Second-order effects: when P-Δ stops being negligible

Every frame deflects under load, and that deflection adds moment that a first-order analysis never sees. A column carrying axial load $P$ that sways sideways by $\Delta$ picks up an extra moment $P\Delta$ - and that extra moment causes more sway, which adds more moment. The question for any frame is only whether this feedback loop is big enough to matter.

Two effects, one family

It helps to separate the two second-order effects, because codes treat them differently:

• P-Δ (big delta) is the global sway effect: the whole storey leans, and the gravity loads acting on that lean generate additional storey moments. This is the one that governs the overall stability of a frame.
• P-δ (small delta) is the member effect: an individual compression member bows between its ends, and the axial load acting on that bow amplifies the member moment. This is closely related to the column buckling check.

For frame design it is usually the global P-Δ effect that decides whether you need to do anything special.

The αcr screen

EC3 gives a single, elegant gate for deciding whether second-order effects can be ignored. Compute the ratio of the elastic critical buckling load of the frame to the applied design load:

$\alpha_{cr} = \frac{F_{cr}}{F_{Ed}}$

If $\alpha_{cr} \ge 10$ for an elastic analysis, the frame is stiff enough to be classed as non-sway, and first-order analysis is acceptable - the amplification is small enough to neglect.

Below 10, you must account for the effect: either amplify the first-order sway moments, or run a true second-order analysis.

The beauty of the check is that it tells you whether you have a problem before you spend any effort solving one. A heavily braced frame typically sails past 10; a slender unbraced frame often does not.

The cheap fix: amplified sway moments

For frames that fail the screen but are not wildly slender, you rarely need a full non-linear analysis. The amplified-sway-moment method multiplies the sway moments from a first-order analysis by:

$\frac{1}{1 - 1/\alpha_{cr}}$

It is a one-line correction applied to the sway component of the moments, and it captures the P-Δ amplification well for moderately slender frames. As $\alpha_{cr}$ approaches the lower limit the factor grows quickly, which is the method's own warning that you are getting close to where a rigorous second-order analysis becomes necessary.

What this means in practice

Three habits keep frame stability honest:

1. Always compute $\alpha_{cr}$. It is cheap and it is the difference between knowing your frame is non-sway and assuming it.
2. Watch unbraced and tall frames. They are the ones that drop below 10, and the ones where ignoring P-Δ is genuinely unsafe, not just inaccurate.
3. Treat a low $\alpha_{cr}$ as a stiffness message. Adding bracing or stiffening columns raises $\alpha_{cr}$ and often removes the need for amplification altogether - usually a better answer than simply factoring up the moments.

The related flexural buckling article covers the member-level stability that sits alongside this frame-level check.