Virtual work and deformation exercises

Module 10Level 3reactions and load path

Where does the unit load go?

You want the deflection at the quarter-point of a beam. Where should the unit load of the virtual system be applied?

Predict first, then reveal

Reason from the supports, the movement and equilibrium — not from what looks familiar.

AAt the quarter-point, in the direction of the deflection you want — the unit load represents the movement you are asking about.BBeside the real load, so the two systems line up.CAt mid-span, because that is where beams deflect most.

Module 10Level 2reactions and load path

A unit force or a unit moment?

You want the rotation (slope) of a beam at a support, not its deflection. What unit action should the virtual system use?

Predict first, then reveal

Reason from the supports, the movement and equilibrium — not from what looks familiar.

AA unit moment applied at that point — a unit moment pairs with a rotation, just as a unit force pairs with a translation.BA unit force, because the unit load is always a force.CEither works, since they give the same answer.

Module 10Level 3which impossible

A unit load at a support

By mistake, the unit load is applied at a simple support of the beam (which cannot move vertically). What contribution does the virtual system compute?

Predict first, then reveal

Reason from the supports, the movement and equilibrium — not from what looks familiar.

AZero — the virtual moment m is zero everywhere for a load at a support, so its overlap with M is zero. That correctly says the support does not deflect.BThe full mid-span deflection, because the unit load is on the beam.CAn error, because you cannot put a load at a support.

Module 10Level 4change and sensitivity

Which region contributes most?

For the mid-span deflection of a uniformly loaded beam, which part of the span contributes most to the movement?

bending moment (tension side)

Predict first, then reveal

Reason from the supports, the movement and equilibrium — not from what looks familiar.

AThe middle of the span, where both the real moment M and the virtual moment m are large; the ends, where m is nearly zero, contribute almost nothing.BThe supports, because that is where the reactions are.CEvery part contributes equally, because the load is uniform.

Module 10Level 4change and sensitivity

Does the biggest real moment always dominate?

A region of a structure has the largest real bending moment M, but the virtual moment m there is almost zero for the movement you want. How much does that region contribute to the movement?

bending moment (tension side)

Predict first, then reveal

Reason from the supports, the movement and equilibrium — not from what looks familiar.

AVery little — the contribution is M·m, so a large M paired with a near-zero m gives almost nothing. The virtual diagram decides which regions matter.BThe most, because it has the largest real moment.CExactly half, regardless of the diagrams.