This last summer I was contacted my father who needed to layout the pattern for a saddle cut for an irrigation system he was installing. A saddle cut is a way of cutting one pipe to match the outside curved surface of another pipe. The resulting pattern will look like the profile of a saddle, a curved center section that goes over the second pipe and two curved sections that extend down on either side.
Dad's project had a smaller HDPE pipe that would be used as a form to pour concrete around an orchard valve over the larger cement supply pipe. He knew there had to be a mathematical formula to do it and knew I might know where to find it. Well, I didn't remember the formula right off, but I did remember how to figure it out without a formula.
In this instructable I will show how to layout the pattern to cut a pipe for a saddle cut. I will cover both the first way I learned to do it by drafting out the pattern and using the mathematical formula I eventually found to determine the measurements and creating the pattern.
Step 1: Drafting the Old Way
When I first learned drafting it was by using paper and mechanical pencil. Not even the nice new mechanical pencils with the fine leads that come in different sizes and hardness, but with the old style mechanical pencils with the long heavy leads that had to be sharpened and were placed in mechanical pencils with extendable mechanical jaws to hold them.
It's not that I am THAT old, because the new style draft pencils were available but the teacher didn't like them. He still liked the old ways, and he still had sets of actual drafting pencils with wood on them and loved them. Drafting computers were around but few were had by schools. The year after this I changed schools and was able to use the first of four in my new school that had the latest AutoCAD which was in its third update. So yes I am at least that old.
For this I will be using a simple program that creates vectored images with standard English measurements. The program is not even intended to be used for this but it will do what I need without having to download or purchase another program.
Step 2: Starting the Layout of the Pattern
First I needed the sizes of the pipes. The length is not important but the diameter of the pipe is what is important. The lower cement pipe and the larger of the two has an outside diameter of 23 1/4" (23.25"). The upper pipe is HDPE which needs to extend from the top of the lower at a 90 degree angle straight up is corrugated on the exterior but smooth on the interior. This meant we couldn't reliably layout the measurements on the exterior but could do so on the interior, so the interior measurement was used 21" (21.00").
Since the important contact between the two pipes was the interior edge of the HDPE pipe to hold the wet cement in until it cured we wouldn't need to worry about the cut angle as much as if it was necessary that the entire pipe wall was needed to be in contact with the mating surface of the lower pipe to connect them properly.
Your drawing scale will depend on your work area. If you are doing this on a computer then it doesn't always matter since things can be rescaled and moved around, but if you are on paper the size of the paper, the overall size of what you are working on in the real world, and where you start your drawing will affect your scale. This example would fit on standard printer paper at 1/4" scale. This is all part of basic layout from a drafting class, so I'm not going to go into all of that here.
Establish a vertical line on your work area at 90 degrees. On the lower half draw a second shorter line intersecting the first at 0 degrees. Use this intersection as the center point and draw a circle to represent the larger cement pipe's outer edge with a diameter of 23.25". The area to the right of this will be area where the pattern will be laid out and were most your measuring will be done to establish the pattern. (*This lower intersecting line is not really needed unless your program requires it for circles or you just want to use it as reference.)
Measure up your first vertical line the distance of one half the diameter of the top pipe plus .25", and draw another short line intersecting the vertical at 0 degrees. Use this intersection as the center point and draw a circle to represent the inner edge of the upper pipe with a diameter of 21.00". The .25" gap is just personal preference in this example, you can make it larger or smaller. I did mine this way to give me some room to work and make it possible to see the different lines to come clearly.
Draw two more vertical lines at 90 degrees down from the outer edges of the upper circle, or tangent to the upper circle's outer edge, down to intersect the lower circle. The arc of the lower circle between these two lines is where the upper pipe will contact the lower one. Just make sure the vertical center line and these two lines are all parallel to each other.
Draw a line through the intersections of these two lines and the lower circle extending it far out to the right of the circles, and one from the top edge, tangent, from the top of the lower circle far out to the right side as well. These are the base line or bottom and top of the cut pattern.
Step 3: Determine the Number of Layout Points
Here is where you have to determine how many point to establish around the pipe. The number of points can determine how accurate you pattern and cut will be. More points means more work but also more corrections to your cuts, fewer points means there are fewer corrections. That being said the size and material of the pipe you are working on also limits the number of points you can accurately or need to make.
