multinail hub - trigonometry theory€¦ · dual pitch roof figure a2-08-01-01 l1 = left top chord...

Multinail Australia

Multinail Hub - Trigonometry Theory25 May 2020

Due to continual product improvement Multinail Australia Pty Ltd. reserves the right to change the product/s depicted - both in description and specification.This document has to be read in conjunction with Multinail’s Technical Manual.

Trigonometry Theory

Introduction

Trigonometry is a branch of mathematics that studies triangles, particularlyright angle triangles. Trigonometry deals with the relationship between thesides and the angles of triangles. Roof Trusses are comprised of manytriangulated members which enables them to span over larger distances.

Following are some examples of how trigonometry can be used to work outrequired pitches or heights of existing buildings so that additional Trusses canbe made. These details can also be used to confirm member lengths andangles before manufacturing Trusses.

Note that the calculator used for the following examples is based on a genericcalculator and is for demonstration purposes only. Individual calculators maydiffer in exact functionality

H = Hypotenuse = side opposite to the 90° angle.A = Adjacent Side = side next to the angle we are consideringO = Opposite side = side opposite to the angle we are considering

SINE … sin

α = arc sin ⁄

O = H sin α

H = ⁄

O H

O sin α

Figure A2-01-01-01

COSINE … cos

α = arc cos ⁄

A = H cos α

H = ⁄

A H

A cos α

Figure A2-01-01-01

TANGENT … tan

α = arc tan ⁄

O = A tan α

A = ⁄

O H

O tan α

Figure A2-01-01-01

https://www.multinail.net/technical-manual-au/estimating-detailing/a2-truss-trigonometry-theory/a2-01-01?

Multinail Australia

Multinail Hub - Bell Trusses25 May 2020


Bell Trusses

Figure A2-04-01-01

L1 - Lower Top Chord LengthL2 - Upper Top Chord LengthD1 - Lower Top Chord DepthD2 - Upper Top Chord DepthP1 - Lower Plum CutP2 - Upper Plum CutH1 - Underside Bottom Chord to Underside Lower Top ChordH2 - Underside Bottom Chord to Overall Lower Top ChordH3 - Underside Bottom Chord to Underside Upper Top ChordH4 - Underside Bottom Chord to Overall Upper Top ChordS2 - Horizontal Length of upper Top Chord


Multinail Australia

Multinail Hub - Creepers (Supported by Hip, double mitred)25 May 2020


Creepers (Supported by Hip, double mitred)

a) Bottom Chord

Figure A2-07-01-01

Figure A2-07-01-03

For 45º case only:

C = B1 / cos (45°) / 2

L1 = Station – C + B/2

On calculator use:

C B1, ÷, cos, 45, ÷, 2, =

L1 Station, -, C, +, (, B, ÷, 2, =

ie: C = 35 ÷ cos 45 ÷ 2 = 25, L1 = 1200 – 25 + (35÷2) = 1193

b) Top Chord

Figure A2-07-01-02

L = Creeper Top Chord Length

L = (Creeper Bottom Chord Length + Overhang) / cos (Roof Pitch)

On calculator use:

On calculator use: L1, +, Overhang, =, ÷, Roof Pitch, cos, =

ie: L = 1193 + 750 = ÷ cos 20 = 2068


Multinail Australia

Multinail Hub - Dual Pitch Roof25 May 2020

Dual Pitch Roof

Figure A2-08-01-01

L1 = Left Top Chord LengthL2 = Right Top Chord LengthPL,R = Left, Right Plum Cut DepthHL,R = Rise Left, Right(Height Underside Bottom Chord to Underside Top Chord)H = Overall Height, Underside Bottom Chord to Overall Top ChordeaveH = Constant - Underside of Eave to Top of Wall

Figure A2-08-01-02

Case (A)

Known:Left Pitch : 12ºRight Pitch : 20ºSpan : 8000mmLeft Overhang : 750mmStandard Upstand : 6mmTop Chord : 120 x 35

Calculating Sequence:

PL = TC / cos (Left Pitch)HL = BreakL x tan (Left Pitch) + UpstandBreakR = Span – BreakLH1 = PL + HL – UpstandAngle1 = arc tan (H1 / BreakR)Side1 = BreakR / cos (Angle1)Angle2 = arc sin (TC / Side1)Right Pitch = Angle1 – Angle2eaveH = Left Overhang x tan (Left Pitch) – UpstandPR = TC / cos (Right Pitch)Right Overhang = (eaveH + Upstand) + tan (Right Pitch)L1 = (BreakL + Left Overhang) / cos (Left Pitch)L2 = (BreakR + Right Overhang) / cos (Right Pitch)HR = BreakR x tan (Right Pitch) + UpstandH = HL + PL or HR + PR

