Castigliano's method for calculating displacements is an application of his second theorem, which states:

If the strain energy of a linearly elastic structure can be expressed as a function of generalised forceP_{i}then the partial derivative of the strain energy with respect to generalised force gives the generalised displacementw_{i}in the direction ofP_{i}.

In general, this is given by;

*w*

_{i}= ∂U/∂

*P*

_{i}

The strain energy stored in a linear elastic system due to bending is given by;

**Solved Example**

For the frame loaded as shown below, let us find the vertical deflection at point C due to bending using Castigliano's theorem.

**Solution**

*Section BC*

M

*x*= -20

*x*

U

_{1}= ∫[(-20x)

^{2}/2EI]

*dx*= ∫ -400x

^{2}/2EI = -400x

^{3}/6EI

Knowing that the limit

*x*= 1.5m;

U

_{1}= 225/EI

*Section AB*

M

*y*= (-20 × 1.5m) = 30 kNm

U

_{2}= ∫[(-30)

^{2}/2EI]

*dy*= ∫ -900/2EI = -900

*y*/2EI

Knowing that the limit

*y*= 2.5m;

U

_{2}= 1125/EI

Total strain energy = U

_{1}+ U

_{2}= (225/EI) + (1125/EI) = 1350/EI

Let the vertical deflection at point C be Î´

_{vM/sub> Work done by the externally applied load = 1/2(P × Î´) Work done = Strain energy stored in the system 1/2(20 × Î´) = 1350/EI}

_{ Î´v = 135/EI metres (adsbygoogle = window.adsbygoogle || []).push({}); }

## No comments:

## Post a Comment