Module 8 · Lesson 8.2
Floors and cores
Four models of one floor panel against the exact answer — and when a core needs more than a stick.
Why this matters
A floor panel and a shear core are the two places where the cheap model and the expensive model differ most, and where the choice between them is most often made by habit.
Both have an exact answer available to compare against, so this is one of the few places in modelling where 'how wrong is the simple model?' has a precise answer rather than an argument.
By the end of this lesson you should be able to
- Compare a one-way model against the exact plate solution
- Say how the error depends on the panel's aspect ratio
- Recognise a peak moment that is a singularity
- Decide when a core stick model needs shear deformation and openings
What you should already know
- Module 6's shear deformation — the core lesson is the same idea at building scale
- Module 12's mesh convergence, for the singularity
One panel, four models
A 5 m × 5 m simply supported panel, 250 mm thick, 10 kN/m². Four models, all compared against the exact series solution of the plate equation:
| Model | Moment, kN·m/m | Error |
|---|---|---|
| One-way strip | 31.25 | +183 % |
| Orthogonal grillage | 15.63 | +41 % |
| Plate theory (exact) | 11.05 | — |
| Shell finite elements | 11.05 | converges to plate theory |
A square panel spans equally in both directions, so treating it as one-way makes a single strip carry all the load: nearly three times the moment that is actually there.
Now stretch the panel to 5 m × 15 m:
| Model | Moment, kN·m/m | Error |
|---|---|---|
| One-way strip | 31.25 | +5.6 % |
| Orthogonal grillage | 30.87 | +4.4 % |
| Plate theory (exact) | 29.58 | — |
The one-way moment has not changed — wL²/8 does not know about the other direction. What changed is the truth it is compared with.
The one-way idealisation is always conservative, and its error falls steeply with aspect ratio: 183 % at 1:1, 5.6 % at 3:1.
The share carried in the short direction tells the same story: 50 % at 1:1, 65 % at 1.5:1, 73 % at 2:1, 80 % at 3:1. Past about 2:1 the panel really is behaving one-way, and modelling it as such costs little.
The peak that is not a result
A flat slab meshed with shell elements will report a very large moment directly over a point support, and it will report a larger one every time the mesh is refined.
That is not a result. It is a singularity: the mathematical model has a point support, a point support has infinite stress, and the mesh is converging towards infinity exactly as it should. Module 12 covers how to tell a singular sequence from a converging one — it grows at a roughly constant ratio rather than settling.
What to do instead is integrate. The total static moment across a panel width must equal wL²/8, however the model distributes it. Integrating the model's moments over a width and comparing with that figure is a check that works regardless of the singularity, and it is the standard way of turning a shell result into something a slab can be designed from.
Cores
A shear core is usually modelled as one vertical line element with the gross second moment of area. Under 12 kN/m of wind, a 30 m tall core 6 m across with 300 mm walls:
| Model | Top deflection | Ratio |
|---|---|---|
| Stick, bending only | 1.07 mm | 1.00 |
| Stick, with shear deformation | 1.31 mm | 1.22 |
| Coupled walls, openings admitted | 2.38 mm | 2.22 |
Shear deformation adds 22 %, and admitting the openings that make it a usable core more than doubles the movement. Neither is a refinement; both are the difference between a plausible answer and a right one.
The threshold is computable: for this core, shear exceeds a fifth of the total deflection below a height/width ratio of 4.73. Squat cores need the shear term; slender ones do not.
All three models give the same base moment, because statics does not care how the core is modelled. It is only the stiffness — and therefore the deflection, the sway, the load share between core and frame, and the period — that depends on the choice.
Try it
Slab model comparison
One panel, four models, compared against the exact series solution of the plate equation.
The short span is fixed at 5 m.
| Model | Moment | Error | Deflection |
|---|---|---|---|
| One-way strip | 31.25 | +182.8 % | 2.00 mm |
| Orthogonal grillage | 15.63 | +41.4 % | 1.00 mm |
| Plate theory (exact) | 11.05 | — | 0.62 mm |
| Shell finite elements | 11.05 | — | 0.62 mm |
At 1.00:1 the panel spans substantially both ways, and a one-way model over-predicts the moment by 183 %. Conservative, and expensive on every panel of the floor.
What each model is blind to
- One-way strip: Two-way action, corner torsion, and the reaction that actually goes onto the long-side supports.
- Orthogonal grillage: Twisting moment, which is why it over-predicts the span moments.
- Plate theory (exact): Cracking, reinforcement, creep, and the supports actually being beams that deflect.
- Shell finite elements: Nothing plate theory does not — but it will happily report a peak moment at a point support that is a singularity, not a result.
What this shows: The one-way moment never changes — what changes is the truth it is compared with.
Try it
Core model comparison
A stick, a stick with shear deformation, and coupled walls — the same core, three models.
1 is fully continuous walls; 0 is two independent walls.
| Model | Deflection | vs bending only |
|---|---|---|
| Stick model, bending only | 1.071 mm | × 1.000 |
| Stick model, with shear deformation | 1.311 mm | × 1.224 |
| Coupled walls | 2.383 mm | × 2.224 |
- Height / width
- 5.00
- Shear passes a fifth of the movement below
- H/D = 4.73
- Base moment
- 5400 kN·m
identical in all three — statics does not care how it is modelled
At this slenderness bending dominates and the stick model is a reasonable first estimate. The openings are still not in it.
What each model is
- Stick model, bending only: One vertical line element with the core's gross second moment of area. The cheapest possible model of a core.
- Stick model, with shear deformation: The same element, with the shear area included. For a squat core this is not a refinement.
- Coupled walls: The openings are admitted: the walls act together only as far as the coupling beams make them.
What this shows: Shear deformation is not a refinement for a squat core, and openings more than double the movement.
Worked example
Is a one-way model good enough for this panel?
Given
- A 5 m × 10 m simply supported panel, 250 mm thick, 10 kN/m²
- The designer proposes to model it as a one-way strip spanning 5 m
Find
The error, and whether it is acceptable
Practice
A one-way strip model of a square panel gives 31.25 kN·m/m and the exact plate solution gives 11.05. By what percentage does the one-way model over-predict?
Practice
A core's top deflection is 1.07 mm from bending alone and 1.31 mm with shear included. What percentage of the total does the bending-only stick model omit?
Check yourself
A one-way model of a square panel over-predicts the moment by 183 %. Why is that still not simply 'safe'?
Check yourself
Why does over-estimating a core's stiffness put load in the wrong place?
Summary
- A one-way model of a square panel over-predicts by 183 %, of a 3:1 panel by 5.6 %
- The one-way moment never changes; the truth it is compared with does
- Past about 2:1 a panel really is one-way and the idealisation costs little
- A peak over a point support is a singularity — integrate rather than refine
- Shear deformation adds 22 % to a squat core, and openings more than double it
- All core models give the same base moment; only the stiffness differs
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