Module 11 · Lesson 11.1
Three kinds of nonlinearity
Material, geometric and directional — how to recognise each, and why the solver menu is not what decides.
Why 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
The stress–strain relationship stops being a straight line. Steel yields. Concrete cracks in tension and crushes in compression. Soil shears. Timber crushes across the grain.
What it costs you: the stiffness now depends on the strain, so it depends on the load, so it changes during the analysis. Unload and the structure may not return to where it started — a plastic hinge is permanent, and the structure that carries the next load case is not the one that carried the last.
Geometric nonlinearity
Equilibrium is satisfied on the deformed shape rather than the original one. Module 10 handled the mild version — P-delta, where the deformed geometry is used but the small-displacement assumption survives inside each element. Full geometric nonlinearity abandons that too, and is needed for cables, membranes, and anything whose shape changes enough to alter how it carries load.
What it costs you: the same as above. The stiffness depends on the displaced shape, which depends on the load.
Directional nonlinearity
The structure behaves differently depending on which way the load goes. A tie carries tension and goes slack in compression. A gap closes and then bears. A friction interface sticks and then slips. Cross-bracing modelled with tension-only members changes its load path entirely when the wind reverses.
This is the one that catches people, because the elements look ordinary and the model runs. It is also the most common, because nearly every real structure has some of it — foundations that can push and not pull, bracing that buckles out of action, bearings that lift off.
Why superposition fails, in all three
Superposition needs one thing: that the response to A + B is the response to A plus the response to B. That holds only if the structure is the same in all three analyses.
In every kind of nonlinearity it is not. Under load A the ties on one diagonal are active; under load B the others are. Under A the beam is elastic; under A + B a hinge has formed. The three analyses are of three different structures, and adding their answers adds answers to different questions.
The only post-processing operation that survives nonlinearity is the envelope — taking maxima and minima across combinations that were each analysed separately. Everything else must happen before the solve.
The trap
Selecting 'linear analysis' does not make a model linear. It makes the solver treat it as though it were, which for a model containing tension-only elements means those elements are silently converted to ordinary bars — carrying compression they cannot carry, in a load path the real structure does not have.
Some packages warn. Many do not, because from the solver's point of view nothing unusual has happened. The check is on you: list the nonlinear features in the model before choosing the analysis. Tension-only, compression-only, gaps, contact, plastic materials, large displacement. If that list is not empty, the analysis type is decided for you.
Check yourself
A basement is held down against flotation by ground anchors modelled with tension-only elements. A linear analysis of the uplift case runs; the gravity case fails. What is happening?
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 does a nonlinear analysis need the load applied in increments?
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
- The envelope is the only post-processing operation that survives
- A linear solver on a nonlinear model silently linearises it, usually without warning
- List the nonlinear features before choosing the analysis type
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint