Module 4 · Lesson 4.2

Where every stability effect belongs

Four effects, three possible homes, and two symmetrical errors that no individual calculation will ever reveal.

Why this matters

Every calculation in a stability check can be individually correct while the set of them is wrong. An effect can be missed because each person assumed someone else had it, or counted twice because two methods both include it. Neither error produces a strange number. The only defence is to decide, explicitly, where each effect lives — and that decision depends on which analysis was run.

By the end of this lesson you should be able to

List the four stability effects and their possible homes
Choose an analysis route and allocate each effect to exactly one place
Recognise the omission error and the double-counting error
Explain why double-counting survives review

Four effects, three homes

The effects:

1.Global sway imperfection — the frame is not plumb.
2.Member bow imperfection — the member is not straight.
3.P–Δ — vertical load acting through the frame's sway.
4.P–δ — axial load acting through the member's own bow.

The possible homes:

The analysis, if it is second-order and the imperfection is modelled.
The member check, through the buckling reduction factor χ.
Nowhere, which is legitimate only when the effect is genuinely negligible — and for global sway, it never is.

The normal route

For most buildings:

Effect	Home
Global sway	Analysis, as equivalent horizontal forces
P–Δ	Analysis if α_cr < 10, otherwise neglected
Member bow	Member check, inside χ
P–δ	Member check, inside χ

That works because χ is derived from an imperfect member's second-order response. The buckling curve already contains a bow, a residual-stress pattern and progressive yielding, rolled together into the imperfection factor α.

The alternative route

Model the member bow explicitly in a second-order analysis. Then the analysis produces the member's real amplified moments, and the member is checked on cross-section resistance only.

Both routes are correct. Mixing them is not.

Take care

The two errors, and why neither announces itself

Omission. A second-order analysis is run on a symmetric frame with no imperfections. There is no sway, so P–Δ contributes nothing, and the analysis reports a stable frame with unamplified moments. Every number is correct. The frame has simply never been asked the question.

The giveaway would be a global sway imperfection with no home — and since no member check covers it, nothing downstream will notice. Global sway is a property of the frame, not of any member, so if the analysis does not carry it, nothing does.

Double-counting. Member bow is modelled explicitly in the analysis, and the member is then checked with χ, which already contains an allowance for bow. The member is penalised twice.

This one is worse to catch, because it errs on the safe side. Nobody queries a member that passes with room to spare, so the error survives every review and simply makes the building heavier. On a large frame that is expensive, and it buys nothing whatever.

Omission produces a design that is unsafe and looks fine. Double-counting produces a design that is wasteful and looks careful. Neither shows up in any single calculation.

Try it

Where does each effect live?

Four stability effects, three possible homes. Choose a route and see where each one ends up. Some combinations leave an effect with no home at all; one counts an effect twice — and neither would be noticed in any individual calculation.

Critical multiplier α_cr6.0

global sway analysis
Modelled as an initial lean or as equivalent horizontal forces. Correct — this effect has no home in a member check.
P-Delta no home
NOT accounted for.
member bow member check
Carried by the buckling reduction factor χ, whose imperfection factor α stands for initial bow, residual stress and progressive yielding together. This is the normal route.
P-delta member check
Also inside χ — the buckling curve is derived from an imperfect member's second-order response.

Second-order effects required?: yes — α_cr below 10
Second-order analysis run?: no
Global sway imperfection: modelled
Member bow: inside χ
Buckling reduction χ: applied
Effects with no home: 1
Effects counted twice: 0

α_cr = 6.0 is below the threshold of 10, so second-order effects matter, and no second-order analysis was run and no amplification applied.

Things worth trying

Start with the defaults: first-order analysis, global imperfection modelled, bow inside χ, α_cr = 6. One effect has no home — P–Δ — because α_cr is below 10 and no second-order analysis was run.
Now turn on the second-order analysis. Everything has a home and the route is clean. This is the normal design route for most buildings.
Turn OFF the global sway imperfection while leaving the second-order analysis on. This is the omission error: on a symmetric frame there is nothing to amplify, so the analysis reports a stable frame it never actually tested.
Turn ON member bow AND leave χ applied. This is the double-counting error. The member is penalised twice, so it passes with room to spare — which is exactly why nobody catches it.
With member bow modelled, turn χ off. Now the route is clean again, and the member is checked on cross-section resistance. Both routes work; mixing them does not.
Raise α_cr above 10 with a first-order analysis. P–Δ becomes legitimately unnecessary — but note the wording: that is a calibrated threshold, not a physical boundary. Nothing changes about the frame at α_cr = 10.

Worked example

Two designers, two routes, and one mistake each

Given

An unbraced two-storey frame with α_cr = 5
Designer A runs a first-order analysis with equivalent horizontal forces, then member checks with χ
Designer B runs a second-order analysis with member bow modelled, then member checks with χ

Find

Whether each route is complete, and what each designer has got wrong.

Assumptions

Both designers' individual calculations are arithmetically correct
The α_cr threshold of 10 for elastic analysis, code-calibrated and unverified here

Practice

A frame has α_cr = 5. If the first-order sway moment at a base is 240 kNm, what is the amplified moment, in kNm?

Practice

A frame has α_cr = 12 and is being analysed elastically. By what factor must its moments be amplified?

Practice

That same frame with α_cr = 12 is to be analysed PLASTICALLY instead. It is a clad structure, so in the UK the plastic threshold is 10. What amplification factor now applies?

Check yourself

Why does double-counting member imperfections survive design review so reliably?

Summary

Four effects: global sway, member bow, P–Δ, P–δ
Three homes: the analysis, the member check, or nowhere
Normal route: sway and P–Δ in the analysis; bow and P–δ inside χ
Alternative route: model the bow, then check cross-section resistance only
Both routes work; mixing them counts the bow twice
Global sway has NO home in a member check — omitting it simply loses it
Second-order analysis and imperfections are independent choices, and software does one without the other
Write the allocation down at the start of the job — it is a one-minute check for a whole frame

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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