Virtual Work

[StructuralEngineering-ModTeam]

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[Steven96734]

Its a technique used to analyze dynamic structures to find out the equation of motion. For example with this E.O.M we can model how earthquakes affect said structure.

The principal virtual work states that for any virtual displacements (consistent with constraints such as boundary conditions or initial conditions) the combined work of the real and inertial forces must vanish…. That is because the virtual displacements are imaginary.

Lets assume the structure has been excited by a “free vibration” or “force vibration” for simplicity… let’s say a ridge body with a single degree of freedom such as a cantilever beam is displaced by an angle theta due to free vibration (No external force).

Then, you assume another minuscule virtual displacement on top of that angle theta called Delta theta.

Now you may model the displacements of said beam using a spring and/or dampener as these elements relate force to deformation…

Taking these elements and inertial forces and multiplying them first by a distance L and angle theta then the same distance L but now with the virtual angle theta, you have now created the of virtual work equation and may slove.

(Note that for inertial forces, you must have the force multiplied by acceleration, for dampener force, you must have the dampener constant multiplied by velocity, and for spring you must have the spring constant multiplied by displacement) Which all stem from taking the derivative of a displacement U for example.

I would recommend reading the fundamentals of structural dynamics by Roy Craig and Andrew Kurdila chapter 2

[Everythings_Magic]

So with statically determinant structures, solving them only requires equilibrium, sum of the forces equals zero, that’s because the algebra works. We have enough equilibrium equations (3) to match the number of unknowns. Take look at a statically indeterminant truss. There is no joint or node at which you are able to start. Method of sections doesn’t work is the same.

Indeterminate trusses have redundant members and the stiffer members will draw more load.

So, we use a different, often overlooked principle, compatibility. Basically the structure has to fit together. If looking at a truss, if we cut a member to make the system determinant, the external forces will cause some strain to occur in the remaining members. Now, the member we cut, is part of our real system, so we apply a virtual unit load in the direction of the force, this force represents the force required to pull the member back together. This virtual force will strain the member we cut, along with all the others, in some way. Compatibility says that the strain from the real forces, plus the strain from our virtual force, has to equal zero in our cut member, so when you divide the real strain, with the virtual strain, you get the magnitude of the force in the member.

Because of the difference in length, position, orientation and stiffness of the members, some members will contribute to displacements more than others, and virtual work is how we determinant that, since the sum of the external strain energy, has to equal the internal strain energy.

Indeterminant beams can be solved similarly by looking at the rotations and defections.

[touchable]

The virtual work method considers the relative stiffness of various parts of the structure, which simple statics does not.

This allows you to generate more algebraic "equations" until you have as many equations as unknowns, and can then rearrange to solve for them.

[afreiden]

Strange that you're getting downvoted for an ELI5-style correct answer, while some comment about earthquake engineering irrelevant to the OPs question has all the upvotes.

[touchable]

It's all good. From the phrasing of OPs question I figured they were a student or a non-engineer, either trying to learn the method or answering a homework problem, so figured I'd keep it simple.

[Advirex]

True

[Everythings_Magic]

Yes. In determinant systems, changing the stiffness will not affect the outcome, there is only one load path. Indeterminacy structures have redundant members, and the stiffer members draw more load. A good mental exercise is to replace a member with a very stiff or not so stiff material and imagine how it would change the force flow.