Module 14 · Lesson 14.1

What the answer depends on

Elasticity, swing, and the ranking that surprises people the first time they compute it.

Why this matters

Every model has inputs you are confident about and inputs you guessed. A single run treats them identically and gives one number with no indication of which is which.

A handful of extra runs fixes that, and tells you two things a single run never can: which assumptions actually matter, and whether the model behaves the way physics says it should when you push it. Both are worth more than another decimal place on the base case.

By the end of this lesson you should be able to

Define elasticity and compute it by central difference
Check a computed elasticity against the exponent in the underlying formula
Distinguish elasticity from swing and rank by the right one
Justify a plausible range from something other than habit

What you should already know

Beam deflection formulas and their exponents
The diagnostic checks of Module 12 — a sensitivity study is a check as well as an exploration

Two kinds of input

Design variables are things you choose: span, depth, section, grid, material. You control them, and varying them is exploring the design.

Uncertain parameters are things you do not: the modulus of the concrete actually poured, the imposed load actually present, the stiffness of the ground, the fixity a nominally pinned connection actually provides. You estimate them, and varying them is testing your exposure.

Both get varied in a sensitivity study and they mean different things. Sensitivity to a design variable is an opportunity. Sensitivity to an uncertain parameter is a risk.

Elasticity

The natural measure is the percentage change in the output per percentage change in the input:

elasticity = (∂y/∂x)(x/y) — evaluated at the base case

It is dimensionless, which is what makes it useful: it compares a span in metres with a modulus in gigapascals with a load in kilonewtons per square metre.

And it has a check built in. For anything with a power-law relationship, the elasticity is the exponent:

Relationship	Elasticity
δ ∝ L⁴	+4
δ ∝ 1/d³	−3
δ ∝ 1/E	−1
f ∝ 1/√m	−0.5

So a numerical sensitivity study on a beam deflection must return +4 for span and −3 for depth. If it does not, the model is not doing what beam theory says — which makes a sensitivity study a verification tool as well as an exploration tool.

Swing, and why it beats elasticity

Here is the part that surprises people.

A structural steel modulus has an elasticity of −1 on deflection, which sounds significant. But E for steel is known to within about ±2 %, so its whole plausible range moves the deflection by 4 %.

The imposed load actually present on a floor has an elasticity well under 1. But its plausible range spans a factor of six — from a nearly empty floor to a densely occupied one — so it moves the answer by far more.

Swing = elasticity × range. A variable matters in proportion to how much the output moves across the range it could plausibly take, not in proportion to its exponent.

Ranking by elasticity alone is a common and expensive mistake. It directs attention to the inputs with the strongest mathematical leverage rather than to the ones you are actually uncertain about.

Ranges need reasons

A range of ±10 %, applied to everything, is a guess wearing a number. Every range should have a stated basis:

Span: the grid the architect has set, plus construction tolerance, plus the possibility of a late grid change. Narrow.
Section depth: the serial sizes available within the agreed floor zone. Discrete, not continuous.
Steel modulus: one of the genuinely fixed numbers. ±2 %.
Imposed load present in service: what is actually on the floor when it vibrates, not the ultimate design value. Very wide.
End restraint: a nominally pinned connection behaves as continuous at vibration strains and as a pin at ultimate. The range spans both, and it is enormous.

Writing the basis down is what makes the study reviewable. A reviewer can disagree with a stated range; they cannot engage with an unstated one.

Elasticity and swing

E_{y, x} = \frac{\partial y}{\partial x} \cdot \frac{x}{y}, swing = \frac{y ( x _{high} ) - y ( x _{low} )}{y ( x _{base} )}

What it calculates: How much an output responds to an input, and how much that matters

$E_{y, x}$: Elasticity — percentage change in y per percentage change in x (—)
swing: Fractional change in y across x's plausible range (—)
x_low, x_high: The ends of the plausible range, with a stated basis (varies)

This assumes

The elasticity is evaluated at the base case and may differ elsewhere
The swing assumes the response is monotonic between the ends — check it

In plain terms: Elasticity is a property of the physics; swing is a property of the physics and your ignorance. A design decision responds to the second. That distinction is the whole content of this lesson.

