Slope and Hip factors

[Rich]

Just posting this here as I think I've posted in a total of 10 times or more in different forums :)

To figure the slope factor for common rafters
Take the slope and square it - add 1 - and take the square root of it.
SqRt((X/12^2)+1)
When X = 5
5/12^2 = .17366
.17366 + 1 = 1.17366
SqRt(1.17366) = 1.08

To figure the slope factor for hip rafters
Take the slope and square it - add 2 - and take the square root of it.
SqRt((X/12^2)+2)
When X = 5
5/12^2 = .17366
.17366 + 2 = 2.17366
SqRt(2.17366) = 1.47

[Cole]

That is good info for newbies, great post!!!

[Joe Bartok]

Rich: I've got a question. Bear with me, despite all my math solutions for complex roof joinery, sometimes I'm unfamiliar with more standard terminology.
I assume a slope factor is the magic number needed, given the common run and pitch, to calc an actual length for the rafter. I looked at your formula for the common rafter, and recognised it as being the Square Root(tangent Pitch Angle squared plus one), equal to 1/cos(Pitch Angle). In other words, I would have done the same calculation as Common Run/cos(Pitch Angle), that is, Common Run/cos(22.61986). Using a value of one for the Common Run gives an answer of 1.08. For two 5/12 pitches meeting at a 90 degree deck angle, I get 16.41644 for a Hip pitch angle using my calculators. Then, using 1/(sin 45*cos 16.41644), the slope factor for the Hip comes to about 1.47. (The 45 in sin 45 is the deck angle, half of the total 90 degrees as the eaves meet).
So, it seems I've got the definition correct. I tried your glossary, but can't seem to get in for some reason (blank page).
Just out of curiosity, how would most framers deal with b@$t@rd pitches and an irregular deck angle? The slope factor formula for hips would only work for regular (equal) pitches where the eaves meet at 90 degrees (90 degree deck angle).
Hey, this is fun. I had to edit this post because I think I ran afoul of your censor. I thought a b@$t@rd pitch was correct terminology, but my original post says it's a bastard pitch. Gee, I learned something new today!

[Rich]

LOL - sorry about the censor - I'll delete that from the censor catcher list. I'm also in the process of getting the glossary re-written - the script I was using to make it a little easier on me broke when I upgraded php the last time.
Yes - that is what the slope factor is - I use it more for roof areas. For differing slope angles I've used the hawkindale angle spreadsheet - I actually think you posted it in another post at one point. Maybe I'm mis-reading your mention of b@stard pitches.

[Joe Bartok]

Actually, I thought it was pretty funny. :lol: Though I was a bit surprised when I read my post and it said "monkey" pitches! My first thought was to blame it on my lack of typing skills, and then I thought, no way! I'm not that bad! :lol:
Anyway, if my b@$t@rdisation of the word b@$t@rd was unsuccessful, there was always the term irregular pitches to fall back on.

[Rich]

:)
Bastard should be ok now.. bastard bastard

[Rich]

Yep - working.

[speaker]

Well I have learned one thing here today "never mess with a mathematician with steel toe boots" (lol)

I had to test your (or see if I could) formula with trig. which by the way I learned today (the basics anyhow) from,

http://www.ex.ac.uk/cimt/mepres/allgcse/bka4.pdf,

and here is what I came up with

in red (your post)


To figure the slope factor for common rafters
Take the slope and square it - add 1 - and take the square root of it.
SqRt((X/12^2)+1)
When X = 5
5/12^2 = .17366
.17366 + 1 = 1.17366
SqRt(1.17366) = 1.0834

example: PYTHAGORAS THEORY
7/12 pitch 34 span
run 17.5ft (unadjusted)
rise 10.208 (unadjusted)
(a^)+(b^) = c^
(17.5^)+(10.208^) =hypotenuse^
410.453 = sq/rt hyp
hyp = 20.2596 (unadjusted i.e. (hap + ridge thickness))

USING FORMULA ABOVE
SQ/RT ((7/12^)+1) = .3403
.3403 + 1 = 1.3403
SQ/RT (1.3403) = 1.1577
SO............
1.1577 * 17.5 = 20.2597
WORKS


