Sunday, May 6, 2012

Timber Porn



Billy Dillon called my copy of "Traité Théorique et Pratique de Charpente" by Louis Mazerolle Timber Porn and it pretty much stayed a porn book on Stereotomy geometry until I read through Tim Moore's blog on Stereotomy. The Trapezoidal Ground Plan on page 52 was meant to be an exercise in Theorie des  dévers  de pas et des niveaux dévers.  level cant ?? or level slope?? 


The end view of a hip rafter or common rafter without elevation? Draw the end view of a rafter without any slope?
http://compagnonnage.info/blog/blogs/blog1.php/2011/11/21/remarques-sur-le-cachet-des-indiens-de-buenos-aires


I thought the exercise was pretty straight forward until I got to the skewed rafter in the Trapezoidal Ground Plan. The angles that the skewed rafter had for the miter angle at the foot and the miter angle at the peak of the skewed rafter didn't make any since mathematically. So I extracted a tetrahedron from the peak and foot of the skewed rafter to find out where these angles came from.


This first drawing is from the Google Sketchup plugin I'm writing for the exercise. I know it doesn't have much to do with drawing a 3D object in 2D space, but it helps me understand the geometry  better.

Here's a screen shot the of the Google Sketchup plugin.



To understand the miter and bevel angles at the peak of the skewed rafter I extracted a tetrahedron from the peak of the skewed rafter. 


The tetrahedron extracted from the peak of the rafter does not have an right angle corner in each of the faces of the tetrahedron so I needed to treat it like a regular polyhedron and find the footprint of the polyhedron like we do for the platonic solid  tetrahedron with 3 equilateral triangles at each vertex. 


http://www.sbebuilders.com/tools/geometry/tetrahedron_platonic_soild.php




By drawing a line perpendicular to the line EA through C-B2 it develops the common rafter run that we can use for the slope of the extracted tetrahedron. The slope angle of the extracted tetrahedron is the same slope  angle as the slope angle from the point G to K. So, now it's beginning to make since mathematically.

The angle at B2 is similar to  a jack rafter side cut angle. The miter angle at the skewed rafter peak can be calculated as the Purlin rafter slope angle.

Skewed Rafter Slope =     43.70889 °
Skewed Rafter Plane Tilt =   8.68257°

Purlin Rafter Slope Angle = arcsin ( sin ( Pitch Angle ) × cos ( Jack Rafter Side Cut Angle ))
Miter Angle at Peak of Skewed Rafter = arcsin ( sin ( 43.70889° ) × cos ( 8.68257° )) = 43.08446°

To find the bevel angle at the peak of the skewed rafter 

Plan Angle at Peak of Rafter = arctan( sin ( Pitch Angle  ) ÷ tan( Plan Angle))
Plan Angle at Peak of Rafter = arctan( sin ( 43.70889 °  ) ÷ tan( 73°)) = 11.9288°


Purlin Bevel Angle = arctan( tan ( Pitch Angle ) × sin ( Plan Angle At Peak of Rafter ))
Purlin Bevel Angle = arctan( tan ( 43.70889 ° ) × sin ( 11.9288° )) = 11.17679°





A similar process can be used to fine the miter and bevel angles at the skewed rafter foot.

Skewed Rafter Slope =     90° - 43.70889 ° = 46.29111°
Skewed Rafter Horizontal Rotation Angle =   8.30528°

Purlin Rafter Slope Angle = arcsin ( sin ( Pitch Angle ) × cos ( Jack Rafter Side Cut Angle ))
Slope Angle at Foot of Skewed Rafter =  arcsin ( sin (  46.29111° ) × cos ( 8.30528° )) = 45.66606° 

Miter Angle at Foot of Skewed Rafter = 90° - 45.66606°  = 44.33393°

To find the bevel angle at the foot of the skewed rafter 


Purlin Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Purlin Bevel Angle  = arcsin( cos ( 43.7088 )  *  cos( 73 )) = 12.20111°









Here are some of the Google SketchUp files that were generated by the Google SketchUp Plugin



Two Google SketchUp files with the tetrahedrons extracted.




8 comments:

  1. I haven't come up with an entirely satisfactory translation of "dévers de pas." Part of my confusion has to do with its spelling. In Mazerolle it is written as "devers de pas" (without the accent) which would mean "footprint shape," "footprint region," or "horizontal section region." This would fit with another name for the method: "occupation de bois," which translates as "timber sections." On the other hand, more recent books call the method "dévers de pas," and a compagnon told me that this is how it should be written. That means "footprint skew." Part of the issue is that "devers" is an archaic word that only survives in some French locutions, whereas "dévers" is very common; I suspect a linguistic shift has happened in which speakers have seized on a common word whose pronunciation is close to the correct, obscure word, thereby losing the original meaning.

    "Niveaux de dévers" refers to the scribing method of twisting timbers about their axes to reflect their alignment in the structure. "Twist amount?" "skew level?" "skew alignment?" I really don't know. What would you call this operation in English?