On calculator use:

PL TC, ÷, cos, Left Pitch, =HL BreakL, x, tan, Left Pitch, +, Upstand, =BreakR Span, -, BreakL, =H1 PL, +, HL, -, Upstand, =Angle1 H1, ÷, BreakR, arc (or inv), tan, =Side1 BreakR, ÷, cos, Angle1, =Angle2 TC, ÷, Side1, =, arc (or inv), sin, =Right Pitch Angle1, -, Angle2, =eaveH Left Overhang, x, tan, Left Pitch, -, Upstand, =PR TC, ÷, cos, Right Pitch, =L1 BreakL, +, Left Overhang, =, ÷, cos, Left Pitch, =L2 BreakR, +, Right Overhang, =, ÷, cos, Right Pitch, =HR BreakR, x, tan, Right Pitch, +, Upstand, =H HL, +, PL, = or HR, +, PR, =

Example:

PL = 120 ÷ cos 12 = 123HL = 5059 x tan 12 + 6 = 1081BreakR = 8000 – 5059 = 2941H1 = 123 + 1081 – 6 = 1198Angle1 = 1198 ÷ 2941 = arc tan = 22.163°Side1 = 2941 ÷ cos 22.163 = 3176Angle2 = 120 ÷ 3176 = arc sin = 2.165°Right Pitch = 22.163 – 2.165 = 20°eaveH = 750 x tan 12 – 6 = 153PR = 120 ÷ cos 20 = 128Right Overhang = 153 + 6 = ÷ tan 20 = 437L1 = 5059 + 750 = ÷ cos 12 = 5939L2 = 2941 + 437 = ÷ cos 20 = 3595HR = 2941 x tan 20 + 6 = 1076H = 1081 + 123 =1204 or 1076 + 128 = 1204

Case (B)

Known:Left Pitch : 12ºBreakL : 5059mmSpan : 8000mmLeft Overhang : 750mmStandard Upstand : 6mmTop Chord : 120 x 35


PL = TC / cos (Left Pitch)HL = BreakL x tan (Left Pitch) + UpstandBreakR = Span - BreakLH1 = PL + HL - UpstandAngle1 = arc tan (H1 / BreakR)Side1 = BreakR / cos (Angle1)Angle2 = arc sin (TC / Side1)Right Pitch = Angle1 - Angle2eaveH = Left Overhang x tan (Left Pitch) - UpstandPR = TC / cos (Right Pitch)Right Overhang = (eaveH + Upstand) + tan (Right Pitch)L1 = (BreakL + Left Overhang) / cos (Left Pitch)L2 = (BreakR + Right Overhang) / cos (Right Pitch)HR = BreakR x tan (Right Pitch) + UpstandH = HL + PL or HR + PR

On calculator use:

PL TC, ÷, cos, Left Pitch,, =HL BreakL, x, tan, Left Pitch, +, Upstand, =BreakR Span, -, BreakL, =H1 PL, +, HL, -, Upstand, =Angle1 H1, ÷, BreakR, arc (or inv), tan, =Side1 BreakR, ÷, cos, Angle1Angle2 TC, ÷, Side1, arc (or inv,) sin, =Right Pitch Angle1, -, Angle2, =eaveH Left Overhang, x, tan, Left Pitch, -, Upstand, =PR TC, ÷, cos, Right Pitch, =Right Overhang eaveH, +, Upstand, =, ÷,tan, Right Pitch, =L1 BreakL, +, Left Overhang, =, ÷, cos, Left Pitch,, =L2 BreakR, +, Right Overhang, =, ÷, cos, Right Pitch,, =HR BreakR, x, tan, Right Pitch, +, Upstand, =H HL, +, PL, = or HR, +, PR, =

Example:

