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Computational Engineering / module 10

Second-order, buckling and dynamic methods

What the geometric stiffness matrix does, why a buckling load factor is not a factor of safety, and what a dynamic analysis needs from the model.

What you will understand

  • Explain geometric stiffness physically and say why it depends on the load
  • Compute the amplification a compression load applies to deflections and to imperfections
  • State why P-delta results may not be superposed
  • Set up linear buckling as an eigenvalue problem and read its output correctly
  • Back-calculate an effective length, and say what it is meaningless without
  • Distinguish a local buckling mode from a global one
  • Say what a dynamic analysis needs from a model that a static one does not

Lessons

  1. Geometric stiffness: why load changes stiffness

    The second matrix. Where it comes from, why it scales with force rather than with EA, and what that costs you.

  2. Buckling as an eigenvalue problem

    Finding the load factor that cancels the stiffness, and reading what comes back without over-claiming.

  3. What these analyses need from the model

    Mesh density for buckling, mass for dynamics, and the modelling decisions that a static analysis lets you get away with and these do not.

Module summary
  • Tension stiffens, compression softens, and the geometric stiffness matrix is that statement
  • Kg depends on the load, so nothing derived from it may be superposed or scaled
  • A buckling load factor is an amplifier, not a margin
  • Deflections are amplified by α/(α−1); imperfections by 1/(α−1) — different numbers
  • The 10 % error line is at α = 11; the convention quoted in practice is 10
  • A buckling mode has arbitrary magnitude — only the ratios and the location mean anything
  • One element per member reports a buckling load about 22 % too high
  • A dynamic model needs mass, and mass is a load-case decision, not a material property
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

What does a buckling load factor of 8 tell you about a frame?

Practice

A frame has αcr = 5 and a first-order sway of 24 mm, with an initial out-of-plumb of 10 mm. Compute the total second-order sway including the amplified imperfection, in mm.

Practice

A column's buckling analysis gives Pcrit = 5 200 kN. It has EI = 21 000 kN·m² and a storey height of 4 m. Compute the effective length in metres.