Sunday, 8 April 2018

Geometry of Joist Bridging

If you look at typical wood floor framing, you will often see rows of cross-bridging between the floor joists. The bridging helps prevent joists from twisting when they're loaded. It also helps make sure concentrated loads are shared between multiple joists. A floor with bridging is also little less bouncy when people walk across than an identical floor without the bridging.

View of some floor joists with cross-bridging.

In this post I analyze the geometry of bridging to answer questions like "what is the angle of the plumb cuts?" and "what is the minimum length needed to cut a piece of bridging?"

Let's look at the bridging in a joist bay and define some common floor framing dimensions:

A typical joist bay with bridging.

S is the centre-to-centre spacing of the joists. Typically 16" [406 mm] in Canada and the U.S., though other spacings are sometimes used.

b_1 is the width of the joist. Typically 1-1/2" [38 mm] in Canada and the U.S. where modern dimensional lumber is used. Historically, dimensional lumber was thicker than it is today, so if you're in an old house you might find 1-5/8" [41 mm], 1-7/8" [48 mm], 2" [51 mm], or even thicker joists.

d_1 is the depth of the joist. Typical joist depths correspond to the standard dimensional lumber sizes readily available. In Canada and the U.S. that generally means 7-1/4" [184 mm], 9-1/4" [235 mm], or 11-1/4" [286 mm]. Historically, dimensional lumber depths were 1/4" to 3/4" [6 to 19 mm] greater than they are today.

d_2 is the depth of the bridging member. Bridging is typically cut from nominal 2x2s (true dimensions 1-1/2" x 1-1/2" [38 mm x 38 mm]).

Let's delve into the geometry now and define a few more quantities:

L_2 is the minimum length piece you could theoretically cut the bridging from.

p is the length of the plumb cut face of the bridging.

Δ is the minimum length of the bit that would be cut from the top or bottom edge in making the necessary plumb cuts.

θ is the angle the bridging makes measured from horizontal. It is also the angle of the plumb cut. 

From basic trigonometry, we can get the following relations:

Expression for tanθ

Expression for Δ

Expression for p

Expression for L_2

So everything depends on , which is the most important quantity we need to know. And we can try to solve for theta. The problem is, we get this:

Expression for θ

That's not fair, making θ a function of itself like that. And with some significant effort, we might even be able to get the exact solution. Probably not in a general form, but for each specific case when the values of S, b_1, d_1, and d_2 are known (as they usually are). But carpenters usually aren't also mathematicians and don't go around with powerful computing software to do that for them. We need to be a bit more pragmatic. An exact solution isn't necessary. Real surfaces aren't perfectly flat or perfectly square or perfectly plumb. If you can make a cut accurate within 0.5°, you're doing pretty well. So we just need a way to get a good approximation of θ

If we look at this again, there is an angle that we can calculate without difficulty.

From basic trigonometry:

Expression for θ + Δθ

Well that's something we can solve with a calculator. Now comes for the approximating.
θ + Δθ is not close enough to use for our plumb cuts. But it should be close enough to give us a first approximation of p. In other words:

First approximation of p

If we substitute into our expression for θ above, that gives us an approximate solution for θ which can be calculated without much difficulty:

First approximation of  θ

So how good is our approximation? Well here's a table to compare the first approximation of θ to the exact solution for common sizes of floor framing. Bridging

Comparison of approximate and exact solutions of θ for nominal 2x2 [38 mm x 38 mm] cross-bridging and common floor joist arrangements.

So it's a pretty good approximation. Error was only 0.62° in the worst case (nominal 2x12s spaced 12" apart on centre) and where spacing was at least 16" the error was always less than 0.27°. That should easily be good enough to cut and install the bridging properly. 

Well what if for some reason you want an even better approximation? Let's look at the expression for
θ again.

Expression for θ

In this form the expression lends itself well to the fixed-point iteration method. We don't need any fancy software. We start with a reasonable initial guess to input theta on the right side and come out with  a closer solution on the left hand side. Then we take our better approximation and use it as input again on the right hand side to calculate an even better solution. The cycle can be repeated as many times as needed to converge on the solution. 

Fixed-point iteration scheme for θ

The first iteration of the fixed-point iteration is exactly how I arrived at the first approximation of θ. The initial guess used as input on the right hand side was θ + Δθ, because it was easy to calculate and isn't too far off the mark. If we use the iterative method here we converge on the solution fairly quickly, and should be accurate to less than 0.1° with just three iterations. Of course, it helps to have a good guess to start with, but we could've pulled a number almost out of thin air like θ = 0° or 45° and we still would have quickly converged on the correct answer. 

With θ you can calculate L_2, the theoretically shortest piece of lumber you could cut a piece of bridging from for your given floor joist arrangement. Using the approximate θ gets you pretty close to the exact solution. You could simply compensate by adding a small nominal amount (say 1/4" [6 mm]).

Comparison of approximate and exact solutions of L_2 for nominal 2x2 [38 mm x 38 mm] cross-bridging and common floor joist arrangements.