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Structural Analysis Fundamentals / module 15

Virtual work and energy methods

Strain energy, the principle of virtual work, and the unit-load method — one technique that finds the deflection of almost anything.

What you will understand

  • Calculate strain energy stored in axial members and in bending
  • State the principle of virtual work and explain what makes a virtual system valid
  • Use the unit-load method to find the deflection of a pin-jointed truss
  • Use the unit-load method to find beam deflections and rotations
  • Apply Maxwell's reciprocal theorem and know why it is useful

Lessons

  1. Work, strain energy and the missing half

    Where the ½ in ½Pδ comes from, and what the structure does with the energy.

  2. Virtual work and the unit-load method

    One idea that finds the deflection of any pin-jointed structure.

  3. Unit load for beams, and reciprocal theorems

    The same idea with an integral instead of a sum — plus a result that saves real work.

  4. Strain energy in shear and torsion

    The other two ways a member stores the work you do on it.

  5. Castigliano's theorems

    Differentiate the strain energy, and the deflection falls out.

Module summary
  • Work done on an elastic structure is stored as recoverable strain energy
  • A gradually applied load does work ½Pδ, which is why the ½ appears in every strain-energy expression
  • Virtual work: for a structure in equilibrium, external virtual work equals internal virtual work
  • The virtual system must satisfy equilibrium; the real system must satisfy compatibility. They are otherwise unrelated
  • Unit-load method for trusses: δ = Σ NnL/AE
  • Unit-load method for beams: δ = ∫ Mm/EI dx
  • Put a unit moment instead of a unit force and the same integral gives a rotation
  • Maxwell: the deflection at B due to a load at A equals the deflection at A due to the same load at B
Teaching basis and review status

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

Check your understanding

Check yourself

Why does the factor of ½ appear in U = ½Pδ?

Practice

A bar 2.00 m long, area 1200 mm², E = 205 000 N/mm², carries a gradually applied 50.0 kN load. What is the strain energy stored, in joules?

Check yourself

In the unit-load method, what must the virtual (unit-load) system satisfy?

Practice

A pin-jointed member has N = 60 kN, n = 1.5, L = 4000 mm, A = 800 mm² and E = 205 000 N/mm². What is this member's contribution to the joint deflection, in mm?