L-Shaped Stair with Landing (Quarter-Turn)

About L-Shaped Stair with Landing Calculation

This L-shaped stair with landing calculator performs the geometric calculation of a quarter-turn staircase with a 90° turn. Using the staircase plan dimensions and floor-to-floor height, it determines step dimensions, landing level, stair angle, step-comfort ratio, and the geometric dimensions of stringers and risers.

The calculation uses a uniform rise height and a uniform calculated going for both flights. When automatic step selection is enabled, the calculator tests possible combinations of step counts and chooses the option that best matches the specified comfort ranges.

Guidelines and recommendations

Number of rises and rise height

Total number of rises. The lower flight contains the specified number of steps plus one additional rise onto the landing. The position of the upper step determines whether another rise is required to reach the second-floor level.

N = nlow + 1 + ntop + δ

Here, nlow is the number of lower steps and ntop is the number of upper steps. δ = 1 when the upper step is below the second-floor level, and δ = 0 when it is level with the second floor.

Rise height. The total floor-to-floor height H, mm, is divided equally among all rises.

h = H / N

Therefore, every rise has the same calculated height. Changing the position of the upper step changes the number of rises, so with the same floor-to-floor height it also changes the rise height, stair angle, and step-comfort ratio.

Calculation of the going

Available length of the upper flight. The landing and flight width B, mm, is subtracted from the total staircase length L, mm. The remaining length is divided by the number of upper steps.

atop = (L - B) / ntop

Available length of the lower flight. The same calculation is performed using the second plan dimension of the staircase W, mm.

alow = (W - B) / nlow

Calculated going. The same going is used for both flights. Therefore, the final value is the smaller of the two calculated values.

a = min(atop, alow)

This principle ensures that both flights fit within the specified dimensions. If one flight has more available horizontal space, that extra length does not increase the step going because the calculator keeps the step size uniform throughout the staircase.

Full tread depth

Calculated going and physical tread depth are different values. The calculated going a defines the horizontal spacing between adjacent steps. The full tread depth additionally includes the nosing projection e, mm, and, when risers are used, the riser thickness tr, mm.

d = a + e + tr

If risers are disabled, tr = 0. The nosing projection and riser thickness increase the physical tread depth but do not change the calculated going, flight angle, or the 2h + a comfort ratio.

Landing level

Landing height. The number of rises up to the landing equals the number of lower steps plus one additional rise onto the landing.

Hpl = (nlow + 1) × h

The value Hpl is given in millimetres relative to the lower floor level.

Stair angle and step comfort

Flight angle. Because both flights use the same rise height and calculated going, one inclination angle is used.

α = arctan(h / a)

The calculator uses 30-40° as a practical comfort range. The angle is displayed to the nearest 0.1°.

Rise height. A range of 150-200 mm is used for evaluating the result.

Calculated going. A range of 270-320 mm is used for evaluation.

Comfort ratio. The relationship between rise height and going is also checked:

2h + a

The calculator uses 600-660 mm as the comfort range, with a recommended target of 630 mm. The formula uses the calculated going without the tread nosing projection.

Automatic selection of the number of steps

Combination search. Automatic selection tests every combination from 1 to 13 upper steps and from 1 to 13 lower steps. For each combination, the calculator recalculates the rise height, going, inclination angle, and 2h + a ratio.

Primary selection. Priority is given to the combination with the smallest total deviation from the ranges 150-200 mm for rise height, 270-320 mm for going, 30-40° for angle, and 600-660 mm for 2h + a. The deviation of each parameter is normalised relative to the width of its permitted range and then squared. A value within its range receives no penalty.

Selection between similar options. If the main scores are equal, the calculator then considers the number of parameters outside their ranges, how close the results are to the centres of the recommended ranges, how evenly the available lengths of the two flights are used, and the total number of steps.

For this reason, automatic selection cannot always produce four values within the recommended ranges for every set of dimensions. If the available geometry does not allow this, the calculator selects the closest valid combination and separately highlights values that fall outside the recommended ranges.

Stringer calculation

Inclined length of one step. The basic length along the flight is calculated using the Pythagorean theorem.

l = √(h2 + a2)

Upper stringer. In the adopted geometric model, the upper flight stringer length is calculated as the number of upper steps multiplied by the inclined length of one step.

Ltop = ntop × l

The upper and lower edges of the upper stringer have the same calculated length in this model.

Lower stringer. First, the length of its upper edge is determined from the number of lower steps, rise height, tread thickness ts, and flight angle.

Llow,top = (nlow × h - ts) / sin(α)

The length of the lower edge is corrected for the stringer width k, mm.

Llow,bottom = Llow,top - k × (tan(α) + 1 / tan(α))

Riser calculation

Riser height. When riser calculation is enabled, the tread thickness is subtracted from the rise height.

hr = h - ts

Number of risers. It is taken as equal to the total number of rises N. The length of each riser is taken as equal to the flight width B.

Calculation assumptions

• The two flights are arranged at 90° to each other.
• The landing width is taken as equal to the specified flight width.
• All rises have the same height.
• The calculated going is the same in the lower and upper flights.
• The calculation is performed in millimetres. The main linear results are rounded to the nearest millimetre.
• Automatic selection uses unrounded intermediate values, so rounding of displayed results does not affect the selected number of steps.

European regulatory references

EN 17210:2021 "Accessibility and usability of the built environment - Functional requirements". This European standard provides functional requirements for the accessibility and usability of elements of the built environment, including steps and stairways.

CEN/TR 17621:2021 "Accessibility and usability of the built environment - Technical performance criteria and specifications". This technical report supplements EN 17210 and provides technical criteria used when implementing its functional requirements.

Mandatory staircase dimensions can vary between EU countries and depend on the type and use of the building. Therefore, the ranges 150-200 mm, 270-320 mm, 30-40°, and 600-660 mm used by this quarter-turn staircase calculator are geometric guidelines for evaluating the resulting layout, while final dimensions should be checked against the national building requirements of the country where the staircase will be installed.

FAQs

Why does the rise height change when the upper step is level with the second floor?

In this mode there is one fewer rise than when the upper step is positioned below the second-floor level. The same total floor-to-floor height is divided by fewer rises, so each rise becomes higher. The stair angle and the 2h + a ratio change at the same time.

Why does the calculator use the smaller of the two calculated going values?

The upper and lower flights are limited by different plan dimensions of the L-shaped staircase. A uniform going must fit in both directions, so the shorter available run becomes the controlling value. This keeps the step spacing consistent throughout the quarter-turn staircase.

Why can one result remain outside the recommended range even when automatic selection is enabled?

Automatic selection can only change the whole-number step count in each flight. For some combinations of floor-to-floor height, staircase length, width, and landing size, no integer combination can simultaneously satisfy the recommended rise height, going, angle, and 2h + a ratio. In that case, the calculator selects the option with the smallest overall deviation.

What is the difference between calculated going and full tread depth?

The calculated going is the horizontal distance between adjacent rises and is used to determine the stair slope and walking comfort. The full tread depth additionally includes the tread nosing and, when risers are installed, their thickness. Therefore, the physical tread depth is usually greater than the value used in the 2h + a formula.

Why should the recommended dimensions be checked against the requirements of a specific country?

European documents EN 17210 and CEN/TR 17621 provide a common framework for accessibility and technical performance in the built environment, while mandatory staircase geometry is also governed by national regulations. Requirements for the same L-shaped stair with landing can therefore differ depending on the country and the intended use of the building.