U-Shaped Stair with Landing
About U-Shaped Stair with Landing Calculation
This U-shaped stair with landing calculator is designed for a double quarter-turn staircase with a 180° turn and an intermediate landing. Based on the specified dimensions, it calculates riser height, tread depth, stair angle, the 2h + a comfort value, landing level, stair flight width, stringer dimensions and, when required, riser dimensions for wooden and metal stairs.
The number of steps can be entered manually or selected automatically. In automatic mode, the calculator checks possible step-count combinations and selects the option that best matches the defined geometric ranges for a comfortable staircase.
Guidelines and Recommendations
Number of rises and riser height
Number of rises. The vertical calculation is based on the number of rises, which does not always equal the number of horizontal treads. One additional rise is always added to the entered number of steps in the lower flight. For the upper flight, an additional rise is added when the top step is below the upper-floor level.
Nrises = nlower + 1 + nupper + k
The coefficient k = 1 when the top step is below the upper-floor level, and k = 0 when the top step is at the upper-floor level.
Riser height. The total stair height H in millimetres is divided by the total number of rises. All rises are assumed to have the same height.
h = H / Nrises
The calculator uses 150-200 mm as a practical reference range for riser height. Values outside this range are highlighted in the results.
Tread depth and full tread depth
Available flight length. The landing width Blanding is subtracted from the total staircase length L. The remaining horizontal distance is used to position the steps in both flights.
Calculated tread depth. The same tread depth is used for both flights. The available length is divided by the greater of the upper and lower flight step counts. This allows the flight with more steps to fit within the specified length while keeping the same tread depth in the second flight.
a = (L - Blanding) / max(nupper, nlower)
The recommended range used for calculated tread depth is 270-320 mm.
Full tread depth. The tread nosing is added to the calculated tread depth. If risers are included in the calculation, their thickness is also added.
G = a + S + triser
Here G is the full tread depth, S is the tread nosing, and triser is the riser thickness. If risers are not used, triser = 0.
Stair angle and comfort value
Stair angle. The inclination angle is calculated from the riser height h and the calculated tread depth a.
α = arctan(h / a)
The calculator uses 30-40° as a practical reference range for a comfortable stair angle.
Comfort value. The relationship between riser height and tread depth is additionally evaluated using two riser heights and one tread depth.
2h + a
The range used is 600-660 mm, with a recommended target of approximately 630 mm. This value evaluates riser height and tread depth as a combined proportion rather than as two independent dimensions.
Automatic step count selection
Combination search. In automatic mode, integer values from 1 to 13 steps are checked independently for the upper and lower flights. For every combination, the calculator determines riser height, tread depth, stair angle and the 2h + a comfort value.
Deviation assessment. For each of the four parameters, the calculator determines a normalized deviation from the recommended range. If the value lies within the range, the penalty is 0. Outside the range, the deviation is divided by the width of the corresponding range.
p = 0 when vmin ≤ v ≤ vmax
p = (vmin - v) / (vmax - vmin) when v < vmin
p = (v - vmax) / (vmax - vmin) when v > vmax
Final score. The main selection criterion is the sum of the squared normalized deviations of the four parameters.
P = ph2 + pa2 + pα2 + p2h+a2
Squaring the deviation gives greater weight to strongly unfavorable values, so one large deviation is not hidden by several small improvements in other parameters. The combination with the minimum P is selected.
Additional selection criteria. If several options have the same primary score, preference is given in sequence to the option with fewer parameters outside the recommended ranges, then to the option closer to the centres of those ranges: 175 mm for riser height, 295 mm for tread depth, 35° for stair angle and 630 mm for 2h + a. After that, preference is given to a more even distribution of steps between the two flights.
If the specified staircase dimensions do not allow all four recommended ranges to be satisfied at the same time, automatic selection chooses the combination with the smallest total deviation. Therefore, a red value after automatic selection indicates a limitation of the specified geometry rather than a failure of the automatic calculation.
Landing level
Landing height. The lower flight contains one more rise than the entered number of its treads. The landing level is therefore calculated by multiplying the height of one rise by the number of rises in the lower flight.
Hlanding = (nlower + 1) × h
The result gives the elevation of the landing above the lower floor level in millimetres.
Stair flight width
Symmetrical arrangement. The upper and lower flights are assumed to have the same width. The gap between the flights Z is first subtracted from the total staircase width B, and the remaining width is divided equally between the two flights.
Bflight = (B - Z) / 2
For example, with a total staircase width of 2000 mm and a gap of 50 mm, the calculated width of each flight is 975 mm.
Stringer calculation
Inclined step length. The basic distance between adjacent steps along the inclined line of a stringer is calculated using the Pythagorean theorem.
Lstep = √(h2 + a2)
Lower stringer. The upper edge length of the lower stringer takes into account the number of lower-flight steps, riser height, tread thickness and stair angle.
Llower,upper = (h × nlower - ttread) / sin(α)
The lower edge length is additionally adjusted according to the stringer width Bs.
Llower,lower = Llower,upper - Bs × tan(α) - Bs / tan(α)
Upper stringer. For the upper flight, both displayed lengths are calculated as the number of upper-flight steps multiplied by the inclined step length.
Lupper = nupper × Lstep
Risers
Riser panel height. When riser calculation is enabled, the visible height of the riser panel is calculated as the difference between the rise height and the tread thickness.
hriser = h - ttread
Quantity and length. The number of risers is taken as the total number of rises Nrises. The length of each riser is equal to the calculated stair flight width Bflight.
European standards and design references
General design principles. When the calculated staircase geometry is used in a project, the requirements of the country where the stair is built should be considered together with EN 1990 “Eurocode - Basis of structural design”.
Actions on stairs. For imposed loads and other actions on building elements, EN 1991-1-1 “Eurocode 1: Actions on structures - Part 1-1: General actions - Densities, self-weight, imposed loads for buildings” is applicable.
Structural material. For wooden stringers, EN 1995-1-1 “Eurocode 5: Design of timber structures - Part 1-1: General - Common rules and rules for buildings” is used. For steel members, EN 1993-1-1 “Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings” is used.
FAQs
Why can some values remain red after automatic step selection?
Automatic selection chooses the best step-count combination for the specified staircase dimensions, but it does not change those dimensions themselves. If it is impossible to achieve a riser height of 150-200 mm, tread depth of 270-320 mm, stair angle of 30-40° and 2h + a = 600-660 mm simultaneously, the calculator selects the option with the smallest overall deviation.
Why can the two flights have different numbers of steps while using the same tread depth?
The calculated tread depth is determined from the flight with the greater number of steps and is then applied to both flights. The flight with fewer steps therefore occupies a shorter horizontal distance while the step geometry remains consistent between the two flights.
How does the position of the top step affect the staircase calculation?
If the top step is below the upper-floor level, an additional vertical rise is included in the calculation. With the total height unchanged, this reduces the height of each individual rise and affects the landing level, stair angle and dimensions of the inclined members.
Why is the 2h + a value checked separately from riser height and tread depth?
Riser height and tread depth can each fall within their individual recommended ranges while their combination still produces a less comfortable walking rhythm. The 2h + a formula links the two dimensions, with approximately 630 mm used as an additional reference for comfortable stair geometry.
Why is each stair flight narrower than half the total staircase width?
A specified gap is left between the two parallel flights. This gap is first subtracted from the total staircase width, and only the remaining width is divided equally between the flights using (B - Z) / 2.