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sbebuilders
10-11-2007, 10:52 AM
Our next commercial building has 6x12 hips and valleys on a 10:12 pitch. I was calculating the height of the hip backing angle rise and was puzzled by the difference in the backing angle rise versus the hip drop adjustment. I checked Joe Bartok's web site and the calculations were correct. I then proceed write a script that printed out all of the hip backing angle, hip backing angle rise and hip drop adjustment height. However, I'm still a little puzzled by the height I came up with for the plan angle height on top of the hip rafter after it is backed out.

Here are the formulas I used to calculate the backing angle rise.

Evae Angle = 90°
Plan Angle = 45°
Pitch Angle = arctan ( Pitch ÷ 12)
Hip Pitch Angle = arctan( tan (Pitch Angle) * sin ( Plan Angle))
Hip Backing Angle = arctan ( sin ( Hip Pitch Angle) ÷ tan ( Plan Angle) )
Hip Backing Angle Rise = ( Material Width * .5 ) * tan ( Hip Backing Angle) ) or Valley Trough Depth
Plan Angle Unit Rise = tan ( Plan Angle )
Hip Unit Rise Run = 12 ÷ sin ( Plan Angle )
Unit Hip Drop = Pitch ÷ ( Hip Unit Rise Run * (Plan Angle Unit Rise * 2 )
Hip Drop Adjustment = ( Material Width * .5 ) * Unit Hip Drop

-----------------------------------------------------------

Plan angle rise on top of the hip backed out hip formula.

A = ( Material Width * .5 ) / cos(Hip Backing Angle)
C = sqrt (A² + A²)

Hip Plan Angle Rise = C * tan ( Hip Pitch Angle)

On a 10:12 pitch with 5 ½ x 11 ½ hips:

A = 2.75 / cos(26.92) = 3.08
C = sqrt ( 3.08² + 3.08²) = 4.36
Hip Plan Angle Rise = 4.36 * tan ( 30.51) = 2.57

This 2.57 inches seems to be incorrect.

Got any suggestions for the correct formula?

http://www.sbebuilders.com/cgi-bin/rafters/hip-backing.cgi Hip Backing Angle Script

http://www.sbebuilders.com/rfi/images/backing_angle.html Bigger image of the Hip Backing Angle Rise

Sim

Joe Bartok
10-11-2007, 11:16 AM
Sim, I don't have time to peruse your calculations in detail until tomorrow but I suspect the problem might lie in that if we know the Backing Angle rise in cross-section view this rise must be rotated to the dimension viewed in plumb section.

Plumb Backing Angle Rise = Backing Angle Rise ÷ cos Hip Pitch Angle

The run remains the same for both angles. (And I find the name "Plumb Backing Angle" misleading, this angle is not used as a blade bevel like the Backing Angle).

This is the difference in dimension I'm talking about:

Development of the Backing Angle (http://ca.geocities.com/web_sketches/backing_angle_development/backing_angle_development.html) ... the Backing Angle rise is the altitude of the triangle representing the Hip Pitch Angle, the altitude in question being perpendicular to the hypotenuse of the triangle.
Development of the Plumb Backing Angle (http://ca.geocities.com/web_sketches/plumb_backing_development/plumb_backing_angle_development.html) ... the Plumb Backing Angle rise is the rise of the Hip Pitch triangle, or the side opposite the Hip Pitch Angle.

If the width of the Hip rafter is significant I offset the Hip (the rafter itself, not the reference line of the run ... this is fixed!) to produce equal shoulder heights, this process also locates the "plane-in" or intersection points of the jack rafters. This offset is proportional to the runs of the backing angles in the two preceding slideshows. In other words, I don't necessarily use .5 × Hip rafter width to find the Hip drop. Absolutely not if it's a 12" wide rafter and we're dealing with unequal roof slopes.

This is a drawing of the Hip Rafter Offset (http://ca.geocities.com/web_sketches/framing_math_notes/hip_rafter_shift/hip_shift.html) by "ollee" (William Dillon).

