Straight Single-Flight Stair
About Straight Single-Flight Stair Calculation
This calculator determines the geometry of a straight single-flight stair to the next floor from the given opening dimensions, number of treads, and the parameters of the stringer, treads, risers, and handrail. It is suitable for preliminary proportioning of the staircase, checking walking comfort, and obtaining the main dimensions for manufacturing a timber or steel structure.
The calculation is focused on geometry and layout. It helps coordinate rise height, tread depth, slope angle, stringer dimensions, and handrail length, but it does not replace a separate verification of load-bearing capacity, fixings, guards, and connection details in the final design.
Guidelines and recommendations
Calculation sequence
Base scheme. The calculation is based on the floor-to-floor height H in mm, the horizontal opening length L in mm, the staircase width B in mm, the number of upper treads n, the tread thickness t in mm, the tread nosing o in mm, and the stringer or side member width k in mm. If risers are included, their thickness r in mm is also taken into account.
Number of rises. First, the number of rises N is determined. If the top tread is at the level of the upper floor, then N = n. If the top tread is below the upper floor level, then N = n + 1. This affects both the height of one rise and the riser height.
Step rise. The height of one step is calculated by dividing the total height by the number of rises:
h = H / N
The meaning of this formula is that the full floor-to-floor height is distributed evenly among all staircase rises.
Tread depth and slope angle
Horizontal run per step. The horizontal step run is determined as the opening length divided by the number of upper treads:
a = L / n
This is the basic horizontal projection of one step without the nosing and without the riser thickness.
Final tread depth. The value used to assess walking comfort is calculated as the sum of the basic projection, the nosing, and the riser thickness when risers are included:
b = a + o + r
The meaning of this result is that the calculator shows the actual construction tread depth that is visible and used in the finished staircase.
Stair slope angle. The slope angle is determined from the ratio between the step rise and the horizontal run:
α = arctan(h / a)
The greater the rise for the same horizontal run, the steeper the staircase. The greater the horizontal run for the same rise, the flatter the ascent.
Calculation of tread and riser dimensions
Tread length. In this calculator, the tread length is taken as equal to the staircase width B in mm. This is the tread blank dimension across the staircase.
Riser height. If risers are included, their clear height is determined by subtracting the tread thickness from the step rise:
hr = h - t
The meaning is that part of the total rise is already occupied by the tread thickness itself, so the visible riser height is smaller than the full height of one rise.
Number of risers. The same count as the number of rises, N, is used for risers. If the top tread is below the upper floor level, the number of risers is one greater than the number of upper treads.
Calculation of the stringer or side member
Inclined step spacing. To build the toothed line of the stringer, the distance between adjacent steps along the slope is first calculated:
s = √(h2 + a2)
This is the geometric length of one repeated segment along the staircase slope.
Length of the upper side of the stringer. The upper side of the stringer is calculated from the total height of the step set with the tread thickness taken into account:
ltop = (h × n - t) / sin α
The meaning of this formula is that the total vertical rise is converted into an inclined length using the actual staircase angle.
Length of the lower side of the stringer. The lower side is obtained from the upper side by subtracting the segments associated with the width of the stringer or side member:
lbot = ltop - k × tan α - k / tan α
In this way, the calculator accounts for the cuts at the bottom and the top that appear because the load-bearing member has a finite width.
Handrail calculation
Handrail length. The handrail is calculated as the sum of the inclined segments along the staircase and an additional top extension:
lhandrail = n × s + k × tan α + Δ
Here Δ = 0 if the top tread is at the level of the upper floor, and Δ = b if the top tread is below the upper floor level. This means that when the staircase ends below the upper floor, the calculator adds one more horizontal segment equal to the tread depth.
Practical comfort guidelines
Slope angle. In this calculator, the range 30-40° is highlighted as a comfortable reference. A flatter staircase usually requires more space, while a steeper one makes climbing and descending less comfortable.
Step rise. For comfortable use, a range of 150-200 mm is often used. At lower values, the staircase becomes more extended in plan, while at higher values the step becomes steeper and more tiring.
Tread depth. A common reference range for residential staircases is 270-320 mm. This is a range in which the foot usually lands securely without making the walking rhythm too short or too stretched.
Related European standards
General design basis. For design verification, the general reference is usually EN 1990 Eurocode. Basis of structural design. This standard sets the overall approach to reliability, combinations of actions, and limit states.
Loads. For permanent and imposed loads, the usual reference is EN 1991-1-1 Eurocode 1. Actions on structures. General actions. Densities, self-weight, imposed loads for buildings. This is the standard used to define design loads on stair flights, landings, and guards as part of a full project.
Load-bearing material. For a timber staircase, load-bearing capacity is usually checked according to EN 1995-1-1 Eurocode 5. Design of timber structures. General rules and rules for buildings, and for a steel staircase according to EN 1993-1-1 Eurocode 3. Design of steel structures. General rules and rules for buildings. This calculator does not perform those verifications and instead provides the geometry for subsequent engineering checks.
FAQs
Why can the number of rises differ from the number of upper treads?
This depends on the position of the top tread relative to the upper floor level. If the last tread is below the finished floor level, one additional rise is added up to the upper floor, so the step rise is calculated from n + 1 rather than only from the number of upper treads.
Why is the tread depth greater than the opening length divided by the number of treads?
Because the calculator adds the tread nosing and, when relevant, the riser thickness to the basic horizontal projection. As a result, it outputs the construction tread depth of the finished staircase, not only the geometric run inside the opening.
What do the lengths of the upper and lower sides of the stringer show?
These are not two different parts, but two characteristic dimensions of the same load-bearing side member after the staircase angle is set and its width is taken into account. These dimensions are useful for marking out the blank, making the cuts, and checking whether the stringer fits into the available opening.
Can this calculation be used for both timber and steel staircases?
Yes, for the geometry of a straight single-flight staircase the calculation is suitable in both cases, because the logic of step layout and slope determination is the same. However, strength, stiffness, deflection, connection details, and guard requirements must be checked separately for the actual structural material.
Is this calculator sufficient for a full staircase working design?
For selecting dimensions, proportions, and preliminary detailing, it is usually sufficient. For a final design, especially with long spans, high loads, or non-standard details, a separate structural verification and compliance check against the applicable local building requirements are still needed.