Download Developments of Hip and Valley Roof Angles based

Developments of Hip and Valley Roof Angles based on the Roof Lines
Overview of Tetrahedra and Triangles defining a Hip or Valley Roof
An ALTERNATE MODEL for HIP / VALLEY RAFTERS:
Valley orientation:
The deck is a level plane,
with the plane of the bottom
shoulder of the valley
dropping away from the deck.
D
DD
90 – DD
SS
Hip run = 1
cos DD
D
Dihedral angle
= 90 – A5Bm
DD
Dihedral angle =
90 – A5Pm
2
cos DD
1 / sin DD
Plan
Dihedral angle =
90 – A5Pa
Isometric
Projection:
Dihedral angle =
90 – A5Ba
Hip orientation
R1 is the dihedral
angle between the
deck and the
bottom face of the
hip/valley rafter.
R4Pa
R4Pm
R4Ba
tan R1
R4Bm
P6m
R5Bm
SS
R1
2
cos DD tan R1
2
cos DD
R5Pm
2
tan R5B = (cos DD tanR1) / cos DD = tanR1 cos DD
tan R5P = tanR1 / (1 / sin DD) = tanR1 sin DD
Same R5 angle formulas as the previous analysis.
Also: P6 = (90 – R5P) – (90 – SS) = SS – R5P
An ALTERNATE MODEL for HIP / VALLEY RAFTERS:
Kernels of R4, R5 and A5 Angles
Kernels re-scaled to “Hip run” = 1: Compare the drawings below
to the models extracted directly from the Valley rafter in the
previous section, as well as the kernels extracted from the stick.
R4P
R1
90 - DD
Dihedral angle =
90 – A5P
1
R5P
R4Pa
R4Pm
R4Ba
R4Bm
R5Bm
SS
R1
Dihedral angle =
90 – A5B
R5Pm
R5B
R4B
1
R1
DD
GENERAL HIP / VALLEY MODEL:
VALLEY meets RIDGE or HEADER
VALLEY meets MAIN COMMON RAFTER
VALLEY meets MAIN PURLIN
Dihedral angle between side face
and bottom face of Valley rafter =
90 degrees.
Dihedral angle between Roof
plane and bottom face of Valley
rafter = C5.
The triangle showing angles labeled Q2
and P1 defines the plane of the Main Purlin
perpendicular to the Roof plane.
Other angular values to expect at the
intercept of the Purlin plane and the bottom
face of the Hip/Valley: R3, P3, C2, C1, R2
Q2
Working point for purlin related angles.
C5
R1
R4B
P1
R5B
DD
SS
Angles to
expect at intercept
of Valley Peak
meets Ridge or
Header:
DD, R4B, R5B,
A5B
MAIN SIDE VIEW
Anticipated angles
where Valley Foot meets
Main Common Rafter
or Eave:
90 – DD, R4P, R5P,
A5P, P6
For the sake of clarity, only some of the major angles on the
model faces have been labeled. Exploded views will show the
remaining angles in more detail.
The planes that form the boundaries of the model (planes of R4
and R5 angles) are the same planes created by cutting a plane on the top
of a post to conform to the bottom face of a Hip/Valley rafter. Expect
R1, A5, R4, and R5 angles at Valley meets Post.
GENERAL HIP / VALLEY MODEL:
Kernels extracted from the general model
Valley peak meets
Main Purlin
Valley foot meets
Main Common
Rafter
Mutually perpendicular lines
90-R1
90-R2
R4P
90-R5P
Dihedral
angle =
90-A5P
* Dihedral
Q2
90-P6
angle = P2
90-R3
90-P3
* Note: See alternate
C5
C5
SS
diagram of kernels
extracted from model.
Dihedral
angle =
90-DD
P1
* Dihedral
Dihedral
angle =
90-C2
angle =
90-P2
Dihedral
angle =
90-C1
Q2
C5
R1
R4B
P1
R5B
DD
SS
Note the lines that represent Common
Rafter or Purlin Depth : Valley Depth. The
ratio cosC5 = cosSS / cosR1 = cosP1 / cosR2
plays a role in determining the location of
angle C5.
