Staircases step-by-step: a geometry lesson

from Staircases step-by-step: a geometry lesson

Mark Milner delivers an expert, step-by-step (pun intended) guide on how to get the maths right and build a well fitting straight-flight private staircase to recommendations laid down in Approved Document Part K.

The pinnacle of a joiner’s craft, designing and building stairs requires a level of expertise which, once acquired, commands respect amongst colleagues and increases employability and earning potential. With such attributes comes responsibility – mistakes can be very costly and so the importance of ‘getting it right’ cannot be overstated.

Before we look at the steps to getting it right, it will be useful to familiarise ourselves with the main components of a straightflight staircase and basic setting out terms, together with recommendations that apply to a private stair as laid down in The Building Regulations 2010: Approved Document Part K: Protection from falling, collision and impact, 2013 edition (ADK),1 illustrated in Figures 1 and 2 respectively and described below.

Strings: the inclined structural members – the sides of the stair which support the treads and risers. A string can be identified as a ‘wall string’, where the side of a staircase abuts a wall; or an ‘outer string’, where the side of a staircase is exposed.

Newel post: a heavy, vertical member, to which the handrail is connected and the string (usually) tenoned.

Tread: the horizontal part of a step – the part upon which stair users tread.

Riser: the vertical part of a step that provides support to the tread

Nosing: the moulded front edge of a tread (or landing) which overhangs the face of the riser.

Rise: the vertical distance from the top surface of one tread to the top surface of the next. The maximum permitted rise is 220mm.

Going: the horizontal distance from the tip of one nosing to the tip of the next. The minimum permitted going is 220mm.

The pitch line is an imaginary line which touches all the nosings in a flight. The maximum pitch, measured between a horizontal plane (the floor) and the pitch line, is 42°. Headroom is measured vertically from the pitch line and must be not less than 2m.

As well as specifying maximum and minimum sizes for the rise and going, ADK also stipulates the relationship between the >> two in order to make walking up and down stairs as comfortable as possible. This relationship is expressed as a formula which states that twice the rise plus the going should be not less than 550mm or greater than 700mm, and can be written:

2R + G = 550 to 700 (where R = rise and G = going)

For example, a staircase with a rise of 200mm and a going of 225mm would satisfy this requirement because the sum of twice the rise plus the going does fall between 550mm and 700mm:

2 x 200 + 225 = 625mm

Getting it right: maths for staircase design

Seldom will the answer to this calculation be a whole number. Although our answer is very close to 9, rounding down to the nearest whole number will give us a rise greater than 220mm:

1995 ÷ 9 = 221.7mm

But when we round up to 10:

1995 ÷ 10 = 199.5mm

The answer of 199.5mm is below the permitted maximum of 220mm.

So, having performed two simple calculations we have learnt two things: firstly, how many risers (or steps) there will be (10) and secondly, how high each rise will be (199.5mm). A quick double-check will confirm this:

10 x 199.5 = 1995mm R

Step 2:

Next, we must determine a suitable going to go with our rise of 199.5mm. My method of choice (there are several) is trigonometry because it’s quick, easy, and totally accurate. Using a scientific calculator, the rise is divided by the tangent of 42°:

199.5 ÷ 42 tan = 221.6mm (round to 222mm)

An inverse calculation, using our rounded answer of 222mm should, if correct, produce an answer of just under 42°:

199.5 ÷ 222 = 0.8986 tan –1 = 41.9° R

The secret to a perfect-fitting staircase? Calculations – combining accurate site measurements with the recommendations given in ADK to calculate suitable staircase dimensions.

A new straight-flight staircase from the ground floor up to an intermediate landing is required for a client’s holiday home in Cornwall, and the following describes the design process. In addition to the width, the following site measurements have been recorded and are illustrated in Figure 3:

1. Finished floor to finished floor rise: 1,995mm

2. Limit of horizontal distance at lower floor: 2,642mm

3. Length of stairwell opening: 1,874mm

4. Finished floor to underside of ceiling height: 2,340mm

Step 1:

To work out the rise, we’ll start by dividing the total floor to floor rise (1,995mm) by 220. Why 220? Because 220mm is the maximum permitted rise:

1995 ÷ 220 = 9.1

Space in the client’s holiday cottage is limited and a staircase with a pitch of 42° takes up the least amount of room while complying with ADK’s recommendation.

Step 3: Checking the relationship between the dimensions of the rise and going is quick and easy. Remember the formula:

2R + G = 550 – 700

So, when we substitute the letters R and G in the formula for the rise and going that we have now worked out, we get the following sum:

2 x 199.5 + 222 = 621mm R

621mm is more than 550mm and less that 700mm and ADK’s recommended relationship between the dimensions of the rise and going is satisfied. What’s more, 621mm hits the ‘sweet spot’ of being very close to the midpoint between 550 and 700mm. >>

Checking that there’s sufficient headroom – a minimum of 2m – is probably the trickiest part of the process. We’ll start with what we know: we know the staircase can rise up to 340mm and still provide sufficient headroom, because when we take 2,000mm away from the floor-to-ceiling height we get:

2340 – 2000 = 340mm

340mm represents a little less than two risers:

340 ÷ 199.5 = 1.7

As can be seen in Figure 4, at the point where the staircase rises to 340mm, there is around ¾ of one tread (or going), meaning there are a further 8¼ treads between this point and the landing trimmer or trimming joist. If the combined horizontal length of these 8¼ treads, ie 8¼ goings, is less than the length of the stairwell opening (1,874mm), then there will be sufficient headroom. If greater than 1,874mm, the headroom height will be less than the required 2m.

The going for our staircase is 222mm, therefore:

8.25 x 222 = 1831.5mm

The combined horizontal length of the remaining 8¼ steps is less than the stairwell length of 1,874mm, meaning that the available headroom will be greater than the required 2m. Figure 4 also shows that there will be a clear 400mm between the foot of the new staircase and the doorway.

In designing our client’s 10-rise staircase we have demonstrated compliance with the Building Regulations because:

• The rise, at 199.5mm, is less than the maximum 220mm.

• The going, at 222mm, is greater than the minimum 220mm.

• The angle of pitch is (slightly less than) 42°.

• Twice the rise plus the going, at 621mm, is greater than 550mm and less than 700mm.

• Headroom is greater than 2m. n the author

Milner

Author of Simply Stairs: The Definitive Handbook for Stair Builders, ISBN 978-1-84995-149-4, Whittles Publishing, 2015, available from the BM TRADA bookshop.

Staircases step-by-step: a geometry lesson

Getting it right: maths for staircase design

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Staircases step-by-step: a geometry lesson