I need help with the miter and bevel angles for a fence rail that ends at a stone wall.
The fence: Picket fence with two horizontal rails, each rail a 3 by 3( 2.5 by 2.5 inch). The top of bottom rail is 8 inches above ground and the top of the top rail is 34 inches above ground.
The wall: Quarried granite, 18 inches high. From the front of the fence, the wall angles to the right at an angle of 35 degrees. The wall also is wider at the bottom than the top, it is out of plumb by 4 degrees.
The saw: I have a compound miter saw that miters 60 degrees to the right and 55 degrees to the left. The saw bevels 45 degrees both left and right.
The bottom rail will run horizontally into the wall, so I would think that I can set my miter angle to 35 degrees and the bevel to 4 degrees.
The top rail will angle downward from the post closest to wall with a downward pitch of 40 degrees. What should my miter and bevel angles be for this top rail?
I would be appreciative of either the answers or someone pointing me to the place where I can figure out the answer.
Thanks in advance.
Replies
bump
ajs,
The low tech solution would be to just grab some scrap and make a few cuts. With each iteration you'll get closer to the right measure.
Even if you figure out the angles mathematically ahead of time, you'll most likely need to do a little "fine tuning" because it's just so hard to get a true read on angles.
If you could somehow post a picture, it would help a lot to visualize the problem. Lots of guys here (myself included) like geometry problems, so you'd likely get some takers if you could get a picture up.
Thanks for the reply. I will try to attach a front, back, and top view of the project. The cut of interest is the one on the rail that angles down from the gate post, where it runs into the wall. I have changed some of the dimensions since drawing this in sketchup a few months ago, so please do not be distracted if all of those dimensions are not consistant with what I posted earlier. The pertinent numbers are:
Rail from gatepost to wall tilts downward at an angle of 40 degrees from the horizontal
Wall is 35 degrees from being perpendicular to the fence ( top view)
Wall is 4 degrees from being plumb, and is wider at the base than the top. This is my first try at attaching files, so wish me luck.
Alan
ajs,
Thanks for the pictures -- I totally understand what you're doing now.
I think the numbers I gave you in the previous post will work. But those are hard cuts to make. It's just about impossible to measure the length of the rail, and unless the board is cut to the actual needed length, you can't really try the angles on for size. If I were you, I would most definitely be using some pieces of scrap to dial things in.
Alan I don't have time right now but I made copies of your pics to take with me and will try to get back later today with an answer. Hopefully a correct answer ... :)Joe Bartok
I didn't have time to type up the full analysis but here is a quick summary in case anyone disagrees with the results.
The fence-wall intersection was treated as a Hip roof. The 40° fence slope forms the "Hip Slope Angle", 35° and 55° are the "Plan Angles". Back calculating returned the "Common Slope Angles". On the drawing these angles are sloping away (down) from our viewpoint.
Starting with these values as givens the question is, what is the side cut angle at the peak of an unbacked Hip rafter at its intersection with a plumb wall, and the saw blade bevel required to cut such a Hip from the upper shoulder. The side cut angle would be 47.57094° ... this is the angle measured on the stick. The corresponding blade angle is 31.77203°.
Since the wall slopes out 4° toward its foot, we subtract this value from the "natural" 31.77203°. (The blade bevel along this face normally creates an obtuse dihedral angle on the Hip, so we subtract).
The remaining angle on the stick on the adjoining face (54.49355°) and the saw blade bevel to cut along this line (36.65247°) were solved using the standard Compound Angle Formulas.
I think there is another approach to solving this compound angle and if I have time tonight I will try it and post the results tomorrow.
Alan ... I didn't have time to make my own drawing so I modified yours. If you have any objection to my using your image file just give me a shout and I will remove it (or the mods can delete it).
EDIT: Some of the angles on the drawing are incorrect. See post further down on this page.
View Image
Joe Bartok
Edited 8/16/2007 3:33 pm ET by JoeBartok
Edited 8/16/2007 3:42 pm ET by JoeBartok
Edited 8/16/2007 5:10 pm ET by JoeBartok
Joe,
I'm actually pretty good with geometry, but when it comes to hip angles, you make me look like a fifth grader. ;)
I think there is another approach to solving this compound angle...
I think vector calculus might work great here, but I couldn't remember how to do it if my life depended on it.
