Geometry for Arched Trellis Design

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Dear Staff,

I am building an arched trellis for my apple plants.

Materials:
. . .1x4 cedar boards 8' long (2)to make the arch out of 1/2" plywood 4x8' as support behind cedar cut for arch
. . .2x4 cedar boards 8' long (2)as the supports spaced 8 feet apart.

Desired dimensions:
. . .2x4s placed two feet underground in poured concrete blocks (leaving six feet above ground to work with)
. . .2x4s spaced 8' apart to provide the diameter of the arch. I want the apex of the arch to be 4', so the radius is 4' also.

Question: What are the cuts (measurements) of the 1x4 8' cedar planks so that they will make an arch four feet high at the top and eight feet at the base? Lets say there are 12 or 18 pieces of cedar to the arch, whatever provides the best stability and gives the arch appearance. (1/2 Plywood will be glued or nailed to the backside (not visible)for support. Is this enough support? Then the arch will be screwed into the top 8" of the 8' 2x4s.

My design assumption is that I need one piece that I can cut 12 or eighteen times, but I need to be sure about the angles and measurements otherwise it will not meet the 4' arch height requirement and the 8' diameter.

Any help? Thanks
Bryan
:D
 
This sounds like a very involved class project! With what formulas were you provided? How far have you gotten? Also, are you doing this geometry project just according to the geometry, or is your teacher wanting "real world" results?

Please provide specific details, along with a clear listing of all of the steps you (or your class group) have completed or attempted thus far. Also, please clarify the sort of answer (you think) your instructor wants regarding the "is this enough support" kinds of questions. (This is an engineering/physics question, and is naturally well beyond the usual purview of math tutoring.)

Thank you.

Eliz.
 
Trellis Dimension Calculation part 2

Dear Staff,

This is a trellis for my apple tree, so it requires a real solution. The steps I have taken to get this far.

I tried to do the math and came out with a diameter that was too small. I cut eight pieces and found them to only create an apex of 2 feet. ...LOL :shock:

I need a solution that I can apply, but I do not have any formulas.... :cry:

I have a guess and a form right now of :
4" 8/16 for the top
4" 5/16 for the bottom
the angle is 2.5 degrees for a 87.5 degree application on either side of the wood. Supposedly if 32 pieces are cut, then this will give me 180 degree diameter covering the 8 feet.

I am now cutting out a few squares out of paper to see if this will give me what looks like a correct calculation. I lack a formula, I just divided four times.

Hope this helps. Thanks
Bryan
 
Is this your trellis?.

trelliswr1.gif



Do you what the length of the arch?. It is \(\displaystyle \frac{2{\pi}r}{2}=\frac{2{\pi}(4)}{2}=4{\pi}=12.57\;\ feet\).

Or 12 feet 6-7/8 inches.

Break that up into however many pieces you need to get around the arch.

If you use 12 pieces to get around the arch, they'd be 1 feet 5/8 inches each. If you use 18, they'd be 8-3/8 inches each. You'll have to 'fudge' accordingly.
 
Re: Trellis Dimension Calculation part 2

bcpeistrup said:
This is a trellis for my apple tree, so it requires a real solution.
Click to expand...
You are aware that we're math tutors, and not engineers, architects, or even carpenters, right? I mean, we might prove things about "ideal" geometric objects, but not many of us create the blueprints that you seek.

If you've got a homework question that needs tutoring assistance, we're here for you. But to get a structure designed and built, it might be better to hire someone with relevant professional experience and training, such as a carpenter, a building contractor, or a construction engineer.

Good luck!

Eliz.
 
bcpeistrup said:
Dear Staff,

I am building an arched trellis for my apple plants.

Materials:
. . .1x4 cedar boards 8' long (2)to make the arch out of 1/2" plywood 4x8' as support behind cedar cut for arch
. . .2x4 cedar boards 8' long (2)as the supports spaced 8 feet apart.

Desired dimensions:
. . .2x4s placed two feet underground in poured concrete blocks (leaving six feet above ground to work with)
. . .2x4s spaced 8' apart to provide the diameter of the arch. I want the apex of the arch to be 4', so the radius is 4' also.

Question: What are the cuts (measurements) of the 1x4 8' cedar planks so that they will make an arch four feet high at the top and eight feet at the base? Lets say there are 12 or 18 pieces of cedar to the arch, whatever provides the best stability and gives the arch appearance. (1/2 Plywood will be glued or nailed to the backside (not visible)for support. Is this enough support? Then the arch will be screwed into the top 8" of the 8' 2x4s.

My design assumption is that I need one piece that I can cut 12 or eighteen times, but I need to be sure about the angles and measurements otherwise it will not meet the 4' arch height requirement and the 8' diameter.

Any help? Thanks
Bryan
:D
Click to expand...
If you divide an arch of radius \(\displaystyle \L r\) into \(\displaystyle \L n\) pieces, the cut angle for both ends of each piece is \(\displaystyle \L 180^{\circ}/2n\)and the smaller (inner) length of a piece is \(\displaystyle \L 2r \sin(180^{\circ}/2n ) .\) So for a 4 foot radius and 12 pieces the cut angle is \(\displaystyle \L 7.5^{\circ}\) and the inner length is 1.0442 feet. The total length of 12 of these comes out to 12.53 feet, which as it should be, is about 1/2 inch less than the arch length galactus calculated.

To get the inner length, I used the chord length calculation from here.
 
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