Calculating mitre and bevel angles for panel backers?

[Kenneth Rhodes]

How are  backer strut miter and  bevel angles calculated?

Consider for example  a 3v Class 1, Method 1 dome.


BACKERS Parallel to the Pentagon Base:

I have a set of PANEL dome plans which specify that Pentagon backers which are parallel to the base strut have an 8.82 degree miter and 35.04 degree bevel on each end.  To be precise the plans call for 8.82 "S"way and 35.04 "T"ilt.  These are 2X4 or 2X6 backers are struts which connect the Pentagon legs, of length A, which run parallel to the base which is length B.

Here is the dihedral/bevel data for the A struts:
Length A Dihedral = 165.56 - 180.00  = -14.44 / 2 = -7.22


If I have used Gerry's radial angle spread sheet calculator these are  the relevant factors
      Pentagon Triangle BAA:
                        CF      Central Angle       Face Angle      Radial Angle
                    B: 0.403548     23.28           70.73           72.00 - 90 = -18.00
                    A: 0.348615     20.08           54.63           55.69 - 90 = -34.31
                    A: 0.348615     20.08           54.63           55.69 - 90 = -34.31
 
Now it seems that half of the Face Angle for side B would be the correct for the bevel, but 70.73 / 2 =  35.365  = 35.37, not 35.04 degrees, a difference of 0.325 of a degree . Could be an error in the plans? Or, could it represent a difference somehow related to the panel struts being twice as wide as a single strut dome?

Likewise, the 8.82 degree miter/Sway setting is puzzling. it is 1.6 degrees larger than the 7.22 degree bevel calculated for the Pentagon legs. 


BACKERS Perpendicular to Pentagon Base:

In regard to backers that are perpendicular to the the Pentagon base, length B, the plans call for  a 4.13 degree miter on the end that connects to the base and 12.34 miter/Sway and 54.00 bevel/ tilt on the end connecting to strut A, the Pentagon leg. I do not fully understand why, but I can see that  since the Length B Dihedral = 168.64  - 180.00  = -11.36 /2 =  -5.68 degrees. But the plans specify a bevel of 4.13 degrees on the Pent side of B strut shared with a B strut Hexagon base.  Conveniently 11.36 - 4.13  leaves  a bevel of  7.23 degrees which matches well with the 7.22 degree bevel given by Domebook 2 for the Hex/Hex B length. 
 
But the angles for the ends of the backer connecting to the Pentagon legs are given as 12.34 miter/Sway and 54 bevel/Tilt.  I have no idea how these angles are calculated.

Any assistance will be deeply appreciated.

Regards,
Kenneth Rhodes

                                    
    Hexagon Triangle BCC:
                        CF      Central Angle       Face Angle      Radial Angle
                    B: 0.403548     23.28           58.58           60.00 - 90 = -30.00
                    C: 0.412411     23.80           60.71           62.15 - 90 = -27.85
                    C: 0.412411     23.80           60.71           62.15 - 90 = -27.85

[Gerry in Quebec]

Hi Kenneth,

With one exception, the mitre and bevel angles you cited for cutting the backers for the AAB triangle, whether horizontal or vertical to the B strut, look correct to me. The exception: I get 8.83 degrees instead of 8.82 for the mitre angle of the backer parallel to the B strut.

I'm attaching an Excel calculator, including a few drawings. The calculations reflect several assumptions:

1) The corner brace between two struts is parallel to the third strut of the triangle.

2) The intermediate and outside studs are perpendicular to that third strut.

3) The construction method is either hub & strut using dimensional lumber or a panel method, such as the Pease system, but not the Oregon Dome panel method (which eliminates the dihedral angle on the inner faces of struts).

4) All three vertices of the dome triangle are the same distance from the spherical centre. In other words, the dome radius is constant.

These calculations, which I did quite a few years ago, were pretty complex and time-consuming. The resulting equations in the spreadsheet are most likely not in their simplest form, but they give the right values. I leave it to you to inspect the details of the equations if you're so inclined. 