Edited 8/16/2007 3:43 pm ET by Ragnar17
Is there a specific reason to end the fence rail on the stone wall instead of another post?
I went through the math kind of fast and the comparison to a Hip roof could be totally wrong. In fact the first time I attempted this I was using the side cut angle for the jack rafter (i.e., treating the fence like a backed Hip rafter) ... wrong!
Vector math would work. I was also thinking of treating the as a "layover" type rafter where the dormer roof intersects the main roof at an irregular angle ...
Hopefully Alan will take your suggestion to heart and try this compound angle on a piece of scrap first.
Joe Bartok
Edited 8/16/2007 3:49 pm ET by JoeBartok
Methinks I goofed. I forgot my point of view for that drawing is upside down. This means that the obtuse dihedral angle on the bottom shoulder of the Hip rafter actually decreases by the 4° slope of the wall ... so the "natural" saw blade bevel for the upper shoulder increases to 31.77203° + 4° = 35.77203°.
The remaining angles are as follows:
Angle on the Stick on the Adjoining Face = 45.69259°
Saw Blade Bevel for Adjoining Face = 33.18873°
Supplementary angles, cross section of the stick projected on the wall:
61.88342° and 118.11658°
Sorry about that folks! Testing this compound angle on an expendable stick sounds like a really good idea.Joe Bartok
Joe and Rez,
Thanks for your responses, here is a single reply to the comments and suggestions you both have made.
Rez, Good suggestion about the post next to the wall but there is one negative. The fence/gate is a Christmas present from me to my daughter and she wants the fence to taper to the wall.
Joe, its fine to use my drawing, thanks for asking. I consider you to be the guru of this kind of thing, so I feel better about not finding it to be an easy problem. One of Rez's early suggestions led me to an approach which seems to work. I used Sketch-up to draw the rail and wall in their proper dimensions and orientation and then dragged the rail until it "went through" the wall. Then it was easy to use the Sketch Up tape measure to measure the length of each edge of the rail, from the rail end to the intersection with the wall. I transferred those lengths to the rail, drew my lines, and used the miter saw to cut to the lines. Then off to my daughter's house this morning ( two hour drive) to find out that the fit is just what I needed. Thanks to you both.Al
Alan it seems you already have an answer, but I thought about this some more last night and don't want to feel the time was wasted, so here goes. Thanks for posting this question. It was a good one and I'm going to be making a record of the solution to this.
I tried making a model of this compound angle last night and it was close but not quite right. Another look at the math showed that the numbers in the original diagram were all out by about a degree.<!----><!----><!---->
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There was another dumb mistake, this time a false assumption about the hypothetical “47.57094° Hip Rafter Side Cut Angle”. The edge of this angle doesn’t lie on a level line. Since the fence is sloping at 4° it is biting into this angle more and more as the fence goes downslope, increasing the “side cut” angle. So much for the hip roof math. I took a hint, played it safe and tried vector products to find the dihedral angles between the planes of interest.<!----><!---->
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The diagram shows only the projections of the unit vectors in plan view. Taking the arccosine of the dot products solves the dihedral angles (Vector Product Calculator). Once the dihedral angles (and hence saw blade bevels) the Compound Angle Equations can be applied to calculate the remaining angles.<!----><!---->
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Summary of Compound Angle on the Fence:<!----><!---->
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Angle on the Stick, Upper Shoulder = 49.53719°<!----><!---->
Saw Blade Bevel for Upper Shoulder = 28.15265°<!----><!---->
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Angle on the Stick, Plumb Face = 54.87917°<!----><!---->
Saw Blade Bevel for Plumb Face = 34.90233°<!----><!---->
<!----> <!---->
This makes three different answers … third time lucky? There is another way of looking at this intersection.
