4' Torsion

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4. Torsion

• ENGR 310 Mechanics of Materials
• Fall, 2007
• Tomasz Arciszewski

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Our Focus

• Analysis of stresses and deformations in members
  subjected to pure torsion
• Pure torsion occurs when internal forces in a
  given member can be reduced to a single torsional
  moment whose vector is positioned on the
  longitudinal axis

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Torsional Moment Torsion

• A couple vector situated on the longitudinal axis
  of a member
• Also called a torque
• Torsion is caused by action of two oppositely
  directed torsional moments applied to a given
  member (both active or one active and one
  reactive)

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Example of Torsion
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Assumptions

• Perfectly linear elastic behavior
• Circular x-section (solid or hollow)
• No warping effects
• Small rotations
• Constant length
• Constant diameter
• (First order theory)

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Warping First Case

• A highly deformable member
• A circular x-section
• A grid of parallel circles and longitudinal lines
  on the outer surface

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Warping First Case

• Two torques are applied
• Circles remain circles
• Longitudinal lines become twisted
• All angles are equal
• End x-section remains flat

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Warping Second Case

• A highly deformable member
• A square x-section
• A grid of parallel circles and longitudinal lines
  
• on the outer
• surface

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Warping Second Case

• Two torques are applied
• Grid lines become curves
• End x-section becomes a surface
• All x-sections
• become surfaces (a plane becomes a surface)

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Warping of X-sectional Area
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Warping

• is understood as complex deformations of elastic
  members subjected to torsion which include
  deformations of the x-sections, which are
  transformed from planes into surfaces. It is
  accompanied by the occurrence of bi-moments and a
  complex distribution of both normal and shear
  stresses

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Angle of Twist

• A prismatic circular member
• Fixed at x 0
• Free end
• Loaded by a torque at the free end
• Angle ?(x) - angle of twist
• It depends on x

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Sign Convention
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Deformations

• A slice of shaft (?x length)
• A small element inside the member in radial
  distance ? is considered
• Both rear face ?(x) and front face rotate ?(x)
  ??
• Rotation difference ?? causes shear strain

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Shear Strain
After
Before
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Shear Strain

• Initial angle between AB and AC is ?/2 (90)
• After deformation, this angle is reduced to ?
• Shear strain is
• ? ?/2 - lim ? when
• C approaches A and
• B approaches A

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Shear Strain

• Consider triangles BDA and BDC
• Assume that ?x ? dx and ?? ? d?
• ?BDC ? BD ? d?
• ?BDA ? BD dx ?
• Result
• ? d? dx ?
• ? ? d? /dx

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Shear Strain

• For a given x-section
• d? /dx is constant
• Shear strain varies only with the radial distance
  ?
• Linear shear strain distribution
• Maximum shear strain occurs on the surface

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Shear Strain Distribution

• Rmax c
• ? max - occurs on the outer surface
• ? (?/c) ? max
• Result valid also for circular tubes within the
  material

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Torsion Formula

• Linear-elastic material
• Hookes law applies
• ? G ?
• ?max occurs on the outer surface
• ? (?/c) ?max

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Torsion Formula

• Equilibrium of external torque and internal
  torque (resultant
• of shear stresses)
• T ?A ?(?dA) ?A ?(?/c) ?max dA
• ?max /c constant
• T ?max /c ?A ?2 dA
• ?A ?2 dA - J, polar moment of inertia
• ?max Tc/J
• ? T?/J

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Polar Moment of Inertia

• A differential/ elementary ring is considered
• J ?A ?2 dA p/2 c4
• Units - mm2
• or m2

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Shear Stress Distribution

• X-section linear distribution
• Complimentary stresses in action
• Shear stress occur on axial planes (normal to
  the x-section

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Shear Stress Distribution

• Shear stress occur on axial planes (normal to
  the x-section
• Linear variation

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Wood Shaft Failure

• Tendency to split along the axial plane or along
  fiber
• Anisotropic material

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Hollow Shaft

• Inner radius ci
• Outer radius co
• J p/2 (co4 - ci4)
• Linear distribution of stresses within
• material

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Hollow Shaft Failure by Yielding
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Absolute Maximum Torsional Stress

• Maximum shear stress occurs at the outer surface
• When the torque distribution is not uniform, a
  torque diagram is to be drawn along a member

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Absolute Maximum Torsional Stress

• Torque signs are determined using the right hand
  rule
• Individual segments are to be examined to find
  the maximum torque and the related maximum shear
  stress

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Angle of Twist

• Angle of twist is important in
• Car drive shafts
• Oil wells
• Asymmetric complex structural systems

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Angle of Twist

• It is the angle ? of relative rotation of one
  end of a shaft with respect to its other end
• Circular x-section
• X-section gradually changes along x
• Linear-elastic behavior
• Saint-Venants principle in place

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Angle of Twist Calculation

• A differential disk is considered
• T is a function of x, T(X)
• Relative rotation is represented by d?
• Shear strain for an element in distance ? is
• d? ? dx/?
• Hookes law ? G ?

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Angle of Twist Calculation

• d? T(x)dx/J(x)G
• ? ?0LT(x)dx/J(x)G

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Angle of Twist Special Case

• Constant torque and x-sectional area
• ? TL/(JG)

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Shear Modulus Determination
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Angle of Twist Multi-segment Shaft

• Several segments with various x-sections
• Variable torque, but constant for the individual
  segments
• ? ?TL/(JG)

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Sign Convention