Module 12 · Lesson 12.2
Units, signs and magnitude
The most common serious error in structural modelling is a factor of a thousand — and the technique for finding which input it is in.
Why this matters
Ask an experienced analyst what they see most often and the answer is not a subtle modelling misjudgement. It is a unit slip, a wrong sign, or an order-of-magnitude mistake in a dimension, a material property or a load.
What makes them dangerous is not that they are hard to make. It is that the model runs cleanly, produces well-formatted results, and gives no signal at all — and that the auto-fitting view means a model a thousand times too big looks exactly like one that is right.
By the end of this lesson you should be able to
- Localise a numerical error by comparing which outputs moved and which did not
- Explain why determinate moments are independent of stiffness and why that matters
- Compute the factor a given unit slip produces in each output
- Check the sign of a result against what the structure physically does
The technique: which outputs moved
This is the most useful diagnostic idea in the module, and it is not obvious.
In a statically determinate structure, the moments and forces come from equilibrium alone. They do not depend on stiffness at all. So a stiffness error — a wrong modulus, a wrong second moment of area — changes the deflections and leaves the moments exactly alone.
That gives you a two-way test:
| Moments | Deflections | Where the error is |
|---|---|---|
| wrong | wrong | Geometry or load |
| right | wrong | Stiffness: E, I, or a section property |
| right | right | Not a numerical error — look at the modelling |
One comparison narrows a model with thousands of elements to one of three categories. It is worth doing before anything else, because the categories need completely different investigations.
The factors
Unit slips are not all the same size, because different quantities carry different powers of length:
| Slip | Factor | Why |
|---|---|---|
| mm entered as m | 10³ | Length to the first power |
| Area in mm² as m² | 10⁶ | Length squared |
| Section modulus | 10⁹ | Length cubed |
| Second moment of area | 10¹² | Length to the fourth |
| N as kN | 10³ | |
| Pa as MPa | 10⁶ | |
| Density as unit weight | ≈ 102 | The factor is g, between kg/m³ and kN/m³ |
The second moment of area is the largest routine unit error available, and it is the one you meet when section properties are typed by hand from a spreadsheet in different units.
The one that hides
A mass error leaves every static result untouched. Deflections, moments, reactions, stresses: all correct. Only the frequency moves, and it moves by the square root — a factor of a thousand in mass gives a factor of 31.6 in frequency.
So a model with the mass wrong by a factor of a thousand passes every static check you can run. It is caught only by asking a dynamic question, and only if you know what band the answer should be in.
Every static check in this course is blind to a mass error. That is not a gap in the checks; it is a property of statics.
Signs
Sign conventions differ between packages and both conventions are defensible. Building-design software usually treats a gravity load as positive; general analysis software takes a vector in global axes, where the vertical axis points up and gravity is therefore negative.
The check is visual and takes two seconds: switch on the load graphics and look at which way the arrows point. Moments are harder, because some packages use the left-hand rule; find out which yours uses rather than assuming.
The most dangerous sign error is one that runs cleanly and produces the right magnitude. A tank wall loaded inwards instead of outwards reports hoop compression of exactly the right size — and the reinforcement designed from it is minimum steel where the structure needs hoop tension steel.
Try it
Unit error laboratory
A consistent steel cantilever, then one named unit slip applied to it. Read what every output does.
Slip
A length typed in millimetres into a field expecting metres is a thousand times too big.
- Quantity affected
- length
- Factor applied
- 1000.0000
| Output | Correct | With the slip | Ratio |
|---|---|---|---|
| Flexural rigidity EI | 94500.0000 | 94500.0000 | unchanged |
| Tip deflectionloudest | 0.0190 | 1.90e+7 | × 1.00e+9 |
| Root moment | 150.0000 | 150000.0000 | × 1000.0000 |
| Self-weight per metre | 0.8779 | 0.8779 | unchanged |
| Base reaction | 30.2674 | 5292.3814 | × 174.8543 |
| Tip stiffness | 1312.5000 | 1.31e-6 | × 1.00e-9 |
| Modal mass | 0.1266 | 126.5644 | × 1000.0000 |
| First frequency | 16.2074 | 1.62e-5 | × 1.00e-6 |
The moment has moved, which means the error is in the geometry or the load rather than in the stiffness. In a determinate structure the moment comes from equilibrium alone.
The technique
- Compare a quantity that depends on stiffness with one that does not. In a determinate structure the moment does not.
- Moment right, deflection wrong: stiffness. Both wrong: geometry or load. Both right, frequency wrong: mass.
- The powers of length are what set the factors — 10³ for a dimension, 10⁶ for an area, 10¹² for a second moment of area.
What this shows: A slip is detectable if you look at the right number and invisible if you look at the wrong one — and a mass error is invisible to every static check there is.
Worked example
Localising an error in one comparison
Given
- A simply supported beam, 8 m span, 30 kN/m uniformly distributed
- The model reports a maximum moment of 240 kN·m
- It reports a maximum deflection of 47 mm
- The section is a 533 × 210 × 92 UB with I = 55 200 cm⁴, in S355 steel
Find
Whether there is an error, and if so where
Practice
A model's self-weight comes out 102 times too large. The density field was filled with a value taken from a materials table. What is the ratio between concrete's density in kg/m³ and its unit weight in kN/m³, to one decimal place?
Practice
A modal mass is entered a thousand times too large. Frequency goes as 1/√m. By what factor is the reported frequency wrong? Give the factor by which it is too low.
Check yourself
Why does a unit error in mass leave every static result untouched?
Check yourself
A deflection comes out a hundred million times too small. What does the exponent tell you?
Summary
- Moments right and deflections wrong means a stiffness error, and rules out geometry and load
- Unit slips come in powers of a thousand; second moment of area is 10¹²
- A mass error is invisible to every static check there is
- Look at the load arrows; sign conventions differ and both are defensible
- A wrong-sign result with the right magnitude is the most dangerous kind
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