U-Shaped Winder Stair

About U-Shaped Winder Stair Calculation

This U-shaped winder stair calculator performs the geometric calculation of a staircase with a 180-degree turn and winder steps. It can also be used as a double quarter-turn staircase calculator. Based on the specified dimensions and number of steps, it determines the riser height, going of the straight flights, staircase angle, step comfort value, geometry of the winder section, stringer dimensions and, when required, riser dimensions.

The calculation is arranged so that the upper and lower straight flights have the same riser height and the same calculated going. The winder steps divide the complete 180° turn into the specified number of equal angular sections.

Guidelines and recommendations

Automatic selection of the number of steps

When automatic selection is enabled, the calculator checks the permitted combinations of the number of upper, lower, and winder steps: upper and lower straight steps - from 0 to 13, winder steps - from 2 to 8.

For each geometrically valid combination, the calculator determines the step height h, calculated tread depth a, flight inclination angle, and the comfort indicator 2h + a. The combinations are compared using guideline ranges of 150-200 mm for step height, 270-320 mm for tread depth, 30-40° for the angle, and 600-660 mm for the expression 2h + a.

The calculator selects the combination with the smallest deviation from these ranges. If several combinations produce practically identical results, preference is given to the number of winder steps closest to 3, followed by a more even distribution of steps between the upper and lower straight flights.

If the specified staircase dimensions do not allow a combination that satisfies all recommended ranges at the same time, the closest valid combination is selected. In this case, some result values may be highlighted in red.

If the number of upper, lower, or winder steps is changed manually, automatic selection is disabled and the calculator then uses the values entered by the user.

Number of rises and riser height

Total number of rises. It is determined from the number of lower, upper and winder steps. The position of the top step also affects the calculation: if it is below the upper-floor level, one additional rise is included; if it is level with the upper floor, no additional rise is added.

n = nlower + nupper + nwinder + k

Here k = 1 when the top step is below the upper-floor level, and k = 0 when it is level with the upper floor.

Riser height. The total staircase height H, in mm, is divided by the resulting number of rises. All rises are assumed to have the same height.

h = H / n

As a practical guideline, the calculator highlights a riser height of 150-200 mm. A value outside this range does not change the calculation formula, but indicates that the selected combination of dimensions and number of steps should be reconsidered.

Calculated going of the straight steps

Available flight length. If the overall staircase length in plan is L and the width of one flight is B, the length available for the straight sections is L - B. The total plan width of the U-shaped staircase is taken as 2B.

Going of the upper and lower flights. A separate possible going is first calculated for each straight section:

aupper = (L - B) / nupper

alower = (L - B) / nlower

If one of the straight sections has zero steps, that section does not limit the final going. The smaller of the calculated values is selected, and the going is additionally limited to the flight width B.

a = min(aupper, alower, B)

This gives both straight flights the same calculated going while ensuring that both fit within the specified staircase length. The calculator uses 270-320 mm as a practical reference range for this value.

Overall tread depth

Calculated going and construction depth. The calculated going a is the horizontal distance between consecutive rises. The overall tread depth additionally includes the tread nosing e. If risers are included, their thickness tr is also taken into account.

d = a + e + tr

When risers are not included, tr = 0 is used. The staircase angle and step comfort value are calculated from the going a, not from the overall tread depth d increased by the nosing.

Step comfort value

Relationship between riser height and going. After h and a have been determined, the calculator checks the widely used Blondel relationship:

2h + a

The calculator treats 600-660 mm as a comfortable range, with approximately 630 mm as the recommended reference value. The formula evaluates riser height and going together: an adequate going cannot compensate for an excessively high riser, and vice versa.

Staircase angle

Stair angle. It is calculated from the riser height h and the calculated going a.

α = arctan(h / a)

The calculator uses 30-40° as a practical guideline for a comfortable staircase angle. Since the angle is calculated after selecting the common going, it is the same for the upper and lower straight flights.

Geometry of the winder turn

180° turn. The full turn is divided by the number of winder steps nwinder. The angular increment between adjacent boundaries of the winder steps is:

Δφ = 180° / nwinder

The boundaries of each winder tread are constructed using successive rays with this angular increment and their intersections with the limits of the turning area. Changing the number of winder steps therefore changes the shape of the individual treads while preserving the complete 180° turn.

