Section Downloads Download & Print TTT I Sec 04 Slides TTT I Sec 04 Handouts Course binders are available for purchase Not required Version.1 Section 04: Truss Math Before You Start Get a scientific calculator - one that has SIN, COS, and TAN functions. Section 04 -Truss Math involves formula calculations using arithmetic and trigonometry. Get some paper and a pencil or pen. Research ARITHMETIC & TRIGONOMETRY online for additional help. 3 Truss Math Outline Order of Operations Dimensions Fraction to Decimal Feet Conversion Decimal Feet to Fraction Conversion Trigonometry Right Angle Triangles Labeling Right Angle Triangles Trigonometric Equations Trig Functions Solving Right Angle Triangles Pythagorean Theorem Pitch Triangles Right Angle Triangles Properties Pitch Table Heel Height Calculations Plumb Cut Conventional Rafter Heels Truss Geometry Problems 4 Truss Math Outline Presentation Practice Problems Examples Answers Quizzes TTT I Sec 04 Handouts Operation Division Multiplication Math Symbols a b a x b Symbol a/b (a)(b) a b a b a(b) Addition a + b Subtraction a - b 5 6 014 SBCA 1
Math Symbols square root symbol exponent or power notation xxxx x or x 3 Order of Operations Consider the following equation: 1 + x 3-4 5 + 6 nd SHIFT 7 8 Incorrect Order of Operations If you calculate the operations in the order they appear, left to right, then you will get the following answer: 1 + x 3-4 5 + 6 1 + = 3 3 x 3 = 9 9-4 = 5 5 5 = 1 1 + 6 = 7 Correct Order of Operations The order of operations gives precedence to division and multiplication, moving from left to right: 1 + ( x 3) (4 5) + 6 1 + (6) (0.8) + 6 7 0.8 + 6 6. + 6 = 1. 9 10 Order of Operations (1 + ) x 3-4 (5 + 6) (3) x 3-4 (11) 9 0.3636 = 8.6364 Order of Operations 1. Solve operations inside of parenthesis.. Solve for the exponent (n x ) or square root. 3. Solve multiplication &/or division from L to R. 4. Solve addition &/or subtraction from L to R. One way to remember the order is: Please Excuse My Dear Aunt Sally. parenthesis multiplication or division exponents (or roots) addition or subtraction 11 1 014 SBCA
Dimensions Quiz 1 The truss industry uses a unique fractional 14-0-0 14-0-0 notation 5-5-1 that represents 4-4-0 4--4 4--4 dimensions 4-4-0 5-5-1 with feet, 6-8 inches & sixteenths of 5an inch: ft in sx 7-10- 0-10-0 4-5 4-5 18.5 inches 3 would be represented 7 as: 1-4-0 6 1 4-5 4-5 1 4 6 1 15 14 11 10 10-14 3-7 6-8 8-1 10-10 3-7 1 13 61 8 8-1 4-5-1 or 4-5-1 8-0-0 10608 5-4-0 4--4 4--4 5-4-0 500 LB 6 8 9 10-14 1-4-0 0-10-0 7-10- 14 Dimensions Construction Calculators can perform calculations in ft in sx. All dimensions will have to be converted to decimal feet notation. 1 6 8 = 1.5417 ft. Take answers out to 4 decimal places (0.0000). Rounding errors may occur. Fraction to Decimal Feet Conv. The steps to convert ft-in-sixteenths to decimal feet. 1 6 8 8/ or 8 = 0.5 in. 6 + 0.5 = 6.5 in. 6.5 in. 0.5417 ft. 1 in./ft. 1 + 0.5417 = 1.5417 ft. 15 Decimal Feet to Fraction Conv. Quiz 17.3581 ft. 17 in sx 0.3581 ft. x 1 17 4 sx 0.97 in. x in. ft. = 4.97 in. sixteenths in. = 4.755 sixteenths 0.755 is more than 0.5 so 4.755 becomes 5 17 4 5 18 014 SBCA 3
Angles Quiz 3 90 90 90 90 0 Right Triangles Σ angles = 90 + angle A + angle B = 180 angle A + angle B = 180-90 angle A + angle B = 90 Labeling Right Triangles The side directly across from the 90 right angle is always the longest side, it is called the otenuse. Tangent Sine Cosine Hypotenuse 90 1 Labeling Right Triangles The side of the triangle directly across from θ is called the osite side. The side that intersects with the otenuse to make the angle θ is called the acent side. Labeling Right Triangles Hypotenuse Opposite 90 Adjacent 3 4 014 SBCA 4
