Stair layout & construction over sloped or uneven surfaces:

Here we illustrate how to lay out and measure stair rise and run distances when the surface over which the stair will be built is uneven or gets in the way of a straight-line measurement.

We include stair layout calculations where the stair passage is over uneven ground or surfaces such as a bulged hill (page top) or a depression in the surface. If you don't adopt a way around these obstructions, your stair measurements and stair build plan will be wrong and the stairs won't fit the site.

This article includes example stair building calculations and warns that inconsistent step rise or tread size anywhere in the stairway may be a trip/fall hazard. We also explain how to include landings and platforms in stair design calculations.

How to Measure & Layout a Stair Rise & Run over Uneven Ground

However some terrain shapes are irregular enough or are humped so that you cannot just take off measurements directly.

It's better to make an accurate and direct measurement of the stairway run and rise so that your calculated tread depths and riser heights will be proper and uniform.

Many builders use a transit to make accurate angle and distance measurements.

Certainly designing and building stairs with uniform tread depth and riser height over rough ground like that in our photo (left) can be a serious challenge.

Here we provide an alternative crude but effective approach to stair rise and run measurement using string, plumb line, and level, along with a stick or board (or two of them) of sufficient length.

Basically we project a horizontal line out from the upper elevation over the landing spot for the stairway, using that line to measure both stair run and stair rise accurately.

[Click to enlarge any image]

If the horizontal run distance is small, you can get by with straight lumber tacked in place temporarily to permit measurement of horizontal run projection and vertical rise.

Typically we use our stringer 2x lumber (before it has been cut to length) for this purpose since by definition the diagonal stringer will be longer than the horizontal stair run.

This approach works to measure and design a stairway over any irregular surface.

 

Make certain that the horizontal boards are straight and that they are held level and that your vertical board, useful for holding up the outer end of that upper horizontal member is also vertical.

Take off the rise and run measurements as we show in the sketch.

For longer distances than the length of boards you have on hand, you will do fine with string to set out the horizontal distance, a vertical pole, and a plumb line and measuring tape.

Below, for completeness, we include how to obtain stair rise and run lengths using sine and cosine tables - not a process we are seriously recommending.

Stairs built over both an uneven hillside & uneven bottom or top landing

Our photo shows a stairway during construction by the author & an associate (Arthur Cady).

We faced a double difficulty of constructing a (replacement) exterior stairway over a rough hillside and stairs that had to land on a sloping surface.

By supporting the stringer above the hillside we bypassed the irregularities of the slope itself,but as you can see from the photo, because we had to land on a sloping surface the two stringers could not be identical, yet they had to line up properly for the treads to be level.

Using the longer (downhill) stringer as the main pattern, we measured and laid out the tread positions along the stringer surface, then aligned the two stringer tops and mapped the master layout onto the inner side of the uphill, shorter stringer.

The treads were placed between the stringers, supported by both cleats and end-nailing through the stringers from outside.

Best design would have insisted on a level landing platform at the stair bottom but in the real world we can't always meet that criteria. In this case, the stairs had to land on a narrow driveway.

Had we installed a level landing platform the drive would have become unusable by the client's vehicles. The result is a first step with uneven riser height depending on where you enter the stairway.

Where it is impossible to avoid a step with riser height that varies from others in a stairway or where a stair ends on an uneven surface,

1. Provide secure hand railings
2. Provide a color cue that is unique for the irregular step or riser
3. Provide a warning sign (at least at public stairways)

Example building code citation for irregular risers ending at a sloping surface

Normally a stairway walking surface for treads and landings is expected to be close to level.

R311.5.3 excerpt from Treads & Risers:

The walking surface of treads and landings of a stairway shall be sloped no steeper than one unit vertical in 48 units horizontal (2% slope).

But in unavoidable circumstances of sloped surfaces or irregular riser heights exceptions are permitted provided measure are taken to reduce falls.

IBC 1009.3.2 Dimensional uniformity.

Stair treads and risers shall be of uniform size and shape. The tolerance between the largest and smallest riser height or between the largest and smallest tread depth shall not exceed 0.375 inch (9.5 mm) in any flight of stairs.

The greatest winder tread depth at the 12-inch (305 mm) walk line within any flight of stairs shall not exceed the smallest by more than 0.375 inch (9.5 mm) measured at a right angle to the tread’s leading edge.

Exceptions:

1. Nonuniform riser dimensions of aisle stairs complying with Section 1025.11.2.

2. Consistently shaped winders, complying with Section 1009.3, differing from rectangular treads in the same stairway flight.

Where the bottom or top riser adjoins a sloping public way, walkway or driveway having an established grade and serving as a landing, the bottom or top riser is permitted to be reduced along the slope to less than 4 inches (102 mm) in height, with the variation in height of the bottom or top riser not to exceed one unit vertical in 12 units horizontal (8-percent slope) of stairway width.

The nosings or leading edges or treads at such nonuniform height risers shall have a distinctive marking stripe, different from any other nosing marking provided on the stair flight.

The distinctive marking strip shall be visible in descent of the stair and shall have a slip-resistant surface. Marking stripes shall have a width of at least 1 inch (25 mm) but not more than 2 inches (51 mm). 

