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. CE 331, Spring 2003 HW 2a 1 / 6 Beam Analogy -Theory

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1 E 331, Spring 2003 HW 2a 1 / 6 We need to make several decisions in designing trusses including truss shape, height and member sizes. lthough we will likely use a computer program to select the final 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. When using the beam analogy, we basically look at a truss as if it were a beam, and set the external bending moment at a section (moment due to loads and reactions, Mext ) equal to the internal bending moment at the section (moment due to bar forces, Mint). You will derive the equations and procedures for using the beam analogy to determine preliminary member sizes (cross-sectional areas) and to check the results from a computer-aided analysis of the truss. In step 1, you will first lay the ground work by calculating the external and internal moments at several sections of the truss. In step 2, you will derive the procedure for 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 = 10 ksi 6 ft RIS bar forces + ve = compression - ve = tension k k k k k k 10 k 10 k

2 E 331, Spring 2003 HW 2a 2 / 6 1. alculate average chord forces for each panel. ut the truss at either Section -, Section - or Section - (assigned by teacher). Write an equilibrium equation for the moments about the middle of the panel where the diagonals cross (point a ). Separate terms involving external forces (loads and reactions) and internal forces (bar forces) draw braces under these terms and label them Mext and Mint. alculate the average chord force magnitude, f_avg = (abs(f_top) + abs(f_bot)) / 2. alculate the average magnitude of the chord forces from RIS (on page 1). Example FD for Section - a f_top f_1 f_2 f_bot 10 k ΣMa = 0, -(10k)(4 ) - (f_top)(3 ) + (f_bot)(3 ) = 0 Mexternal Minternal 40k ft = 40k ft = f _ top + f _ bot 2 f _ avg (6') 40k ft f _ avg = = 6.67 k 6' (6')

3 E 331, Spring 2003 HW 2a 3 / 6 2. Write out a procedure for checking the computer-generated chord forces in a parallel chord truss.

4 E 331, Spring 2003 HW 2a 4 / 6 3. Write a procedure to calculate the minimum cross-sectional area for the chords (assuming all chords are the same size). The allowable tensile and compressive stresses, Ft and Fc respectively, are given on page 1. Draw the analogous beam for the truss directly below the truss. alculate and draw the reactions, shear and moment diagrams for the beam directly below the sketch of the analogous beam. 8 ft 6 ft 10 k 10 k 10 k V, k 5 k -5 k -10 k M, k-ft 80 k-ft 120 k-ft 40 k-ft 100 k-ft 120 k-ft

5 E 331, Spring 2003 HW 2a 5 / 6 alculate the moment at the midpoints of panels 1, 2 and 3 (at 4, 12 and 20 from the left support, respectively). ompare these moments with the external moments from step 1. Write the procedure for calculating the minimum chord cross-sectional area (min chord).

6 E 331, Spring 2003 HW 2a 6 / 6 4. Truss shape. 4a. Draw the shape you would use for a six-panel simply supported truss supporting one concentrated load at midspan if you wanted all panels to have the same average chord force? 4c. Draw the shape you would use for a six-panel simply supported truss supporting a uniform load if you wanted all panels to have the same average chord force? 4d. What shape would you use for an eight-panel truss with a cantilever overhang supporting a uniform load? ssume six of the eight panels are between the supports and the remaining two panels overhang the right support. 4e. Draw the shape you would use for of a plate-girder with a cantilever overhang supporting a uniform load assuming the flanges have constant area along the girder length. 4f. Draw a constant-depth plate-girder with a cantilever overhang supporting a uniform load assuming the flanges have variable area along the girder length. Indicate the flange thicknesses you would use at different locations along the girder.