Trigonometric Equations Quiz 4 Tan θ Sin θ 6 Trigonometric Functions The Tangent function is represented on your calculator as a button. To find the tangent of 3, enter 3 into your calculator & hit the TAN button. The answer should be 0.649. Since it is the ratio there are no dimension units to this number. 7 Different Answer?...Try This Make sure that the calculator is in degree mode. The radian or gradient modes will produce the wrong answers. You may have a button marked DRG to control this setting. Reverse the information input to the calculator. Some calculators require the TAN button to be pressed before you enter the number. You may also have to hit the enter key to get the final answer. 8 Trigonometric Functions Examples: Tan 1 = 0.3839 Tan 58 = 1.6003 Sin 44 = 0.6947 Sin 36 = 0.5878 Cos 15 = 0.9659 Cos 6 = 0.4695 Inverse Trigonometric Functions Tan θ 0.8430 Tan θ = 0.8430 Sin θ 9 30 014 SBCA 5
Inverse Trigonometric Functions Tan θ = 0.8430 Tan -1 Tan -1 (Tan θ) = Tan -1 (0.8430) nd Quiz 5 θ = Tan -1 (0.8430) SHIFT θ = 40.13 31 Trigonometric Equations Solving Right Triangles Tan θ Sin θ Tan θ EXAMPLE: If you knew the otenuse and θ of a right-angle triangle, and you wanted to calculate the acent side which equation above would you use? Sin θ 33 34 Solving Right Triangle Examples Example problems have been included in the presentation - each followed by the solution. When Example 1 displays, stop the course with the PAUSE button. Solve the example on scratch paper. Use the NEXT button to continue to the solution & repeat through the examples. Example 1 GIVEN: acent = 14 ft. & osite = 8 ft. FIND: The value of θ TTT I Sec 04 Handouts 35 36 014 SBCA 6
Answer 1 GIVEN: acent = 14 ft. & osite = 8 ft. FIND: The value of θ Example GIVEN: otenuse = 3 ft. & osite = 9 ft. FIND: The value of θ This answer involves acent, osite and θ. The trig equation that applies is the tangent equation: Substitute the two known values into the formula: Perform the division calculation: Tan θ 0.5714 Calculate θ: θ = Tan -1 (0.5714) = 9.744 Tan θ Tan θ 8 14 37 38 Answer GIVEN: otenuse = 3 ft. & osite = 9 ft. FIND: The value of θ Example 3 GIVEN: θ = 37 & otenuse = 17 ft. FIND: The value of acent This answer involves otenuse, osite and θ. The only Trig equation that applies is the sine equation: Substitute the two known values: Perform the division calculation: Calculate θ: θ Sin Sin θ 9 3 Sin θ 0.3913 0.3913 3.04-1 Sin θ 39 40 Answer 3 GIVEN: θ = 37 & otenuse = 17 ft. FIND: The value of acent Example 4 GIVEN: θ = 4 & otenuse = 15.7 ft. FIND: The value of osite This answer involves θ, otenuse and acent. The only Trig equation that applies is the cosine equation: Substitute the known values: Cos 37 17 Isolate the 'acent' term on one side of the equation by multiplying both sides of the equation by 17: Cos 37 17 17 17 The resulting equation is: 17Cos 37 or 17Cos 37 Calculate acent: 17 0.7986355 13.5768 feet or 13-6 - 15 41 4 014 SBCA 7
