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The faces of square cut fascia are equivalent to the faces of purlins at right angles or perpendicular to the plane of the roof. Unlike the related planes on the purlins which terminate at the side or plumb face of a Hip-Valley rafter, the planes of the fascia continue until they intercept one another. Therefore the Deck Angle or Plan Angle from the adjoining span is required to calculate the angles on the fascia. The formulas are listed below for comparison. MAIN SPAN ANGLE on FACE of PURLIN perpendicular to PLANE of ROOF arctan (tan Main Deck Angle ÷ sin Main Common Pitch Angle) 90° - P1m = arctan (tan DD ÷ sin SS) MAIN SPAN ANGLE on FASCIA arctan (tan Adjacent Deck Angle ÷ sin Main Common Pitch Angle) arctan (tan D ÷ sin SS) ADJACENT SPAN ANGLE on FACE of PURLIN perpendicular to PLANE of ROOF arctan (tan Adjacent Deck Angle ÷ sin Adjacent Common Pitch Angle) 90° - P1a = arctan (tan D ÷ sin S) ADJACENT SPAN ANGLE on FASCIA arctan (tan Main Deck Angle ÷ sin Adjacent Common Pitch Angle) arctan (tan DD ÷ sin S) Compound angles must be cut where the Hip-Valley foot meets the eaves to accommodate the planes of the square cut fascia. Cutting the compound angle forms a plane on the Hip-Valley at this juncture, creating an angle measured with respect to the edge on the upper or lower shoulders of the Hip-Valley. The angle on the fascia, complementary to the Fascia Miter Angle, is calculated and measured with respect to the level plane of the deck. The bottom shoulder of the Hip-Valley slopes at angle R1 with respect to the deck. This creates a difference between the angle measured on the fascia and the angle created on planes of the Hip-Valley faces that match the fascia at the eaves. Angle formed on the plane as the Hip-Valley foot meets Eave: Angle on Fascia + P3, or 90° - Fascia Miter Angle + P3 This angle may be solved by two methods using the LAG calculator. Initial parameters for the following examples: (select Rise/Run Entry Mode) Main Span = 7 ÷ 12 Adjacent Span = 12 ÷ 12 Total Deck Angle = 90° Note the following returns: P3m = 13.19257° P3a = 26.14568° From the Square Cut Fascia Calculator: 90° - Main Span Fascia Miter Angle = 49.18024° 90° - Adjacent Span Fascia Miter Angle = 67.58488° Angles on planes of Hip-Valley foot meets Eave: 90° - Main Span Fascia Miter Angle + P3m = 49.18024° + 13.19257° = 62.37281° 90° - Adjacent Span Fascia Miter Angle + P3a = 67.58488° + 26.14568° = 93.73056° The supplementary angle 180° - 93.73056° = 86.26944° is shown in the sketch. The angle traced on the bottom shoulder of the Hip-Valley as it meets the face of a purlin perpendicular to the plane of the roof, measured with respect to the long axis of the Hip-Valley, is defined as 90° - R3. The dihedral angle measured between the bottom shoulder of the Hip-Valley as it meets the face of a purlin perpendicular to the plane of the roof is defined as 90° - C2. Enter the following values in the calculator (select Angle Entry Mode): Main Span = 90° - C2m = 90° - 6.55996° = 83.44004° Adjacent Span = 90° - C2a = 90° - 28.10911° = 61.89089° Total Deck Angle = 180° - (R3m + R3a) = 180° - (25.98281° + 43.82500°) = 110.19219° Select the following returns: 90° - P2m = 62.37281° 90° - P2a = 93.73056° The supplementary angle 180° - 93.73056° = 86.26944° is shown in the sketch. The remaining two angles on the faces under consideration are angles Q2. Enter the following values in the calculator (select Rise/Run Entry Mode): Main Span = 7 ÷ 12 Adjacent Span = 12 ÷ 12 Total Deck Angle = 90° Select the following returns: Q2m = 