chapter 3: torsion - union collegeminerva.union.edu/bucinelr/mer214/lecturenotes/mer214l10.pdf ·...

Union CollegeMechanical Engineering

ESC023: Mechanics of Materials 1

Chapter 3:Torsion

Stress ConcentrationsNoncircular MembersThin-Walled Hollow Shafts



Stress Concentrations



Determination of Stress Concentration

JrTK

JcTK max

max⋅

⋅=⋅

⋅=τ



Example 1The stepped shaft shown is supported by bearings at A and B.

Determine the maximum stress in the shaft due to the applied torques.

How can this stress be reduced?

The fillet of the junction of each shaft has a radius of 6mm



Shafts with Non-Circular Cross-Sections



Stress in Non-Circular Shafts



Stress Concentrations Factors



Example 2The 6061-T6 aluminum shaft shown has a cross-sectional area in the shape of a equilateral triangle.

Determine the largest T that can be applied to the end of the shaft.

The allowable shear stress is 8ksi

The angle of twist at the end is restricted to .02 radians.

How much torque can be applied to the shaft of circular cross section made from the same amount of aluminum.



Thin Walled TubesClosed Cross Sections

• The member is cylindrical• The cross section does not vary along the length

of the member• The cross section is closed• The wall thickness is small compared with the cross-

sectional dimensions of the member• The member is subjected to end torques only• The ends are not restrained from warping



Thin Walled TubesClosed Cross Sections

( ) ( )flowshear

0

====

Δ−Δ==∑qttt

xtxtF

BBAA

BBAAx

τττ

ττ



Average Shear Stress

( ) ( )

tAT

qAdAqdMT

dAqpdsqdstpdFpdM

2

22

2

0

0

=

===

====

∫∫

τ

τ



Angle of TwistFrom an Energy Solution

∑∫Δ

⋅⋅⋅

⋅=⋅

⋅⋅⋅

=i

i

ts

GALT

tds

GALT

22 44φ



Example

chapter 3: torsion - union collegeminerva.union.edu/bucinelr/mer214/lecturenotes/mer214l10.pdf ·...

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