Bending and Torsion
August 2, 2007
Tiffany Xu and Robin Young
COSMOS- Cluster 3
Outline
•
•
•
•
•
•
Engineering Mechanics
Axial stress and strain
Poisson’s Ratio
Young’s Modulus
Bending
Shear Stress and Torsion
Engineering Mechanics
• Statics
Š The study of forces
and moments on
stationary objects
• Dynamics
Š The study of forces
and moments on the
motion of objects
• M = Px
Stress
• σ (axial stress)
• τ (shear stress)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Strain
• A measurement used to define the
deformation of a material when subjected
to stress
• ε = ΔL/L
Poisson’s Ratio
• Ratio of transverse strain to
longitudinal strain
• ν= - εtrans/εlongitudinal
Young’s Modulus
•Hooke’s Law:
F=kΔL
• E (Young’s Modulus)
Š A measure of the stiffness of a given material
Š Stiff materials have large E
Š Compliant materials have small E
• E=σ/ε (stress-strain relationship)
Outline
•
•
•
•
•
•
Engineering Mechanics √
Axial stress and strain √
Poisson’s Ratio √
Young’s Modulus √
Bending
Shear Stress and Torsion
Properties of Bending
• Compressive Stress
Š Shrinks in the axial
direction
Š Causes negative ΔL/L
• Tensile Stress
Š Expands in the axial
direction
Š Causes positive ΔL/L
Stress- Strain Curve
• Elastic
deformation
• Plastic
deformation
Stress- Revisited
• σ (axial stress) √
• τ (shear stress)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Shear Modulus
• Measure of rigidity
• Ratio of shear stress to shear strain
• Recall:
Š σ=F/A
• ε= ∆L/L
• G= FL
A∆L
Shear Stress and Torsion
• When a shaft is placed in torsion, shear
stress occurs:
Š τ = Tr/J where J is the shaft’s polar moment
of inertia
ƒ For a solid round shaft, J = πr4/2
ƒ For a hollow round shaft, J = π/2[r04 - ri4]
More about Torsion
• To find the angle of
twist…
• φ = TL
GJ
Š G= Shear Modulus
Š T= torque
Š J= polar moment of
inertia
Why is torsion important?
• Stress and stiffness
are key components
that affect a shaft’s
performance
• Shafts and beams in
cars and machinery
need to be designed to
deform or resist
deformation
Shaft 1
Shaft 2
More applications of bending
and torsion!
Left ‘I’ Beam - Solid
Stiff in bending, flexible in torsion
Flanges can be different sizes
Right ‘I’ Beam - foam filled
Adds torsional rigidity
Hard inserts can be placed in foam as
attachment points.
http://www.performance-composites.co.uk
Conclusions
• There are two types of stress:
Š Axial
Š Shear
• Young’s modulus, Poisson’s ratio, and
Shear modulus are interrelated
• These equations are used to design
efficient and successful tools and
machines
Acknowledgements
Thank you to Professor Horsley and Mike
Paskowitz for their great knowledge and
assistance with this project!
Bibliography
ƒ Beams Under Load. Feb. 2001. QUT. 28 Jul.
2007.<http://olt.qut.edu.au/bee/civil/gen/static/content/theory/context .htm>.
ƒ Crandall, Stephen H. Dahl, Norman C. Lardner, Thomas J. An Introduction
Mechanics of Solids. Edition 2. U.S: McGraw-Hill, Inc. 1978.
ƒ Engineering Stress-Strain Curve. 1999-2005. Key to Metals Task Force.
26 Jul. 2007. <http://www.key-to-steel.com/Articles/Art43.htm>.
ƒ Meaning of Poisson’s Ratio. Roderic Lakes. 19 Jul. 2001. University of
Wisconsin. 26 Jul. 2007. <http://silver.neep.wisc.edu
/~lakes/PoissonIntro.html>.
ƒ Shear and Bending. 26 Jul. 2007. <http://staffweb
.itsligo.ie/staff/bmulligan/ce1/shear.htm>.
ƒ Torsion: Deformation-Angle of Twist. 20 Jan. 1998. University of
Wisconsin-Stout. 28 Jul. 2007. <http://physics.uwstout.
edu/StatStr/Strength/Torsion/tors62.htm>.
to