of the shaft as

7
Shearing Strain

• We can also apply the equation for
  maximumsurface shear strain to a hollow circular
  tube.
• This applies for all types of materials elastic,
  linear, non-linear, plastic, etc.

c
c
8
Elastic Shearing Stress

• Calculate shear stress in a bar made of linearly
  elastic material.
• Recall Hookes Law for shearing stress tG?

9
Torque

• We still need to relate t to the applied torque
  T, which is generally the known, applied load.
• First, find the resultant moment acting on a
  cross-section and set this equal to T.

c
10
Torque

• Continuing from previous slide
• Where J is the polar moment of inertia of the
  cross section of the bar (see Appendix A.3 in
  your textbook).
• Plug this into the equation for tmax.

11
Torque

• For a non-uniform bar
• For a continuously varying bar

12
Inclined Plane

• Cut a rectangular element along the plane at an
  angle ?.

13
Inclined Plane
x
y

• Sum forces in x-direction.
• Sum forces in y-direction.

14
Inclined Plane

• tmax occurs at ? 0º, 90º
• smax occurs at ? 45º
• tmax smax
• When s? is max, t? 0, and when t? is max, s?
  0.

15
Example Problem

• Part 1. For the 60 mm diameter solid cylinder and
  loading shown,
• determine the maximum shearing stress.
• Part 2. Determine the inner diameter of the
  hollow cylinder , of 80 mm
• outer diameter, for which the maximum stress is
  the same as in part 1.

16
Example Problem

• Part 1. For the aluminum shaft shown (G 27
  GPa), determine the torque
• T that causes an angle of twist of 4o.
• Part 2. Determine the angle of twist caused by
  the same torque T in a solid
• cylindrical shaft of the same length and
  cross-sectional area.

17
Torsion Statically Indeterminate Problems and
Transmission Shafts (3.6-3.8)

• MAE 314 Solid Mechanics
• Yun Jing

18
Statically Determinate Problems

• Find the maximum shearing stress in each bar.

T3
T2
T1
19
Statically Indeterminate Problems

• Method for torsion is the same as the method for
  statically indeterminate axial load deflection
  problems.
• Apply what youve already learned
• M R N
• M number of compatibility equations needed
• R number of unknown reactions (or internal
  stresses)
• N number of equilibrium equations
• Compatibility equations for a torsion problem are
  based on angle of twist.

20
Statically Indeterminate Problems

• Find the largest torque T0 that can be applied to
  the end of shaft
• AB and the angle of rotation of the end A of
  shaft AB. Allowable shearing stress is

21
A circular shaft AB consists of a 10-in.-long,
7/8-in.-diameter steel cylinder, in which a
5-in.long,5/8-in.-diameter cavity has been
drilled from end B. The shaft is attached to
fixed supports at both ends, and a 90 lb.ft
torque is applied at its midsection. Determine
the torque exerted on the shaft by each of the
supports.
22
Transmission Shafts

• In a transmission, a circular shaft transmits
  mechanical power from one device to
  another.
• ? angular speed of rotation of the shaft
• The shaft applies a torque T to another device
• To satisfy equilibrium the other device applies
  torque T to the shaft.
• The power transmitted by the shaft is

Generator
Turbine
23
Transmission Shafts

• Units for PT?
• ? rad/s
• T Nm (SI)
• T ftlb (English)
• P Watts (1 W 1 Nm/s) (SI)
• P ftlb/s (1 horsepower hp 550 ftlb/s)
  (English)
• We can also express power in terms of frequency.

24
Example Problem

• A 1.5 meter long solid steel shaft of 22 mm
  diameter is to
• transmit 12 kW. Determine the minimum frequency
  at which the
• shaft can rotate, knowing that G 77.2 GPa, that
  the allowable
• shearing stress is 30 MPa, and that the angle of
  twist must not
• exceed 3.5o.

25
Stress Concentrations in Circular Shafts

• Up to now, we assumed that transmission shafts
  are loaded at the ends through solidly attached,
  rigid end plates.
• In practice, torques are applied through flange
  couplings and fitted keyways, which produce high
  stress concentrations.
• One way to reduce stress concentrations is
  through the use of a fillet.

Fitted keyway
Flange coupling
26
Stress Concentrations in Circular Shafts

• Maximum shear stress at the fillet
• Tc/J is calculated for the smaller-diameter shaft
• K stress concentration factor

Fillet
27
Example Problem

• The stepped shaft shown rotates at 450 rpm.
  Knowing that r 0.25 in,
• determine the maximum power that can be
  transmitted without
• exceeding an allowable shearing stress of 7500
  psi.