Module 11
Nonlinear and explicit analysis
What changes when superposition fails: equilibrium paths, limit points, Newton–Raphson, arc-length, dynamic relaxation and the explicit time step.
What this module covers
- Distinguish material, geometric and directional nonlinearity, and recognise each in a model
- Explain Newton–Raphson through the residual rather than through the formula
- Say why the modified method is sometimes the better choice despite needing more iterations
- Read an equilibrium path and identify limit points, snap-through and bifurcation
- Explain why load control cannot pass a limit point and what can
- State what dynamic relaxation computes and what its intermediate states mean
- Compute an explicit stable time step and say what sets it
Lessons
Material, geometric and directional — how to recognise each, and why the solver menu is not what decides.
Start lesson →Guess, measure the out-of-balance, divide by the tangent, correct, repeat — and what each of those four words is doing.
Start lesson →Where load control runs out, what displacement and arc-length control can see that it cannot, and what snap-through actually is.
Start lesson →
Module checkpoint
Check what you have taken in
3 questions
Question 1
A load-controlled nonlinear analysis of a shallow dome fails to converge at 96 % of the design load. What is the most useful interpretation?
Question 2
A structure has tangent stiffness kT = 80 − 40u + 8u² kN/m. At what displacement is its tangent stiffness least, in metres? Then check whether it ever reaches zero.
Question 3
An explicit analysis of a 60 ms impact uses 25 mm concrete elements. Concrete has E = 30 × 10⁶ kN/m² and ρ = 2.4 t/m³. Using 90 % of the critical step, how many time steps does the run need? Give the answer to the nearest hundred.