Catenary Arch Math for an 11 cubic foot kiln

Lets waste no time in digging into the math for a Catenary arch. First I will explain what we are building and then do the brick math and simplify the formulas needed to calculate whats needed.

This kiln is tall enough and wide enough to work with single shelves in the middle and stacked vertically. The shelves measure 12" x 24" which gives you 288 square inches per shelf. Picture this:

Our interior opening is 32" wide and 32" high and the kiln will be 28" deep.

Our formulas for determining catenary volume is "L * ((4/3*H) * (1/2 * W)) The best way to use the formula is to input the numbers in inches. With our numbers above we find that our arch volume is equal to 19114.66. To covert to cubic feet we divide this number by 1728 which is how many cubic inches are in a cubic foot. We now get 11.06 cubic feet. You can adjust the dimensions as needed to get the cubic footage you want

The next question I wanted to answer was how many bricks do I need for the front and back faces and how many bricks do i need for the arch? We now need to do two calculations for surface area. The faces are easy since we already did that calculation when finding the volume. (4/3*H) * (1/2 * W) gave us 682 square inches.

We should quickly identify the face edges of a standard firebrick. Standard bricks are 9" long by 4.5" wide and 2.5" thick. If we do the quick math for face edges we get 11.25" for the end, 22.5" for the side edge and 40.5" for the top edge

Now to identify how many bricks for the front brick wall we simply take the arch area which is 682 square inches and divide by the face area for the side edge 22.5 square inches. We get 30.31 bricks. Now an important note is that this brick wall will be inset into the arch so that its flush. Like the picture below.

To get the surface area of the arch I hung a string 32" wide and 32" low. The string is what i copied to make the arch for the form. I was then able to measure the string length and got 80". **THIS IS IMPORTANT** Because we want the arch to cover the front and back walls we need to add a brick width to each end. We know that we want the interior to be 28" so we add 4.5" to each end. We now know that we want the arch surface to be 37" long. Now its a simple area formula 80" * 37" and we get 2,960" divide this by the brick side face edge and we get a 131.55. So it will take approximately 132 bricks mounted on its side edge to cover the entire arch form.

Now something to keep in mind is that the arch is not straight and the brick edges are. So we will still need to either cut brick or use refractory mortar with the bricks to seal the arch. You can see in the images that there will need to be some cutting so that the edges are flush