Solving an Indeterminate Beam: Virtual Work Approach

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Discussion Overview

The discussion revolves around solving an indeterminate beam for various deflections using the virtual work approach. Participants explore different methods and concepts related to this topic, including compatibility equations and the application of virtual work in structural analysis.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about modeling the virtual model to find the moment diagram and integrate for deflections.
  • Another participant suggests drawing the bending moment diagram for each force and calculating deflection using the area of the moment diagram, height, and material properties (I and E).
  • A later reply challenges the previous suggestion, indicating that it does not align with the virtual work method and instead resembles the conjugate beam method.
  • One participant proposes writing the compatibility equation and finding deflections and flexibility coefficients using virtual work.
  • Another participant claims to have figured out the problem by applying virtual work to various methods, indicating that it can be used for any indeterminate structure released to a stable state.
  • A participant mentions three energetic methods based on virtual work: II theorem of Castigliano, Maxwell-Mors integrals, and the theorem of Vershchagin.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to solving the problem, with multiple competing views and methods being discussed. Some participants agree on the use of virtual work, while others suggest alternative methods.

Contextual Notes

There are unresolved aspects regarding the definitions and applications of the methods discussed, as well as the specific conditions under which each method is applicable.

stinlin
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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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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
 
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 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.

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):

[tex]\Delta_{end} + P_{end}f_{end} = 0[/tex]
 
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)
 

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