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.

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  20. I have been playing the lottery since i was 19 years and now i am 47 meaning i have been playing the lottery for 28 years. The biggest amount I have ever won in my life was 1200 bucks. But one day my story turns to history after I find this man's name on the Internet that he is the best when it comes to winning the lottery. This man is a very strong voodoo Dr who gives out the lottery winning numbers that can never fail. After all my years of laboring and struggling to win the lottery i finally won ($12,000,000) Dr Winner, is the name, EMAIL him now: (Winnersspellcast@gmail.com this is the only way to win the lottery and the best way via his website:https://winnersspellcast.blogspot.com) Call or text him: +2348138289852,,,

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  21. GET YOUR EX LOVER BACK. This is my testimony of life that I will tell everyone. I have been married for 20 years Another woman had a spell to get me away from my lover, My husband left me and the children we suffered for 4 years until I saw post about where this man Dr Victor has helped so many people recover their broken marriages relationships to get their ex lovers back. I decided to give him a message about my family how my husband left me for another woman so he can help me bring back my loving husband home, This great man told me what to do and I followed him as he instructed. After 48 hours, as he told me, I saw a car enter the compound. I was shocked when I saw my husband. As I share my testimony with you my husband came back to me and the kids that's why I'm happy to put all of you in the same situations to meet this man for solutions to your problems bring your lover back to you mend your broken marriage your relationship restored back to happiness as you wished. Contact this great man, you can message him on his email... drvictorspellcaster01@gmail.com or message him on whatsapp with this number... +2348112935937

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  22. How to Play the Satta King Game
    The Satta King Game is a lottery game in which players bet on collections of numbers ranging from 0 to 99. This game has a lot of similarities with lottery games. It is a popular type of lottery that can be played both online and offline. The Satta King is a kind of betting lottery. The players choose a number from the hundred available and bet on it. If they win, they are awarded a sum of 90 multiple rupees. This game is popular throughout India.
    To start playing this game, you need to first log in to the satta matka King website. Click on the play now button and then create a username and password. Remember to choose a username that is not public and is not your family member's name. Once you've created a username and password, you can choose a game and click on the register button. This way, you'll be able to log in and start playing right away.
    After logging into the Satta King website, you can begin playing the matka result King game. To register, click the play now button. To create your account, enter your username and password. Then, select the game you want to play and click the register button. This will open your profile for other players to see. satta result You should always be careful when choosing your user name and password. This is important because you don't want your friends and family to know who you are and what you're doing.
    The Satta King Game has two main elements - the Satta kalyan matka result and the Satta king record chart. You can use the record chart to predict the Satta result that's about to open. There is also a fixed timing for each Satta kin game. Once you've played your Satta game, you'll be able to see the Satta khana and record chart in advance.
    After you've chosen your numbers, you'll need to enter them. In the satka matka King game, players choose one number from one hundred. If they're right, they'll receive ninety multiple rupees. If they don't win, they'll be given eighty times the original amount. As a result, this game is a form of lottery. While the Satta King game may seem to be illegal in some countries, it is legal in India.
    When you play the Satta King game online, you'll be satta matka result for winning. However, you'll be required to invest at least 30% of your money to win. This is an essential aspect of this game because you don't want to lose your money. You need to be able to quit when satka matka you're near the maximum amount of money. The more you play, the higher your chance of winning. The Satta King Game is a fun way to spend your free time, and it is a fun way to pass the time.
    The Satta King Game is a great way to keep your satta result active. You can spend hours playing the game with friends and family, and the Satta King game is a great way to do so. The online version of the game is free to play and there are many ways to play. The only real disadvantage is that you have to be patient and have a lot of patience. If you want to win, you should invest some time to learn the basics of the game.
    You can play the

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  23. Ranthambore National Park is Best Place to visit in holidays For enjoying Real Nature View and tigers kingdom You can see real tigers infront of your eyes by gypsy safari and you can click epic pictures of wild life animals.

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