- Gerry in Québec

[Kenneth Rhodes]

WOW! Thanks again for sharing your hard work and  expertise with us.  I now have perhaps a dozen graphic files and  spreadsheets which you have posted to the Geodesic Help Group. Gerry, I really think you should publish a book.  I'm sure to be spending a few hours with this spread sheet.  :-)

Regards,
Ken Rhodes

[Gerry in Quebec]

Hi again Ken,

You're welcome.

If you know the partial dihedral angle across each strut of a dome triangle, plus the face angles of the triangle, you can easily compute the mitre and bevel angles for cutting the ends of a corner brace. The attached spreadsheet is more mathematically transparent (i.e., it has shorter equations) than the one I posted earlier today. That's because it uses face angles and deriviatives of dihedral angles as inputs rather than chord factors.

You mentioned the idea of writing a book. I did so many years ago but never got around to publishing it! For me, a journalist/science writer by training and profession, it was too much of a busman's holiday! Maybe I'll get around to it if and when I get to "retire".

You reminded me that the equations for the corner brace and stud angles were buried in that old manuscript. I found the Word file and reorganized some of the equations. The results are in the spreadsheet. I haven't yet pulled out the equations for the vertical studs... but may do so later, time permitting and spirit willing.

Good luck with your ongoing math adventure.

- Gerry

[Ashok Mathur]

Dear Gerry,

My teacher of Operations Research taught that whenever I use a mathematical tool to calculate an output, I should also see how sensitive is the output to the accuracy of input.

For example, how much would the two mitre angles differ, if the input angles were given rounded to the nearest integers?

Would the actual implementation suffer if the mitre angles were set to the nearest integer?

What would be the effect if either the input or the output as an average of the various available values?

Regards

Ashok

Regards

Ashok

--
--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp+unsubscribe@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp+unsubscribe@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[Kenneth Rhodes]

Double WOW! Thanks again, Gerry.

I would certainly love to see any calculations you have regarding struts perpendicular to the triangle base. Yet, I am very impressed as to how useful struts/corner braces parallel to the base can be. A strut or struts perpendicular to the base can be attached to the parallel strut without having to resort to compound angles, making them easy to measure and easy to cut. The 3v Panel Dome Plans I have show Hexagon struts perpendicular to the base with a Miter of 7.22 and a bevel of 60.00 - I have never found a Miter or Radial arm saw that can bevel 60.00. I am sure someone who know what they are doing can pull it off though. But the angles for struts parallel to the base, the angles that I have seen are within the range of most compound/sliding Miter saws so internal bracing for Skylights, vents, cupolas and what not can be more easily accomplished.
( a dome without what not is a dome without joy).

Gosh, even sub-triangle(s) can be built with struts parallel to each side.  Years ago, a fellow sold me some 2v hub and strut plans.  He noted glibly in his plans that for larger 2v dome each triangle could be sub-divided with additional triangles.  Not a word or hint as to the how to determine the compound angles involved.  Those plans were basically useless paper.  Actually another fellow did virtually the same thing. He said for larger domes additional bracing would have to be installed.

Your assistance and instructions are deeply appreciated. My adventures continues...

:-) Ken Rhodes

[Gerry in Quebec]

Here are a few examples of the use of corner braces and studs (backers) in triangular framing of domes and similar buildings. Two photo collages attached.

[Gerry in Quebec]

Ken, Ashok & others,

In the last spreadsheet I posted, there was a typing error in the text version of the equation for angle BSy. There is one too many open square brackets. The one before "arctan [..." should be deleted. The Excel equation that does the actual calculation, along with the value displayed in the red output box, are correct.

Here's a new version of the Excel file with that and other typos corrected.