To figure the slope factor for hip rafters
Take the slope and square it - add 2 - and take the square root of it.
SqRt((X/12^2)+2)
When X = 5
5/12^2 = .17366
.17366 + 2 = 2.17366
SqRt(2.17366) = 1.47

example: trig (hip)
7/16.97 pitch 34 span
rise = 10.208
7/16.97 angle = (7/16.97 (hit =) then (inv) (tan)
22.416 degrees
now if we use sin = op/hy
sin22.416 = 10.208/x
(to cancel x, times both sides by x) =
xsin22.416 = 10.208
now to isolate x, / both sides by (sin 22.416) =
x = 10.208/sin22.416
x = (10.208/22.41sin = 26.7696

example above:
sq/rt(7/12^2) +2
sq/rt =.3403 +2
sq/rt=2.3403
1.5298
so...............
(1.5298)*(17.5)
=26.7715

WORKS

it took me ages to do this, but hey I didn't know a thing about trig when I got up this morning. I was on the above web site all day.

could one of you guys tell me is this high school math, I just don't remember????

[Rich]

Yes - Trig and Geom are High School maths.

[speaker]

I guess I have a good memory It's just short

[Joe Bartok]

Just what everyone wanted, right?
First of all, I had to edit my math in yesterday’s post. The principle of the 3:4:5 triangle was familiar to ancient civilizations thousands of years ago, and Pythagoras mathematically proved the theorem that bears his name for all right angled triangles. Yesterday, I destroyed it all with a few strokes on a computer keyboard. Duh! Anyway, for the sake of anyone reading these posts and actually paying attention to the math, it’s now corrected.
Here’s a graphic interpretation of the math for the slope factors that Rich posted, solved using the Pythagorean theorem (speaker: check this out, no "trig"!). I've also added some trig that might clarify where the equations I posted were coming from. And this is how I would have dealt with BASTARD pitches meeting at an irregular deck angle. Sorry, it’s in Word again. Best I can do for now, and lucky, at that. We've got a bit of an ice storm going on here that took out the Internet. Tomorrow, I’ll try to get this in pdf format.
Also: Rich, or Tom, whoever makes the decisions on what is or isn’t posted. I’ve noticed others posting diagrams, and I’ve been doing the same. Math, especially applied math of this nature, is a very visual thing; a picture really is worth a thousand words. Not to mention fun (well, for me, anyway). I hope I haven’t been overdoing it with posting diagrams, and if I have, let me know and I’ll keep it to a minimum.
Comments and suggestions are welcome; I'm trying to learn something here too!

[Rich]

No problem on the postings - we're getting close to the maximum allowed - so I'll override it :)

[Cole]

Are we really?

[Rich]

All told the attachments alone are 28mb. Default maximum is 10mb :)

[Joe Bartok]

Here's the pdf file, as promised yesterday.
Just a few minor changes; I've added some right angles wherever they won't clutter the diagrams. Otherwise, the balance of the trig and math is basically the same.
Hope there aren't going to be two copies of this post. Had a problem with the computer and/or the 'net; this is my second attempt.

[Roy J.]

Hi Joe,
I couldn't get your theory for bastard pitches to load on my computer. I'm doing a 7x9 gazeboo with all exposed framing and the owner wants a pitch close to a 7/12. Do you or anybody know what the angle of the hipboards would be?
thank you,
Roy

[Rich]

I'm assuming your gazebo is square...
7/17 is the pitch for a hip with a common pitch of 7/12.. this essentially equates into the rule of 17.. whereas taking a common rafter into a hip rafter turns it 45 degrees. Think of a box as 12 by 12... so common rafter runs would be 12 - to figure the diagonal take the square root of 2 (a shortcut) and then times 12 - comes up with 16.96 or something like that (from memory) .. equates closely to 17 anyway.

[Rich]

Should also say that I'm assuming that all hips do not meet in the same place.. otherwise you end up with the dreaded bastard hips :)

[Roy J.]

Hi Rich,
I'm trying to make this work without a ridgeboard--hips all meeting in the same place. So, yes, I guess this is a dreaded bastard hip...ugh.

[Rich]

haha.. it's not all that bad. The easiest way, without a bunch of voodoo math, is to snap it on the floor of the gazebo. Then using the 7/12 as your slope figure out your rise - which is 2'0-1/2" for the 7' side (plus your HAP). This then gives you a 5-7/16/12 pitch on the 9' side giving you the same rise so all the rafters will hit.
Hopefully that's making a little sense.. I can post a drawing if need be... just don't have time now.

[Rich]

R. Birch or Joe Carola probably have a much better explanation than I do.

[Roy J.]

Thanks Rich,
That's very helpful. I'll snap it out today and mess around with it a bit.
R J.

[Dragon]

Or just go buy a construction master pro. Enter your rise, enter your run, click on the Hip button, click on jack rafter button repeatedly, make cuts, done.

