8 ft. 5 k 5 k 5 k 5 k. F y = 36 ksi F t = 21 ksi F c = 10 ksi. 6 ft. 10 k 10 k

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Sherilyn Nicholson
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1 E 331, Fall 2004 HW 2.0 Solution 1 / 7 We need to make several decisions when designing a truss. First, we need to determine the truss shape. Then we need to determine the height of the truss and the member sizes. lthough we will likely use a computer program to select the final member sizes, we need to make initial estimates 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 tool that helps us understand how forces are distributed through a truss. The beam analogy produces exact results for parallel chord trusses (horizontal top and bottom chords) but is still useful for understanding the behavior of other types of trusses. In this exercise, 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. alculate the sum of the chord and diagonal forces in the 1st, 2nd and 3rd panels 2 2. Draw the shear and moment diagram for the analogous beam. 4 3a,b Write an equation relating the chord forces in a panel to the external moment and the diagonal forces in a panel to the external shear 5 3c,d. Write a procedure to calculate preliminary chord and diagonal bar sizes 6 4. Write a procedure for calculating the preliminary chord and diagonal sizes 7 Example: 5-panel parallel-chord truss. The truss dimensions and loads as well as the reactions and bar forces are shown below. 8 ft F y = 36 ksi F t = 21 ksi F c = si 6 ft RIS bar forces + ve = compression - ve = tension k k k k k k

2 E 331, Fall 2004 HW 2.0 Solution 2 / 7 1. alculate the sum of the chord forces and the sum of the diagonal forces in each panel indicated below. Mid-Panel #1. ut the truss at Section - (see figure on previous page). hord Forces: Sum moments about Point a where the diagonals cross. ssume the chord forces are about equal magnitude. Then the top chord must be in compression (as indicated in the figure at right) and the bottom chord must be in tension (as indicated). Σ M a = 0, (-10k)(4 ) + (3 ) + f bot (3') = 0 ( + f bot )(3 ) = 40 k-ft + f bot = 13.3 k a f1 FD for Section - Diagonal Forces: Sum forces in the vertical direction. ssume the diagonal forces are about equal magnitude. Then f 1 must be downward (as indicated in the sketch) and f 2 must be upward (as indicated). Σ Fv = 0, - f 1,v f 2,v = 0 (where f 1,v = the vertical component of f 1 ) /10 f 1 6/10 f 2 = 0 f 1 + f 2 = k Mid-Panel #2: ut the truss at Section -. hord Forces: Same procedure as for Mid-Panel #1. a f1 Σ M a = 0, (-10k)(12 ) + (4') + (3 ) + f bot (3') = 0 ( + f bot )(3 ) = 100 k-ft FD for Section - Diagonal Forces: Same procedure as for Mid-Panel #1. + f bot = k Σ Fv = 0, - f 1,v f 2,v = 0 6/10 f 1 6/10 f 2 = 0 f 1 + f 2 = 8.33 k

3 E 331, Fall 2004 HW 2.0 Solution 3 / 7 Mid-Panel #3: ut the truss at Section -. hord Forces: Same procedure as for Mid-Panel #1. Σ M a = 0, (-10k)(20 ) + (12') + (4') + (3 ) + f bot (3') = 0 ( + f bot )(3 ) = 120 k-ft FD for Section - a f1 + f bot = 40.0 k Diagonal Forces: Same procedure as for Mid-Panel #1. Σ Fv = 0, - - f 1,v f 2,v = 0 f 1 + f 2 = 0 k

4 E 331, Fall 2004 HW 2.0 Solution 4 / 7 2. Draw the shear and moment diagrams for a beam with the same span and loading as the truss (shown below). Indicate the moment and the shear at each of the mid-panel points. 8 ft 6' V, k - - M, k-ft k-ft k-ft 40 k-ft 100 k-ft 120 k-ft

5 E 331, Fall 2004 HW 2.0 Solution 5 / 7 3a. Use the results on the preceding pages to write a procedure (including necessary formulas) for calculating the maximum average chord force in a parallel-chord truss. Observations: f chord_avg = ( + f bot ) / 2 For any panel, ( + f bot )(truss height / 2) = M midpanel where M midpanel = the moment from the M diagram at midpanel Rearranging the second equation: ( + f bot ) / 2 = M midpanel / truss height f chord_avg = M midpanel / truss height Therefore, to calculate the maximum average chord force, f chord_avg : 1. Draw the M diagram for the analogous beam. (This represents the moments due to external forces loads and reactions.) 2. Find the maximum mid-panel moment on the moment diagram, max M midpanel 3. alculate the max. average chord force from the following equation: max f chord_avg = (max M midpanel ) / truss height 3b. Use a similar procedure to write a procedure for calculating the maximum average diagonal force in a parallel-chord truss. Observations: f diag_avg = (f 1 + f 2 ) / 2 For any panel, (f 1 + f 2 )(truss height / diagonal length) = V midpanel where V midpanel = the shear from the V diagram at midpanel Rearranging the second equation: (f 1 + f 2 ) = V midpanel (diagonal length / truss height) f diag_avg = (½) V midpanel (diagonal length / truss height) Therefore, to calculate the maximum average diagonal force, f diag_avg : 1. Draw the V diagram for the analogous beam. (This represents the shears due to external forces loads and reactions.) 2. Find the maximum mid-panel shear on the shear diagram, max V midpanel 3. alculate the max. average diagonal force from the following equation: max f diag_avg = (½) (max V midpanel ) (diagonal length / truss height)

6 E 331, Fall 2004 HW 2.0 Solution 6 / 7 3c. Write a procedure to calculate the min. cross-sectional area of the chords (min. chord ) given the following: max f chord_avg (calculated using procedure on previous page) max allowable tensile stress = F t as specified on Page 1 max allowable compressive stress = F c as specified on Page 1 axial stress (σ) due to axial force (P) with cross-sectional area () from Strength of Materials: σ = Soln: σ = P / min chord based on allowable tensile stress, F t : set allowable stress = load stress F t = P / = f chord_avg / = f chord_avg / F t min chord based on allowable compressive stress, F c : set allowable stress = load stress F c = P / = f chord_avg / = f chord_avg / F c ut since F c is always F t min chord = f chord_avg / F c 3d. Write a similar procedure to calculate the min. cross-sectional area of the diagonals (min. diag ). Soln: ll of the same equations apply. Therefore min diag = f diag_avg / F c

7 E 331, Fall 2004 HW 2.0 Solution 7 / 7 4. Write a procedure to check the reasonableness of the bar forces calculated by RIS (shown on the first page). Soln. ompare f chord_avg calculated using the beam analogy with the avg chord forces from the RIS output. Panel f chord_avg from beam analogy, k ½[ + f bot ] from RIS, k = = = Note: f chord_avg from beam analogy will equal with f chord_avg from RIS to within the accuracy of the beam analogy calculations (e.g. if you carried four significant figures throughout your calculations, the hand-calc results should = the RIS results to within three significant figures. To check the signs of the bar forces from RIS, compare the sign of one of the chords with the direction of the chord force in the FD used to calculate the f chord_avg. For example, in Panel 1: must be in compression, which is positive in RIS. The bar force from RIS was k, which checks. a f1 FD for Section -

Example: 5-panel parallel-chord truss. 8 ft. 5 k 5 k 5 k 5 k. F yield = 36 ksi F tension = 21 ksi F comp. = 10 ksi. 6 ft.

Example: 5-panel parallel-chord truss. 8 ft. 5 k 5 k 5 k 5 k. F yield = 36 ksi F tension = 21 ksi F comp. = 10 ksi. 6 ft. 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