You may not be able to mark by hand all the points you establish on small pipe with a pen or pencil. The mark alone may be larger than the accuracy of your measurements. Or steel pipe might not need high accuracy of the cut if any high spots can be ground down and gaps filled by welding.
We were working with large pipe and ran a few calculations to find out how many points we might need and could accurately measure. For this we need the circumference of the upper pipe's interior calculated:
C = Pi * 21.
C = 65.9736"
Once you have the circumference you can calculate the distance between points depending on the number of points used. Circles have 360 degrees and can be broken down in some easily measured points. For small pipe you may only need 8 points cut into 45 degrees each. For larger pipe 36 points cut into 10 degrees each may not be enough. We figured that 24 points would give enough accuracy at 15 degrees each.
Dividing the circumference by 24 gives a distance between each point of 2.7488" or 2.75" This measurement is something that can easily be laid out by hand.
Now we know the angles and distance between points, back to the drawing.
Step 4: Back to the Layout
We are going to draw the 15 degree angles we determined in the last step in the upper circle. We really only need them in the lower right quadrant since the other side will be matched, and computers are typically more accurate than by hand. We don't actually need all 24 angle lines drawn around the circle, since you will see we don't use all of them to draw the pattern we only need to know how many there will be at a given angle and the spacing they would be around the circumference. Just make sure they extend out and intersect the circle.
If you are doing this by hand with straight edge, compass and protractor doing them in both lower sides can help to give more indicators to keep you straight and the next lines we do parallel. And doing them all the way around will help you make sure the angle lines all pass through the center.
Once the angle marks are intersecting the upper circles line draw vertical lines parallel to the center vertical line coming down through each intersection of each angle line and the circle to intersect the lower circle. You will see that even though the distance between each angle line is the same the vertical lines you just drew have different spaces between them. The further from the center line you go the closer they get.
Now draw horizontal lines coming from the intersections of the vertical lines and the lower circle out to the right between the top edge line and the bottom line. All should be parallel to each other. You can see were the distance between line here are different. The ones nearest the top and bottom are closer together than those in the middle. This has to do with the angles of the circles as they curve towards and away from each other.
I found that with the program I was using that it didn't have a good way to fix and guarantee the spacing between the next lines I need next so I used one of the polygons available to have a fixed distance. If you have a better program to do his use it, or if you are doing it by hand you just need to measure each distance and mark the bottom line as a reference as where each line should be drawn. Just far warning my next drawing may look a little off.
In the last step we figured out there needs to be 2.75" between each point on the circumference of the upper circle to get 24 points at 15 degrees each and that the total circumference was 65.9736" (66.0"). So starting somewhere near the lower circle you need to start marking the lowest line at 2.75" distances starting with a 0.0" point and continue out to the full circumference length.
I used a rectangular polygon with a width of 2.75" as my spacer. I moved it along from point to point as I drew lines marking the spacing and the height to each horizontal line. You can also see I placed my notes on my drawing to reference for when I get interrupted I don't have to look for them again or recalculate.
I placed the spacer with its edge lined up with the 0 mark and drew a vertical line up from the bottom line to intersect the next horizontal line. I moved the spacer to the align with this new line, and drew the next vertical line from the bottom line up to intersect next horizontal line higher than the last one. I continued until I intersect the top horizontal line, then draw vertical lines up to the horizontal line lower than the one before until reaching the lowest horizontal line again. From there I again started making them taller than the one before until I reach the top, and again made each one shorter until I was below the lowest line again. You can see how having the ability to space these automatically can save s lot of time.
If you are doing this by hand it is actually much faster than this program, but not as fast as the real good drafting programs. Make your first mark near the lower circle on the lowest line. Set your compass to your scaled width of 2.75". Put the compass point on the mark and make a small mark where the compass crosses the lowest line to the right of the point, move the point to this new mark, and make a new mark where the compass crosses the line to the right of this mark. Continue until you have 25 marks across the lowest line. (Why not 24 because the first mark is your starting point, or point 0.
Once each mark is set draw a vertical line at each mark after point 0 that extends up at 90 degrees until it intersects the line above the one the last line intersected. Your pattern should look like supports for a roller coaster.
Step 5: Making the Wave
Up to this point you have only been using your drawing to establish the height and marker points so you can draw the actual cut line. This is where we will actually draw the cut line.