GENERAL HIP / VALLEY MODEL:
Kernels rotated and re-scaled
Valley foot meets Main
Common Rafter
Valley peak meets Main
Purlin (opposite hand)
Dihedral angle = 90-DD
Dihedral angle = 90-C1
90-R2
1
C5
90-R1
Q2
R4P
90-R3
C5
1
90-R5P
Dihedral angle = 90-C2
Dihedral angle = P2
Dihedral angle = 90-A5P
Dihedral angle = 90-P2
NOTES:
Three mutually perpendicular lines form one of the vertices (see “Purlin kernel
extracted from Hip kernel”); the kernels may positioned with any face containing a
right angle as the “deck”. The angle arrangements shown above match those of the
kernels extracted from the stick.
The kernels may be split along their respective dihedral angles: Valley peak
meets Main Purlin along 90-C2, 90-C1 or P2, and Valley foot meets Main
Common along 90-A5P, 90-DD or 90-P2. Each split produces an arrangement of
angles as per a standard Hip kernel. The alternate exploded view depicts the kernels
split along dihedral angles P2 and 90-P2.
Angles that upon casual inspection seem to have no direct connection to one
another are now related through their respective kernels, which may be used for
dimensioning as well as simply producing formulas. In addition, since the groups of
angles are now defined, and a given angle occurs in more than one kernel, angular
values may be determined empirically by developing the kernels using compass and
straightedge only.
The sample equations given on the following page cover only a very few
possible formulas.
VALLEY ANGLE FORMULAS:
Valley Peak meets
Main Purlin
tanR2 = tanC5 tanR3
tanR3 = tanR2 / tanC5
KNOWN
EQUATION
(From “Extracting the Purlin
kernel from the Hip kernel”)
tanP1 = tanSS tanP2
Valley Foot meets
Main Common Rafter
tanR1 = tanC5 tan(90-R4P)
tanR4P = tanC5 / tanR1
Divide the kernel along dihedral angle
Divide the kernel along dihedral angle
90–C2, producing two standard Hip
90–A5P, producing two standard Hip
kernels. Consider the kernel on the left
kernels. Consider the kernel on the left
hand side:
hand side:
cosC1 = sinC2 / sinR2
sinC2 = sinR2 cos C1
cosC1 = sinC5 / sinP1
cosDD = sinA5P / sinR1
sinA5P = sinR1 cos DD
sinQ2 = cosR3 / cosC1
cosR2 = cosP2 / cosC1
cosR5P
= cos(90-R4P) / cosDD
tan(90-R2)
= tan(90-C2)sin(90-R3)
(From Standard Hip kernel)
cos(90- Q2) = cosR3 / cosC1
tanR1 = tanSS sinDD
tanC2 = tanR2 cosR3
cosR5P =sin R4P / cosDD
tan(90-R1)
= tan(90-A5P) sinR4P
tanA5P = tanR1 sinR4P
Consider the standard Hip kernels created on the right hand side:
tan(90-C5)
= tan(90-C2) sinR3
tanC2 = tanC5 sinR3
tan(90-P2)
= sin(90-C5) / tanR3
tanR3 = tanP2 cosC5
tanR1 = tanSS sinDD
tan(90-C5)
= tan(90-A5P) sin(90-R4P)
tanC5 = sinR1 / tanDD
tanP2
= sin(90-C5) / tan(90-R4P)
tanA5P = tanC5 cosR4P
tanR4P = tanP2 / cosC5
The process may be continued, using any known formula as a
template (for example, tanP2 = cosSS / tanDD), and substituting
cognate angles from the unsolved kernel. Remember to
compensate for trig functions of complementary angles. The next
diagram depicts an alternate method of extracting Hip kernels
from the general model. Rotate any appropriate face to the deck,
and apply the methods outlined above to obtain solutions.
