Module 16 · Lesson 16.2

Every check, and the one that governs

The design is not a list of passes. It is a set of utilisations of which exactly one is largest — and in this building it was not the one anybody expected.

Why this matters

Ask an engineer what governed a beam and you learn something about the design. Ask whether it passes and you learn almost nothing. This lesson designs the four members of the building, records every check, and finds that bending — the thing the whole course has spent most of its time on — governed absolutely nothing.

By the end of this lesson you should be able to

Design each member of the building and record every check
Identify the governing check on each member and on the structure
Recognise when two checks are so close that the section is the problem
Follow a change through the structure it was not intended to affect

A design is a set of utilisations

Each check compares a demand against a capacity, and their ratio is the utilisation. A design is therefore a set of numbers between zero and one, and exactly one of them is largest.

That largest one is the governing check, and it is the single most useful thing to know about a design, whether or not anything is failing:

It is the first thing that will fail if the loading rises.
It is the only thing worth changing if the member has to be improved.
It tells a reviewer what the designer was actually up against.

A report that says all checks pass has computed all of this and then thrown it away.

There is a second question worth asking straight afterwards: how far ahead of the next check is it? A member at 98% in bending and 35% in shear is a bending problem, and deepening it will help. A member at 98% and 95% is a section problem — relieve bending and shear immediately takes over, so nothing much improves until the section itself changes.

Worked example

Stage 1 — the slab

Given

200 mm one-way slab, 5.0 m span, C30/37
w = 14.63 kN/m² from the take-down
25 mm cover, H12 bars

Find

The reinforcement, and which check decides it.

Worked example

Stage 2 — the beam, first attempt

Given

350 × 550 beam, 7.5 m simply supported span, C30/37
w = 77.26 kN/m from the take-down; M = 543 kNm, V = 290 kN
35 mm cover, H10 links, H32 main bars

Find

The reinforcement, and whether the section works.

Predict first

The beam failed deflection at 112%. It will be deepened from 550 mm to 650 mm. What happens to the beam's own design moment?

Worked example

Stage 3 — the revision, and what it did to everything else

Given

The beam deepened from 350 × 550 to 350 × 650
Everything else unchanged

Find

Whether the beam now works, and what the change cost elsewhere.

Try it

The whole building, in one picture

Eight checks on four members, ranked. The bar at the top governs. Change any dimension and watch the consequences travel down the load path — including to the members you were not trying to change.

Beam depth550 mm

Slab thickness200 mm

Column size400 mm

Pad side2,600 mm

Imposed load q_k3.0 kN/m²

Slab permanent load g_k: 7.50 kN/m²
Beam design load w: 77.3 kN/m
Beam moment M_Ed: 543 kNm
Beam effective depth d: 489 mm
Steel required / provided: 2620 / 3217 mm²
H10 link spacing: 298 mm
Column N_Ed: 1576 kN
Saved by the imposed reduction: 84 kN
Characteristic load on the pad: 1142 kN
Bearing pressure: 184 kN/m²

1 check fails. Span/depth is the worst at 112%. Span/depth governs clearly, 18 percentage points ahead of Span/depth. There is a single thing to fix.

Things worth trying

Start at the defaults: a 550 beam FAILS span/depth at 112% while its bending sits at 82% and its shear at 53%. Nothing in the strength calculation would have warned you.
Raise the beam to 650. Deflection drops to about 71% and the whole beam falls quiet — but watch the column load rise at the same time, because the deeper beam weighs more and three storeys of it reach the ground.
Now thicken the slab instead, and leave the beam at 550. It helps the slab, and it makes the beam WORSE — a thicker slab is a heavier load on the beam, and the beam was the member in trouble.
Shrink the column towards 250 mm. Its own axial check tightens, but so does the pad's bearing pressure — a lighter column is less load, yet the pad is sized for a load that barely moved.
Shrink the pad. Bearing pressure climbs steeply, since it goes with the square of the side, and it takes over as the governing check long before punching shear becomes interesting.
Raise the imposed load towards 7.5 kN/m². Watch which check reaches 100% first — it is not the same one at every set of dimensions, and that is what the governing check is telling you.
Find a set of dimensions where the top two bars are within a few points of each other. That is a well proportioned structure: no single change improves it, because whatever you relieve, the next check takes over.

The finished design, and what governed it

Every check on every member, as utilisations:

Member	Check	Utilisation
Slab	Span/depth	0.94
Slab	Bending	0.87
Foundation	Bearing pressure	0.93
Beam	Span/depth	0.71
Beam	Bending	0.69
Column	Axial	0.45
Beam	Shear	0.43
Foundation	Punching shear	0.18

The structure is governed by the span/depth ratio of the slab — the first member designed, the simplest one, and the one nobody worries about.

Three observations are worth more than the table itself.

Bending governed nothing. Not the slab, not the beam, not the column. The course spends most of its time on bending because it is where the mechanics lives, but in an ordinary floor it is rarely the constraint.

Deflection governed the slab and failed the beam. Both are span/depth checks, and both are empirical rules rather than mechanics. That is uncomfortable and it is true.

The foundation's 0.93 is not really a finding. The pad was sized from the bearing pressure, so it could hardly report anything else — round the 2.51 m up to 2.6 m and you get 0.93 rather than 1.00. A member sized directly from a limit will always sit near that limit, and quoting it as the governing check is close to circular. The informative governing check is the largest one among the members whose size was chosen for some other reason, and that is the slab at 0.94.

Note

Where the two closest checks are within a few points

On the finished beam, span/depth is at 0.71 and bending at 0.69 — two percentage points apart.

That is a different situation from a single dominant check. If the loading rose, both would fail at nearly the same moment, and relieving either one alone would achieve almost nothing because the other would immediately take over.

It is also a sign of a well proportioned section. A beam at 0.71 and 0.69 is using its depth about as efficiently as its steel. A beam at 0.95 and 0.30 is not — it is deeper than it needs to be in one respect and stretched in another.

So the spread between the top two checks is worth reporting alongside the governing one. It answers a different question: not what fails first, but is this section the right shape at all.

Practice

At the base of the column, G_k = 937 kN and Q_k = 281 kN, and the imposed reduction factor is 0.8. What is N_Ed, in kN?

Practice

The pad carries a characteristic column load of 1162 kN and weighs 101 kN itself. It is 2.6 m square. What is the bearing pressure, in kN/m²?

Practice

Deepening the beam from 550 to 650 mm adds 0.875 kN/m of self-weight. Two beam half-spans of 3.75 m frame into each column, on three storeys. How much factored load does that add to the column, in kN?

Check yourself

The slab's bending utilisation is 0.87, yet K = 0.053 is only a third of K_lim. How can both be true?

Check yourself

A beam is at 0.95 in bending and 0.92 in shear. What does that tell you that a pass/fail would not?

Summary

A design is a set of utilisations, and exactly one is largest
The governing check is what to report — it decides what happens when loading changes
The spread to the second check says whether the SECTION or the check is the constraint
Bending governed nothing in this building; deflection governed the slab and failed the beam
A downstand beam in sagging is a T-section, and the flange is worth a factor of eight in K
Bars are sometimes chosen for fit rather than area — 6 H25 would not go in a 350 web
A footing's steel is often set by minimum steel, not by the moment
Fixing the governing check loads every member below the one you changed

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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