- Gerry

On Tuesday, March 7, 2017 at 4:26:26 PM UTC-5, Gerry in Quebec wrote:

[Gerry in Quebec]

Hi Asok,

I find it difficult to know or guess what effect the accuracy of inputs will have on the accuracy of outputs. As a rule of thumb, I don't round numbers until I have to -- for example, when I find myself standing in front of the compound mitre saw with a 2x6 in my hands. In the old days when people did calculations with a hand-held calculator or with pencil and paper (and a big eraser), there was a price to pay for using highly accurate inputs: it was really time consuming. But nowadays, tools like Excel have no problem dealing with highly precise inputs and spit out the answers almost instantaneously.

I did a little test using the most recent Excel calculator in which the mitre and bevel angles of corner braces are calculated using the face angles of triangles and the partial dihedral angles of the triangle edges. First I input angles, in degrees, to five decimal places and recorded the outputs. Then, as you suggested, I rounded the inputs to the nearest integer and compared the results with the first set of outputs. 

Scenario 1

Inputs: 6.22793; 8.04462; 59.69587; 49.19321

Outputs: 8.20; 9.30; 40.52, 29.98

Scenario 2

Inputs: 6; 8; 60; 49

Outputs: 7.93; 9.22; 40.73; 29.68

So, even with heavy rounding up front, the two sets of mitre and bevel angles are pretty close -- maybe close enough for use in the shop.

- Gerry

Regards

Ashok

To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com

--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.

To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.

[Ashok Mathur]

Dear Gerry,

Operations Research has known for decades that most complex calculations give results that are parabolic near the optimum values. To clarify, this statement means that the results do not change very much near the optimum just as a parabola's Y values do not change much near the center when X values change.

With today's tools, as you mention. calculations to a large degree of precision is very easy; however 'Thinking' with round numbers is very much easier to do and understand.

CNC machines and laser cutters are changing the degree of precision that is easily achievable - so go ahead and use such machines.

Sorry for the diversion.

Regards

Ashok

Regards

Ashok

To unsubscribe from this group, send email to GeodesicHelp+unsubscribe@googlegroups.com

--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.

To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp+unsubscribe@googlegroups.com.

[Gerry in Quebec]

Hi Ken,

Take a look at this discussion 5 years ago about cutting steep angles on a compound mitre saw -- especially the post by Blair Wolfram of Dome Inc., dated 5/13/12.

https://groups.google.com/d/topic/geodesichelp/G5vvG2gmays/discussion

I've often had to cut vertical studs by hand, because the bevel angles were too steep for my compound mitre saw. For one dome, I made a special mitre box and that worked okay. But usually I have used a protractor to draw lines on both the narrow and wide faces of say, a 2x6. I put the board in a vise then cut along the line of sight created by the two lines. That board serves as a template for cutting others.

If you know the mitre and bevel angle settings of a compound angle, you can easily calculate a third angle, the one on the narrow face of the board. In the attached image, that angle is labeled SA. SA = arctan (cos MS / tan BS).

- Gerry in Québec

On Tuesday, March 7, 2017 at 9:46:36 AM UTC-5, Kenneth Rhodes wrote:

[Gerry in Quebec]

Hi Ken,

I've added the equations for the compound angles of vertical studs to the spreadsheet I posted earlier on corner brace angles. I've also added the studs to the diagram. I've tried to reduce the equations in their simplest form -- or at least simpler than what's in my Word manuscript.

Note: One of the inputs -- 90 minus the partial dihedral angle at edge x -- is also an output, i.e., the mitre setting angle (MS) of the lower ends of the vertical studs (simple mitre cuts rather than compound cuts).

- Gerry in Québec

On Tuesday, March 7, 2017 at 4:26:26 PM UTC-5, Gerry in Quebec wrote:

.... I haven't yet pulled out the equations for the vertical studs... but may do so later, time permitting and spirit willing.

