Lecture 18 - Computer Analysis of Trusses Systems of Linear...

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In Lecture 16, we used the method of joints to solve for the forces in the members of the truss shown below. This approach is generally cumbersome when done by hand. However, it has the advantage that it can be used to obtain all unknowns (i.e., the forces in all members and the reactions). To avoid the cumbersome hand analysis, the method of joints can be implemented within a computer program. This leads to a system of non-homogeneous linear equations which can be solved by employing a matrix inverse. Computer Analysis of Trusses: Systems of Linear Equations and Matrix Inverse R/S Text Prob. 7-3
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T AB T BD Joint B T BC y x From free-body diagram of joint B: Σ F x = T BC sin 30º - T AB sin 60º = 0 Σ F y = -T BC cos 30º - T AB cos 60º - T BD = 0 T CD T AD T BD Joint D y x Σ F y = T BD - 3000 = 0 T BD = 3000 lb = 3000 lb (T) T AB = -1500 lb = 1500 lb (C) BC 3000 lb Note: All forces on free-body diagram of joint assumed to act away from joint (i.e., all members are assumed to be in tension). Negative result will indicate compressive force. 30 o 60 o From free-body diagram of joint D: Review of Solution using Method of Joints
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Known T CD C y Joint C T BC y x Σ F x = - T CD - T BC sin 30 ° = - T CD - (-2598) sin 30 ° = 0 Σ F x = -T AD + T CD = - T AD + 1299 = 0 From free-body diagram of joint D: T CD T AD T BD Joint D y x T AD = 1299 lb = 1299 lb (T) T CD = 1299 lb = 1299 lb (T) 3000 lb 60 o Note: Reactions were not obtained. Alternative solution would be to first determine reactions from equilibrium of complete truss and then apply equilibrium to joints A, C, and D. This would have involved 8 equilibrium equations instead of 5 as used here. From free-body diagram of joint C: (note that, although member BC is in compression, the force T BC is drawn in tension on subsequent FBD's and the sign is accounted for when the value of T BC is substituted into an equation)
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Alternate Solution using Matrix Inverse For each joint, 2 equilibrium equations can be written. Thus, in this problem, 8 equilibrium equations are available for solving for the 8 unknowns (5 member forces + 3 reactions). By writing all of the 8 equilibrium equations and assembling them into a matrix equation, the solution for all unknowns can be obtained via a matrix inverse computation.
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Alternate Solution using Matrix Inverse From free-body diagram of joint B: Σ F x = T BC sin 30º - T AB sin 60º = 0 Σ F y = -T BC
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