Example Hip Offset and Hip Drop Calculation (http://ca.geocities.com/web_sketches/framing_math_notes/hip_valley_shift_drop/hip_shift_drop_calcs.html)

Joe Bartok
10-11-2007, 11:41 AM
Plan angle rise on top of the hip backed out hip formula.

A = ( Material Width * .5 ) / cos(Hip Backing Angle)
We need the Backing Angle rise first, and have to use cos Hip Pitch Angle in the divisor, not cos Hip Backing Angle.

This calculation is for two 10/12 slopes with a 90° corner angle (true? or are we talking about an octagon or some irregular corner?). Since the pitches are equal there will be no offset to worry about.

½ Hip Rafter Width × tan Backing Angle
= 2.75 × tan 26.91568° = 1.396101

Plumb Backing Angle Rise = Backing Angle Rise ÷ cos Hip Pitch Angle
= 1.396101 ÷ cos 30.50896° = 1.620453 ... there's the infamous Hip drop.

Check ...
Plumb Backing Angle = arctan (1.620453 ÷ 2.75) = 30.50896°

The same value obtained by using strictly trigonometric methods. Also, where equal pitches at a 90° eave angle are concerned, the Plumb Backing Angle equals the Hip Rafter Pitch Angle.

PS ... this post was edited. I was using 3 inches as half the Hip rafter width rather than 2.75. I'm spoiled; I get to cut the material to size and often forget that stock lumber dimensions aren't exact ... :)

Joe Bartok
10-11-2007, 12:13 PM
Sim, hopefully you can make heads or tails of this. The graphics suck big time and someday I'll get around to editing them so they look more three dimensional. This is another way of looking at the relationship between the Backing Angle and the Plumb Backing Angle (and a couple of other angles):

Revolution of the Cutting Plane (http://ca.geocities.com/web_sketches/backing_angle_revolution/backing_angle_revolution.html)

Rather than working out the angles in a roof from the dimensions of the material, I prefer to find the angle first and do the dimensioning afterward. As usual there are a whole bunch of formulas for finding a given angle. Here are two methods of solving the Plumb Backing Angle:

Plumb Backing Angle ... Solution #1 (http://joe.bartok.googlepages.com/plumb_backing_angle_trig.jpg) ... arctan (tan Hip Pitch Angle ÷ tan Plan Angle)
Plumb Backing Angle ... Solution #2 (http://joe.bartok.googlepages.com/plumb_backing_angle_trig2.JPG) ... arctan (tan Common Pitch Angle × cos Plan Angle)

The Plumb Backing Angle produced by either formula is 30.50896°.

Hip Drop = ½ Hip Rafter Width × tan Plumb Backing Angle
= 2.75 × tan 30.50896° = 1.620453

To reiterate, if this were an irregular roof there would be different Plumb Backing Angles for the Main and Adjoining sides of the roof, and the Hip offset rather than ½ Hip Rafter Width would have to be considered.

sbebuilders
10-11-2007, 01:05 PM
Sim, hopefully you can make heads or tails of this. The graphics suck big time and someday I'll get around to editing them so they look more three dimensional. This is another way of looking at the realtionship between the Backing Angle and the Plumb Backing Angle (and a couple of other angles):

Revolution of the Cutting Plane (http://ca.geocities.com/web_sketches/backing_angle_revolution/backing_angle_revolution.html)


The Plumb Backing Angle produced by either formula is 30.50896°.

Hip Drop = ½ Hip Rafter Width × tan Plumb Backing Angle
= 2.75 × tan 30.50896° = 1.620453



Joe, your Revolution of the Cutting Plane web page cleared up my misunderstanding about the hip drop height versus the backing angle rise. I added the plumb backing angle to the scripts and it produced the same results as your pages.

http://www.sbebuilders.com/cgi-bin/rafters/hip-backing-octagon.cgi Octagon Backing Angle

However, I guess I'm still in the dark about the triangle I was using for the 2.57 inches rise.


Sim

Joe Bartok
10-11-2007, 01:38 PM
Joe, your Revolution of the Cutting Plane web page cleared up my misunderstanding about the hip drop height versus the backing angle rise.

