Scaffold Base Calculator: formulas and assumptions

This article explains the method behind the Scaffold Base Calculator. Every formula is shown and every assumption stated, in the same order as the calculator: bearing pressure first, then base plate thickness.

Part A: Bearing pressure

1. Notation and default values

P: load per standard (kN), default 20
g: gravitational acceleration, 9.81 m/s²
Bx, By: base plate plan dimensions (mm), default 150 × 150
t: sole board thickness (mm), default 38
w: sole board width (mm), default 225
Lb: sole board length (mm), default 450
n: number of boards laid side by side in a layer, default 1 (2 in the lower layer of a cruciform)
rg: load spread with the grain (horizontal : vertical), default 2
rp: load spread across the grain (horizontal : vertical), default 1
qa: allowable bearing pressure of the ground (kN/m²), default 150 (stiff clay)

2. Load input

The load can be entered in kN, kg or tonnes. It is always converted to kN first:
- Entered in kN: P = value
- Entered in kg: P = m × 9.81 ÷ 1000
- Entered in tonnes: P = m × 9.81
So 1 tonne = 9.81 kN and 1,000 kg = 9.81 kN.
Assumptions:
- P is the vertical load in one standard, applied centrally to the base plate. Eccentricity and moments are not considered.
- P is a working (unfactored) load compared with an allowable bearing pressure. No partial safety factors are applied.

3. Allowable bearing pressure

qa is picked from a list of typical presumed allowable bearing pressures, or entered from a site investigation (Custom). Where a range is given, the value used is shown last.

Concrete slab, sound, at least 150 mm: assumed, uses 1,000 kN/m²
Tarmac on sound sub-base: assumed, uses 200 kN/m²
Strong igneous / gneissic rock: uses 10,000 kN/m²
Strong limestone / sandstone: uses 4,000 kN/m²
Schists and slates: uses 3,000 kN/m²
Strong shales, mudstones, siltstones: uses 2,000 kN/m²
Dense gravel / sand and gravel: over 600, uses 600 kN/m²
Medium dense gravel: 200 to 600, uses 200 kN/m²
Loose gravel: under 200, uses 100 kN/m²
Dense sand: over 300, uses 300 kN/m²
Medium dense sand: 100 to 300, uses 100 kN/m²
Loose sand: under 100, uses 50 kN/m²
Very stiff / hard clay: 300 to 600, uses 300 kN/m²
Stiff clay (default): 150 to 300, uses 150 kN/m²
Firm clay: 75 to 150, uses 75 kN/m²
Soft clay and silt: under 75, uses 40 kN/m²
Custom: your value from a site investigation

Assumptions:

Where a range is given, the lower end is used; for "under" categories a value below the limit is used. This is deliberately cautious.
These are typical values for preliminary use only and must be confirmed by site investigation.
Concrete and tarmac values assume sound, well-supported surfaces. Slab thickness, edges, joints, voids and the sub-base must be checked separately.
The ground is assumed level and uniform under the footprint, and not affected by water, frost, nearby excavations or slopes.

4. Pressure under the base plate alone

Base plate area: A0 = Bx × By ÷ 1,000,000 (m²)
Pressure: p0 = P ÷ A0 (kN/m²)

The base plate is treated as a rectangle Bx × By bearing evenly. The spigot, the plate shape (round or square) and plate flexibility are ignored. This is also the result for the "Base plate" arrangement with no sole boards.

5. Load spread through sole boards

Load spreads outward as it passes down through each layer of timber. For each layer, the footprint arriving at its top (from the plate or the layer above) is split into the size along that layer's grain and the size across it:

Along the grain: a′ = a_in + 2 × rg × t
Across the grain: c′ = c_in + 2 × rp × t
Limited by board length: a = smaller of a′ and Lb
Limited by the width of boards side by side: c = smaller of c′ and n × w

The factor 2 allows for spread on both sides of the load. The footprint leaving the bottom of one layer is the input for the next layer down, turned to suit that layer's grain direction.