Try it

Sensitivity explorer

A floor beam with five inputs, each with a range justified from the project rather than assumed. Read the elasticities against the exponents you already know.

Output

Low bound: 0.00055
Base case: 0.00182
High bound: 0.00490
Spread high/low: 8.97
Bracket valid?: yes

Tornado ordering — by swing, not by elasticity.
Variable	Elasticity	Range	Swing	Monotonic?
Imposed load present in service	0.51	0.40–2.50 kN/m²	88.7 %	yes
Section depth	-2.87	400.00–533.00 mm	-78.7 %	yes
Span	3.96	8.70–9.30 m	26.4 %	yes
End restraint stiffness	-0.04	0.00–60000.00 kN·m/rad	-24.9 %	yes
Young's modulus	-0.96	205.00–215.00 GPa	-4.6 %	yes

Every variable moved the output the same way across its whole range, so pushing them all to their worst ends really does bracket the answer.

Where each range comes from

Span: Grid set by the architect; construction tolerance and a possible late grid shift of one column line.
Section depth: The serial sizes available within the agreed floor zone.
Young's modulus: Structural steel. One of the few genuinely tight inputs, and its narrow range is the point.
Imposed load present in service: What is actually on the floor when it vibrates, not the ultimate design value. The widest range here, and usually the one that decides the answer.
End restraint stiffness: A nominally pinned connection behaves as continuous at vibration strains and as a pin at ultimate. The range spans both.

Two things this study is doing at once

Verifying: for deflection the elasticities must be +4 for span, −3 for depth and −1 for modulus, because those are the exponents in the formula. Anything else means the model is not doing what beam theory says.
Exploring: the tornado ordering directs effort at the inputs with the largest swing, which are not the ones with the largest elasticity. Young's modulus has an elasticity of −1 and almost no swing, because it is known to ±2 %.
And it cannot find interactions. Varying one input while holding the others at base values is silent about combinations, and no amount of finer sampling fixes that.

What this shows: Swing = elasticity × range, and the ranking follows swing — so a tight input with a big exponent can matter less than a loose one with a small exponent.

Worked example

The tornado for a floor beam

Given

A floor beam whose deflection is being assessed
Five inputs: span, section depth, Young's modulus, imposed load present, end restraint stiffness
Each with a plausible range justified from the project rather than assumed

Find

Which inputs the deflection actually depends on

Practice

Deflection goes as span to the fourth power. If the span increases by 5 %, by what percentage does the deflection increase? Use the elasticity approximation.

Practice

Variable A has an elasticity of 4 and a plausible range of ±3 %. Variable B has an elasticity of 0.8 and a plausible range of ±40 %. Which produces the larger swing in the output, and by what factor? Give the ratio of B's swing to A's.

Worked example

Ranking two variables that both matter

Given

A floor's deflection depends on span with an elasticity of +4 and on the imposed load with an elasticity of +1
The span is fixed by the grid to within ±2 %
The imposed load actually present ranges from 0.5 to 2.5 kN/m² against a nominal 1.5

Find

Which variable the answer is more exposed to

Worked example

A sensitivity study used as a verification check

Given

A beam model is perturbed: span +1 %, depth +1 %, modulus +1 %
The deflection changes by +4.06 %, −2.97 % and −0.99 %

Find

Whether the model is behaving as beam theory says it should

Practice

A variable has an elasticity of 0.5 and a plausible range of ±40 %. What swing does it produce in the output, as a percentage?

Check yourself

Why is elasticity a more useful measure than a raw derivative?

Check yourself

A sensitivity study reports an elasticity of exactly zero for a parameter. What are the two possible explanations?

Check yourself

Why should a plausible range have a stated basis rather than being ±10 %?

Summary

Design variables are opportunities; uncertain parameters are risks
Elasticity is dimensionless and equals the exponent for a power law
A sensitivity study verifies the model as well as exploring it
Swing = elasticity × range, and the ranking follows swing
Every range needs a stated basis, or the study is not reviewable

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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