GENERAL HIP / VALLEY MODEL:
Standard kernels extracted from General Model
(kernels re-scaled for clarity)
R4P
Dihedral angle = 90-C2
P2
C5
P6
C5
Dihedral angle = 90-A5P
90-R3
P3
90-P2
Q2
Note that the two kernels above
may be combined along the
plane of C5 to create a new
kernel.
C5
R1
R4B
P1
R5B
DD
SS
P2
90-SS
90-R2
90-P2
90-P1
C5
Dihedral angle = 90-C1
The two kernels may
be combined along the
plane of C5 to create
the kernel examined in
“Extracting the Purlin
kernel from the Hip
kernel”.
Dihedral angle = 90-DD
C5
90-R1
EXTRACTING KERNELS from “the STICK”:
Valley Peak meets Main Purlin
DD projected to bottom shoulder of Valley is 90-R3
Dihedral Angle = 90-C1
90-R3
Valley Peak
90-R2
90-R2
Q2
90-R3
90-C2
DD
D
Valley Peak
90-DD
Plan
Main Common Rafter
Main Purlin
For the purpose of clarity,
backing angles and other cuts are
not shown.
Valley Foot meets
Adjacent Common Rafter: same
kernel as Valley Foot meets
Main Common Rafter,
substitute adjacent side values.
Also anticipate these
angular values at Valley meets
Post or Eaves.
Dihedral Angle = 90-DD
Valley Foot
90-R1
R4P
90-R1
90-R5P
R4P
90-A5P
Valley Foot meets Main Common Rafter
90-DD projected to bottom face of Valley is R4Pm
(90-D projected is R4Pa at Adjacent Common Rafter)
EXTRACTING KERNELS from “the STICK”:
Valley Peak meets Header
DD projected to bottom plane of
Valley is R4Bm
Dihedral Angle = DD
Valley Peak
90-R1
90-R5Bm
R4Bm
90-R1
R4Bm
90-A5Bm
Header
The same angles
occur at Valley meets
Post.
DD
D
Valley Peak
Plan
Dihedral Angle = D
Valley Peak
R4Ba
90-R1
90-R5Ba
R4Ba
90-R1
90-A5Ba
Valley Peak meets Valley Peak
D projected to bottom face of Valley is R4Ba
Images of 3D Model ... Right Triangles composing the Tetrahedra
modeling the Hip Rafter Side Cut Angles at the Peak of the Hip Roof
Main Slope = 10/12 Adjoining Slope = 7/12
Corner Angle measured between Eaves in Plan View = 90°
Development of Tetrahedron modeling the 10/12 Side Angles
(Trigonometric Scale)
Development of Tetrahedron modeling the 7/12 Side Angles
(Trigonometric Scale)
Plan View Lines and Angles
Views of the Tetrahedra modeling the Hip Side Cut Angles
juxtaposed along the Triangle of the Hip Slope
Hip Side Cut Angles at Peak of Hip Rafter
Juxtaposed Tetrahedra viewed from
Triangles representing upper shoulder of unbacked Hip Rafter
The triangles follow the plane of the upper shoulder of an unbacked Hip Rafter. This plane
passes through a line perpendicular to the Hip Run, and the slope of the plane, measured from
level, is equal to 25.54245°, the Hip Slope Angle. In other words, the Dihedral Angle between
the plane containing the Hip Side Cut Angles and the plane following level is the Hip Slope
Angle.
Juxtaposed Hip Side Cut Tetrahedra viewed from 10/12 Side
Juxtaposed Hip Side Cut Tetrahedra viewed from 7/12 Side
Juxtaposed Hip Side Cut Tetrahedra viewed from the line of the Rise
Views of the Tetrahedra modeling the Hip Side Cut Angles
separated along the Triangle of the Hip Slope
Oblique View of Hip Side Cut Tetrahedra
View of Hip Side Cut Tetrahedra from the Triangles of the Hip Slope
Cutting the Backing Angle and comparing 3D Geometry Notes ...
