Castiglianos Theorem: Elastic Body Displacements & Swing Analysis

[umarfarooq]

Homework Statement

a) State Castiglianos theorem for translational and rotational displacements of an elastic body, stating precisely the meanings of the terms.

b) A swing in a childrens play area is constrcuted from a steel tube bent into a quarter circle of radius R. One end is rigidly fixed to the ground with the tangent to the circle vertical, and the swing attached to the other end. Assuming that the beam has a section constant EI, derive experssions for the vertical and horizontal displacements of the swing when a downwards load P is applied to it.

Homework Equations

dV=dU/dL=d(int(M^2/(2*E*I)*r*dTheta,Pi->0)/dP

The Attempt at a Solution

sorry but I am completely baffled

 

[Pyrrhus]

Where are you stuck??

You need to show your work, may i recommend to use your moment equation in function of the angle the radius makes with the vertical.

 

[umarfarooq]

okay, i think my answer is wrong but this is what I've got.
the moment is M(theta) is PRCos(theta) + F(R-RSin(theta)) where f is ficticious i know so do i disregard that.

Therefore M^2(theta) = R^2(P^2Cos^2(theta) + F^2 - F^2Sin^2(Theta).

Therefore i use that in the formula d(int(M^2/(2*E*I)*r*dTheta,Pi->0)/dP
I use the trig identities for Cos^2(theta) and Sin^2(theta) and integrate. If i ignore the ficticous force F the value of the integral is P^2(Pi/2)

This gives me a deflection of (P^2*Pi*R^3)/(4*E*I)
Is this correct
Would appreciate it a lot, Thanks

 

[Pyrrhus]

okay, i think my answer is wrong but this is what I've got.
the moment is M(theta) is PRCos(theta) + F(R-RSin(theta)) where f is ficticious i know so do i disregard that.

Therefore M^2(theta) = R^2(P^2Cos^2(theta) + F^2 - F^2Sin^2(Theta).

Therefore i use that in the formula d(int(M^2/(2*E*I)*r*dTheta,Pi->0)/dP
I use the trig identities for Cos^2(theta) and Sin^2(theta) and integrate. If i ignore the ficticous force F the value of the integral is P^2(Pi/2)

This gives me a deflection of (P^2*Pi*R^3)/(4*E*I)
Is this correct
Would appreciate it a lot, Thanks

You need to read castigliano's theorem again, in this case the fictitious force is equal to the applied load, unless the applied load is not at the free end, if that's the case you must specify where it is, so we can actually help out!