CE 331, Spring 2004 Beam Analogy for Designing Trusses 1 / 9 We need to make several decisions in designing trusses. First, we need to choose a truss. Then we need to determine the height of the truss and the member sizes. Although we will likely use a computer program to select the final truss height and member sizes, we need to make initial estimates of these parameters for our first analysis. We can make better decisions about truss design if we understand the patterns of internal forces (bar forces) caused by the external loads on trusses. The "beam analogy" is a conceptual tool for understanding how forces are distributed through a truss. The beam analogy works best with parallel chord trusses (horizontal top and bottom chords) but still provides insight for other types of trusses. You will derive the simple equations and procedures for using the beam analogy by working through an example. You will perform the following steps: page 1. Calculate chord and diagonal forces in the 1 st, 2 nd and 3 rd panels 2 2. Draw the shear and moment diagram for the analogous beam 4 3. Develop equations to calculate the chord and diagonal forces in any panel 5 4. Write a procedure for spot-checking the RISA-generated bar forces 6 5. Calculate preliminary chord and diagonal bar sizes 6 6. Write a procedure for calculating the preliminary chord and diagonal sizes 8 7. Discuss efficient truss shapes for given loads 9 Example: 5-panel parallel-chord truss. 5 @ 8 ft F yield = 36 ksi F tension = 21 ksi F comp. = 10 ksi A 5 k 5 k 5 k 5 k B C 6 ft A B C 5 k 5 k 5 k 5 k RISA bar forces 6.064 k C 16.311 k C 19.628 k C 7.580 k T 3.722 k T 0.465 k C 9.087 k C 7.269 k T 4.611 k C 17.022 k T 0.465 k C 20.372 k T 10 k 10 k
CE 331, Spring 2004 Beam Analogy for Designing Trusses 2 / 9 1. Calculate the sum of the chord forces and the sum of the diagonal forces for each panel. Mid-Panel #1. Cut the truss at Section A-A to write an equation for the chord forces and an equation for the diagonal forces in the first panel. Chord Forces. Write an equilibrium equation for the moments about the middle of the first panel where the diagonals cross (point a ). Assume chord forces have about equal magnitude. f Then top chord must be in compression and bottom top a f chord must be in tension for equilibrium. 1 f 2 Σ M a = 0, (-10 k f )(4 ) + f top (3 ) + f bot (3 ) = 0 bot (f top + f bot )(3 ) = 40 k-ft f top + f bot = 13.3 10 k k Diagonal Forces. Sum forces in the vertical direction. Assume the diagonal forces have about equal magnitude. Then f 1 must be compressive and f 2 must be tensile. Σ F v = 0, 10 k 10 f 1_v f 2_v = 0 6 (6/10) f 1 + (6/10) f 2 = 10 k f 1 + f 2 = 16.67 k 8 Mid-Panel #2. Cut the truss at Section B-B. Chord Forces. 5 k a f 1 f top f 2 f bot 10 k Σ M a = 0, -(10 k )(12 ) + (5 k )(4 ) + f top (3 ) + f bot (3 ) = 0 (f top + f bot )(3 ) = 100 k-ft f top + f bot = 33.3 k Diagonal Forces. Σ F v = 0, 10 k 5 k f 1_v f 2_v = 0 (6/10) f 1 + (6/10) f 2 = 5 k f 1 + f 2 = 8.33 k 5 k 5 k a f 1 f top f 2 f bot
CE 331, Spring 2004 Beam Analogy for Designing Trusses 3 / 9 Mid-Panel #3. Cut the truss at Section C-C. Chord Forces. Σ M a = 0, -(10 k )(20 ) + (5 k )(12 ) + (5 k )(4 ) + f top (3 ) + f bot (3 ) = 0 (f top + f bot )(3 ) = 120 k-ft f top + f bot = 40.0 k Diagonal Forces. Σ F v = 0, 10 k 5 k -5 k f 1_v f 2_v = 0 (6/10) f 1 + (6/10) f 2 = 0 k f 1 + f 2 = 0 k