That's kind of funny. I was dubious about posting a link to the "Revolution of the Cutting Plane", the diagrams needed to be tidied up and I was afraid it might cause confusion.

I did a quick random sample of your chart values and they look O.K. to me.

If anyone wants to compare notes here's a link to my Hip-Valley Sections Calculator (http://ca.geocities.com/web_sketches/hip_valley_dimensioning/hv_sections.html).

Richard Birch
10-11-2007, 02:24 PM
Sim,

I assume this is about marking and cutting 5 ½” thick Hip/Val rafters.

Another approach would be to calculate the length of the H/V rafter at its physical side, or short to short of the cheek cuts/bevels, (as seen in plan view). For common pitches remove the hip from the effective run and calculate.

With a CM Calculator: enter the Hip’s thickness as a Hypotenuse for a 12/12 pitch, press run to get the dimension subtracted from the effective run of the King Common. For the 5 ½” hip I get 3 -7/8”. This will work for a hip, backed or not. And again, it eliminates the need to do the confusing “Infamous Hip Drop” calculation.

This trick will also work with irregulars too. The hip shifts proportionately.

I use an excel spreadsheet for my rafter cut lists calculator.

Richard

Joe Carola
10-11-2007, 02:47 PM
Sim,

Years ago, I used to do this using your 5-1/2" thick Hip:

(Pitch÷16.9705 x Half Ridge Thickness) = Hip Drop

10/16.9705 x 2.75 = 1.620459"

sbebuilders
10-11-2007, 03:27 PM
Sim,

I assume this is about marking and cutting 5 ½” thick Hip/Val rafters.

Richard

Richard, What I was really after was the backing angle height, so I could snap the line on the hip rafter for the hip rafter backing bevel saw cut. I just got side tracked with the hip backing angle rise versus the hip drop. I added the hip drop to the scripts so I could see the difference in the two dimensions.

The 6x12 hip rafters are DF #1 SDS KD with 19% max moisture content so their costing me about $250.00 each, so I can't afford to back them out incorrectly.

This is kind of funny about the “Infamous Hip Drop” calculation.

Instead of buying a new speed square so I could look at the Swanson Blue Book, I looked though all my rafter books and this is what I found.

1: Roof Framer's Bible -- gives the hip drop on all the rafter tables,but does not show how to back out a hip, even though it does give the hip backing angle on each page. The one diagram it does have of a backed out hip, it's label hip drop when it should be label plumb backing angle rise.

2: A Roof Cutter's Secrets -- 1989 edition -- not one word about backing out a hip or dropping the hip in the table of contents. I thought this edition was a waste of money. But I here the latest version is ok.

3: The Steel Square -- shows backing out and dropping the hip, but a very poor diagram for the example.

4: Roof Framing by H.H Siegle -- nothing in the table of contents about backing out a hip or dropping the hip.

5: The Rafter Book -- this book have the best examples of backing out the hips & valley and gives backing angles on each page of the book, but it does not say anything about dropping the hip.

6:Full Length Roof Framer -- good examples of backing out the hips & valley and gives backing pitches on each page of the book, and talks about dropping the hip when the hip rafter is not backed out.

7: Roof Framing by Marshall Gross -- I can't fine my copy. Must have lent to one of my boys, but I'm pretty sure it talks about backing out the hip.


Sim

sbebuilders
10-11-2007, 03:44 PM
Sim,

Years ago, I used to do this using your 5-1/2" thick Hip:

(Pitch÷16.9705 x Half Ridge Thickness) = Hip Drop

10/16.9705 x 2.75 = 1.620459"

Joe, That's why we need cheat sheets. Now that you remined me of the formula, I remeber using it all the time back in the 80's before we switched over to marking the plate line intersection on the hip rafter and using that mark for the HAP.