Arrangements:

Base plate: no boards
Single board: one board, grain along X
Double board: two boards stacked, both with grain along X
Cruciform: one board along X, over two boards side by side along Y (n = 2)
Custom: up to 4 layers, each with its own thickness, grain direction, number of boards, length, width and spread ratios

Assumptions:

Spread of 2:1 along the grain and 1:1 across it, as commonly used for timber sole boards.
The spread cannot go beyond the timber: it is limited to the board length along the grain and to n × w across it.
Boards laid side by side are tight together with no gaps, so their combined width is n × w.
Boards are stiff enough to spread the load evenly, so the pressure under the final footprint is uniform. Board bending, crushing and moisture condition are not checked.
Board self-weight is ignored.
The load is centred on the boards, so the spread is the same on both sides.
In Custom mode, a newly added layer turns its grain 90° to the layer above, and defaults to enough boards to cover the footprint arriving from above (footprint width ÷ w, rounded up).

Base plate area: A0 = Bx × By ÷ 1,000,000 (m²)
Pressure: p0 = P ÷ A0 (kN/m²)

The base plate is treated as a rectangle Bx × By bearing evenly. The spigot, the plate shape (round or square) and plate flexibility are ignored. This is also the with no sole boards.

6. Bearing pressure and utilisation

Bearing area: A = x × y ÷ 1,000,000 (m²), where x × y is the footprint leaving the bottom layer
Pressure on the ground: p = P ÷ A (kN/m²)
Utilisation: u = p ÷ qa
Minimum area needed: A_req = P ÷ qa (m²)
Equivalent square side: s = √A_req × 1000, rounded up to the next 5 mm

Result shown:

u of 0.85 or less: OK
u above 0.85 up to 1.00: OK, close to limit
u above 1.00: Fails, spread further

The 0.85 band is a visual warning only, not a code limit. The "Compare arrangements" panel repeats sections 4 to 6 for each standard arrangement using the same inputs. The section drawing is a sketch; the height of the pressure block is not to scaleBearing area: A = x × y ÷ 1,000,000 (m²), where x × y is the footprint leaving the bottom layer

Pressure on the ground: p = P ÷ A (kN/m²)
Utilisation: u = p ÷ qa
Minimum area needed: A_req = P ÷ qa (m²)
Equivalent square side: s = √A_req × 1000, rounded up to the next 5 mm

Result shown:

u of 0.85 or less: OK
u above 0.85 up to 1.00: OK, close to limit
u above 1.00: Fails, spread further

7. Worked example (default inputs)

Inputs: P = 20 kN, base plate 150 × 150 mm, cruciform of 38 × 225 × 450 mm boards, stiff clay qa = 150 kN/m².

Base plate alone: 20 ÷ (0.150 × 0.150 = 0.0225 m²) = 888.89 kN/m²
Layer 1 (along X), along the grain: 150 + 2 × 2 × 38 = 302 mm, within 450, so 302 mm
Layer 1 (along X), across the grain: 150 + 2 × 1 × 38 = 226 mm, limited to 1 × 225 = 225 mm
Layer 2 (along Y), along the grain: 225 + 2 × 2 × 38 = 377 mm, within 450, so 377 mm
Layer 2 (along Y), across the grain: 302 + 2 × 1 × 38 = 378 mm, within 2 × 225 = 450, so 378 mm
Bearing area = 0.378 × 0.377 = 0.1425 m²
Pressure = 20 ÷ 0.1425 = 140.34 kN/m²
Utilisation = 140.34 ÷ 150 = 94%, OK, close to limit
Minimum area = 20 ÷ 150 = 0.133 m², so the equivalent square side is 370 mm

Comparison of arrangements:

Base plate: 150 × 150 mm, 888.89 kN/m², 593%, fails
Single board: 302 × 225 mm, 294.33 kN/m², 196%, fails
Double board: 450 × 225 mm, 197.53 kN/m², 132%, fails
Cruciform: 378 × 377 mm, 140.34 kN/m², 94%, OK

For the double board, layer 2 along the grain is 302 + 152 = 454 mm, limited to 450 mm; across the grain it is 225 + 76 = 301 mm, limited to 225 mm.