PL = 120 ÷ cos 12 = 123HL = 5059 x tan 12 + 6 = 1081BreakR = 8000 - 5059 = 2941H1 = 123 + 1081 - 6 = 1198Angle1 = 1198 ÷ 2941 = arc tan = 22.163ºSide1 = 2941 ÷ cos (22.163) = 3176Angle2 = 120 ÷ 3176 = arc sin = 2.165ºRight Pitch = 22.163 - 2.165 = 19.998 = 20ºeaveH = 750 x tan 12 - 6 = 153PR = 120 ÷ cos 20 = 128Right Overhang = 153 + 6 = ÷ tan 20 = 437L1 = 5059 + 750 = ÷ cos 12 = 5939L2 = 2941 + 437 = ÷ cos 20 = 3595HR = 2941 x tan 20 + 6 = 1076H = 1081 + 123 = 1204 or 1076 + 128 = 1204


Case (C)

Left Break (BreakL) : 5059mmRight Break (BreakR) : 2941mmSpan : 8000mmOverall Height (H) : 1204mmStandard Upstand : 6mmLeft Overhang : 750mmTop Chord : 120 x 35


H1 = H - UpstandAngle1L = arc tan (H1 / BreakL)Angle1R = arc tan (H1 / BreakR)Side1L = BreakL / cos (Angle1L)Side1R = BreakR / cos (Angle1R)Angle2L = arc sin (TC / Side1L)Angle2R = arc sin (TC / Side1R)Left Pitch = Angle1L – Angle2LRight Pitch = Angle1R - Angle2RPL = TC / cos (Left Pitch)PR = TC / cos (Right Pitch)HL = H - PLHR = H - PReaveH = Left Overhang x tan (Left Pitch) – UpstandRight Overhang = (eaveH + Upstand) / tan (Right Pitch)L1 = (BreakL + Left Overhang) / cos (Left Pitch)L2 = (BreakR + Right Overhang) / cos (Right Pitch)

On calculator use:

H1 H, -, Upstand, =Angle1L H1, ÷, BreakL, =, arc (or inv), tan, =Angle1R H1, ÷, BreakR, =, arc (or inv), tan, =Side1L BreakL, ÷, cos, Angle1L, =Side1R BreakR, ÷, cos, Angle1R, =Angle2L TC, ÷, Side1L, =, arc (or inv), sin, =Angle2R TC, ÷, Side1R, =, arc (or inv), sin, =Left Pitch Angle1L, -, Angle2L, =Right Pitch Angle1R, -, Angle2R, =PL TC, ÷, cos, Left Pitch, =PR TC, ÷, cos, Right Pitch, =HL H, -, PL, =HR H, -, PR, =eaveH Left Overhang, x, tan, Left Pitch, -, Upstand, =Right Overhang eaveH, +, Upstand, =, ÷, tan, Right Pitch, =L1 BreakL, +, Left Overhang, =, ÷, cos, Left Pitch, =L2 BreakR, +, Right Overhang, =, ÷, cos, Right Pitch, =

Example:

H1 = 1204 – 6 = 1198Angle1L = 1198 ÷ 5059 = arc tan = 13.323°Angle1R = 1198 ÷ 2941 = arc tan = 22.163°Side1L = 5059 ÷ cos 13.323 = 5199Side1R = 2941 ÷ cos 22.163 = 3176Angle2L = 120 ÷ 5199 = arc sin = 1.323°Angle2R = 120 ÷ 3176 = arc sin = 2.165°Left Pitch = 13.323 – 1.323 = 12°Right Pitch = 22.163 – 2.165 = 20°PL = 120 ÷ cos 12 = 123PR = 120 ÷ cos 20 = 128HL = 1204 – 123 = 1081HR = 1204 – 128 = 1076eaveH = 750 x tan 12 – 6 = 153Right Overhang = 153 + 6 = ÷ tan 20 = 437L1 = 5059 + 750 = ÷ cos 12 = 5939L2 = 2941 + 437 = ÷ cos 20 = 3595


Multinail Australia

Multinail Hub - Hip Rafter (case 45˚ only)25 May 2020


Hip Rafter (case 45˚ only)

Figure A2-05-01-01

L - Raking Hip Rafter Length = L3/Cos αL3 - Horizontal Hip Rafter Length = L1 + L2L2 - Hip Span = (Span / 2) / cos (45°)L1 - Hip Overhang = Overhang / cos (45°)α - Hip Pitch = arc tan (tan (Roof Pitch) / √2)

On calculator use:

L1 span, ÷, 2, ÷, cos, 45, =L2 Overhang, ÷, cos, 45, =L3 L1, +, L2, =α tan, Roof Pitch, ÷, 2, √, =, inv, tanL L3, ÷, cos, α, =

ie:

L1 = 8000 ÷ 2 ÷ cos 45 = 5657L2 = 750 ÷ cos 45 = 1061L3 = 5657 + 1061 = 6718α = tan 20 ÷ 2 √ = inv tan = 14.43°L = 6718 ÷ cos 14.43 = 6937

Figure A2-05-01-02

Figure A2-05-01-03


Multinail Australia

Multinail Hub - Jacks (Perpendicular to Trusses, double mitred)25 May 2020


Jacks (Perpendicular to Trusses, double mitred)

Figure A2-06-01-01

L - Raking Hip Rafter Length = L3/Cos αL3 - Horizontal Hip Rafter Length = L1 + L2L2 - Hip Span = (Span / 2) / cos (45°)L1 - Hip Overhang = Overhang / cos (45°)α - Hip Pitch = arc tan (tan (Roof Pitch) / √2)

On calculator use:

L1 span, ÷, 2, ÷, cos, 45, =L2 Overhang, ÷, cos, 45, =L3 L1, +, L2, =α tan, Roof Pitch, ÷, 2, √, =, inv, tanL L3, ÷, cos, α, =

ie:

L1 = 8000 ÷ 2 ÷ cos 45 = 5657L2 = 750 ÷ cos 45 = 1061L3 = 5657 + 1061 = 6718α = tan 20 ÷ 2 √ = inv tan = 14.43°L = 6718 ÷ cos 14.43 = 6937

Figure A2-05-01-02

Figure A2-06-01-03


Multinail Australia

Multinail Hub - Standard Truss - Member Lengths25 May 2020


Standard Truss - Member Lengths

Figure A2-02-01-01

Top Chord Length

L = (Half Span + Overhang) / cos (Roof Pitch)

On calculator use: Half Span, +, Overhang, = , ÷, cos, Roof Pitch, =

ie: L = 4000 + 750 = ÷ cos 20 = 5055

Rise (Underside of Bottom Chord to Underside of Top Chord)

H = Half Span x tan (Roof Pitch) + Upstand

On calculator use: Half Span, x, tan, Roof Pitch, + Upstand, =

ie: H = 4000 x tan 20 + 6 = 1462

Plumb Cut Depth

PCD = TC / cos (Roof Pitch)

On calculator use: TC, ÷, cos, Roof Pitch,, =

ie: PCD = 90 ÷ cos 20 = 96

Overall Height (Underside of Bottom Chord to Overall TopChord)

H1 = H + PCD

On calculator use: H, +, PCD, =

ie: H1 = 1462 + 96 = 1558

Heel Cut Horizontal Distance

HX = (BC – Upstand) / tan (Roof Pitch)

On calculator use: BC, -, Upstand, =, ÷, tan, Roof Pitch, =

ie: HX = 90 – 6 = ÷ tan 20 = 231

Heel Cut Length

HL = (BC – Upstand) / Sin (Roof Pitch)

On calculator use: BC, -. Upstand, =, ÷, Sin, Roof pitch, =

ie: HL = 90 – 6 = ÷ sin 20 = 246


Multinail Australia

Multinail Hub - Truncated Trusses25 May 2020


Truncated Trusses

Figure A2-03-01-01

Truncated Truss Depth

D = Station x tan (Roof Pitch) + Upstand

On calculator use: Station, x, tan, Roof Pitch, +, Upstand, =

ie: D = 2400 x tan 15 + 6 = 649

Jack Rafter Depth at Station Point

HR = TC / cos (Roof Pitch)

On calculator use: TC, ÷, cos, Roof Pitch,, =

ie: HR = 90 ÷ cos 15 = 93

https://www.multinail.net/technical-manual-au/estimating-detailing/a2-truss-trigonometry-theory/a2-03-00/a2-03-01?

Multinail Australia

Multinail Hub - Truncated Bell Truss25 May 2020


Truncated Bell Truss

Figure A2-03-06-01

Calculating Sequence

HR = Rafter Plumb Depth / cos (Upper Pitch)L1 = (Overhang + Break) / cos (Lower Pitch)A = Rafter Plumb Depth / tan (Upper Pitch)S2 = Station – Break - AL2 = S2 / cos (Upper Pitch)L3 = Span – 2 x (Station – A)PL1 = Lower Top Chord Depth / cos (Lower Pitch)H1 = Break x tan (Lower Pitch) + S2 x tan (Upper Pitch)+ Upstand + PL1

Examples:

HR = 120 ÷ cos 20 = 128L1 = 750 + 1000 = ÷ cos 12 = 1789A = 128 ÷ tan 20 = 352S2 2400 – 1000 – 352 = 1048L2 1048 ÷ cos 20 = 1116L3 8000 – 2 x (2400 – 352) = 3904PL1 90 ÷ cos 12 = 92H1 1000 x tan 12 + 1048 x tan 20 + 6 + 92 = 692

On calculator use:

HR Rafter Plumb Depth, ÷, cos, Upper Pitch, =L1 Overhang, +, Break, =, ÷, cos, Lower Pitch, =A Rafter Plumb Depth, ÷, tan, Upper Pitch, =S2 Station, -, Break, -, A, =L2 S2, ÷, cos, Upper Pitch, =L3 Span, -, 2, x, (, Station, -, A), =PL1 Lower Top Chord Depth, ÷, cos, Lower Pitch, =H1 Break, x, tan, Lower Pitch, =, +, (, S2, x, tan, Upper Pitch, =,+, Upstand, +, PL1, =


Multinail Australia

Multinail Hub - Truncated Trusses - Type A25 May 2020


Truncated Trusses - Type A

Figure A2-03-02-01

A = Jack Rafter Plumb Depth at Station Point / tan (Roof Pitch)

On calculator use: Jack Rafter Depth at Station Point, ÷, tan, Roof Pitch, =

ie: A = 93 ÷ tan 15 = 347

B = Station - A

On calculator use: Station, -, A, =

ie: B = 2400 – 347 = 2053

L1 = Span - 2 x B

On calculator use: Span, -, 2, x, B, =

ie: L1 = 8000 – 2 x 2053 = 3894

L = (B + Overhang) / cos (Roof Pitch)

On calculator use: B, +, Overhang, =, ÷, cos, Roof Pitch,, =

ie: L = 2053 + 750 = ÷ cos 15 = 2902



ie: D = 2400 x tan 15 + 6 = 649


Multinail Australia

Multinail Hub - Truncated Trusses - Type B & C25 May 2020


Truncated Trusses - Type B & C

Truncated Truss – Plan View

Figure A2-03-03-01

Detail

Figure A2-03-03-02

Figure A2-03-03-03

C = B1 / 2 / cos (45)

On calculator use: B, ÷, 2, ÷, cos, =

ie: A = 35 ÷ 2 ÷ cos 45 = 25

Lx = Truncated Station – C

On calculator use: Station, -, C, =

ie: Lx = 2400 – 25 = 2375


Multinail Australia

Multinail Hub - Truncated Trusses - Type B25 May 2020


Truncated Trusses - Type B

Figure A2-03-04-01

A = Horizontal Top Chord Depth / 2 / tan (Roof Pitch)

On calculator use: Horizontal Top Chord Depth, ÷, 2, ÷, tan, Roof Pitch, =

ie: A = 90 ÷ 2 ÷ tan 15 = 168

B = Station – A


ie: B = 2400 – 168 = 2232

L1 = Span – 2 x B


Ie: L1 = 8000 – 2 x 2232 = 3536

C = B1 / 2 / cos (45)

On calculator use: B, ÷, 2, ÷, cos, =

ie: A = 35 ÷ 2 ÷ cos 45 = 25

L = Station + Overhang – C / Cos (Roof Pitch)

On calculator use: Station, +, Overhang, -, C, =. ÷, Cos, Roof Pitch, =

ie: L = 2400 + 750 – 25 = ÷ Cos 15 = 3235



ie: D = 2400 x tan 15 + 6 = 649


Multinail Australia

Multinail Hub - Truncated Trusses - Type C25 May 2020


Truncated Trusses - Type C

Figure A2-03-05-01

A = Horizontal Top Chord Depth / 2 / tan (Roof Pitch)

On calculator use: TC, ÷, tan, Roof Pitch, =

ie: A = 90 ÷ tan 15 = 336

B = Station – A


Ie: B = 2400 – 336 = 2064

L1 = Span – 2 x B


ie: L1 = 8000 – 2 x 2064 = 3872

L = Station + Overhang – C / Cos (Roof Pitch)

On calculator use: Station, +, Overhang, -, C, =. ÷, Cos, Roof Pitch, =

ie: L = 2400 + 750 – 25 = ÷ Cos 15 = 3235



ie: D = 2400 x tan 15 + 6 = 649


multinail hub - trigonometry theory€¦ · dual pitch roof figure a2-08-01-01 l1 = left top chord...

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