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    1. Tim,

      This what I get from Google Translate from some of the text on page 50:
      Pour obtenir le devers de pas de la demi-ferme biaise GK, on fait l`elevation d`un chevron d`emprunt KL, d`equerre a` la sabliere BC, en portant la hauteur de ferme de K en M, et en joignant ML qui sera le rampant du chevron d`emprunt a` la tete M; on fait un trait carre` au rampant ML prolonge jusqu` a` la rencontre de la ligne de trave ou de niveau, au point Q on tire QG qui sera l`alignement du devers de pas.
      For the cant of no half-GK strong biases, it is the elevation `d` a `loan chevron KL` s bracket to the sabliere BC, bringing the height of farm K in M, and ML will be joining the rampant borrowing of chevron `s head to M, it is a square` relate to crawling ML extends for up to meet the trave line or level, to the point Q is pulled HQ which will be the `alignment cant not.

      `alignment cant not?? I guess it could mean "footprint alignment?

      "Niveaux de dévers" refers to the scribing method of twisting timbers.

      I don't see any twisting timbers in the drawing. Skewed to the outside eave-line of the footprint, yes, but not twisted.

      When I think of twisted rafters I envision something like this.
      http://www.en.charpentiers.culture.fr/node/546

      After thinking more about the skewed rafter in the drawing, today, it's more like a purlin rafter, rather than a hip rafter. A purlin rafter is perpendicular to the roof surface and so is this skewed rafter. If we ran a purlin rafter from the hip rafters located at B & C, we would not call it a twisted rafter. Just a purlin rafter running parallel with the eave line.

      So, I guess this
      "Niveaux de dévers" refers to the scribing method of twisting timbers

      would translate in English to

      "Niveaux de dévers" refers to the scribing method of rotating timbers about their axes to reflect their alignment in the structure.

      The translation from French to English is difficult and sometimes misleading, but the geometry in the book is fun to learn.

      Sim

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    2. Let's see if I can do a bit better than Google:

      To obtain the footprint of the slanted half-truss GK one draws the elevation of a dummy rafter KL, square to the plate BC, by drawing the height of the truss from K to M and then drawing ML, which will be the slope of the dummy rafter to the peak at M. Next, one draws a line perpendicular to ML extending it to the intersection with the cross piece or ground level [i.e., KL], finding point Q. Then, draw QG which will be the direction of the footprint [parallel to its edges].

      The French carpentry authors like to use construction terms metaphorically when they describe geometry, so you see "rafter," "dummy rafter" etc. in what is just a line drawing.

      The "chevron d'emprunt" is an important idea that is used throughout. It is always drawn from the peak perpendicular to the plate.

      There are no "niveaux de dévers" in the drawing you are looking at. That is explained in the drawing of the model on pg 53, which I am working through in my most recent posts, so stay tuned.

      I mean "twist" in the sense of rotation about the member's principal axis, to distinguish it from rotation in space relative to other members.

      Delete
    3. Thanks Tim,

      I think Google's translation isn't even close after reading your translation.

      Do you have a labeled drawing of:

      The "chevron d'emprunt" is an important idea that is used throughout. It is always drawn from the peak perpendicular to the plate.

      "chevron d'emprunt"
      Are talking about the triangle (KLM), since it's drawn perpendicular to the plate(BC) and drawn to the peak(M)?


      Sim

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    4. I have a bit of an advantage; I've lived in France for 9 years. The French carpentry texts, and Mazerolle is far from the most opaque, are a bit like reading Shakespeare.

      KL is the chevron d'emprunt. KLM is an elevation view of it.

      I haven't made a labeled drawing yet showing the chevron d'emprunt. Any of the Mazerolle drawings that show a roof slanted relative to the principal elevation view will use a chevron d'emprunt to construct a true elevation of the roof.

      Tim

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    5. Tim,
      Looking at Jean-Michel's website I came across this page.
      A beautiful guitarde in Vendôme (41)
      or in French
      Une belle guitarde à Vendôme (41)
      http://compagnonnage.info/blog/blogs/blog1.php/2012/02/08/une-belle-guitarde-a-vendome-41

      Any thoughts on this translation of French to English for one of the comments on the page?

      its top is a square flag on a circular plan

      Sim

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  2. Sim,
    Could you give a brief explanation of how you "extracted a tetrahedron from the peak of the skewed rafter" ?
    Did you create an SU model by hand (without plug-ins) and then measure the angles ?

    PS I wonder if Google Translate has a parameter "Shakespearean English" - even if it did, I doubt it has a parameter "francais charpente ancien" or some such :-)
    Rob

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    Replies
    1. Rob,

      I did extract the tetrahedron from a plugin drawing.

      I'll draw up a couple of Sketchup drawings showing how I extracted the tetrahedron from the skewed rafter.

      Sim

      Delete