Starting from point 0 draw a line from where it is on the lowest line up to the where the next vertical line crosses the line above the lowest one, then from that one to where the next vertical line crosses the line above that one. This continues until you connect each one to the one after it reaching the top horizontal line, then back down to the lowest line, then it all repeats it self until you reach the end. If doing it my hand you can use a French curve connecting three of the intersecting point at a time to make it less angled and more flowing. If you have a good drafting program it should have a curved line function.
On mine I did it with a different color so it really stands out. Once you reached the end this line should look like a standing wave or sine wave. You can see what the cut line will look like laid out flat. If the size of what you are working on is not too big you can actually draw this out full size on some cardboard or poster board to be used as a template to draw your cut line on the pipe.
If it is bigger than what could be done by template then you can use the measurements from a properly scaled drawing to lay it out on the pipe to be cut. Simply measure each line from the base line up to where it intersects its individual top line and record the real world scale in a table for each point on the drawing. The included table are the measurements I got from this program. Not exactly high tolerance fit but it can get me close enough to hold cement or weld on.
Step 6: An Easier Way
I had an acceptable pattern layout for my father to work with for his project, but I still wanted to find the real mathematical formula to calculate the cut pattern with just so I could know for next time. So off to the internet to take a look around.
I found lots of websites that would do the calculations for you, even print you out a life size template for a nominal fee of course and lots who sold programs that would do it for you and output the code in any format or language you like. But none of them would show you how it was done or show you the code. I persevered and finally found a website with what I had been looking for.
On a blog about CNC I found this formula:
X = (R1/Sin (JA)) - (((R1 - Sqr(R1^2-(Cos ( AP ) * R2)^2)) / Sin(JA)) + ((Tan (90 - JA) * (Sin(AP) * R2))))
R1 = large radius
R2 = small radius
JA = Joint Angle
AP = AnglePosition
- The discussion blog also had a link to Wolfram MathWorld page on Steinmetz Solids. This was a much, much more in-depth explanation on the intersection of two shapes and the area within that intersection. If you're bored and need some light reading check it out.
Now I had what I needed, a formula to work with and a lot of experience in Excel. I could turn out all the calculations I needed to make a pattern and layout the cut lines. I could quickly check the outcome of using different angles and get the heights I needed and the spacing between each.
You can see from the base calculation column the height at point 0 starts at almost 5 inches from my edge. On this type of joint angle it should be 0.0". Cutting off five inches of pipe when it should be none is way too much waste of material when the pipe is expensive. I had to bring the lowest height down to the edge of the pipe. So in the decimal column I used the lowest base height value and subtracted it from all the height values. Now the edge of the pipe is my starting height and I only need to cut out the sections that go over the bottom pipe saving a lot of pipe and money.
I also included a graph to check my calculations on. I can quickly see the approximate cut pattern from my calculations. The formula also lets you calculate cut patterns for joints that are not 90 degrees.
You can see on the graph that with a joint of 45 degrees how much different the saddle pattern is with a large section extending out on one side and only small saddle sections on each side. Much faster than drawing it all out which can also be done for the 45 degree joint but with a lot more steps and drafting than what I have shown here. No I'm not going to do the 45 degree one in this instructable.
Step 7: Use It
Now you have all the measurements you need to layout the pattern on your project. On my father's pipe it was large enough to get inside and work. This was good because the the outside of the pipe to be cut was corrugated. The layout was done on the smooth inside surface. You also need to have a square cut end from which to measure from. If you don't have that the measurements won't give you a correct cut line. So you have to either square the end you are measuring from or make a full size template that can be squared to the pipe wall and moved to meet the edge.
If you make a template you need to use enough material so there can be at least a 2" base to hold it all together. Measure up from the 2" line to layout your cut line. You can see the cut line in my initial template was laid out with just a ruler, when I got serious I used a French curve to lay it out and make it a much better layout.
Sorry no photos of the final template it was used and tossed be for I thought to take a picture. I still need to find a photo of the final cut forms to post, but you can see how to make the pattern and get your measurements from what I have given you so far.
I hope this will help someone with their projects and give them the answers I had to work so hard to find.
*UPDATE: Found the picture of the actual saddle cut made from this. You can see a split in the side of the form so the form can be opened a little to remove it from the set cement. The corrugations on the outside ended up being useful as a perfectly sized guide for ratchet tie down(s) to pull it together and hold it closed while the cement cured. A piece of wood can be fitted into them also as another guide to hold them aligned if needed.