L-Shaped Concrete Stair with Landing (Quarter-Turn)
About L-Shaped Stair with Landing Calculation
The calculator performs a geometric calculation of an L-shaped quarter-turn concrete staircase with a landing and a 90° turn. Based on the specified dimensions, it calculates the step dimensions, stair flight angle, landing level and concrete volume separately for the lower flight, upper flight and landing.
Both flights are calculated with the same rise height and the same horizontal going. The landing is assumed to be square, with each side equal to the stair flight width.
Guidelines and recommendations
Calculation of steps and stair flights
Number of rises. The floor-to-floor height is divided by the total number of vertical rises. The lower flight includes an additional rise from its last step to the landing level. The position of the top step determines whether there is one more rise before reaching the upper floor level.
N = nlower + 1 + nupper + k
Here nlower and nupper are the numbers of steps in the lower and upper flights. 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.
Rise height. All rises are assumed to have the same height. With a floor-to-floor height H, mm, the rise height is calculated by dividing the total height by the number of rises.
h = H / N
Step going. The horizontal length of the upper flight is obtained by subtracting the landing width from the total staircase length. The resulting distance is divided by the number of steps in the upper flight.
Lupper = L - B
a = Lupper / nupper
Here L is the specified staircase length, mm, B is the stair flight and landing width, mm, and a is the calculated step going, mm. The same a value is used for the lower flight, so both flights have the same step proportions.
Lower flight footprint. The horizontal length of the lower flight equals the number of its steps multiplied by the calculated step going. The actual transverse dimension of the entire staircase is the sum of this length and the stair flight width.
Llower = nlower × a
Wactual = B + Llower
The geometry is considered valid when Wactual does not exceed the specified width W. Therefore, the actual staircase width may be slightly smaller than the maximum available dimension.
Stair angle. The angle of both flights is calculated from the ratio between the rise height and the horizontal step going.
α = arctan(h / a)
Comfort value. The relationship between two rise heights and one step going is used to assess the proportions of the steps.
2h + a
The calculation uses practical guidelines of 150-200 mm for rise height, 270-320 mm for step going, 30-40° for stair angle and 600-660 mm for the 2h + a value. The midpoints of these ranges are 175 mm, 295 mm, 35° and 630 mm respectively.
Landing level. The landing height is determined by the number of rises in the lower flight. One additional rise onto the landing is added to the number of lower-flight steps.
Hlanding = (nlower + 1) × h
How automatic step selection works
Checking alternatives. The calculator checks integer combinations of step counts for the upper and lower flights. For each alternative it recalculates h, a, the stair angle, the 2h + a value and the actual staircase footprint.
Deviation assessment. For each of the four parameters, the relative deviation from the recommended range is calculated. If a value is within its range, its penalty is zero. Outside the range, the deviation is divided by the width of the corresponding range.
p = deviation / range width
The primary selection criterion is the sum of the squares of these four deviations. This means that one large deviation affects the result more strongly than several small deviations.
S = ph2 + pa2 + pα2 + p2h+a2
Selecting the best alternative. Preference is given to the combination with the lower S value. The number of parameters outside their recommended ranges is considered next. Among comparable alternatives, the calculator selects the solution closer to 175 mm, 295 mm, 35° and 630 mm, then the alternative that makes better use of the available staircase width, and finally the one with the smaller total number of steps.
Concrete volume calculation
Stair flight slab. The stair flight slab thickness is treated as the distance between the stepped upper profile and the inclined underside, measured perpendicular to the direction of the flight. For each step, the length of the inclined segment is calculated using the Pythagorean theorem.
s = √(a2 + h2)
For the upper flight, the longitudinal cross-sectional area consists of the inclined slab area with thickness t and the concrete area that forms the stepped profile.
Aupper = nupper × s × t + nupper × a × h / 2
The volume is obtained by multiplying the longitudinal cross-sectional area, mm2, by the stair flight width B, mm, and converting cubic millimetres to cubic metres.
Vupper = Aupper × B / 109
Lower flight. Its longitudinal cross-section is constructed from the actual stepped upper profile, the inclined underside of the slab and the connection with the landing. The part where the flight overlaps the landing volume is subtracted from the cross-sectional area, so the same concrete is not counted twice. The resulting area is also multiplied by the stair flight width and converted from mm3 to m3.
Vlower = Alower × B / 109
Landing. In plan, the landing is assumed to be square with side length B. With a landing slab thickness tlanding, mm, its volume is calculated as the volume of a rectangular slab.
Vlanding = B × B × tlanding / 109
Total concrete volume. The final value is the sum of the three geometric volumes. No additional waste or reserve factor is applied to the result.
V = Vlower + Vupper + Vlanding
European standards
EN 1992-1-1, Eurocode 2 "Design of concrete structures - Part 1-1: General rules and rules for buildings". This standard sets out general principles for the design of concrete and reinforced concrete structures. The step geometry ranges used by the calculator are practical comfort criteria and do not replace requirements established for a particular building by applicable national regulations.
EN 206 "Concrete - Specification, performance, production and conformity". This standard applies to the specification of concrete properties and requirements. The calculator performs a geometric volume calculation, so the concrete class and mix composition do not change the calculated structural volume.
FAQs
Why can the actual staircase width be smaller than the specified width?
The specified width is treated as the maximum available dimension. The lower flight consists of a whole number of steps with the same going as the upper flight, so its length changes in discrete increments. The quarter-turn staircase calculator selects a number of steps that allows the L-shaped concrete staircase to fit within the specified dimensions.
Why can automatic step selection leave some values outside the recommended ranges?
With fixed staircase length, width and height, there is not always an integer number of steps that simultaneously provides a rise height of 150-200 mm, a going of 270-320 mm, an angle of 30-40° and a 2h + a value of 600-660 mm. In this case, the calculator selects the combination with the lowest overall deviation from these guidelines.
How does the position of the top step affect the calculation?
If the top step is below the upper floor level, there is an additional vertical rise to the floor. With the same total floor-to-floor height, this increases the number of rises and reduces the calculated height of each step. The change in rise height also affects the stair angle, the 2h + a value and the geometry of the concrete stair flights.
How is double-counting of concrete avoided where the flight meets the landing?
The full geometric longitudinal cross-section of the lower flight is calculated first. The area that also belongs to the landing slab is then excluded from it. This allows the lower flight volume and landing volume to be added without counting the overlapping region twice.
Does a left-hand or right-hand turn affect the concrete volume?
With identical dimensions, left-hand and right-hand configurations are mirrored versions of the same geometry. Step dimensions, stair angle, landing level and calculated concrete volume do not change when the direction of the turn is mirrored.