Be back soon …
<!----><!----> View Image
Joe Bartok
Edited 8/17/2007 8:11 am ET by JoeBartok
Edited 8/17/2007 8:12 am ET by JoeBartok
Edited 8/17/2007 8:41 am ET by JoeBartok
Why not think of the fence as the foot of a dormer common or jack rafter meeting the main roof slope in a layover type of Valley roof?<!----><!----><!---->
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Calculator: Layover Rafters intersect slope of Main Roof … scroll down the page for diagrams and formulas.<!----><!---->
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Main Roof Pitch Angle = 90°– 4° = 86°<!----><!---->
Adjoining Roof Pitch Angle = 40°<!----><!---->
Angle between Ridge Lines in Plan View = 35°<!----><!---->
<!----> <!---->
Set the calculator in “Pitch Angle” mode and enter those numbers. The results agree with the vector product - compound angle calcs to four decimal places … that’s close enough for me. To compare the two different methods here are the Layover Rafter Calculator returns:<!----><!---->
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Defined (entered) Plan Angle = 35°<!----><!---->
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Angle on the Stick, Upper Shoulder of Fence …<!----><!---->
Jack Rafter Side Cut Angle = 49.53719°<!----><!---->
Saw Blade Bevel for Upper Shoulder …<!----><!---->
Sleeper Bevel = – 28.15261°<!----><!---->
<!----> <!---->
Angle on the Stick, Plumb Face of Fence …<!----><!---->
Angle on the Stick, plumb face of Adjoining Rafter = – 54.87922°<!----><!---->
Saw Blade Bevel for Plumb Face …<!----><!---->
Saw Blade Bevel Angle = 34.90233°<!----><!---->
<!----> <!---->
Footprint or Mortise Angle = – 68.07796°<!----><!---->
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If we were mortising this fence there’s another piece of information in that calculator.<!----><!---->
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Main Roof Sheathing Angle = 24.71838° … we lay out this angle with respect to the level upper edge of the fence. This is the reference line for the housing (or mortise) angle of 180° – 68.07796° = 111.92204°.
Joe Bartok
Edited 8/17/2007 8:18 am ET by JoeBartok
Edited 8/17/2007 8:19 am ET by JoeBartok
Edited 8/17/2007 8:19 am ET by JoeBartok
Edited 8/17/2007 8:21 am ET by JoeBartok
O.K., I have edited that last message several times and "Message Truncated" keeps appearing on the board. So here's the rest of the post.
If we were mortising this fence there’s another piece of information in that calculator.
Main Roof Sheathing Angle = 24.71838° … we lay out this angle with respect to the level upper edge of the fence. This is the reference line for the housing (or mortise) angle of 180° – 68.07796° = 111.92204°.
P.S. ... Alan, post some pics of your SketchUp solution. It's always interesting to compare notes.
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Joe Bartok
Edited 8/17/2007 10:26 am ET by JoeBartok
Thanks, Joe,I will try that on Sunday when I am back home and compare the lines with the ones I generated with the other method.Al
The fence/gate is a Christmas present from me to my daughter and she wants the fence to taper to the wall.
Just scribe it and move on or its gonna be a late present. Sometimes simpler is mo betta.
I don't know a single carpenter that would use math to cut an angle up against a rock,... but I know 10 that would cut that angle and make it fit like it grew there in less than 2 min. :)
dug
Man do I agree with that. I'm not going to engineer a masonry scribe. Just get it and go.
Dug, If you guessed that I am a retired engineer doing carpentry and not a carpenter, you would be right. I was not able to take my miter saw to my daughter's house to do the cuts so I wanted to do them ahead of time at home, but scribing the lines on-site and doing the cuts with a hand saw would have worked just fine and been a lot faster. Thanks for the input.Al
Scribing would work better where irregular surfaces are concerned. I do a lot of my work with log builders ... you can't get much more irregular than that and there sure is a lot of scribing going on.
We do use a lot of math exactly as Alan intends to use it - precutting. Especially where the roof is concerned. Most of the roof can be calculated and laid out quickly and accurately working on the relative safety of the ground.
Here is a link to a very brief summary of the fence-sloped wall intersection formulas. Some of the terms have been redefined in terms of roof components and to "fit" other formulas on my web pages. Hopefully it's correct this time ... I've already done this wrong twice. :)
Rafter intersects sloped Wall
Joe Bartok
Edited 8/18/2007 11:03 am ET by JoeBartok
Actually, maybe this problem doesn't need a picture after all.
The bottom rail will run horizontally into the wall, so I would think that I can set my miter angle to 35 degrees and the bevel to 4 degrees
Seems right to me.
The top rail will angle downward from the post closest to wall with a downward pitch of 40 degrees. What should my miter and bevel angles be for this top rail?
Seems you could use the same approach here as with the bottom rail. Position the rail on the saw such that the top face of the rail is against the vertical saw fence. Then set your primary angle on the saw to 40 degrees, and the bevel to 35.
I'm not 100% sure this would work, so it'd be a good idea to use a piece of scrap first. ;)