CE 331, Spring 2004 Beam Analogy for Designing Trusses 4 / 9 2. Draw the shear and moment diagram for a beam with the same span and loading as the truss (an analogous beam). Indicate the moment and the shear at each of the midpanel points. 5 @ 8 ft A 5 k 5 k 5 k 5 k B C 6 ft A B C 5 k 5 k 5 k 5 k 10 k 10 k 10 k 10 k 5 k V, k M, k-ft k-ft 120 80 k-ft 0 k 40 k-ft 100 k-ft 120-5 k -10 k k-ft M drawn on compression side of beam
CE 331, Spring 2004 Beam Analogy for Designing Trusses 5 / 9 3. Develop general equations for the chord and diagonal forces in any panel. Compare the equations you wrote for each panel in Step 1 with the shear and moment diagrams from Step 2 and the RISA-generated bar forces from page 1. Fill in the tables below and write general equations for the chord and for the diagonal forces. Loads f top h a f 1 f 2 f bot L panel Reaction Chord Forces. Panel 1 f top + f bot from Step 1 13.3 k f top + f bot from RISA (Page 1) 6.06 k + 7.27 k = 13.3 k M from moment diagram 40.0 k-ft 2 3 33.3 k 40.0 k 16.31 k + 17.02 k = 33.3 k 19.63 k + 20.37 k = 40.0 k 100.0 k-ft 120.0 k-ft (f top + f bot )(h/2) = M ext where M ext = moment due to external forces (loads & reactions) Diagonal Forces. Panel 1 f 1 + f 2 from Step 1 16.67 k f 1 + f 2 from RISA (Page 1) 7.580 k + 9.087 k = 16.67 k V from shear diagram 10.0 k 2 3 8.33 k 0.0 k 3.722 k + 4.611 k = 8.333 k -0.465 k + 0.465 k = 0.0 k 5.0 k 0.0 k (f 1 + f 2 )(h/l diag ) = V ext where V ext = shear due to external forces
CE 331, Spring 2004 Beam Analogy for Designing Trusses 6 / 9 4. Write a procedure to check the RISA output. Use sentences and equations to carefully describe how to check the RISA output for: a. the critical chord force 1. Find the max. mid-panel moment for the analogous beam. 1.1 Draw the moment diagram for the analogous beam due to the loads and reactions (M ext ). 1.2 Calculate the maximum moment at the middle of a panel. 2. Determine the sign and the magnitude of the chord forces from the beam analogy. 2.1 Determine the sign of the chord forces based on whether Mext at the midpanel point is drawn on the tension side or on the compression side. 2.2 Calculate the sum of the chord forces from: (f top + f bot )(h/2) = M ext 3. Determine the sign and the magnitude of the RISA chord forces for the same panel. 3.1 The RISA sign convention is compression is positive, tension is negative. 3.2 Calculate the sum of the chord forces from the RISA output. 4. Compare the results of Step 2 with the results of Step 3 to determine if the RISA output is reasonable. b. the critical diagonal force 1. Find the max. mid-panel shear for the analogous beam. 1.1 Draw the shear diagram for the analogous beam due to the loads and reactions (V ext ). 1.2 Calculate the maximum shear at the middle of a panel. 2. Determine the sign and the magnitude of the diagonal forces from the beam analogy. 2.1 Determine the sign of the diagonal forces based on whether Vext at the midpanel point is drawn on the positive side or on the negative side. The sign convention for shear is: + ve V 2.2 Calculate the sum of the diagonal forces from: (f 1 + f 2 )(h/l diag ) = V ext 3. Determine the sign and the magnitude of the RISA diagonal forces for the same panel. 3.1 The RISA sign convention is compression is positive, tension is negative. 3.2 Calculate the sum of the diagonal forces from the RISA output. 4. Compare the results of Step 2 with the results of Step 3 to determine if the RISA output is reasonable.
