docDevelopments of Hip and Valley Roof Angles based
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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