[Joe Bartok]

Hi Roy:
I'm not sure which "theory" isn't loading. If it's the slope factors pdf posted in this thread don't worry too much about it. I still don't understand how the slope factors apply to an irregular roof and don't use them anyway.
Here's a link to the web based Hip-Valley Roof Framing Calculator (http://ca.geocities.com/web_sketches/hip_valley_dimensioning/framing_working_points.html) I generally use. I heavily biased in favour of log building and timber joinery and the dimensioning is only along the theoretical centerlines and intersections of the roof members.
This Excel worksheet also solves only the Unadjusted Hip-Valley Dimensions (http://ca.geocities.com/web_sketches/framing_math/hip_valley_imperial.xls).
Enter any pitch, any pitch, any plan angle between zero and 180°, and you're all set.

EDIT: What shape is the 7×9 gazebo footprint? Rectangular, hexagonal, octagonal? Or something really interesting?

[Joe Bartok]

Roy, I’m going to assume your gazebo is rectangular. The plan angles are:
arctan (7/9) = 37.87498°
arctan (9/7) = 52.12501°

For the Hip rafters to intersect at one point requires two different intersecting common slopes. There are two possible solutions.

If your 7/12 common pitch is the steeper of the two pitches the Hip rafter angle is:
arctan (7 × sin 37.87498° / 12) = 19.70407° (4 5/16 over 12)
This makes the common pitch on the adjoining side:
arctan (tan 19.70407° ÷ sin 52.12502°) = 24.40397° (5 7/16 over 12)

If your 7/12 common pitch is the shallower of the two pitches the Hip rafter angle is:
arctan (7 × sin 52.12501° / 12) = 24.72396° (5 1/2 over 12)
This makes the common pitch on the adjoining side:
arctan (tan 19.70407° ÷ sin 37.87498°) = 36.86990° (9 over 12)

The easiest way, without a bunch of voodoo math ...
I had to sacrifice a couple of chickens to my computer. :D

[Rich]

LMAO.. wtg Joe.

[Roy J.]

Thank you all very much. I snapped it out like you said, Rich and it started to look sort of do-able finally but we didn't start it yet. I wrote your figures down Joe, thanks a lot. (It's a rectangle shape) And I appreciate the link to the online framing calculator. Because all the framework on this gazebo is exposed I am a bit concerned that from an asthetic point of view that the jack rafters won't be lining up on the hips like a regular roof and won't look very symetrical from below. We might have to talk to the owner about that aspect. I will keep you all posted.
Roy J.

[Dragon]

Keep in mind that with a bastard pitched hip the hip will not sit in the center of the angle. If it does your overhangs will not be the same on each side and will extend further out on the shallower pitched sides. You have to allow for that in your layout.

[Rich]

Good point Dragon.. went through that on a project last year. Homeowner asked why the hip was off center.. because you wanted equal overhangs on 2 different pitched roofs.

[Dragon]

If Joe wants to sacrifice more fowl he can figure that offset.

:D

[Joe Bartok]

I should have sacrifed a virgin. Not to the computer; this would have to be dealt with personally. And of course the "sacrifice" doesn't involve any killing.
Roy, the jack rafters can be aligned along their centerlines as in these (not very clear, sorry!) Valley Roof Images (http://ca.geocities.com/xpf51/gallery/GALLERY_2.html). If there is a huge differece in common pitches this "alignment" may not produce aesthetic results as the spacing of the jacks on one of the spans may become a bit crowded. The Match Spacing O.C. (feet) field in the lower right-hand corner of the web based calculator gives the value.
The other guys are right about the overhangs being unequal. This is something I have rarely done but here are some Soffit Overhang Adjustment Diagrams and Formulas. (http://ca.geocities.com/web_sketches/framing_math_notes/valley_soffit_overhang/valley_soffit_overhang_shift.html) Unless there's a serious difference such as the 3/12 and 10/12 pitches illustrated in that link we just leave the overhangs unequal.
Since the work is exposed there is another Hip shift you may want to consider. To make the Hip rafters present faces of equal depth the Hip rafter may be offset in proportion to the plan angles. This Hip-Valley Section Calculator (http://ca.geocities.com/web_sketches/hip_valley_dimensioning/hv_sections.html) will solve the offset for you. (That and a few other things ... I think I'm going to have to "re-train" myself on what that calculator does).

[Roy J.]

