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Act as if what you do makes a difference. It does.

— William James.

Introduction

Figure 1: Example of a Beautiful Balustrade. (Source)

I am about to begin some construction on my cabin in northern Minnesota, and the design of the stairs has been on my mind. I consider well-designed stairs to be both beautiful and functional. A major factor in making a stairway beautiful is its balustrade – a railing with supporting spindles (aka balusters). The balustrade for a stairs is often referred to as a banister.

While I have discussed stair design in a number of other posts (here, here, and here), I have never discussed the design of the balustrade. Since Fine Homebuilding magazine had a short, but interesting article this month on a formula for designing a balustrade, now is a good time for a post on balustrade design.

My plan here is to present the Fine Homebuilding formula and then to augment the formula for use in my stair-oriented balustrade application. My Mathcad source and its PDF are here.

Background

Figure 2 shows the commonly used terms for the parts of a balustrade. The balustrade can be horizontal or part of a stairway.

Figure 2: Definition of Balustrade Terms. (Source)

Model

Figure 3 shows the variables that I will be using for my balustrade exercise. The Finehomebuilding article uses different variable names (e.g. A, B, C),  and I wanted something a little easier for me to remember.

Formula

Equation 1 is my form of the formula presented in Fine Homebuilding. I altered the formula to reflect the terms that I normally use in my engineering life (i.e. length, pitch, spacing, width). I also incorporated the ceiling function, which is symbolized using $ceiling(x)\equiv \left\lceil x \right\rceil$.

 Eq. 1 $\displaystyle P=\frac{{L+W}}{{\left\lceil {\frac{{L+W}}{{{{S}_{{Max}}}+W}}} \right\rceil }}$

where

• L is the length of the balustrade.
• SMax is the maximum baluster spacing the building code allows for balusters.
• P is the center-to-center spacing of the balusters, which is the actual pitch.
• W is width of the balusters.

Analysis

Derivation

Figure 4 shows my derivation of the formula. Note that the code allows a maximum spacing (SMax) between balusters, which drives the minimum number of balusters – you can add more. The variable M represents the number of balusters plus 1. The derivation assumes that the number of balusters is made large enough to ensure that the baluster spacing is less than or equal to the code's maximum allowed value.

Figure 4: Derivation of the Baluster Pitch Formula.

Example

Equation 1 applies to both horizontal and stair balustrades. In the case of stairs, the length becomes the length of the horizontal projection of the balustrade length. Figure 5 shows my calculations for the baluster pitch assuming a stair balustrade with the following characteristics:

• L = 50 in (horizontal length)
• θ = 30° stair angle
• W = 2 in (baluster width)
• SMax = 4 in (code specified maximum spacing)

Figure 5 shows the calculations.

Figure 6 shows a scale drawing of the stairs. The scale drawing and the calculations agree. Figure 6 includes the term "phantom block", which is used in the Fine Homebuilding article. The term refers to treating the newel posts like the zeroth and last baluster.

Figure 6: Railing Example.

Conclusion

I have never built a balustrade, but I plan on building one next summer. I anticipate using this formula for that build.

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3 Responses to Layout Formula for a Balustrade

1. I am also getting a home built right now, and I love really nice looking stairs. I definitely agree with you that the balustrade is one of the biggest key points in making a really beautiful set of stairs. It looks like building one of these can be fairly difficult, with lots of formulas and exact measurements. I am definitely not an expert of any kind in carpentry, so I'll probably see if I can just get some ordered. I hope your stairs and banisters turn out great!

• mathscinotes says:

Hi Skylar,

My staircase will be finished in a couple of weeks. My wife and I have been designing our retirement house for the last few years. We are almost done. I will include pictures when we are done.

mark

2. Lesa says:

Hi Mark, what is the formula to get the the angle I would need to drill in my newel posts for horizontal tubular metal spindles? Are they even called spindles if they are horizontal or are they called inner railings? Thanks for you interesting post...