MECH 398

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MECH 398
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Loading of a Pressurized Cylinder

• TAs Markus Timusk, Wang Fenglin

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Objectives

• To estimate the axial and hoop stress in a thin
  walled cylinder, under different loading
  conditions, using strain gauge measurements.
• Other objectives?

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Stress in cylinder walls

• Function of
• Geometry and dimensions
• Material properties
• Pressure in the cylinder (gauge)

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How to determine stresses in a component

• Analytical calculation
• Numerical methods
• Strain gauges
• Photoelastic strain measurement

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Pressurized cylinders analytical solutions

• Assume thin-walled, r/t gtgt 10
• Assume long w.r.t. radius
• Assume youre not dealing with the corners

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Axial or longitudinal stress

• Take section around circumference and balance
  forces
• saxial 2 p r t p p r 2 0
• Or saxial p r / 2 t
• p pressure, r radius, t wall thickness

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Circumferential or hoop stress

• Similarly, shoop p r / t
• What does it mean?

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Apparatus in lab

• Two modes of operation

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Stresses in this apparatus

• 10 strain gauges see figure in manual
• If piston fixed to cylinder, equations apply,
  saxial and shoop biaxial stress state
• If piston not fixed to cylinder, shoop only.

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Poissons ratio

• Definition
• n - e perpendicular / e parallel
• How to find with this equipment?

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Youngs modulus

• Definition
• s E e
• How to find with this equipment?

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Combined loading and stress states

• Assume plane stress
• Principal stresses s1 and s2 maximum and
  minimum normal stresses
• Element drawn with principal stresses acting on
  sides no shear stress
• In cylinder, hoop and axial stresses are
  principal stresses

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Mohrs Circle

• Graphical method of showing stress
  transformations
• Axes are normal stress and shear stress
• q (not shown) is angle of rotation from an
  element with principal axes on its sides

• Image from http//www.efunda.com/formulae/solid_m
  echanics/mat_mechanics/mohr_circle.cfm

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Maximum in plane shear stress

• Construct square element with sides 45 to
  principal stresses
• This element has max shear stress in that plane
• in biaxial stress, for any element orientation

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Maximum Shear Stress Criterion

• This criterion for failure is somewhat more
  conservative than a criterion of yielding
  occurring when some normal stress exceeds tensile
  yield strength.
• Theorem A ductile material begins to yield at
  some critical shearing stress tcr

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Maximum Shear Stress Criterion (continued)

• For uniaxial stress, s YS
• For biaxial stress state given principal stresses

If both principal stresses have same sign
and
If principal stresses have opposite signs
See plot
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Note

• About the MSS Criterion
• Dont do it using the maximum in-plane shear
  stress.

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Factor of Safety

• What is the failure criterion?
• Maximum load, yield strength, ultimate strength,
  maximum shear stress
• FS Maximum
• Actual

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Error Propagation by RSS

• Recall from MECH 215
• If y f(x1, x2, x3)

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Error Prop. (continued)

• If you have a functional relationship
• If you have instrument errors
• Dont worry about doing multi-step RSS