[Underline emphasis added - Ed.]

The steps shown above are not in compliance in that the same black no-slip stripe is used on both treads rather than just the un-even bottom riser/tread in this hotel stairway.

Regulation on Measuring Riser Height at Sloped Landing Surfaces

OK, so where do we measure the riser height when the landing surface is sloped?

The British Stair Recommendations give this reasonable advice:

2.6 Where the landing of a stair is formed by the ground and slopes across the width of the flight, then the rise of the step shall be measured at the middle of the tread's width.

That means that if you set the mid-width height at a desired riser height - say 7" - then towards one end the riser height will be less than desired and towards the other end of the step the riser height will be more than desired.

That's why special colour cues are required for these steps.

Here is a similar U.S. stair code for handling stair construction and marking at uneven landing surfaces

2007 NYS Stair Code: 1009.3 Stair treads - exception to step riser height is permitted at joining a sloping surface

Where the bottom or top riser adjoins a sloping public way, walkway or driveway having an established grade and serving as a landing, the bottom or top riser is permitted to be reduced along the slope to less than 4 inches (102 mm) in height with the variation in height of the bottom or top riser not to exceed one unit vertical in 12 units horizontal (8-percent slope) of stairway width.

The nosings or leading edges of treads at such nonuniform height risers shall have a distinctive marking stripe, different from any other nosing marking provided on the stair flight.

The distinctive marking stripe shall be visible in descent of the stair and shall have a slip-resistant surface. Marking stripes shall have a width of at least 1 inch (25 mm) but not more than 2 inches (51 mm).

The Mathematical Approach to Building Stairs Spanning Uneven Ground

The geometry approach to calculating stair rise and run over rough terrain when you have only two measurements works correctly provided you know enough triangular geometry to convert between your sloped-line measurement (the hypotenuse of the triangle) and the triangle's other sides that form the rise & run dimensions.

Using a calculator and it's square root function, and the most basic knowledge of geometry is not enough - we also need a table of sine and cosine values, functions readily found these days online. [45]

a² = b² +c² - the square of the length of the hypotenuse (a) equals the squares of the lengths of the opposite sides of a right triangle (b) and (c).

In a geometry or trigonometry text, you'll see that when we refer to the angle ab, we mean the angle formed at the intersection of lines a and b, or in this case the lower left angle in our sketch.

And so the three angles formed by the triangle's sides are ab, bc, and ac.

Knowing that the triangle abc will always include a 90 degree right angle ( i.e. this is always a "right triangle"- angle bc in our sketch) allows use of sine and cosine tables to obtain the lengths of the two unknown sides b and c in the sketch.

Geometry teaches that if we know three pieces of data about a right triangle (one side length and two angles, or two side lengths and one angle) we can obtain all of the other data about a triangle - all of its angles, and the actual lengths of all of its sides.

sin of angle ab = (length of the opposite side c / the triangle's hypotenuse a) = a / c

cos of angle ab = (length of the adjacent side b / the triangle's hypotenuse a) = b / a

But the rub is that we need to know at least two of the angles on this triangle. We know bc is always 90 degrees.

On the ground, we'd have to measure either angle ab or angle ac. You can do this using a transit, a protractor or other angle measuring tool either by placing your angular measuring tool on the sloped surface of the stringer side a, or by actually measuring the angle formed between side a and a horizontal or level surface.

Our carpenter's square at left is one such tool, though there are some neat level tools that, placed on a slope, will simply give you the angle directly as a readout.

Example: if our stairs are to run from the two extreme ends of the 2x stringer we illustrate above,

and if that length (after cutting to "fit" the hill and desired stair run) measured 100 inches exactly,

using just that known length of the hypotenuse of the triangle we would obtain the key measurements of

stair rise c and

stairway run b

by first converting all measurements to inches and then using a table of sine and cosine values.

Suppose we measure angle ab (sketch above) at 30 degrees.

sin of angle ab = (length of the opposite side c / the triangle's hypotenuse a) = a / c = 100 / c

cos of angle ab = (length of the adjacent side b / the triangle's hypotenuse a) = b / a = b / 100

Reading from sine and cosine tables, we'll find that side c (rise) = 50" and that side b (run) = 86.6"

If you don't think this whole sine-cosine process is absolutely horrible for field use in building a set of stairs, then we're not on the same planet.

...

Continue reading at STAIR RISER SPECIFICATIONS or select a topic from the closely-related articles below, or see the complete ARTICLE INDEX.

Or see these

Recommended Articles

• SLIP TRIP & FALL HAZARD LIST, STAIRS, FLOORS, WALKS
• STAIR CODES & STANDARDS - home - you are on this page
• STAIR CONSTRUCTION IDEAL DIMENSIONS
• STAIR CONSTRUCTION, LOW ANGLE SLOPE
• STAIR CONSTRUCTION, SPECIFIC ANGLE or SLOPE
• STAIR CONSTRUCTION, SPECIFIC TOTAL RISE & RUN
• STAIR DESIGNS for UNEVEN / SLOPED SURFACES
• STAIR DIMENSIONS, WIDTH, HEIGHT - home
• STAIR TREAD COLOR CUES REDUCE FALLS for more color cues to reduce falls at steps

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