The Chartership Forum

Structural Dynamics / module 16

Nonlinear structural dynamics

What changes when the restoring force depends on history: hysteresis, tangent stiffness, iterative equilibrium, ductility demand, residual displacement, and why superposition and response spectra stop applying.

What you will understand

  • Describe a bilinear hysteretic system and state what each of its three parameters controls
  • Explain why a nonlinear step needs iteration and what the tangent stiffness is for
  • Compute ductility demand, hysteretic energy and residual displacement from a response
  • Explain why linear superposition does not apply directly to nonlinear response and identify the limits of linear modal and spectral methods.
  • Explain why linear superposition and independent linear modal response spectra cannot directly represent yielding
  • Recognise when a nonlinear analysis is genuinely needed and when it is not

Lessons

  1. When the restoring force remembers

    Hysteresis, the bilinear model and equilibrium iteration in implicit nonlinear integration.

  2. Ductility, and what it buys

    Strength reduction factors, the equal-displacement and equal-energy rules, equivalent damping, and the demand that peak displacement does not describe.

Module summary
  • A yielding structure's restoring force depends on where it has been, not only on where it is
  • The bilinear model needs three numbers: initial stiffness, yield force, post-yield ratio
  • Tangent stiffness is what the solver linearises about; it is not the secant stiffness
  • Implicit nonlinear steps generally need an equilibrium solution; explicit and event-based methods use different procedures
  • Yielding changes stiffness and dissipation; its effect on seismic demand depends on the spectrum and response history
  • Equal-displacement and equal-energy approximations relate strength reduction to ductility under stated assumptions
  • The area of a stable closed hysteresis loop is energy dissipated over that cycle
  • Residual displacement is a separate demand from peak displacement and can govern repair
  • Linear superposition fails; nonlinear time integration or a validated inelastic approximation is needed
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

Practice

A bilinear system has k = 8.0 MN/m and fy = 360 kN. What is its yield displacement, in millimetres?

Practice

The same system reaches 158 mm. What is its displacement ductility demand?

Practice

Using the equal-energy rule, what strength reduction factor corresponds to a ductility of 3?

Check yourself

A nonlinear time-history analysis reports 210 non-converged steps out of 24 000. What should be done?