Computational Engineering / Sensitivity and design-space exploration

What the answer depends on

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

Lesson review: expert verification remains open

This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

A computational engineering reviewer must check algorithms, convergence claims, statistical assumptions and numerical examples beyond the cases explicitly recorded.
Validate the sensitivity lab’s numerical derivatives, units and range conventions independently.
Joint parameter ranges and model interactions need project-specific engineering review.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

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

Variables, uncertainty and sensitivity

Design variables describe choices such as section or grid. Uncertain parameters describe imperfectly known properties or conditions. A sensitivity study can help compare both, but their ranges need different evidence.

Local elasticity is (∂y/∂x)(x/y), where x and y are non-zero. For an exact power law y ∝ xⁿ it equals n. A rectangular Euler–Bernoulli beam under fixed UDL, with width and other properties held constant, gives +4 for span, −3 for depth and −1 for E. Different loading, shear flexibility or coupled parameter changes can give different sensitivities without a modelling error.

Finite-range response

Evaluate the model across justified ranges. The endpoint swing is [y(x_high) − y(x_low)]/y_base; its magnitude helps rank a monotonic response. Elasticity times fractional range is only a local approximation and can be inaccurate across a broad range. Check interior values where the response may turn.

Base parameter ranges on project information, material evidence, available sections and plausible service conditions. Do not assume steel E has a universal ±2% uncertainty or that nominally pinned joints become fully continuous during vibration. Record the range, its basis and any dependencies between inputs.

Elasticity and swing

E_{y,x} = \frac{\partial y}{\partial x}\cdot\frac{x}{y}, \qquad \text{swing} = \frac{y(x_{\text{high}}) - y(x_{\text{low}})}{y(x_{\text{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%, −2.94% and −0.99%

Find

Whether the model is behaving as beam theory says it should

Practice

Using the local linear approximation, what full low-to-high swing is estimated for elasticity 0.5 and a parameter range of ±40%, as a percentage of the base output?

Check yourself

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

Check yourself

A sensitivity study reports zero change for a parameter perturbation. What should be checked?

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