U-Shaped Concrete Stair with Landing
About U-Shaped Stair with Landing Calculation
This U-Shaped Concrete Stair with Landing calculator determines the geometry of a monolithic concrete double quarter-turn staircase with a 180° turn and an intermediate landing. Based on the specified dimensions, it calculates the step rise and tread depth, stair angle, landing level, flight width, and concrete volume separately for each flight and the landing.
All geometric calculations are performed in millimetres. After calculation, the volumes of the monolithic elements are converted from mm3 to m3.
Guidelines and recommendations
Geometry calculation sequence
Number of rises. First, the total number of equal vertical rises is determined. If nl is the number of steps in the lower flight and nu is the number of steps in the upper flight, one additional rise between the lower flight and the landing is always included. If the top step is below the second-floor level, one more final rise from that step to the upper floor is included.
N = nl + 1 + nu + e
Here, e = 1 when the top step is below the second-floor level and e = 0 when the top step is level with the second floor.
Step rise. The total floor-to-floor height H, mm, is divided by the number of rises N. Therefore, all rises in the staircase have the same calculated height h.
h = H / N
Tread depth. The landing depth P, mm, is subtracted from the total staircase length L, mm. The resulting length is the available horizontal run of a flight. The same tread depth a is determined using the flight that has the greater number of steps.
a = (L - P) / max(nu, nl)
This gives both flights the same step rise and tread depth. The flight with the greater number of steps uses the full available length L - P, while the flight with fewer steps occupies a shorter horizontal distance.
Stair angle. The inclination of the flights is determined by the ratio between the step rise h and tread depth a. Because the step dimensions are identical in both flights, the calculated angle is also the same.
α = arctan(h / a)
Comfort formula. The expression 2h + a is used to evaluate the proportion between step rise and tread depth. For automatic step selection, the calculator uses 600-660 mm as the preferred range, with 630 mm as the central reference value.
2h + a = 600-660 mm
Landing level. The intermediate landing is positioned after all steps of the lower flight and the following rise. Its level above the lower floor is calculated as:
Hp = (nl + 1) × h
Flight width. The total staircase width W, mm, is divided between two flights of equal width. The gap g, mm, between the flights is first deducted from the total width.
b = (W - g) / 2
Turn direction. Left-hand and right-hand U-shaped concrete staircases have identical calculated geometry when the same dimensions are used. Changing the direction mirrors the drawings but does not change the step dimensions, flight widths, or calculated concrete volume.
Automatic step selection
Selection criteria. The calculator checks integer combinations of step counts for both flights and calculates the staircase geometry for every valid combination. Four parameters are evaluated simultaneously:
- step rise h - 150-200 mm
- tread depth a - 270-320 mm
- stair angle α - 30-40°
- 2h + a value - 600-660 mm
Choosing the best combination. A value within its corresponding range receives no penalty. For a value outside the range, the relative deviation from the nearest limit is calculated and normalised by the width of that range. The squared deviations for all four criteria are added together, and the combination with the lowest total is preferred first.
If several combinations have the same assessment, the calculator considers the number of parameters that fall outside the preferred ranges. It then prefers the option closest to the centres of the ranges: 175 mm for step rise, 295 mm for tread depth, 35° for stair angle, and 630 mm for the 2h + a value. The next criterion is the smaller difference between the number of steps in the upper and lower flights.
Geometric compatibility. During automatic selection, the calculator also checks how the lower inclined flight slab connects to the underside of the landing. Step combinations in which this connection cannot fit within the specified landing geometry are excluded.
The ranges 150-200 mm, 270-320 mm, 30-40°, and 600-660 mm are used as practical comfort criteria for selecting the staircase geometry. If the specified overall dimensions do not allow all four conditions to be satisfied simultaneously, the calculator selects the combination with the lowest overall deviation, so some results may remain outside the preferred ranges.