Here's the old formula

hip drop = tan(hip pitch angle) * half the thickness of the hip

Here's the new formula after meeting Joe Bartok online

Hip Backing Angle = arctan ( sin ( Hip Pitch Angle) ÷ tan ( Plan Angle) )
Hip Backing Angle Rise = ( Material Width * .5 ) * tan ( Hip Backing Angle) )
Plan Angle Unit Rise = tan ( Plan Angle )
Hip Unit Rise Run = 12 ÷ sin ( Plan Angle )
Unit Hip Drop = Pitch ÷ ( Hip Unit Rise Run * (Plan Angle Unit Rise * 2 )
Hip Drop = ( Material Width * .5 ) * Unit Hip Drop

You can tell Joe Bartok doesn't have to use a CMC on the job site with his ability, but the rest of us do. So I'll add your formula to my cheat sheets, but stick to Joe B's formula on the scripts.


Sim

Joe Bartok
10-11-2007, 03:47 PM
The backing angle is one of the fundamental angles in a Hip-Valley roof. Here's my take on it:

Construction of the Backing Angle (http://ca.geocities.com/web_sketches/framing_math_notes/backing_angle_construction/backing_angle_construction.html) ... there are links to a couple of other diagrams and alternate solutions. (Just a couple ... at last count I had about fifty different formulas relating the backing angle to other angles in a complex Hip-Valley roof).

Extracted from that web page:
Backing Angle Rise (http://ca.geocities.com/web_sketches/framing_math_notes/backing_angle_construction/hip_triangle.gif) ... for the Plumb Backing Angle we would use the rise of the triangle, the line opposite the Hip Pitch Angle. The line of the altitude must be rotated through the Hip Pitch Angle to attain the correct position to serve as the rise of the Plumb Backing Angle. Compare the rises in the solution of the tangent of the Backing Angle and tangent of the Plumb Backing Angle (http://joe.bartok.googlepages.com/backing_angle_trig_models.jpg).
Backing Angle Run (http://ca.geocities.com/web_sketches/framing_math_notes/backing_angle_construction/plan_triangle.gif) ... the same run serves for both the backing bevel and its plumb alter ego.

Although any of the sides in the tetrahedra can be set equal to one, I find that the best procedure is to make the Hip run equal one, as per most of the drawings. The lengths of the sides of the triangles are scaled in terms of the trig functions of the angles and, voila! We have formulas!

Joe Bartok
10-11-2007, 04:02 PM
It's always fun to compare formulas and gain more insight into why they work.

Plumb Backing Angle ... Solution #2 : arctan (tan Common Pitch Angle × cos Plan Angle) (http://joe.bartok.googlepages.com/plumb_backing_angle_trig2.JPG)


(Pitch÷16.9705 x Half Ridge Thickness) = Hip Drop

tan Common Pitch Angle means the same as Pitch.
With respect to a run of 12 and a plan angle of 45°, 1/cos 45° = 16.9706

Since Joe Carola's formula divides by 1/cos 45°, it's the same as multiplying by cos 45° ... or cos Plan Angle in my formula.

Richard Birch
10-11-2007, 05:01 PM
Sim,

I think it’s kind of funny too. As I researched the “Hip Drop” a few years ago I had to buy many of the books on your list so I could read up on it. I found most of them confusing and ambiguous on the HD subject. I can let you know that Gross’s book does have it and his is one of the better examples. My copy of “A Roof Cutter’s Secrets” mentions them, but the author seems to say he feels the term is misleading, “I don’t really like the term because no hip is actually dropped” is how he says it, I believe.

As far as finding the angle height for the backing bevel goes; with a CMC; enter the backing bevel as the pitch, half the thickness as the run, press Rise. (Or equivalent formula)

The problem with the 1/2 thickness method above is that it won’t work well with irregulars because the roof proportions are not 1/1. Using half the hip for two different bevels will produce two different bevel drops/heights. The simplified method I use will automatically offset the Irregular Hip proportionately keeping the HAPs equal. And I don’t need to memorize any special formulas. (I might add that keeping the method simple keeps my spreadsheet calculator simple and uncluttered with information I don’t need.)

Why do you need to back the hip? Could you just back it equally from both sides or add a beveled fur strip to it?

Whether you can or not, either way, I’d still calculate it’s length to mark the short points of a diamond cut hip that thick. My saws won’t cut 45deg compound miters in one pass that deep. If it was a single beveled top cut I’d mark it full length on the side/shoulder. (As I described in the other “Hip Drop” thread.)

