Energy Methods Applied to Stability Problems

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Energy Methods Applied to Stability Problems

• David Begg

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A Strut typical elastic stability problem

• Consider the displacement of the end of a strut
  neglecting axial shortening

Original Position
Displaced Position
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Looking at the End Deflection

• Now Let

• This may be expanded using the binomial series

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End Deflection contd.

• Neglecting terms greater than third order

• In the Limit as

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Strain and Potential Energy

• Strain Energy of Strut (beam of previous week)

• Potential Energy of Load in displaced position

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Total Potential Energy (T.P.E.)

• The total potential energy of a strut system is
  therefore

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Displaced shape of strut

• If we assume a displacement function of Sine type!

• Therefore this displacement function satisfies
  the kinematic boundary conditions.

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Differentials!
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Total Potential Energy

• Substituting

• Integrate using half angle formulae

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You didnt believe this so
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Similarly.

• Therefore T P E

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Equilibrium

• A stationary value occurs when

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Contd

• To check for type of equilibrium

• Substituting for P PCrit

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Buckling Loads PCrit

• Consider the Case when n1

• Consider the Case when n2

• Etc. for n

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• Next week Applications in Dynamics and Matrix
  Notation..