Images of Right Triangles composing the Tetrahedra extracted from the Hip Roof
A Study of the Jack Rafter Side Cut Angles
Developments of Tetrahedra extracted from the Hip Roof
Views of the Tetrahedra extracted from the Hip Roof
juxtaposed along the Triangle of the Hip Slope
Bird's Eye view of juxtaposed Tetrahedra extracted from the Hip Roof
Jack Rafter Side Cut Triangles about centerline of Backed Hip Rafter
Oblique view of juxtaposed Tetrahedra extracted from the Hip Roof
Jack Rafter Side Cut Triangles about centerline of Backed Hip Rafter
Oblique view of juxtaposed Tetrahedra extracted from the Hip Roof showing the
hypotenuses of the Hip Side Cut Triangles (unbacked upper shoulder of Hip Rafter)
Comparison of juxtaposed Hip Rafter Side Cut Angle Tetrahedra (left)
and juxtaposed Tetrahedra modeling the Jack Rafter Side Cut Angles (right)
Juxtaposed Tetrahedra extracted from the Hip Roof
View from side of the 10/12 Common Slope Triangle
Juxtaposed Tetrahedra extracted from the Hip Roof
View from side of the 7/12 Common Slope Triangle
Juxtaposed Tetrahedra extracted from the Hip Roof
View from the line of the Rise
Views of the Tetrahedra extracted from the Hip Roof
separated along the Triangle of the Hip Slope
Oblique view of 10/12 Roof Surface ... Jack Rafter Side Cut Angle
Oblique view of 7/12 Roof Surface ... Jack Rafter Side Cut Angle
Tetrahedra extracted from the Hip Roof
viewed from the Triangles of the Hip Slope
Images of Post Type 3D Model
of the Hip Rafter Side Cut Angles
Main Slope = 10/12 Adjoining Slope = 7/12
Corner Angle measured between Eaves in Plan View = 90°
Development of Post Type Model of the Hip Rafter Side Cut Angles
(Trigonometric Scale)
Comparison of all the Model Types ... Bird's Eye View
Models to left and center ... Side Cut Angles on upper shoulder of unbacked Hip Rafter
The Tetrahedra extracted from the Hip Roof (right) describe a backed Hip Rafter
Comparison of Hip Side Cut Angle Model Types ... Bird's Eye View
Plumb planes passing through the lines on the surface of the
Tetrahedral Model (left) will produce the Post Type Model (right)
Wireframe Sketch illustrating the plumb planes passing
through the Eave Lines and perpendiculars to the Eave Lines
superimposed on the Tetrahedra extracted from the Hip Roof
Tetrahedral and Post Type Hip Side Cut Angle Models
View from 10/12 Side of Roof
Tetrahedral and Post Type Hip Side Cut Angle Models
View from 7/12 Side of Roof
Employing the Post Type 3D Model
of the Hip Rafter Side Cut Angles
to describe the Valley Rafter Side Cut Angles
Hip Model upside down ... Valley Plan Angles
View of the level plane of the Plan Angles
The "eaves" now represent the intersecting ridge lines at the Valley roof peak
Hip Model upside down ... Valley Side Cut Angles
View of the unbacked shoulder of the Valley Rafter
The "eaves" describe the intersecting ridge lines at the Valley roof peak and the
"Hip Length" represents the Valley sloping down from the level plane of the Plan Angles
A Review of the Juxtaposed Tetrahedral Models
... their Relationship to the Post Type Model
Application to describe both the Hip and/or Valley Side Cut Angles
Lines and Angles seen in Plan View
Wireframe Sketch of Plumb Section Planes
superimposed on the Tetrahedra extracted from the Hip Roof
Bird's Eye View of the Juxtaposed Tetrahedra
illustrating the lines and Side Cut Angles formed by the Plumb Section Planes
intersecting the plane of the upper shoulder of an unbacked Hip or Valley Rafter