[Gerry in Quebec]

Ken,

You can cut bevel angles of 60 degrees + on a standard radial arm saw, although it's not easy. I've done it on my Craftsman 10" saw and I know someone who built a large dome using the double-bevel method (sometimes called the Arrowhead method) which involved cutting hundreds of struts this way. This is also the approach taken by Wil Fidroeff for his 4v hubless EconOdomes.

http://econodome.com/

- Gerry

On Wednesday, March 8, 2017 at 12:10:46 AM UTC-5, Kenneth Rhodes wrote:

.... I have never found a Miter or Radial arm saw that can bevel 60.00. 

[Kenneth Rhodes]

Gerry,

Once again, Many, many thanks  for the updated Compound Angles for Braces and Vertical Studs spreadsheet, as well as others you have previously posted..

Brace yourself for a question regarding one of my attempts to think outside the box.  It occurred to me that if struts were cut flat instead of vertically that the "arrowhead" double bevel vertex ends could be cut with just two passes of a circular saw on each end of a 2X4' strut. This would mean that the struts would be severely weakened, but what if another 2X4 strut was centered vertically beneath the top flat strut length forming a T-beam then the strut would be greatly strengthened. I suppose this is  very similar to EconoDomes T-beam concept - I am thinking that EconoDomes T-beam affords a convenient way to  support internal ceiling panels within the inner shell triangles, as well as adding sufficient structural so that expensive 2X6 struts would not be necessary. My notion was/is to bevel/taper the flat part of the outer/inner T-Beam and then connect the vertical part of the struts with a plywood hub/gusset. This would be reversed for the inner shell. Connecting the inner and outer strut layers with plywood gussets makes the two T-Beams into an I-Beam truss of significant cross sectional depth. Each vertex could be strapped. Structural adhesive everywhere.

Using a plywood gusset would allow the arrowhead ends of flat top and bottom 2X4's to be positioned with speed and accuracy. On site and prefabrication construction times would be dramatically reduced, especially for high frequency domes.

So, is this a viable, structurally sound approach to DIY dome construction?
            
Regards,
Ken Rhodes

[Ashok Mathur]

Dear Ken

Make a 3 foot radius double layered icosahedron as a test case.

Regards

Ashok

Regards

Ashok

--

[Ashok Mathur]

Dear Ken,

My previous reply was insensitive and purely based on impulse.

Please ignore it.

Regards

Ashok

Regards

Ashok

[Gerry in Quebec]

Hi Ken,

The arrangement I think you're suggesting might work well, but it's complex and therefore hard to imagine or predict how it would behave structurally in "real life". I do like the idea of I-beam struts that also serve as supports for ceiling panels. See the attached sketch .... Hope I've understood your idea correctly.

- Gerry 

[J Chilla]

Gerry, your maths skills keep me amazed, I feel like I'm watching a rocket scientist at work and he's explaining the orbital velocities of different planets.

But...

Then you add diagrams.

I'm agog.

DK 

[Dick Fischbeck]

What about a dome using only 3 numbers? This has not caught on yet.

Example:

720=N/A, where N is the number of corners and A is the angle of the corner.

Like with 96 corners, the angle is 720/96 which equals 7.5 degrees. For a hemisphere, cut that in half, 3.75 degrees.

Think sheet material; paper to make models.

Just saying, as they say.

[Gerry in Quebec]

Thanks, DK, but it ain't rocket science, just trigonometry. In the early '70s I did study astronomy in Carleton University's physics department where Kepler's laws of planetary motion were essential reading. Our bible was "Principles of Astronomy", by Stanley Wyatt at the University of Illinois. Celestial geometry figured prominently, providing a good basis for later interest in domes and other forms of curvature in architecture.

For anyone here not familiar with the Yahoo dome-home discussion group moderated by DK, check out:

https://beta.groups.yahoo.com/neo/groups/DomeLiving/conversations/messages

- Gerry in Québec

[Kenneth Rhodes]

Not a problem, Ashok.  Building a 3ft radius model is sound advice.   :-)

Thanks, Ken

[Kenneth Rhodes]

Thanks again, Gerry! Your sketch appears perfect.  :-)