Answer 4 GIVEN: θ = 4 & otenuse = 15.7 ft. FIND: The value of osite Example 5 GIVEN: θ = 15 & osite = 8.6 ft. FIND: The value of acent This answer involves θ, otenuse and osite. The only Trig formula that applies is the sine equation: Substitute the known values: Sin 4 15.7 Sin θ Isolate the 'osite' term on one side of the equation by multiplying both sides of the equation by 15.7: Sin 4 15.7 15.7 15.7 The resulting equation is: 15.7Sin 4 or 15.7Sin 4 Calculate: 15.7 0.4067366 6.3858 ft 6-4 - 10 43 44 Answer 5 GIVEN: θ = 15 & osite = 8.6 ft. FIND: The value of acent This answer involves θ, osite and acent. The only Trig formula that applies is the tangent equation: Tan θ Substitute the known values: 8.6 Tan15 Isolate the 'acent' term on one side of the equation by multiplying both 8.6 sides by acent: Tan15 or Tan15 8. 6 Divide both sides by Tan 15: Tan15 8.6 Tan15 Tan15 8.6 The resulting equation is: Tan15 Calculate: 8.6 3.0956 ft 3 1 0.679491 45 Example 6 GIVEN: θ = 44 & acent = 1.7 ft. FIND: The value of otenuse 46 Answer 6 GIVEN: θ = 44 & acent = 1.7 ft. FIND: The value of otenuse This answer involves θ, acent and otenuse. The only Trig formula that applies is the cosine equation: 1.7 Substitute the known values: Cos 44 Isolate the 'otenuse' term on one side of the equation by multiplying both sides by : 1.7 Cos 44 or Cos 44 1. 7 Divide both sides by Cos 44 : Cos 44 1.7 Cos 44 Cos 44 1.7 The resulting equation is: Cos 44 Calculate: 1.7 30.65 30 0 0.7193398 47 Quiz 6 014 SBCA 8
No Angle...Only Sides GIVEN: acent =.7 ft. & osite = 11.3 ft. FIND: The value of otenuse All previous calculations for right-angle triangles have involved θ. No θ! Tan θ Sin θ a + b = c c a Pythagorean Theorem b a + b = c PYTHAGOREAN THOEREM 3 =3 4 =4 5 =5 = 5 3 4 5 49 50 Pythagorean Theorem In terms of the right-angle triangle: Pythagorean Theorem GIVEN: acent =.7 ft. & osite = 11.3 ft. FIND: The value of otenuse 11.3.7 17.69 78.89 406.58 = 0.38 ft. = 0-1-15 51 5 Pitch Triangles Quiz 7 Tan θ 6 Sin θ θ 54 014 SBCA 9
Pitch Triangles Pitch Triangles Adjacent Opposite Hypotenuse Slope Diagonal Tan θ P P 5 Tan θ Tan θ 0.47 1 1 1 θ Tan 1.47.6 55 56 Right Triangle Properties Quiz 8 TTT I Sec 04 Handouts Pitch Table (Blank) Filling In the Pitch Table TTT I Sec 04 Handouts 1 P 1 θ Tan Tan 1 5 1 1 Tan 0.47.6 Now simply use the trig function keys to fill in the remaining: Tan(.6) = 0.47 Sin (.6) = 0.3846 Cos(.6) = 0.931 60 014 SBCA 10
Filling In the Pitch Table Pitch Table (Answers) θ Sin 1 0.3846.6 Using θ, fill in the rest of the trig values: = Cos.6 = 0.931 Tan θ = Tan.6 = 0.46 P 1 Tan θ 1(0.46) 5 61 Truss Heels Standard Heel/ Raised Heel Heel Heights Plumb Cuts Section 0 - Terminology 63 64 Heel Height Calculations Heel Height Calculations = 1 / 4 in. or 3 / 8 in. heel height = plumb cut + butt cut 65 66 014 SBCA 11
Plumb Cut Plumb Cut 67 68 Plumb Cut Plumb Cut Tan θ Sin θ lumber plumb lumber Plumb θ,, &? 69 70 Heel Height heel height = plumb cut + butt cut Quiz 9 lumber heel butt cut Usually know: TC pitch, TC lumber & butt cut Recall the formula to calculate θ: θ Tan 1 P 1 7 014 SBCA 1
Conventional (Rafter) Heels Quiz 10 74 Conventional (Rafter) Heels Conventional (Rafter) Heels Tan θ Tan θ y seat y seat Tan θ rafter heel = plumb - y lumber rafter heel seattan θ rafter heel height = plumb - y 75 76 Math Problems Quiz 11 The rest of this Section will be presented in the following format: 1. Formula - equation page TTT I Sec 04 Handouts. Problem - with the given information TTT I Sec 04 Handouts 1. Answer - displayed on screen only 78 014 SBCA 13
Math Problems When the Problem displays, stop the course with the PAUSE button. Solve the example on scratch paper. Use the NEXT button to continue to the solution & repeat through the examples. 79 Math Problems Outline Truss Heel Conventional Heel Scarf Length Stub Heel Upper Heel Pitch Fascia (Find Heel) Fascia (Matching) Dual Pitch Shed Truss Shed Heel TTT I Sec 04 Handouts 80 Truss Heel Formula Truss Heel Problem Lumber Heel Butt Heel = 8 Truss Heel Answer Conventional Heel Formula Tan θ 8 1 1 8 θ Tan 1 θ 33.69 Lumber Heel Butt 3.50 Heel 1 Cos 33.69 Heel 4.06 1 3 Heel 5.06 5 0 5 3 Lumber Heel Seat Tan θ 014 SBCA 14