CE 331, Spring 2004 Beam Analogy for Designing Trusses 7 / 9 5. Calculate preliminary sizes for the chords and the diagonals. Without using the RISA output, calculate the minimum cross-sectional area of the chords (A ch_min ) and the diagonals (A diag_min ). As part of your solution, indicate the location of the critical members. Hint: use the allowable stresses (F t and F c ) on Page 1.. Prelim. Chord Size. From Step 3: (f top + f bot )(h/2) = M ext Since h is constant for this truss, maximum f top + f bot occurs when M ext is max. Max. M ext = 120 k-ft in the middle of Panel #3. Using eqn. above: (f top + f bot )(6 /2) = 120 k-ft, f top + f bot = 40 k Assume f top f bot, f top f bot 20 k Axial compressive stress = compressive force / axial area: f c = f top / A chord Set compressive stress due to loads (f c ) to the max. allowable compressive (F c ) F c = f c 10 ksi = f top / A ch_min, 10 ksi = 20 k / A ch_min, A ch_min = 2 in 2. Axial tensile stress = tensile force / axial area: f t = f bot / A chord Set tensile stress due to loads (f t ) to the max. allowable tensile stress (F t ) F t = f t 21 ksi = f bot / A ch_min, 21 ksi = 20 k / A ch_min, A ch_min = 1.05 in 2. Largest minimum area controls, therefore A ch_min = 2 in 2 Prelim. Diagonal Size. From Step 3: (f 1 + f 2t )(h/l diag ) = V ext Since h and L diag are constant for this truss, maximum f 1 + f 2 occurs when V ext is max. Max. V ext = 10 k-ft in the middle of Panel #1. Using eqn. above: (f 1 + f 2 )(6 /10 ) = 10 k, f 1 + f 2 = 16.67 k Assume f 1 f 2, f 1 f 2 8.33 k Axial compressive stress = compressive force / axial area: f c = f 1 / A diag Set compressive stress due to loads (f c ) to the max. allowable compressive (F c ) F c = f c 10 ksi = f 1 / A diag_min, 10 ksi = 8.33 k / A ch_min, A ch_min = 0.83 in 2.
CE 331, Spring 2004 Beam Analogy for Designing Trusses 8 / 9 Axial tensile stress = tensile force / axial area: f t = f 2 / A chord Set tensile stress due to loads (f t ) to the max. allowable tensile stress (F t ) F t = f t 21 ksi = f 2 / A ch_min, 21 ksi = 8.33 k / A ch_min, A ch_min = 0.40 in 2. Largest minimum area controls, therefore A diag_min = 0.83 in 2 6. Write a procedure to calculate preliminary sizes for the chords and the diagonals. a. Prelim. Chord Size. 1. Calculate the approximate max. compressive chord force. 1.1 Calculate the sum of the chord forces following Steps 4.a.1 and 4.a.2 above. 1.2 Assume that the chord forces are equal and solve for one of them, say f top 2. Calculate the minimum required area for the chord from Step 1 above. 2.1 Set the axial stress due to loads (f c ) equal to the allowable compressive stress (F c ). 2.2 Use the equation for axial stress below to calculate the min. axial area for the chord, A ch_min. f c = P/A, where P = axial force and A = axial area b. Prelim. Diagonal Size. 1. Calculate the approximate max. compressive diagonal force. 1.1 Calculate the sum of the diagonal forces following Steps 4.b.1 and 4.b.2 above. 1.2 Assume that the diagonal forces are equal and solve for one of them, say f 1 2. Calculate the minimum required area for the diagonal from Step 1 above. 2.1 Set the axial stress due to loads (f c ) equal to the allowable compressive stress (F c ). 2.2 Use the equation for axial stress below to calculate the min. axial area for the diagonal, A diag_min. f c = P/A, where P = axial force and A = axial area
CE 331, Spring 2004 Beam Analogy for Designing Trusses 9 / 9 7. Select truss shape. An efficient design is one in which all members reach their allowable stress simultaneously. Assume in Questions 4a through 4d that all chords of your non-parallel-chord truss have the same cross-sectional area. 7a. Sketch the shape you would use for a simply supported truss supporting one concentrated load at midspan? Explain. 7b. What shape would you use for a simply supported truss supporting a uniform load? 7c. What shape would you use for a truss with a cantilever overhang supporting a uniform load? 7d. Write a procedure for choosing an efficient shape for a truss of specified support conditions and loads, assuming that all chords have the same allowable stress and cross-sectional area. 7e. Sketch an efficient parallel-chord truss for the conditions specified in Question 4c above and indicate the cross-sectional area of each chord using line weight. (Hint: efficient means that all chords reach the allowable stress at the same time).