Part B: Base plate thickness

8. Notation and default values

h: plate thickness (mm), default 5.0
Pt: test load (kN), default 30
L: support span (mm), default 110
b: effective width (mm), default 124 (assumed)
E: elastic modulus (GPa), 210 for steel, 70 for aluminium
δlim: deflection limit (mm), default 5
S: plate size across flats or diameter (mm), default 150

9. Deflection test model

The plate is modelled as a simply supported beam of span L and width b, with a point load Pt at the centre.

Second moment of area: I = b × h³ ÷ 12 (mm⁴)
Central deflection: δ = Pt × L³ ÷ (48 × E × I) (mm)
Load at which deflection reaches the limit: P_lim = δlim × 48 × E × I ÷ L³
Test result: passes if δ is no more than δlim

Units: Pt is converted from kN to N (× 1000) and E from GPa to N/mm² (× 1000), so δ comes out in mm.

Assumptions:

Linear elastic behaviour only. Thin plates may yield before reaching the calculated load, so the result can be optimistic for thin plates.
The spigot is ignored and adds no stiffness.
b = 124 mm is an assumed effective width and should be adjusted to suit the actual plate.
Two-way plate action, shear deformation and support conditions other than simple supports are ignored.
E is 210 GPa for steel and 70 GPa for aluminium. It is set automatically when the material is switched and can be edited.

Example, steel 5 mm plate:

I = 124 × 5³ ÷ 12 = 1,291.67 mm⁴
δ = 30,000 × 110³ ÷ (48 × 210,000 × 1,291.67) = 3.07 mm
3.07 mm is within 5 mm, so it passes

10. Code checks

Minimum thickness (BS EN 74-3): h at least 5 mm for S235 steel, at least 7 mm for aluminium
Plate size (TG20:21): S between 140 mm and 150 mm
Design load against test load: P no more than Pt

Assumptions:

BS EN 74-3 deems S235 steel plates of 5 mm or more, and aluminium plates (0.2% proof stress 195 N/mm²) of 7 mm or more, to be stiff enough.
TG20:21 calls for a 150 × 150 mm (± 10 mm) base plate under each standard. The calculator accepts 140 to 150 mm as OK.
P is the load per standard from Part A. If P is more than the test load it is flagged as a caution only and does not change the overall result.
Switching to aluminium raises the thickness to 7 mm if a smaller value was set.
Plate size is shared between the two parts: entering S sets Bx = By = S in Part A, and Part A shows S as the larger of Bx and By.

11. Overall result

Meets checks: the deflection test passes, the thickness is at least the minimum and the size is 140 to 150 mm.
Below deemed minimum: the deflection test passes but the thickness or size check does not.
Fails test: δ is more than δlim.

The supplier thickness table uses the same deflection formula for 1.5, 2, 3, 4, 4.5, 5 and 6 mm (plus 7 and 8 mm for aluminium) and the selected thickness. The chart plots deflection against thickness from 1.5 to 10 mm.

Guidance only. Ground values are typical presumed allowable bearing pressures and must be confirmed by site investigation. Concrete and tarmac values assume sound, well-supported surfaces; check slab thickness, edges, joints and the sub-base. Load spread uses 2:1 along the grain and 1:1 across it, as commonly used for timber sole boards, limited by board length and the width of boards laid side by side. Spread areas assume the boards are stiff enough to spread load evenly; board self-weight is ignored; g = 9.81 m/s². BS EN 74-3 deems S235 steel of 5 mm or more and aluminium (proof stress 195 N/mm²) of 7 mm or more to be stiff enough; TG20:21 calls for a 150 × 150 mm (±10 mm) base plate under each standard. Designs should be checked by a competent temporary works engineer. Results are for preliminary estimation and educational purposes only. They do not replace professional engineering judgement, applicable codes and standards, or design verification by a qualified engineer. Use at your own risk.