engineering core courses

Solid Mechanics II
 Course homepage
Solid Mechanics II
 Course homepage
C6: Energy Methods
6.1 Elastic Strain Energy for Various Loadings
- Theory - Example
6.2 Conservation of Energy
- Theory - Example - Question 1 - Question 2 - Question 3
6.3 Virtual Work
- Theory - Example - Question 1 - Question 2 - Question 3
6.4 Castigliano’s Theorem
- Theory - Example - Question 1

C6.4 Castigliano’s Theorem

Alberto Castigliano, an Italian railroad engineer, who through a stroke of brilliance derived a new method to calculate displacements and slopes.

His method is simple and can be used to calculate Δ or slope θ at any point in the structure as well, similar to virtual work. However the method is quite difficult to explain using words, so we’ll present the formulas and teach you by working through examples and questions.

Castigliano’s theorem on trusses

The theorem can be applied to find the displacement Δ in trusses via:


Castigliano's theorem on trusses; displacement formula

C6.4 Castigliano’s Theorem

Alberto Castigliano, an Italian railroad engineer, who through a stroke of brilliance derived a new method to calculate displacements and slopes.

His method is simple and can be used to calculate Δ or slope θ at any point in the structure as well, similar to virtual work. However the method is quite difficult to explain using words, so we’ll present the formulas and teach you by working through examples and questions.

Castigliano’s theorem on trusses

The theorem can be applied to find the displacement Δ in trusses via:


Castigliano's theorem on trusses; displacement formula

Note:

  • N, L, A, E all have the same meanings as the axial strain energy.
  • P is a variable (unknown) load that we apply at the point where we want to calculate the displacement Δ of interest.
  • (∂N/∂P) is the 1st partial derivative of the internal normal force with respect to the variable load P.

Complex? Not to worry, we’ll be looking at an example real soon. But first, let’s look at the Castigliano’s theorem applied to beams.

Castigliano’s theorem on beams

The theorem can be applied to find the displacement Δ or slope θ in beams via:


Castigliano's theorem on beams; displacement formula
Note:
  • Again, M, E, I all have the same meanings as the bending strain energy.
  • P is a variable (unknown) load that we apply at the point where we want to calculate the displacement Δ of interest.
  • We apply M' instead as the variable moment if we’re interested in the slope θ at the point of interest.
  • (∂M/∂P) and (∂M/∂M') are the 1st partial derivatives of the internal bending moment with respect to the variable load P and variable moment M'.

Okay! Let’s look at an example now.

Note:

  • N, L, A, E all have the same meanings as the axial strain energy.
  • P is a variable (unknown) load that we apply at the point where we want to calculate the displacement Δ of interest.
  • (∂N/∂P) is the 1st partial derivative of the internal normal force with respect to the variable load P.

Complex? Not to worry, we’ll be looking at an example real soon. But first, let’s look at the Castigliano’s theorem applied to beams.

Castigliano’s theorem on beams

The theorem can be applied to find the displacement Δ or slope θ in beams via:


Castigliano's theorem on beams; displacement formula
Note:
  • Again, M, E, I all have the same meanings as the bending strain energy.
  • P is a variable (unknown) load that we apply at the point where we want to calculate the displacement Δ of interest.
  • We apply M' instead as the variable moment if we’re interested in the slope θ at the point of interest.
  • (∂M/∂P) and (∂M/∂M') are the 1st partial derivatives of the internal bending moment with respect to the variable load P and variable moment M'.

Okay! Let’s look at an example now.

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