I had talked with the owner a couple of weeks ago after I told him the jack rafters wouldn't be lining up and therefore with all-exposed framming the structure would be more appealing from underneath with a short ridgebeam. I hadn't gotten back to the project since then to see if there was some magical trick that could be done to make the rafters line up, in the meantime the owner gradually accepted the idea of the short ridgebeam. So that's how we're going to do it. However I thank you all very very much for all the input and consideration you put into your posts.

Roy

[R Birch]

I saw and read this post a couple of days ago. I was confused by the term “gazebo”. I would call it a Cabana, or Pavilion, since it is a 9x7 rectangle. (I’ll bet/guess it’s for the pool too?) I think the short ridge will work fine for an exposed roof frame. The effective ridge will be 2’ 1-1/2” long and the jacks spaced at 24” o.c. will work out spaced perfectly.

I think that the rectangular diamond hip roof (or irregular diamond hip-roof) is an interesting design and thought a little further discussion on it might be fun. As Joe Bartok pointed out, the plan view ratio of the rectangle is also the ratio of the pitches. Therefore the easiest pitches to put on a 9’ by 7’ irregular diamond hip would be, 9/12 Vs. 7/12. If you wanted your diamond pitched roof’s jacks to “shake hands” at the hips for visual symmetry then You could assign the o.c. layout to the steep side (say 24”) and multiply that by the plan ratio of, the lesser pitch over the steeper pitch (7/9ths, in this case) and the rafter layout for the steep side would be 7/9ths of 24”, or 18-5/8” o.c. (What would the matching step-offs be?)

If you wanted equal overhangs then you would want to work/start from the fascia’s dimensions. (Sides, plus overhangs, equals sides of beginning rectangle/ratio.) It is not a suitable design scenario for an exposed roof frame, in my opinion, since it requires the raising of the bearing plate for the steeper roof. It would probably be a visually awkward build. Box it in with soffit and it would work fine though and rafter o.c. layout concerns vanish too.

Btw Rich, the original “Slope Factors” post is a great subject for roof cutters, for newbie’s and pro’s alike. Thanks for the high praise too, but I think Joe Bartok and you have nailed the answers to the original post and following questions quite well (I couldn’t have done any better, Lol. But I’m sure Joe C would also make some interesting additional comments.) And there is a lot more fun stuff to go with the concepts discussed above too so I threw in a couple of simple and fun observations. There’s plenty more fun to have though.

[Roy J.]

I had kind of figured out how to make it work, if it was a bastard hip. Being that it's 7x9:

1)Run the commons to center on the 7 foot side, then run the commons to center on the 9 foot side.

2)If one pitch is a 7/12 and one pitch is a 9/12 then the hips would be 8/12.

(Just kind of winging it, does that sound right? Whatever the pitches are, the hips are the number in the middle of the two pitches?)

[Rich]

Actually the hip is 7-7/8 / 17

[Roy J.]

Right. 8 (7 7/8) "17". If you guys keep making this sound easier and easier I might end up actually doing it. hmmmm

[Joe Bartok]

1)Run the commons to center on the 7 foot side, then run the commons to center on the 9 foot side.
This correctly solves for the same rise for both sets of Common rafters and the Hip.
2)If one pitch is a 7/12 and one pitch is a 9/12 then the hips would be 8/12.
The Common rafter slopes cannot simply be averaged to find the Hip rafter pitch angle. The value of the Hip rafter slope is determined by the roof geometry or trigonometric equations derived from the geometry.
7/12 Pitch Angle = 30.25644°
7/12 Side Plan Angle = 52.12502°
9/12 Pitch Angle = 36.86990°
9/12 Side Plan Angle = 37.87498°
Hip Pitch Angle = 24.72397° (about 5-17/32 over 12)

There are hundreds of formulas; here's a link to a few Roof Framing Angle Equations (http://ca.geocities.com/web_sketches/framing_math/Framing_Angle_Formulas.html).

EDIT ... typo correction: The Hip rafter pitch I originally posted above was incorrectly given as 5-7/32 over 12.

[Joe Bartok]

Roy: This 3D Hip Roof Models (http://ca.geocities.com/web_sketches/trig_notes/hip_models_and_developments.pdf) pdf may help you out with understanding where the formulas are coming from. If you don't like trigonometry, the formulas can be completely disregarded and the angles solved with compass and straightedge (http://ca.geocities.com/web_sketches/framing_math_notes/hip_valley_roof_ratios/hip_roof_development.html). Lines of equal length in the .pdf file are the same color. The tetrahedrons of the Main and Adjacent sides are developed independently with the triangle representing the Hip pitch angle being common to both sides.
When I first started studing roof framing math I actually developed and constructed 3D models with bristol board and tape, exactly as shown in the linked documents. The dimensions on the last page are for the 9.25/12 side of a Hip roof formed by the intersection of a 9.25/12 and a 6/12 slope, with the angle between the eaves in plan being 90°. The dimensions may be changed to any convenient scale by multiplying all measurements by the same factor.

