Please Solve Using Matlab 1 (137.34 KiB) Viewed 50 times
Please Solve Using Matlab 2 (61.11 KiB) Viewed 50 times
When civil engineers are building a bridge, they must ensure that the structure will withstand the loads on it such as cars, trains and pedestrians and not fail or deform. Fig. 3 depicts a plane truss, (a simplified bridge in 2D), having 13 solid ligaments, or members, (the numbered lines), connecting 8 joints (the numbered circles). The indicated loads, in tonnes, are applied at joints 2, 5, and 6, and we want to determine the resulting force on each member of the truss. 5 7 9 11 12 19 - 1 2 5 10 15 Figure 3: A static truss For the truss to be in static equilibrium, there must be no net force, horizontally or vertically, at any joint. Thus, we can determine the member forces by equating the horizontal forces to the left and right at each joint, and similarly equating the vertical forces upward and downward at each joint. For the eight joints, this would give 16 equations, which is more than the 13 unknown factors to be determined. For the truss to be statically determinate, that is, for there to be a unique solution, we assume that joint 1 is rigidly fixed both horizontally and vertically and that joint 8 is fixed vertically. Resolving the member forces into horizontal and vertical components and defining the constant a = 1/√√2, we obtain the following system of equations for the member forces 2 3 6 10 6 20 8 13
Ji, Joint 2: f2f6, f3 = 10; Joint 3: aff4+ afs, af₁ + fs+afs=0; Joint 4: f4f8, fr = 0; Joint 5: afs+f6=af9+f10, af5+ fr+afg= 15; Joint 6: f10 f13, f11 = 20; Joint 7: fs + afg= af12, afg+f11+af12 = 0; Joint 8: f13+a/12 = 0. (a) Assemble this problem into the standard form of Az = b. How many non-zero terms are in the right hand side b vector? (b) What is the rank and condition number of A? Is this problem ill-conditioned or near singular? (c) Solve this system of equations to find the vector f of member forces. How accurate do you think your answer is? How can you test the quality of your answer?
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