Solving an Indeterminate Beam: Virtual Work Approach

• stinlin
In summary, the conversation discusses how to solve an indeterminate beam for various deflections using virtual work. The person is struggling with their homework and is unsure of how to model the virtual structure and find the moment diagram. They receive help and are told to use the compatibility equation and find deflections and flexibility coefficients. They also mention three different methods for solving these types of problems, all based on virtual work.
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. :(

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• prob_2.jpg
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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 hight 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

Last edited by a moderator:
Nesha said:
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 hight 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.

stinlin said:
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):

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

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!

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)

1. What is an indeterminate beam?

An indeterminate beam is a structural element that cannot be fully analyzed using traditional methods of statics because it has more unknown forces and reactions than there are equilibrium equations available.

2. What is the virtual work approach?

The virtual work approach is a method of solving indeterminate beams by applying the principle of virtual work, which states that the work done by external forces on a system in equilibrium is equal to the work done by internal forces.

3. How does the virtual work approach work?

The virtual work approach involves creating a virtual displacement of the beam and calculating the corresponding virtual work done by external forces and internal forces. The equilibrium equation is then set up by equating the two virtual work terms.

4. What are the advantages of using the virtual work approach?

The virtual work approach offers several advantages, including the ability to solve indeterminate beams with multiple unknowns, the ability to handle complex loading and support conditions, and the ability to account for material non-linearity.

5. Are there any limitations to using the virtual work approach?

One limitation of the virtual work approach is that it assumes small deformations, so it may not be accurate for beams with large deformations. Additionally, it may be more time-consuming and complex compared to traditional methods for simple beam problems.

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