An Introduction to Energy Methods

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An Introduction to Energy Methods

• David Begg

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Some Definitions

• Work Done?

• Components of Work Done?

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Potential Energy

• Is the potential for doing work relative to some
  datum.
• The potential energy of force F _at_ a relative to
  datum _at_ b

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P.E. of Spring
When the spring has been stretched to b it has
a potential for doing work relative to the DATUM
_at_ a
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P.E. of a Mass

• FOR A STABLE EQUILIBRIUM CONDITION A MINIMUM
  TOTAL POTENTIAL ENERGY LEVEL IS REQUIRED.

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P.E. of Structure and Load System

• The potential Energy of a structure in a
  displaced position relative to a DATUM

• The potential Energy of a Load System relative to
  a DATUM

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Total Potential Energy

• The Total Potential Energy (TPE) of a system
  relative to a DATUM

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Rayleigh-Ritz Method

• A method First proposed by Lord Rayleigh in 1877
• Subsequently generalized by W Ritz in 1908

• If the material is assumed to be Linear Elastic

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Contd.

• Note the Stress and the Strain are in the same
  direction

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Application to Beam Bending

• Examining the Strain Energy of Beams in a similar
  manner

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Strain Energy

• Now we have already found the Strain Energy of an
  element

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Lets Integrate! Why not?

• Summing over the whole beam

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Example

• Let the displaced shape of the beam take the form
  of a Sine function SAY!

• This MUST satisfy the kinematic Boundary
  Conditions

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General Expression for SE of Beam
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Strain Energy

• Now

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Youll never believe this!
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Strain Energy.

• The Integral therefore is

• Thus the Strain Energy of a Beam under Central
  Point Loading conditions is

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Potential Energy

• Now we know the Potential Energy of the Load in
  this case is

• Therefore the Total Potential Energy is

• And for Equilibrium

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Displacements

• This technique gives very good answers for
  displacements

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Moments..

• Values for derived actions may lead to acceptable
  values

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• A similar approach can be used for a beam with
  UDL..