Height at the start of the winder section. The lower straight steps plus one additional rise are included between the lower floor level and the base of the winder section.

Hwinder = (nlower + 1)h

The calculator accepts from 2 to 8 winder steps. The upper and lower straight sections may each contain from 0 to 13 steps.

Stringer calculation

Inclined step spacing. The distance between consecutive steps along a straight stringer is calculated as the hypotenuse of a right triangle with riser height h and horizontal going a.

s = √(h2 + a2)

Lower stringer. The upper edge length of the lower stringer is determined from the vertical height of the lower flight, taking into account the tread thickness ts and the staircase angle:

Llower,upper = (nlowerh - ts) / sin(α)

The specified stringer width Bs is additionally taken into account for the lower edge:

Llower,lower = Llower,upper - Bstan(α) - Bs/tan(α)

Upper stringer. Its length is determined from the number of upper straight steps and the inclined step spacing:

Lupper = nuppers

In the geometric model used by the calculator, the upper and lower edges of the upper stringer have the same calculated length. Right-hand and left-hand stair configurations are mirror images: changing the direction changes the drawing but not the numerical dimensions of the components.

Riser calculation

Riser panel height. When risers are included, the tread thickness ts is subtracted from the full rise height.

hr = h - ts

The number of risers is taken as equal to the total number of rises n, and the length of each riser is equal to the flight width B. The riser thickness is also added to the overall tread depth, but it does not change the calculated going a.

Conditions for a valid calculation

Geometric validation. The calculation is performed only for combinations of dimensions that produce positive and consistent staircase geometry. In particular, the staircase length must not be smaller than the flight width, the rise height must be greater than the tread thickness, and the calculated stringer dimensions must remain positive.

• Number of upper straight steps: 0-13.
• Number of lower straight steps: 0-13.
• Number of winder steps: 2-8.
• The tread nosing cannot be negative.
• When risers are included, their thickness must be greater than 0 mm.

Related European standards

Stair dimensions. Permitted tread dimensions and measurement rules depend on the country, building use and type of staircase. In European practice, geometry is checked against national documents such as DIN 18065 "Stairs in buildings - Terminology, measuring rules, main dimensions" in Germany and ÖNORM B 5371 "Stairs, railings and parapets in buildings and outdoor structures - Principles for design and execution" in Austria.

The ranges of 150-200 mm for riser height, 270-320 mm for calculated going, 30-40° for staircase angle and 600-660 mm for 2h + a are used by the calculator as practical guidelines. Final dimensions should be checked against the requirements applicable in the country of construction and for the specific building type.

FAQs

Why does changing the number of steps affect several results at once?

The number of steps determines the total number of rises, so it directly affects the riser height h. Changing the riser height also changes the staircase angle, the 2h + a comfort value, inclined step spacing and stringer dimensions. For this reason, designing a U-shaped winder stair usually involves several successive adjustments to the number of steps.

Why are the calculated going and overall tread depth different?

The calculated going a is the horizontal distance between consecutive rises and is used to determine the staircase slope and step comfort. The overall tread depth additionally includes the tread nosing and, when used, the riser thickness, so it can be noticeably greater.

Why is the same tread going used for both straight flights?

The upper and lower flights may contain different numbers of steps, so a separate possible going is first calculated for each one. The calculator selects the smaller value so that both parts of the double quarter-turn staircase fit within the specified overall length L and maintain the same stepping rhythm.

How does the number of winder steps affect the staircase turn?

The staircase always turns through 180°, but this angle is divided into a different number of sections. For example, three winder steps give a basic angular increment of 60°, while six winder steps give 30°. Increasing the number of winder steps therefore makes the change of direction between adjacent treads more gradual.

Which values should be considered when selecting comfortable stair dimensions?

No single parameter should be assessed on its own. A comfortable combination is commonly obtained when the riser height is approximately 150-200 mm, the calculated going is 270-320 mm, the staircase angle is about 30-40°, and the 2h + a value is within 600-660 mm and close to the reference value of 630 mm.