Second-order analysis
P-Δ and P-δ solved by iteration, geometric imperfections to EN 1993-1-1 §5.3, and why a second-order result cannot be superposed.
Analysis types 4 min read
Second-order analysis solves
by iteration. The structure is solved, the geometric stiffness of the deformed state is formed from the resulting axial forces, and the system is solved again, until the displacement increment between two steps falls below your tolerance.
The answer has the same shape a linear solve produces: displacements, internal forces, reactions and shell resultants. Everything downstream (viewers, tables, design checks, report) reads it unchanged.
The two effects, and why they need different meshes
| What it is | What captures it | |
|---|---|---|
| P-Δ | The whole frame sways, and gravity acting through that sway adds moment | Node displacements alone — no refinement needed |
| P-δ | A member bows between its own two ends, and its axial force adds moment along the span | Interior nodes on the member — a single element has none |
A member modelled as one element cannot bow, so it cannot show P-δ at all. Two elements show it; four is where it is worth trusting. A second-order setup therefore carries its own elements per member, defaulting to four, and raises the analysis mesh to match, including members you have set a coarser count on yourself.
Results do not superpose
This is the one thing that genuinely behaves differently from a linear analysis, and it changes how the work is organised.
Instead:
- A second-order setup solves each combination as its own run.
- A combination that expands to several permutations (an envelope, an EN 1990 rule with each variable action leading in turn) is solved once per permutation.
- Results read the direct answer rather than any sum that could have been formed from load cases.
The practical consequence: a second-order run over forty combinations costs forty solves, not one solve per load case and forty sums. Point the setup at the combinations that matter rather than at everything.
Geometric imperfections — EN 1993-1-1 §5.3
A second-order analysis of a perfectly straight, perfectly plumb frame understates the effects. Real capacity is limited by how far the structure is already out of line before the load arrives. §5.3 covers this in three parts, and all three are available as an initial deformation of the model rather than as equivalent horizontal forces:
| Clause | Imperfection | Applied as |
|---|---|---|
| §5.3.2(3)a | Global sway | Every node moves horizontally by × its height above the base |
| §5.3.2(3)b | Member bow , Table 5.1 | A half sine along each member, about its weak or its strong axis |
| §5.3.2(11) | The elastic critical buckling shape, scaled | Replaces the other two — that is what "a unique global and local imperfection" means |
Deforming the geometry rather than applying notional forces is deliberate: the P-Δ moment then falls out of the second-order solve itself, for every load state, with no per-storey force bookkeeping and no assumption that the structure is a stack of floors.
The mode-shaped imperfection needs a linear buckling result to scale, so run one first.
When the iteration stops
| Outcome | Meaning |
|---|---|
| Converged | The displacement increment fell below the tolerance. This is the answer. |
| Instability detected | The tangent stiffness stopped being positive definite — the structure has passed its critical load under this very load state. |
| Iteration limit reached | Neither of the above happened within the iteration budget. Not an answer; treat it as a failure to converge. |
Instability detection watches the factorisation for negative pivots and is on by default. Without it the iteration either wanders towards a large deflected shape that looks like a result, or fails to converge for a reason nobody can read off the numbers.
Choosing between α_cr and a second-order run
Linear buckling gives you cheaply, and EN 1993-1-1 lets you neglect second-order effects above the threshold. Below it the code permits two options: the amplification rules applied by hand, or this analysis. Where the frame is close to the threshold, or the amplifier is large, running it is faster than justifying the shortcut.
What is still linear here
The geometry is not. Everything else is:
- Material stays elastic — no yielding, no cracking.
- Supports do not gap, lift off or change stiffness with load.
- Sections stay plane and prismatic between nodes.
Related
Linear buckling supplies and the mode shape this analysis can use as an imperfection; Linear static explains the superposition this analysis gives up.
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