Energy Method for Beam Deflection

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Energy Method for Beam Deflection - Castiglianos
Theorem
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Differential equations of the deflection curve

• Prismatic beams

BENDING MOMENT EQUATION
SHEAR FORCE EQUATION
LOAD EQUATION
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Castigilianos Theorem
The strain Energy in Beam
Take the derivative of Strain Energy with respect
to the load P
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Castigilianos Theorem
The partial derivative of the strain energy of a
structure with respect to any load to any load is
equal to the displacement corresponding to that
load.

• Load and corresponding displacement are used in a
  generalized sense. They can be
• Force vs. Translation
• Couple vs. rotation
• Other pairs

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Application of Castigilianos Theorem
To find deflection (?A) and angle of rotation
(?A) at point A
Therefore,
and
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Dummy-Load Method
To find deflection (?C) at center point C, assume
a dummy-load Q at C
Then
When Q 0
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Modified Castigilianos Theorem
We can differentiate the integral by
differentiating under the integral sign
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Example A simple beam AB supports a uniform load
of intensity q 1.5 k/ft and a concentrated load
P 5k. The load P acts at the midpoint C of the
beam. The beam has length L 8.0 ft, modulus of
elasticity E 30?106 psi, and moment of inertia
I 75.0 in4. Determine the downward deflection
at midpoint of the beam by using (1)
Castigilianos theorem and (2) modified form of
Castiligilianos theorem.