Fixing My Stairs Using Mathematics

The Problem

I am a VERY amateur carpenter, but I do love to build things out of wood. One particular specialty area of carpentry that I admire is stair building. I consider stair building (along with roof cutting) to be the brain surgery of carpentry. While occasionally stairs are works of art, they always must be safe and functional. Quite often, I see stairs where the height of either the bottom step or top step is not the same as the other stairs. This is nearly always because the flooring on the bottom or top floors has changed since the stairs were built. Unfortunately, I faced this same situation.

I live in a cold climate and my house has a basement. In order to make the basement livable in the winter, I hired a contractor to build an insulated floor, which is very thick compared with the original carpeting. Unfortunately, the additional lower floor height made the bottom step of my basement stairs very short. This created a safety issue and would be an issue if I were to try and sell the house. The city building inspector noticed the issue, but decided not to force the contractor to fix it. Given that option, the contractor left it short.

A few years after my basement floor was insulated, my wife and I decided to put a wood floor in the basement and re-carpet the stairs. Because the stair framework was exposed, this remodeling gave us an opportunity to equalize all the riser heights with minimal hassle. Unfortunately, we decided to equalize the riser heights the night before the carpet was due to go in. I contacted a carpenter for advice who said that I should just cut a new set of “stringers,” which are the wooden supports that the treads sit on. This would not work, however, because the hour was late and I had neither the lumber nor the equipment to cut the stringers. After thinking about it a bit, the easiest solution was to use shims under each tread to set the unit rise for each tread. This is where a little bit of math came into play.

A Few Definitions

There are many special terms associated with the building of stairs. For this discussion, there are just a few terms that need to be defined. I have illustrated these terms in the following figure.

Illustration of a Few Stairway Terms

The situation that I faced was that my lower floor was now higher by 2.25 inches than when the stairs were originally built. The figure below illustrates the situation. It also shows some of the critical dimensions. Note that my actual stairs has many more treads than the four shown in this drawing but the drawing illustrates the problem just fine.

My Short Bottom Step Makes My Stairway Unsafe

The Fix

Let R be the unit rise before the floor height change, R’ be the unit rise after the floor height change, T be the total rise before the floor height change, T’ be the total rise after the floor height change, and N be the number of stairs. The unit rises before and after the floor height change are computed as follows.

${R} = \frac{T}{N}, {R'} = \frac{T'}{N}$

Our objective here is to derive an expression for the thickness of each shim. In order to easily identify each individual shim height, we can assign a number to each stair tread — assign the bottom tread the number 1 and increment the number for each tread going up (top tread is actually the top finished floor and is assigned N). Each tread will be assigned a specific shim thickness. Let $\tau_i$ be the shim thickness for the ith tread. For the sake of simplicity, we can define $\tau_0$ as the height increment of the newly finished lower floor height from the original floor height.

The following figure is useful in understanding the derivation of an expression for the shim thickness. It shows the relationship between the rise values before and after the shims.

Detail Drawing of An Individual Step

Using the figure above, we can see that the following relationship exists between R, R’, $\tau_{i}$ , and $\tau _{i-1}$ .

$R' + {\tau _{i - 1}} = R + \tau {}_i \Rightarrow R' = R + \left( {\tau {}_i - {\tau _{i - 1}}} \right)$

We can derive a relationship between the heights of the individual shims as follows.

$R - R' = - \frac{{{\tau _0}}}{N} = \left( {\tau {}_i - {\tau _{i - 1}}} \right) \Rightarrow {\tau _i} = {\tau _{i - 1}} - \frac{{{\tau _0}}}{N}$

This expression means that a shim’s height is always $\frac{\tau _0}{N}$ smaller than the shim on the next lower step.

Let’s compute a few shim heights to illustrate the situation.

${\tau _1} = {\tau _0} - \frac{{{\tau _0}}}{N} = {\tau _0} \cdot \frac{{N - 1}}{N}$

${\tau _2} = {\tau _1} - \frac{{{\tau _0}}}{N} = {\tau _0} \cdot \frac{{N - 2}}{N}$

$\vdots$

${\tau _{N - 1}} = {\tau _0} - \frac{{\left( {N - 1} \right) \cdot {\tau _0}}}{N} = {\tau _0} \cdot \frac{1}{N}$

${\tau _N} = {\tau _0} - \frac{{\left( N \right) \cdot {\tau _0}}}{N} = 0$

There is a pattern here and the thickness of the ith shim is given by the following equation.

${\tau _i} = {\tau _0} \cdot \frac{{N - i}}{N}$

We now have all the information we need to cut the shims and fix the stairs. Here is what the stairs looked like after the shimming was completed.

Shimming Equalizes All Unit Rises

My basement steps have been working without problems now for years. The unit rise of each step is exactly the same. All it ended up costing me was some scrap lumber, a little glue, and a little time to derive a solution. What more could I ask for?

Some Practical Issues

IRC Requirements

The main requirement of concern in this situation is the IRC constraint on the allowed variation in riser heights. To understand this requirement, consider the flight of stairs illustrated in the following figure.

Illustration of Riser Height Deviations

The IRC requires that the following requirement be met.

$\max \left( {{R_1},{R_2}, \cdots ,{R_{N}}} \right) - \min \left( {{R_1},{R_2}, \cdots ,{R_{N}}} \right)\; \leqslant \;\frac{3}{8}{\text{ inch}}$

For more information on the IRC stairway requirement, see [1].

Dealing with Thin Shims

When I did my stairs, I worked too hard to make thin shims. If I had it to do over today, I would not have made any shims thinner than 1/8 inch. I would simply ensure that I met the IRC requirement that limits the maximum difference between riser heights to 3/8 inch. Making and installing thin strips is difficult because they are so fragile that they are not worth the effort.

A Reference with a Similar Approach

I did find one reference ([2]) that uses a similar method for fixing a flight of stairs, but the description is is not quite as detailed as presented here.

References

[1] “Visual Interpretation of the International Residential Code:2006 Stair Building Code”, Stairway Manufacturers Association, 2006. Link
[2] B. Abernathy, “Q&A: Your Questions—Pro Answers: Fixing Rough Framed Stairs,” Fine Homebuilding Magazine, Issue 168, pp. 106, Jan 2005. Link

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