That’s just how I do them. You can do them any way you want. In the end they’ll probably turn out the same.

Take some pics. Good luck too.

Joe Fusco
10-11-2007, 05:07 PM
Sim,

Here's some stuff:

http://forums.jlconline.com/forums/showthread.php?t=14998&highlight=dropping
http://www.josephfusco.org/Tips/tip0011.html
http://www.josephfusco.org/Tips/tip0002.html
http://www.josephfusco.org/Flash/8-12_triangle_hipdrop.html

Seems like a lot of posts for a simple topic. . .

Oops one more. This deals with split pitches but you can extrapolate for single pitch.

http://www.josephfusco.org/Articles/Roof_Cutting/split_pitch/Split_Pitch03.html

I believe it's the bevel mark your after.

Joe Carola
10-11-2007, 06:10 PM
Sim,

You can also just punch in the hip angle as the pitch and half the thickness of the hip as the run and press rise for the height.

30.51 [Pitch]

2.75 [Inch] [Run]

[Rise] = 1.62052"

Or,

30.51 [Tan] x 2.75 = 1.62052"

It's the same triangle in this drawing from the other thread but I used 3/4" as the run and half the thickness.

http://forums.jlconline.com/forums/showpost.php?p=324532&postcount=2

sbebuilders
10-11-2007, 07:22 PM
Sim,

Here's some stuff:

http://forums.jlconline.com/forums/showthread.php?t=14998&highlight=dropping
http://www.josephfusco.org/Tips/tip0011.html
http://www.josephfusco.org/Tips/tip0002.html
http://www.josephfusco.org/Flash/8-12_triangle_hipdrop.html

Seems like a lot of posts for a simple topic. . .

Oops one more. This deals with split pitches but you can extrapolate for single pitch.

http://www.josephfusco.org/Articles/Roof_Cutting/split_pitch/Split_Pitch03.html

I believe it's the bevel mark your after.

Joe, Yes, I was looking for more information on the bevel mark. I wrote a script that printed out the hip backing angles, the backing angle rise and the hip drop. I thought that the backing angle rise should match the hip drop, but Joe Bartok set me straight with the Plumb Backing Angle Rise being the rise I needed to compare with the hip drop.

http://ca.geocities.com/web_sketches/backing_angle_revolution/backing_angle_revolution.html Joe Bartok's Revolution of the Cutting Plane

After adding Joe's Plumb Backing Angle Rise formual to the script I can see that the hip drop height matches the Plumb Backing Angle Rise.

Your page on Hip Drop, Mark & Back Bevel Angle is also a good reference for what I was looking for.
http://www.josephfusco.org/Articles/Roof_Cutting/split_pitch/Split_Pitch03.html

http://www.sbebuilders.com/cgi-bin/rafters/hip-backing.cgi The backing angle rise script I was writing.

Sim

Joe Fusco
10-11-2007, 07:24 PM
Joe,

It's still not that easy. You would need to "find" the hip angle either like:

hip_angle=tan-¹(pitch/(12*sqrt(2))) or

[10] [inch] [Rise]
[16.97] [Run]
[pitch] [pitch]

sbebuilders
10-11-2007, 07:25 PM
Sim,

You can also just punch in the hip angle as the pitch and half the thickness of the hip as the run and press rise for the height.



It just doesn't get any eaiser than that, unless CMC comes outs with a single button for it.

Sim

Joe Fusco
10-11-2007, 08:13 PM
Sim,

Joe's Plumb Backing Angle Rise is essentially this:
http://www.josephfusco.org/Flash/hip_bevel_rotated.html

Joe Carola
10-11-2007, 09:27 PM
Sim,

The easiest way to describe the difference between the plumb backing angle height and the backing bevel height is that the backing bevel mark is square off the top of the hip compared to plumb and meets at the same point of the small triangle that makes the mark.

Here's two drawings made way back when using an 8/12 pitch with hips running at 45° and a bastard hip with a 8/12 and 12/12 combination. You will see where the plumb line is drawn from the top of the hip and where the backing bevel line is drawn square offs the top of the hip.