Solving an Indeterminate Beam: Virtual Work Approach

[stinlin]

How do you go about solving an indeterminate beam for various deflections by virtual work? I'm kind of at a loss right now and can't proceed on my homework. I've attached a picture showing the problem I'm confused on - I don't know how to go about modeling the virtual model so I can find the moment diagram and integrate. :(

 

[Nesha]

You must draw bending diagram (moment) of force for each "force".Then you can say that deflection of B is zero. The deflection of B is caculated like this:
Area of moment diagram from outside force x height of diagram in place of centar of gravity in second diagramx1/EI.
I-momentum of inertia, E-Young's modulus (E)
Sorry on bad english, I hope that you undersud what I sad.
http://img216.imageshack.us/img216/6742/prob2ba6.th.jpg

 

[Pyrrhus]

You must draw bending diagram (moment) of force for each "force".Then you can say that deflection of B is zero. The deflection of B is caculated like this:
Area of moment diagram from outside force x height of diagram in place of centar of gravity in second diagramx1/EI.
I-momentum of inertia, E-Young's modulus (E)
Sorry on bad english, I hope that you undersud what I sad.

That's not solving by using Virtual work, and according to what you said looks like using conjugate beam.

How do you go about solving an indeterminate beam for various deflections by virtual work? I'm kind of at a loss right now and can't proceed on my homework. I've attached a picture showing the problem I'm confused on - I don't know how to go about modeling the virtual model so I can find the moment diagram and integrate. :(

Simply write first the compatibility equation, and find the deflections on the primary structure by using virtual work and find the flexibility coefficients by virtual work, too.

For your case there is 1 compatibility equation of the form (considering deflections down positive):

$$\Delta_{end} + P_{end}f_{end} = 0$$

 

[stinlin]

Heh - I figured it out. I had to solve it three different ways to show that virtual work can be applied to any indeterminate structure released to a stable state (i.e. if there's n degrees of indeterminacy, you can release n reactions/supports to make it a determinate structure). :) Thanks for the help!

 

[Nesha]

There are three methods of solving this type of problems, and all three are energetic methods (all of them are based on virtual work). I know them like:
- II theorem of Castillan (not sure how to writte in english his name)
- Maxwell-Mors integrals
- Theorem of Vershchagin (not sure how to writte in english his name)