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TorsionTorsional Deformation of a Circular Shaft
Torque is a type of moment that tends to twist a member about its longitudinal axis.
Engineers need to understand the effects of torque to properly design axles or drive shafts used in vehicles and machinery.
Torque causes shear stresses to develop in shafts.
Before Deformation
After Deformation
T
T
Torque applied to a shaft fixed at one end
Distortion and Shear Strainof an element located at a radial distance ρ from the axis of the shaft.
Shear Strain increases linearly with ρ
The Shear Stress – Strain Diagram
Linear Variation in Shear Stress
For linear-elastic shafts subjected to shear strains, Hooke’s Law applies:
The linear variation in shear strain corresponds with a linear variation in shear stress:
Polar Moment of Inertia for a Tubular Shaft
Due to the complementary property of shear, not only does the internal torqueT develop a linear distribution of shear stress along each radial line in the planeof the cross-sectional area, but also an associated shear-stress distribution isdeveloped along an axial plane.
Power Transmission
Angle of Twist
Angle of Twist – Constant Torque and Cross-Sectional Area
Positive Sign Convention for T and Ф
Statically Indeterminate Torque-Loaded Members
Torsional Stress Concentration
5.68 Known: The A-36 bolt is tightened within a hole so that the reactive torque on the shank AB can be expressed by the equation t = (kx2) N∙m/m where x is in meters.
Find: If a torque of T = 50 N∙m is applied to the bolt head, determine the following:
(a) the constant k, and
(b) the amount of twist in the 50 mm length of the shank.
Assumption: the shank has a constant radius of 4 mm.
0.00 0.01 0.02 0.03 0.04 0.05x, m
0
500
1000
1500
2000
2500
3000D
istr
ibu
ted
To
rqu
e, t
, (Nm
)/m
Problem 5.68 - distributed torque on the shank of an A-36 bolt
t = (kx2) Nm/m
0.00 0.01 0.02 0.03 0.04 0.05x, m
0
10
20
30
40
50
To
rqu
e, T
, Nm
Problem 5.68 - torque on the shank of an A-36 bolt
T = 0.4(106)x3 Nm
5.82 Known: Two shafts are connected using gears fixed to their ends. Their other ends are attached to fixed supports at A and B. They are also supported by journal bearings at C and D, which allow free rotation of the shafts along their axes.
Find: If a 500 N∙m torque is applied to the gear at E as shown, determine the rotation of the gear at B.
Data: shafts are made from A-36 steel, and they each have a diameter of 25 mm.
Schematic:
● Small gear driving a big gear - increases torque and decreases speed of driven gear
● Large gear driving a small gear - decreases torque and increases speed of driven gear
● Car going up a hill.......more torque is needed.......thus a high gear ratio is used (3:1).......but speed is reduced.
● Car traveling on a highway......more speed is desired......thus a low gear ratio is used (1:1)......but torque is reduced.
Gear Basics
2
1
2
1
1
2
r
r