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Queensferry

Module 14 · Lesson 14.2

Bracketing and its limits

Three runs that bound an answer, the assumption they rest on, and what happens when it fails.

Why this matters

A full exploration of five uncertain inputs at eight values each is 32 768 runs. Nobody does that, and nobody needs to — if the response is well behaved, three runs bound the answer.

The phrase 'if the response is well behaved' is doing all the work in that sentence, and this lesson is about checking it rather than assuming it.

By the end of this lesson you should be able to

  • Construct a bracketing set of runs
  • State the monotonicity assumption that bracketing rests on
  • Recognise a non-monotonic response and say what it costs
  • Explain why one-at-a-time studies cannot find interactions

The three runs

Push every input to the end of its range that makes the output worse. Run that. Push every input the other way. Run that. Run the base case in between.

The three results bound the answer — provided each input moves the output the same way across its whole range. That is monotonicity, and if it holds, the worst case really is at a corner of the box.

When it fails

It fails more often than people expect, and the failures are exactly the interesting cases.

Resonance. A floor's response to footfall does not rise or fall monotonically with mass. It peaks where a natural frequency coincides with a walking harmonic. Both more mass and less mass can improve it, and the worst case is in the middle of the range, not at either end.

Support stiffness in a continuous beam. Very soft supports and very stiff supports both give a smooth moment distribution; intermediate values can give the largest differential settlement effects.

Load pattern. Full load on every span is rarely the worst case for moment; a pattern is. That is why codes prescribe pattern loading rather than leaving it to a maximum.

Second-order effects. As αcr falls the amplification rises non-linearly, and near α = 1 it is unbounded. The response to a stiffness reduction is not proportional to the reduction.

In every case, taking three runs and reporting the range would produce a bracket that does not contain the worst case, presented as though it did.

A one-at-a-time study can detect non-monotonicity, because it samples across each range rather than only at the ends. That is a good reason to run it before bracketing rather than instead of it.

Interactions

One-at-a-time studies have a second limitation, and it is not fixable by sampling more finely.

Varying one input while holding the others at their base values cannot find an interaction — a case where the effect of A depends on the value of B. Structures are full of them:

  • The effect of connection stiffness on beam moments depends on the relative stiffness of the columns.
  • The effect of mass on a footfall response depends on the frequency, which depends on the stiffness.
  • The effect of a stiffness modifier depends on which other members also received one.

Detecting interactions needs a design of experiments that varies inputs together — a factorial design, or a sampled global method. Both are beyond what this course covers, and the honest position is:

A one-at-a-time study is a first pass. It finds which inputs matter individually. It does not prove that no combination matters more than any of them alone, and it should not be reported as though it did.

What to report

  • The elasticities, with the exponents they were checked against.
  • The tornado ordering by swing, with each range and its basis.
  • The bracket, with an explicit statement of whether monotonicity was checked and held.
  • Any non-monotonic response found, and where its worst case is.
  • The statement that interactions were not explored.

That last item is not an admission of weakness. It is what makes the rest of the report usable.

Try it

Design space explorer

Sweep two variables together and see the surface. Then compare the cost of doing that for all five.

Horizontal axis

Vertical axis

Section depth (mm) →Span (m) →Deflection1.0 mm2.9 mm

Darker blue is the smallest deflection (0.97 mm); orange is the largest (2.90 mm). The shading is a secondary cue — the range is stated in the label so it does not depend on colour.

Cells sampled
81
Smallest deflection
0.974 mm
Largest deflection
2.898 mm
Ratio across the space
2.98
One-at-a-time runs, all 5 variables
45
Full factorial, all 5 variables
59,049

1312× the cost, and it is what finds interactions

What the surface shows that two sweeps cannot

  • If the contours were straight parallel bands, the two variables would act independently and two separate sweeps would tell you everything.
  • Where the contours curve or fan out, the effect of one variable depends on the value of the other — an interaction, and one-at-a-time sampling is blind to it.
  • Five variables at nine values each is 59 049 runs for a full factorial against 45 for one-at-a-time. That gap is why Module 16 exists.

What this shows: A two-variable sweep shows the interaction that two one-variable sweeps cannot — and a full factorial over five variables is unaffordable.

Worked example

A bracket that does not contain the worst case

Given

  • A floor whose footfall response factor is being assessed
  • The mass present in service is uncertain between 0.4 and 2.5 kN/m² of imposed load
  • At 0.4 the response factor is 6.2; at 2.5 it is 5.8
  • At the base case of 1.2 it is 11.4

Find

What a three-run bracket would have concluded, and what is actually true

    Predict first

    A one-at-a-time sensitivity study finds that no single input moves the output by more than 15 %. What can be concluded about the combined effect of all the uncertainties?

    Practice

    A study varies 4 inputs at 6 values each in a full factorial design. How many runs does that require?

    Practice

    Three inputs each move an output by +12 %, −9 % and +15 % across their ranges. Assuming the effects are additive and can align, what is the largest plausible increase in the output, as a percentage?

    Check yourself

    When does taking the corner combinations fail to bracket the answer?

    Summary

    • Three runs bracket an answer if every response is monotonic
    • Resonance, load pattern and second-order effects are routinely non-monotonic
    • A one-at-a-time study can detect non-monotonicity, which is why it comes first
    • It cannot detect interactions, and no amount of finer sampling fixes that
    • Report that interactions were not explored — it is what makes the rest usable
    Progress is kept in this browser only.

    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