[Roy J.]

Thank you Joe. I bookmarked the second set of drawings--that can be calculated using a compass and straight edge. But I'll need to look at it a bit before I can really understand it. I have no background in trigonometry, though I wish I did.
I don't know for sure but it seems to me that if you could somehow convert this to a series of easy to understand tables you might have the makings for a decent roof cutter's book. I'd been meaning to buy one for a long time.

[R Birch]

Roy,

The best tables to learn to use first can be found on a typical Framing Square. I like the Stanley Squares (mod. 45-120, if you can find one, Stanley discontinued them. Swanson has a fair aluminum version of it available today.) Different squares will have different info on them sometimes. Get the book on using the Steel Square too. It will explain the tables for you. One other thing that will help you understand the square’s tables is the fact that the Steel Square is much older than the circular saw. Some of the tables are for marking compound miter, jack and hip/val, rafter top cuts to be cut with a handsaw. (for regular pitches only though)

Once you have regulars down pat then the irregs will be easier to master. Regular roofs have a one to one plan view ratio. Irregulars are pitch to pitch. (7 to 9 for instance.)

Some rafter books have tables for irregulars, like Barry Mussell’s “Roof Framer’s Bible” (but, I personally don’t care for the way he does his bastards. Unnecessarily complicated, IMO, but worth reading anyway.) Also there’s Marshall Gross’s “Roof Framing”, (kind of “Old School”), “The Full Length Roof Framer” by A. F. Riechers is good, (I carried it on me at work for years, regular pitches only, in half pitches from (point 5, or ½ on 12) .5/12 up to 24/12), “A Roof Cutter’s Secrets”, by Will Holladay is another great book for your construction book library too, but my all-time favorite is Swanson’s “Blue Book” of rafter lengths, it comes with their speed square. (The “One Number Method”. over simplifies things though, makes it easy to miss the point.) These are the ones I’ve read. And I have my own style that I have developed over the years. The “Blue Book” explanation comes closest to my methods, so I like it the most. The problem with all roof cutting books is they don’t take you through the entire process from span, to calculating, to marking, and then to cutting. Properly understanding the modern circular saw, and it’s design features, can eliminate a lot of unnecessary calculations when cutting hip/vals and jacks. But they just don’t address it so it’s up to you to figure out. what they fail to tell you. That’s the real “Roof Cutter’s Secret”.

[Joe Bartok]

I don't know for sure but it seems to me that if you could somehow convert this to a series of easy to understand tables you might have the makings for a decent roof cutter's book.
The log builder I first worked with suggested exactly this. Unfortunately there are an infinite number of possible sets of pitches and plan angles … and each combination of pitch, pitch, plan angle currently produces forty different layout and bevel angles. It would take a while to get the tables in print. :)
The best move in terms of versatility and accuracy is to program a calculator, palmtop or computer solve the angles. Once the angles are known we can work from any given: rise, run or diagonal, and not necessarily "over 12".
... it so it’s up to you to figure out what they fail to tell you. That’s the real “Roof Cutter’s Secret”.
That pretty much says it all. Experience and practice are the best teachers. Books and Internet forums are good resources. Study the geometry and trig, make models to test and relate the theory to the real world, keep a notebook or journal of formulas and successful resolutions of unusual problems.

[Roy J.]

lol...yes, the real secrets are figuring how to make the books work FOR you. Sometimes I've used the Blue Book for finding a hip/valley board--then I won't use it for a year or so and have to remember how to decipher the tables. Other times I've been around guys that go right down the list cutting jackrafters according to the book's calculations (before even trying ONE test fit) and we end up with a pile of scrap to be recut.

I would like to find a book that says something like: "Here's a dutch hip. Here's how to do it." (With easy to understand illustrations and photographs)

Sometimes these foms confuse me though. Like when people talk about "dropping the hip" or "H.A.P". I've never been around guys at work that say things like that. I swear to god I was so confused thinking about dropping a hip that I had to think: "Wait. Are they saying their hipboards have less clearance above the bird's mouth than a common rafter?"

Anyway. I have to go. Thank you as always for the great posts and sharing your abundance of knowledge.
-Roy