86.81320° The supplementary angle 180° - 86.81320° = 93.18680° is shown in the sketch. Q2a = 65.66688° More information may be extracted from the calculations to double-check the validity of the angular values. The Total Dihedral Angle measured between the faces of the square cut fascia is: 180° - (Main Span Fascia Bevel Angle + Adjacent Span Fascia Bevel Angle) 180° - (20.87256° + 48.25487°) = 110.87257° Regardless of how skewed or tilted the planes at the Hip-Valley foot meets Eaves appear they must follow the planes of the fascia. The Total Dihedral Angle between the planes may be checked as follows: Enter the following values in the calculator (select Angle Entry Mode): Main Span = 90° - C2m = 90° - 6.55996° = 83.44004° Adjacent Span = 90° - C2a = 90° - 28.10911° = 61.89089° Total Deck Angle = 180° - (R3m + R3a) = 180° - (25.98281° + 43.82500°) = 110.19219° Select the following returns: C5m = 76.07234° C5a = 6.94490° Note that the "Main Side" calculation returns a value of DD which exceeds 90°. This is significant because it indicates that the "Backing Angles" (the C5 values returned) are oriented in the same direction (that is, both angles viewed in section are measured counter-clockwise) . Therefore the calculation for the Total Dihedral Angle is: Total Dihedral Angle = 180° - C5m + C5a = 110.87256°. The values of any miter angles, bevel angles and the angle on the face created by making the Compound Angle cut may also be tested by more than one calculation. The angles in the following examples are calculated with respect to the lines of the Hip Rafter Ridge and a ridge formed by the intercepting planes of the 12 ÷ 7 and 12 ÷ 12 Reciprocal Planes (in other words, the line created by the intercepting planes perpendicular to the plane of the roof). FASCIA BEVEL ANGLE arctan (cos Fascia Miter Angle ÷ tan Angle between Eave and Hip Ridge) ANGLE between MATCHING FACES of FASCIA arccos (sin Fascia Miter Angle cos Angle between Eave and Hip Ridge) Main (7 ÷ 12) Span Angles: Main Span Fascia Miter Angle = 40.81976° (the miter angle on the wide face of the work) Angle between Eave and Hip Ridge = 63.25779° (the angle on the narrow face of the work) FASCIA BEVEL ANGLE = arctan (cos Fascia Miter Angle ÷ tan Angle between Eave and Hip Ridge) = arctan (cos 40.81976° ÷ tan 63.25779°) = 20.87256° ANGLE between MATCHING FACES of FASCIA = arccos (sin Fascia Miter Angle cos Angle between Eave and Hip Ridge) = arccos (sin 40.81976° cos 63.25779°) = 72.89392° Adjacent (12 ÷ 12) Span Angles: Adjacent Span Fascia Miter Angle = 22.41512° (the miter angle on the wide face of the work) Angle between Eave and Hip Ridge = 39.52120° (the angle on the narrow face of the work) FASCIA BEVEL ANGLE = arctan (cos Fascia Miter Angle ÷ tan Angle between Eave and Hip Ridge) = arctan (cos 22.41512° ÷ tan 39.52120°) = 48.25487° ANGLE between MATCHING FACES of FASCIA = arccos (sin Fascia Miter Angle cos Angle between Eave and Hip Ridge) = arccos (sin 22.41512° cos 39.52120°) = 72.89392° Vector Products offer another approach to solving angles. The following solution for the ANGLE between MATCHING FACES of FASCIA employs the dot product formula with the terms re-arranged. Setting the point where the Hip-Valley foot meets Eaves at (0 , 0 , 0) the co-ordinates of the endpoints of the unit vectors lying on the "ridges" are: Hip Rafter Ridge between 7 ÷ 12 and 12 ÷ 12 Planes: (.503871 , .863779 , .503871) "Ridge" between 12 ÷ 7 and 12 ÷ 12 Reciprocal Planes: (.863779 , .503871 , -.863779) Angle between "Ridges" = arccos ± ( a•b ÷ |a| |b| ) = 72.89392° (or supplementary angle 107.10608°). |
