Stress State ME3180C

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Stress StateME3180C

• Wayne Book
• Fall, 2008

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The Plan

• Analyze loading to get applied load at a cross
  section
• Load at a cross section give applied stress
• Applied stress is analyzed to get stress state
  the principle normal and shear stress
• The stress state will predict failure dependent
  on the material used (Chapter 5).

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Recall the Differential ElementUnder Stress
(Figures from Norton)
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Planar Stressapplied and principle stresses
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3 Mohrs Circlesfor planar stress
snormal stress on surface at angle f tshear
stress at angle ?
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3 Mohrs Circles3 dimensional stress
How do we get the 3 principle normal stresses?
Find plane at angle so that shear stresses are 0
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Matlab finds Principle Stresses
function to find the principle normal (sig) and
shear (tau) stresses sig and tau are 1x3 arrays
Enter data from Example 4-3 sx 40000 sy
-20000 sz -10000 txy 5000 tyz -1500 txz
2500 c0 sxsysz 2txytyztxz - sxtxz2
-sztxy2 c1 txy2 tyz2 txz2 - sxsy -
sysz - szsx c2 sx sy sz form array of
tensor invariants C 1 -c2 -c1 -c0 find
roots of 3rd order polynomial and sort from high
to low this gives principle normal stresses sig
-sort(-roots(C)) evaluate principle shear
stresses tau abs(sig(1)-sig(3))/2,
abs(sig(2)-sig(1))/2, abs(sig(3)-sig(2))/2
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Convenient Applied Strain Calculations

• Pure axial tension/compression
• Direct Shear
• Direct bearing
• Tearout (shear)
• Beams in bending (straight, curved, slender,
  short)
• Beams with shear
• Torsion of shafts
• Columns (slender and short, eccentric)
• Pressurized cylinders (thin/thick, with end
  effects)
• Flywheels

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Tension and Shear
Beware of small separations leading to bending
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Calculate Applied Stress DistributionUniaxial
Tension and Direct Shear
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Instances of ShearWhat area supports shear on
pin? On link and clevis?
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Beam in pure bending
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Bending from transverse (shear) loads produces
shear stress
How is it distributed across height of beam?
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Calculate Applied Stress DistributionBending
normal and shear stress
Shear stress max where normal stress is min
usually ignored. Exception torsion and bending
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Shaft in Torsion
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Calculate Applied Stress DistributionTorsion of
shaft
Complication non-circular cross sections
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Combined Applied StressBending and torsion
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Pressure Vessels
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Other Failure Modes Issuesto be reviewed in
coming classes

• Buckling
• An instability not predicted by static stress
  analysis
• Deformation
• Change in dimension under load precludes function
• Springs deform as part of function
• Stress concentration factors

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A parting thought
"The true measure of a man is how he treats
someone who can do him absolutely no good."
Samuel Johnson (1709-1784)