Conventional Heel Problem Conventional Heel Answer Tan θ 7 1 1 7 θ Tan 1 θ 30.56 Lumber Heel Seat Tan θ 9.5 7 Heel 5.50 Cos 30.56 1 Heel 10.708 3.08 1 Heel 7.500 in 7 in 0 7 8 85 Scarf Length Formula Scarf Length Problem For Flat Bottom Chords For Sloped Bottom Chords BCLumber - Butt BCLumber Scarf - - Seat Tan θ Butt Tan θ 1 Scissor Scarf Tan θ Tan θ Remember 1 Scarf = Scissors Scarf = P1 P Tan θ1 and Tan θ 1 1 88 Scarf Length Answer Stub Heel Formula Flat Bottom Chord Sloped Bottom Chord Scarf Scarf 3.75 Scarf BCLumber - Butt Tan θ 1 3.50-1.50 8 0.667 1 inches 3 3 4 inches or 0-3 - 1 Scissor Scarf BCLumber - Butt - Seat Tan θ Scissor Scarf Tan θ1 Tan θ 3.5 4-1- 4 Cos 18.43 1 Scissor Scarf 8 4 1 1 3.689 1 0.333 1.33 1.356 0.333 4.068 in 4 1 in or 0-4 - 1 H Stub Tan θ H1 P H Stub H1 1 014 SBCA 15
Stub Heel Problem Stub Heel Answer H Stub Tan θ H1 P 8 H Stub H1 89.5 4.437 1 1 15 H 59.50 4.437 63.937 in 5 ft 3 5 3 15 91 Upper Heel Formula Upper Heel Problem This formula assumes elevations at the base of walls 1 and are the same. Same Units! P H W1 H1 D W FloorJoist Decking W3 1 H = 94 Upper Heel Answer Pitch Formula P H W1 H1 D W FloorJoist Decking W3 1 8 H 109.15 14.5 19 97.15 9.5 0.75 97.15 1 H 109.15 14.5 18 97.15 9.5 0.75 97.15 47.375 in 3-11- 6 P 1 OAHeight Heel Rise 1 P 1 D1 D Seat 014 SBCA
Pitch Problem Pitch Answer P 1 = P = 97 P 1 P 1 OAHeight Heel 1 D1 131.565 1.065 1 8.34 159.00 P P Rise 1 D Seat 59.5 1 3.17 30.375 6.375 Fascia (Find Heel) Formula Fascia (Find Heel) Problem P Heel Overhang FasciaWidth Fascia Re veal Frieze 14. 5 1 Heel = 100 Fascia (Find Heel) Answer Fascia (Matching) Formula If H calculates to a negative number, ustments must be made. Bearing Elevations (W1 = W) and OHs (OH1 = OH) are the same: P P1OHFW H H1 1 Heel Heel P Overhang FasciaWidth Fascia Re veal Frieze 14. 5 1 8 0.75 5.5 0.65 7.5 14.5 9.04 in 0 9 1 1 Bearing Elevations (W1 = W) are the same but OHs (OH1 OH) are different: Both Bearing Elevations (W1 W) and OHs (OH1 OH) are different: P1 P H1 OH1 FW1 OH FW 1 1 H P1 P W1 H1 OH1 FW1 OH FW W 1 1 H 014 SBCA 17
Fascia (Matching) Problem 1 Fascia (Matching) Answer 1 P POH FW 17.8 8 0.75 H1 9 1 1 H 1 103 4.15 11 H 9.68 1 ft 10 110 11 1 Fascia (Matching) Problem Fascia (Matching) Answer 105 P1 P 6 10 H1 OH1 FW1 OH FW 6.5 0.75 4 0.75 1 1 1 1 1 H 6.5 8.375 0.65 18.50 1 ft 6 in1 68 H Fascia (Matching) Problem 3 Fascia (Matching) Answer 3 107 P1 P W1 H1 OH1 FW1 OH FW W 1 1 H 7.5 5 H 117.158 0.75 1 0.75 97. 15 1 1 14 H 117.15 810.4685.31597.15.8441 ft10 11014 014 SBCA 18
Dual Pitch Formula Dual Pitch Problem P1 Span H1 H 1 D P1 P 1 1 D1 = Span D D = D1 = 110 Dual Pitch Answer Shed Truss Formula P1 8 Span H1 H 348 14.6875 1 1 D P1 P 8 3 1 1 1 1 39.31 1 D 61.068 in1 ft 9 in1 91 0.97 D1 = Span D D1 = 348 61.068 = 86.93 in. 15 D1 7 ft in715 P1 OAHT D1 H1 1 P3 D H OAHT 1 D P3 P 1 1 D3 = D D Shed Truss Problem Shed Truss Answer D = D3 = 113 P1 8 OAHT D1 H1 156 4.438 108.438 in 1 1 P3 4 D H OAHT 19 4.69 108.44 1 1 39.750 6 D 95.40 in 7 ft 11 711 6 P3 P 4 9 0.47 1 1 1 1 10 D3 D D 19 95.40 96.60 in 8 ft 8 0 10 014 SBCA 19
Shed Heel Formula Shed Heel Problem P D1 H1 H 1 D P P1 1 1 1 Shed Heel Answer P D1 H1 H 1 D P P1 1 1 14 5.5 10.5 1 D 14 8 1 1 Quiz 1 13.667 D 7.3333 in 0.50 5 ft 3 in - 3-5 Feedback 014 SBCA 0