Linear & Non-linear Structural Analysis Notes

Last reviewed 30 Sept 2026 · 6 min read

Linear analysis

A linear analysis assumes:

The material is linearly elastic ( $σ = E ε$ ).
Displacements are small, so equilibrium is written on the undeformed geometry.
Boundary conditions do not change during loading.

Then $[K] {u} = {F}$ with a constant $[K]$ : doubling the load doubles the response and results from load cases can be superposed. It is fast, robust and adequate for most service-level design of buildings and bridge girders.

It fails when the structure is slender (buckling, second-order effects), flexible (cables, membranes), loaded near collapse (cracking, yielding) or when contact opens and closes.

Sources of non-linearity

Type	Cause	Examples
Geometric	Displacements are large enough to change the stiffness	$P$ – $Δ$ in columns, cable-stayed and suspension bridges (cable sag), arches, buckling, membranes
Material	Stress–strain law is non-linear	Yielding of steel, cracking and crushing of concrete, creep, soil plasticity
Boundary / contact	Supports change with the deformation	Bearing lift-off, gaps, sliding friction, tension-only members, soil separation
Force (follower load)	Direction of the load changes with deformation	Wind on a deflected membrane, water pressure

Geometric non-linearity and $P$ – $Δ$

A column with axial load $P$ and lateral displacement $Δ$ has an additional moment $P Δ$ that increases the displacement further. Second-order (P–Δ) analysis includes it, and the stiffness is written as

[K_{T}] = [K_{E}] + [K_{G}]

$[K_{E}]$ = elastic (material) stiffness; $[K_{G}]$ = geometric stiffness, which depends on the axial forces — it reduces stiffness under compression and increases it under tension (as in a cable). Where the axial force is small, $[K_{G}]$ can be neglected.

For a cable the sag makes stiffness depend on the tension (the Ernst equivalent modulus is a simplified geometric correction).

Material non-linearity

Steel: elastic–plastic with strain hardening (bilinear or multilinear curve); yield criteria (von Mises).
Concrete: cracking in tension, crushing in compression, tension stiffening; plastic hinge models or fibre models for members; smeared or discrete cracking models.
Soil: Mohr–Coulomb, Drucker–Prager and more advanced models.
Time-dependent: creep, shrinkage and relaxation (see the construction-stage note).

Solving non-linear equations

Non-linear behaviour means the stiffness depends on the displacement: $[K (u)] {u} = {F}$ . It is solved by incremental–iterative methods:

FormulaNewton–Raphson iteration

Apply a load increment $Δ F$ .
Compute the residual (unbalanced force): ${R} = {F}_{e x t} - {F}_{in t} (u)$ .
Solve $[K_{T}] {δ u} = {R}$ and update $u \leftarrow u + δ u$ .
Repeat until the residual is smaller than a tolerance; then move to the next load increment. The full method updates $[K_{T}]$ every iteration (fast convergence, expensive); the modified method keeps the initial or the increment-start $[K_{T}]$ (cheaper, slower).

Load control, displacement control and arc-length

Control	What is prescribed	Works when
Load control	The load steps	Response is stiffening or gently softening
Displacement control	A chosen displacement	Softening after the peak load (pushover, tests)
Arc-length (Riks) method	The length of the path in load–displacement space	Snap-through and snap-back — where both load and displacement can decrease

Convergence criteria

Iterations stop when the norm of the residual force, the displacement increment or the energy increment falls below a tolerance (often 0.1–1 % of the initial value). Failure to converge signals collapse, an unstable model, too large a step, or a modelling error — not always a numerical problem.

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