Unequal Roof Slopes, Non-Rectangular Corner Angle
Images of Post Type 3D Model of the Irregular Hip Rafter Side Cut Angles
Main Slope = 8.5/12 Adjoining Slope = 10/12
Corner Angle measured between Eaves in Plan View = 120°
Development of Post Type Model of Irregular Hip Rafter Side Cut Angles
(Trigonometric Scale)
Post Type Model of Irregular Hip Rafter Side Cut Angles
View of the upper shoulder of the Hip Rafter
Post Type Model of Irregular Hip Rafter Side Cut Angles
View from 8.5/12 Side of Roof
Post Type Model of Irregular Hip Rafter Side Cut Angles
View from 10/12 Side of Roof
Employing the Post Type 3D Model
of the Irregular Hip Rafter Side Cut Angles
to describe the Irregular Valley Rafter Side Cut Angles
Irregular Hip Model upside down ... Irregular Valley Plan Angles
View of the level plane of the Plan Angles
The "eaves" now represent the intersecting ridge lines at the Valley roof peak
Irregular Hip Model upside down ... Irregular Valley Side Cut Angles
View of the unbacked shoulder of the Valley Rafter
The "eaves" describe the intersecting ridge lines at the Valley roof peak and the
"Hip Length" represents the Valley sloping down from the level plane of the Plan Angles
Overview of the related Hip Side Cut Angle
and Jack Rafter Side Cut Angle Models
Tetrahedra extracted from the Hip Roof
illustrating the Jack Rafter Side Cut Angles
Tetrahedron relating the Hip Side Cut Angle at the Hip Peak (R4P)
the Backing Angle (C5), and the Jack Rafter Side Cut Angle (P2)
tan R4P = tan P2 ÷ cos C5
Main Slope = 10/12 Adjoining Slope = 7/12
Corner Angle measured between Eaves in Plan View = 90°
Comparison of nested Tetrahedra (right)
extracted from the Post Type Model (left)
Development of the Tetrahedron extracted from the
Post Type Model of the Hip Rafter Side Cut Angles
(Trigonometric Scale)
Nested Tetrahedra ... Triangles of Common Slope and P6
Nested Tetrahedra ... Triangle of Backing Angle
Nested Tetrahedra ... Triangle of Hip Side Cut Angle at the Hip Peak
Juxtaposed Tetrahedra relating the
Hip Side Cut Angle at the Hip Peak (R4P)
the Backing Angle (C5), and the Hip Slope Angle (R1)
tan R4P = tan C5 ÷ tan R1
Main Slope = 10/12 Adjoining Slope = 7/12
Corner Angle measured between Eaves in Plan View = 90°
Juxtaposed Tetrahedra extracted from the
Post Type Model of the Hip Rafter Side Cut Angles
Development of the juxtaposed Tetrahedra extracted
from the Post Type Model of the Hip Rafter Side Cut Angles
(Angular Scale)
Juxtaposed Tetrahedra ... Triangles of Common Slope and P6
Juxtaposed Tetrahedra ... Triangles of Backing Angle
Juxtaposed Tetrahedra ... Triangle of Hip Side Cut Angle at the Hip Peak
Juxtaposed Tetrahedra and the Triangles of Hip Slope
... Plumb Section Plane taken through the line of the Hip Run
Application of Juxtaposed Tetrahedra
... Square Beam intersected by Hip Rafter
Main Slope = 12/12 Adjoining Slope = 12/12
Corner Angle measured between Eaves in Plan View = 140°
Model of Nonagonal Footprint Pyramid
Developments of the Juxtaposed Tetrahedra
Development of the Beam
Model of Beam ... Oblique View
Model of Beam ... View from Hip Rafter Foot
Hip Rafter intersects Beam ... View of Upper Surfaces (Roof Planes)
Hip Rafter intersects Beam ... View of Side Faces
Tetrahedron representing Roof Surface Angles
juxtaposed with Model of Hip Rafter intersects Beam
Tetrahedra modeling all angles
juxtaposed with Model of Hip Rafter intersects Beam