A linear finite element analysis assumes that the structure’s stiffness does not change as load is applied. Displacement is proportional to load, so doubling the load doubles every stress, and the part returns to its original shape when the load is removed. A non-linear analysis drops that assumption. Stiffness is allowed to change as the structure deforms, as the material yields, or as parts come into or out of contact, so the load is applied in increments and the stiffness is updated along the way.
Linear analysis is the right tool when stresses stay below yield, deflections are small compared with the part’s dimensions and contacts do not change. Non-linear analysis is needed when any of three things happens: the geometry changes enough to alter how load is carried (geometric non-linearity), the material stops behaving elastically (material non-linearity), or contact between parts opens, closes or slides (contact non-linearity). Many real problems involve more than one.
What a linear analysis assumes
Linear static analysis solves [K]{u} = {F} once, with a stiffness matrix K that stays fixed. That brings two practical advantages. It is fast, because there is only one solution. And it allows superposition: the results of separate load cases can be scaled and added, so a bracket analysed for 1 kN vertical and 1 kN lateral can be assessed for any combination of the two without running again.
The same property is also its blind spot. Suppose a steel bracket shows a peak stress of 120 MPa at 1 kN. A linear model will report 240 MPa at 2 kN and 480 MPa at 4 kN. The first two are believable. The third is not: in S355 steel the material yields at about 355 MPa, the stiffness of the yielded region drops, and load moves to the material around it. The linear result cannot tell you how much permanent deformation follows, or whether the bracket collapses.
The three sources of non-linearity
Geometric non-linearity
Geometric non-linearity arises when deformation is large enough to change how the structure carries load. The equilibrium equations are then written for the deformed shape rather than the original one.
- Membrane stiffening. A flat plate under pressure starts by bending. Once its deflection exceeds roughly half its thickness, it begins to stretch like a membrane and becomes stiffer than linear theory predicts. A linear analysis then overestimates the deflection and misjudges how stress is shared between bending and stretching.
- Stress stiffening. Tension raises lateral stiffness, as in a guitar string or a spinning fan blade; compression lowers it.
- Buckling and snap-through. Slender columns, thin walls in compression and shells under external pressure can lose stiffness suddenly. A domed spring or a clip can snap from one stable shape to another.
- Large rotation. Long flexures, cantilever springs and cables rotate enough that the direction of the load relative to the part changes.
Material non-linearity
Material non-linearity arises when stress is no longer proportional to strain.
- Plasticity. Metals beyond yield follow the stress-strain curve from a tensile test. Non-linear analysis predicts plastic strain, permanent set and the load at which a section becomes fully plastic.
- Hyperelasticity. Rubbers and elastomers in seals, grommets and mounts are non-linear even at modest strain, and nearly incompressible. Material models such as Neo-Hookean, Mooney-Rivlin and Ogden are fitted to test data.
- Time-dependent behaviour. Polymers creep and relax under sustained load, and metals do the same at high temperature.
Contact non-linearity
Contact, sometimes called boundary non-linearity, arises because parts that touch can push on each other but not pull. The area in contact changes with load, gaps close, and surfaces slide against friction. Typical cases:
- A bolted flange with preload, where the joint faces begin to separate as internal pressure rises.
- A travel stop that only engages once a 2 mm gap has closed.
- A pin in a clearance hole, where the bearing area grows with load.
- Interference fits, clamps and snap-fits.
Parts joined permanently, by adhesive or a weld, can be modelled as a bonded, linear connection. Contact only becomes non-linear when surfaces are allowed to separate or slide.
Linear and non-linear analysis compared
| Linear | Non-linear | |
|---|---|---|
| Stiffness | Constant | Changes with deformation, material state or contact |
| Load and response | Proportional | Not proportional; load applied in increments |
| Combining load cases | Results can be scaled and added | Each load case and load sequence solved separately |
| Material data | E and ν | Stress-strain curve, hyperelastic or creep data |
| Deformation | Small displacements and rotations | Large displacements, rotations and strains |
| Contact | Bonded only | Can open, close and slide, with friction |
| Stress above yield | Not physically meaningful | Plastic strain and load redistribution predicted |
| Solve time | One solution, fast | Many increments and iterations, slower |
| Typical questions | Working stress, stiffness, early design comparisons | Collapse load, permanent set, seal compression, insertion force |
Signs you need non-linear analysis
- A linear stress exceeds yield over more than a small, local region, and you need to know what happens next.
- Deflections are large compared with wall thickness, or rotations exceed a few degrees.
- Parts touch, separate or slide under load, or a gap closes during operation.
- The design includes rubber, elastomer, foam or soft plastic that carries load.
- Thin walls or slender members are in compression, so buckling is possible.
- The question is about collapse load, permanent set, insertion force or seal compression rather than working stress.
- The order of loading matters: bolt preload followed by service load, or assembly followed by operation.
Five examples
Snap-fit clip on a moulded housing. The clip deflects a long way, the plastic may yield locally at the root and the hook slides over its catch. All three non-linearities can appear at once. A non-linear analysis gives the insertion and removal forces and the peak strain at the root, set against the material’s allowable strain.
Bolted flange under pressure. Preload clamps the faces together; internal pressure tries to open them. A contact analysis shows whether the faces stay closed, how the bolt load changes as pressure rises and where a gasket loses compression.
Rubber anti-vibration mount. A hyperelastic model gives the load-deflection curve, which is the stiffness a vibration study needs, and the strain levels that govern the mount’s life.
Thin-walled vessel under vacuum. A linear buckling analysis gives the critical pressure for a perfect shell, but thin cylinders and spheres are sensitive to small out-of-roundness and can collapse well below it. A non-linear analysis that includes a realistic imperfection gives a more representative collapse pressure.
Lifting lug at proof load. A proof load sits well above the working load, and local yielding at the edge of the hole is possible. A non-linear analysis shows how far the plastic zone spreads and how much permanent deformation remains, set against the limit in the code or standard you specify.
Impact and drop events combine all three non-linearities with dynamics and short timescales, and are covered by drop test and impact simulation.
When linear is still the right answer
Non-linear analysis takes longer, needs more material data and requires every load case and sequence to be solved separately. For a large share of everyday design work, such as brackets, frames and housings well inside their elastic range, linear analysis gives the same answer far more quickly. It is also the natural first look at a new design, showing the load path and the regions that deserve closer attention.
Local peaks above yield call for judgement. Under static load, a small yielded region at a stress concentration in a ductile metal sheds load to the material around it, and many design codes treat local peak stresses differently from stresses that could cause gross failure. Whether such a peak is acceptable depends on the code or standard you specify, the ductility of the material and whether the load repeats. Under cyclic load, that same peak is where a fatigue crack will start.
When to bring in a specialist
The difficult parts of non-linear work are judgement calls: which non-linearities matter, what material data is good enough, and how contact and load sequence should be represented. SixGate™’s non-linear analysis covers large deformation, plasticity and yield, and changing contact. Each study ends with the change that solves the problem, whether that is a thicker section, a different material or a revised clip geometry, and that change goes into the detailed design, the drawings and the production parts without passing to another firm. For the fundamentals of any model, see what is FEA; for working-stress studies, see stress analysis.