Straight Single-Flight Concrete Stair Calculator
About Straight Single-Flight Stair Calculation
The calculator performs a geometric calculation of a straight single-flight concrete stair. Based on the stair length and height, flight width, number of steps, top-step position, and slab thickness, it calculates the step dimensions, stair slope, comfort indicator, and concrete volume required for the straight single-flight staircase.
The calculation assumes a constant height for all rises, equal tread depth, and constant thickness of the inclined stair slab. The concrete volume is determined from the actual longitudinal profile of the stepped flight without adding any extra allowance for losses.
Guidelines and recommendations
Dividing the stair height and length
Number of rises. The total floor-to-floor height H, mm, is divided by the number of equal rises. If the top step is at the level of the upper floor, the number of rises equals the number of steps N. If the top step is below the upper-floor level, there is one additional rise between the last step and the floor level, so the number of rises equals N + 1.
h = H / N
h = H / (N + 1)
The first formula is used when the top step is level with the upper floor. The second is used when the top step is below the upper-floor level. The resulting value h is the height of one rise in millimetres.
Tread depth. The horizontal stair length L, mm, is divided by the number of steps N. All treads are assumed to have the same depth.
a = L / N
The value a represents the calculated horizontal depth of one tread in millimetres.
Stair slope and comfort indicator
Stair slope. The flight angle is calculated from the ratio of the rise height h to the tread depth a.
α = arctan(h / a)
For automatic step selection, the calculator uses a range of 30-40°. This range is used as a practical guideline for a comfortable straight stair.
Rise height. A range of 150-200 mm is used when evaluating possible layouts. With other parameters unchanged, a lower rise generally produces a gentler stair slope.
Tread depth. The automatic selection uses a range of 270-320 mm. A deeper tread increases the horizontal length of the flight when the number of steps remains unchanged.
Comfort indicator. The calculator also checks the commonly used relationship between rise height and tread depth.
2h + a
The calculator uses a range of 600-660 mm, with approximately 630 mm as the central reference value. This criterion evaluates the combination of rise height and tread depth rather than either dimension separately.
Automatic selection of the number of steps
Checking possible layouts. In automatic mode, the calculator evaluates the number of steps from 1 to 30. For each option, it recalculates the rise height h, tread depth a, stair angle α, and the 2h + a comfort indicator.
Deviation assessment. A value within its recommended range receives zero deviation. A value outside the range is evaluated by the distance to the nearest limit divided by the width of that range. The calculator uses intervals of 50 mm for rise height, 50 mm for tread depth, 10° for the stair angle, and 60 mm for the comfort indicator.
Q = ph2 + pa2 + pα2 + pc2
Here p is the normalized deviation of the corresponding parameter from its recommended range. Squaring the deviations gives greater weight to layouts that exceed a recommended range by a larger amount.
Final selection. The calculator first selects the option with the lowest Q value. If two options have the same score, preference is given to the one with fewer parameters outside their recommended ranges. If these values are also equal, the calculator compares how close the parameters are to the centres of the ranges: 175 mm for rise height, 295 mm for tread depth, 35° for the stair angle, and 630 mm for the 2h + a indicator.
For the specified available dimensions, the automatic mode therefore searches for the most balanced combination of stair parameters. If no whole number of steps can satisfy all four ranges at the same time, the calculator selects the option with the smallest normalized deviations.
Geometry of the monolithic stair flight
Slab thickness. The specified thickness t, mm, is assumed to be constant along the entire flight and is measured perpendicular to the inclined line of the slab. The underside of the slab is constructed parallel to the stair pitch line at a distance t.
Step profile. The upper boundary of the longitudinal section follows the actual sequence of horizontal treads and vertical rises. The lower boundary is the straight inclined underside of the slab. This means that the calculation includes both the inclined slab itself and the additional concrete above it that forms the steps.
Section area. After the longitudinal profile has been constructed, the area S of the closed polygon is calculated in mm2 from the coordinates of its vertices.
S = 1 / 2 × |Σ(xiyi+1 - xi+1yi)|
Concrete volume. The longitudinal section area is multiplied by the constant flight width B, mm. The resulting volume is converted from mm3 to m3.
V = S × B / 1 000 000 000
No additional waste or reserve factor is applied to the calculated volume. The volume multiplier is 1.00, so the result represents the net geometric concrete volume of the stair.
Geometric assumptions used in the calculation
- All rises in the stair flight have the same height.
- All calculated tread depths are equal.
- The flight width is constant along the entire stair.
- The inclined concrete slab has a constant thickness measured perpendicular to its surface.
- Right-hand and left-hand orientations are mirrored versions of the same geometry and do not change the calculated dimensions or concrete volume.
European standards and reference documents
EN 1992-1-1, Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings. This European standard is used for the design of concrete and reinforced-concrete structures together with EN 1990, Eurocode: Basis of structural design, and the applicable National Annex.
Stair dimensions. In EU countries, specific requirements for rise height, tread depth, flight width, and other stair geometry are established by national building regulations. For this reason, the ranges of 150-200 mm, 270-320 mm, 30-40°, and 600-660 mm are used by the calculator as practical comfort and automatic-selection criteria, not as mandatory limits that apply uniformly throughout the EU.
Calculated volume. Partial safety factors used in the Eurocode system for structural verification are not applied to the geometric concrete volume calculation. The volume is calculated directly from the dimensions of the constructed profile and the flight width.
FAQs
Why does automatic selection sometimes produce values outside the recommended ranges?
For a given floor-to-floor height and available stair length, there may be no whole number of steps that simultaneously satisfies the preferred rise height, tread depth, stair slope, and 2h + a value. In this case, the calculator compares layouts from 1 to 30 steps and selects the combination with the smallest total normalized deviation.
How does the position of the top step affect the calculation?
If the top step is level with the upper floor, the number of rises equals the number of steps. If the top step is below the upper-floor level, one additional final rise is created, so the same total height is divided into N + 1 parts. This changes the rise height and therefore also affects the stair slope, comfort indicator, and concrete geometry.
Why is concrete volume not calculated simply by multiplying stair length, width, and slab thickness?
The slab thickness describes only the inclined slab below the steps. Additional concrete above that slab forms the horizontal treads and vertical rises. The calculator therefore determines the area of the complete stepped longitudinal section first and then multiplies it by the flight width.
Should an extra allowance be added to the calculated concrete volume?
The displayed result is the net geometric concrete volume with a multiplier of 1.00. When ordering ready-mix concrete, an additional allowance is often included for actual losses, formwork dimensional variations, and construction conditions, but this allowance is added separately to the calculated result.
Which parameter is most important when selecting the number of steps?
The calculator does not evaluate rise height, tread depth, or stair angle independently. Automatic selection considers four criteria together: 150-200 mm for rise height, 270-320 mm for tread depth, 30-40° for the stair slope, and 600-660 mm for the 2h + a relationship. The final number of steps is therefore selected from their combined balance.