Concrete volume calculation
Inclined length. For one step, the length s, mm, along the slope of the flight is first calculated. It is used to determine the cross-sectional area of the inclined slab with the specified thickness.
s = √(a2 + h2)
Flight slab thickness. The flight slab thickness tm is measured perpendicular to the inclined surface of the slab. The longitudinal cross-sectional area of the flight therefore consists of the inclined slab area plus the triangular concrete sections that form the steps above it.
Upper flight. For the upper flight, the longitudinal cross-sectional area is calculated as the sum of the inclined slab and the stepped portion:
Au = nu × (s × tm + a × h / 2)
The cross-sectional area Au, mm2, is then multiplied by the flight width b, mm. Division by 109 converts the resulting volume from mm3 to m3.
Vu = Au × b / 109
Lower flight. The lower inclined slab extends to the underside of the landing, so its longitudinal profile depends not only on the number of steps but also on the landing slab thickness. The calculator determines the actual area of this profile and subtracts the part that geometrically overlaps the landing volume. This prevents the same concrete from being counted twice at the connection.
Vl = (Al - Aoverlap) × b / 109
Landing. The landing is calculated as a rectangular monolithic slab with the full staircase width W, landing depth P, and landing slab thickness tp.
Vp = P × W × tp / 109
Total volume. The required concrete volume is the sum of the lower flight, upper flight, and landing volumes.
Vtotal = Vl + Vu + Vp
Related European standards
EN 1992-1-1, Eurocode 2 - Design of concrete structures. Part 1-1: General rules and rules for buildings, bridges and civil engineering structures. This standard establishes principles for designing concrete and reinforced concrete structures for strength, serviceability, and durability. Its provisions are relevant when determining the structural flight and landing slab thicknesses, concrete class, and reinforcement.
EN 1991-1-1, Eurocode 1 - Actions on structures. Part 1-1: General actions - Densities, self-weight and imposed loads for buildings. This standard provides rules for determining self-weight and imposed loads, including loads on stairs and landings. The applicable national version of the standard and its National Annex should be taken into account for a specific country.
EN 206 series - Concrete. Specification, performance, production and conformity. These European standards are used when specifying and controlling the properties of concrete for monolithic structures.
The numerical ranges of 150-200 mm, 270-320 mm, 30-40°, and 600-660 mm are the geometric selection criteria used by this U-shaped staircase calculator. They are considered separately from structural verification under the Eurocodes and from staircase geometry requirements established in individual countries.
FAQs
Why can some values remain outside the recommended range after automatic step selection?
The available staircase dimensions do not always make it possible to achieve a 150-200 mm step rise, 270-320 mm tread depth, 30-40° stair angle, and a 2h + a value of 600-660 mm at the same time. In this case, the double quarter-turn staircase calculator selects the combination of step counts with the lowest total relative deviation from these ranges.
Why can the upper and lower flights have different numbers of steps?
The intermediate landing level depends on the number of steps in the lower flight, while the remaining height is distributed over the upper flight. The tread depth remains the same for both flights and is calculated using the flight with the greater number of steps, so equal step counts are not required.
How does the position of the top step affect the U-shaped concrete staircase calculation?
If the top step is below the second-floor level, an additional rise between the step and the upper floor is included. If the top step is level with the second floor, this additional rise is omitted, which changes the step rise, stair angle, 2h + a value, and related staircase dimensions.
Why are the concrete volumes of the lower and upper flights calculated differently?
The upper flight consists of a conventional inclined slab with steps and is calculated directly from its longitudinal cross-section. The lower flight must connect to the underside of the landing, so its actual profile is extended to that connection and any overlap with the landing volume is subtracted to avoid counting the same concrete twice.
How does the gap between the flights affect the required concrete volume?
The gap reduces the width of each flight according to b = (W - g) / 2, so increasing the gap reduces the concrete volume of both flights. The landing is calculated using the full staircase width W, so its concrete volume is not affected by the gap between the flights.