Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 1
Chapter 3:Torsion
Stress ConcentrationsNoncircular MembersThin-Walled Hollow Shafts
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 2
Stress Concentrations
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 3
Determination of Stress Concentration
JrTK
JcTK max
max⋅
⋅=⋅
⋅=τ
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 4
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
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 5
Shafts with Non-Circular Cross-Sections
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 6
Stress in Non-Circular Shafts
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 7
Stress Concentrations Factors
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 8
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.
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 9
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
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 10
Thin Walled TubesClosed Cross Sections
( ) ( )flowshear
0
====
Δ−Δ==∑qttt
xtxtF
BBAA
BBAAx
τττ
ττ
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 11
Average Shear Stress
( ) ( )
tAT
qAdAqdMT
dAqpdsqdstpdFpdM
2
22
2
0
0
=
===
====
∫∫
τ
τ
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 12
Angle of TwistFrom an Energy Solution
∑∫Δ
⋅⋅⋅
⋅=⋅
⋅⋅⋅
=i
i
ts
GALT
tds
GALT
22 44φ
Union CollegeMechanical Engineering
ESC023: Mechanics of Materials 13
Example