Computational Engineering / Nonlinear and explicit analysis
Three kinds of nonlinearity
Material, geometric and directional — how to recognise each, and why the solver menu is not what decides.
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Read the lesson register and outstanding review notesWhy this matters
The most consequential misunderstanding in this area is a simple one: that the difference between a linear and a nonlinear analysis is which solver you picked.
It is not. A model containing tension-only elements is nonlinear whatever solver is running, and everything that follows from nonlinearity — no superposition, no scaling, no combining results afterwards — applies to it. Module 12 has a basement that failed to solve for exactly this reason, and the engineer's diagnosis was that the gravity load case was broken.
By the end of this lesson you should be able to
- Name the three kinds and give a structural example of each
- Identify which are present in a given model description
- Explain why superposition fails for all three
- Recognise that element and material choices make a model nonlinear regardless of solver
What you should already know
- Linear analysis and superposition (Module 9)
- Geometric stiffness and second-order effects (Module 10)
Material nonlinearity
A nonlinear constitutive law changes the stress–strain relationship as loading develops. Yielding, cracking, crushing and soil yielding are examples. Unloading may leave permanent deformation, so some models depend on loading history.
Geometric nonlinearity
Equilibrium is written on a displaced configuration. Small second-order approximations and large-displacement formulations have different scopes; cables and membranes often need the latter.
Contact and unilateral behaviour
Tension-only ties, compression-only supports, gaps and frictional interfaces change the active stiffness or load path as their state changes. Several forms of nonlinearity may occur together.
Analysis and combinations
General nonlinear responses cannot be obtained by adding separately solved case results. Apply the intended combined loading and, where relevant, its sequence. Envelopes of valid separately analysed states remain useful, but other post-processing such as equilibrium checks is also possible.
Check how the selected solver treats each nonlinear feature. A linear analysis may reject it, ignore it or use a specified linearisation; the behaviour is software-dependent. A justified linear approximation can be useful, provided its limitations and range of validity are established.
Check yourself
A basement model has tension-only hold-down anchors but omits ground-bearing contact. Its nonlinear uplift case solves; a gravity case loses downward restraint. What should be checked?
Practice
A model contains: elastic steel members, tension-only cross-bracing, compression-only soil springs, and a linear solver has been selected. How many of the three kinds of nonlinearity are present in the model? Give a number from 0 to 3.
Worked example
Which nonlinearity is which
Given
- Three structures: a cable net under increasing load, a steel beam loaded past yield, and a base plate lifting off its foundation
Find
Which kind of nonlinearity each shows
Check yourself
A base plate is modelled with springs that can carry tension. What is the consequence?
Check yourself
Why are load increments often useful in a nonlinear analysis?
Summary
- Material: the stress–strain curve bends. Geometric: equilibrium moves onto the deformed shape. Directional: behaviour depends on which way the load goes
- All three make the stiffness depend on the load, so all three kill superposition
- Analyse valid combinations and histories before forming response envelopes
- Check whether the chosen solver honours, rejects or linearises each nonlinear feature
- List the nonlinear features before choosing the analysis type