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Fundamentals of Structural Analysis P2

For the second truss: 0 0 − 7.20 − 9.60 − 33.33 0⎤ ⎡40.53 9.60 ⎥ ⎢ 9.60 37.80 0 − 25.00 − 9.60 − 12.80 0 0 ⎥ ⎢ ⎢ 0 0 9.60 − 12.80 0 − 25.00 − 9.60 37.80⎥ ⎦ ⎣ Results will be summarized at the end of the example. (7) Compute the member elongations and forces. For a typical member i: ⎧u1 ⎫ ⎪v ⎪ ⎪ ⎪ S ⎦i ⎨ 1 ⎬ ⎪u 2 ⎪ ⎪v 2 ⎪ ⎩ ⎭i ∆i = ⎣− C − S C EA ∆)ι L Fi=(k∆)ι = ( (8) Summarizing | Truss Analysis Matrix Displacement Method by S. T. Mau r oi 0 33.33 0 0 0 25.00 0 0 0 9.60 0 25.00 -12.80 0 0 - 33.33 0 0 0 0 - 25.00 - 9.60 0 0 37.80 u 2 v 2 u 3 v3 Px1 Pyi p Py 4 u 4 0 For the second truss r 0i 0 40.53 9.60 0 0 - 7.20 - 9.60 - 33.33 0 9.60 37.80 0 - 25.00 - 9.60 - 12.80 0 0 0 0 9.60 -12.80 0 - 25.00 - 9.60 37.80 u 2 v 2 u 3 v3 p Py11 P Py 4 u 4 0 Results will be summarized at the end of the example. 7 Compute the member elongations and forces. For a typical member i S i j u1 V1 u V2 Fi kA A A t L 8 Summarizing results. 25 Truss Analysis Matrix Displacement Method by S. T. Mau Results for the First Truss Node Displacement m Force MN x-direction -direction x-direction -direction 1 0 0 -0.50 0.33 2 0.066 -0.013 0.60 -1.00 3 0.067 0 0 0 4 0.015 0 0 0.67 Member Elongation m Force MN 1 -0.013 -0.33 2 0 0 3 0 0 4 0.015 0.50 5 -0.042 -0.83 Results for the Second Truss Node Displacement m Force MN x-direction -direction x-direction -direction 1 0 0 -0.50 0.33 2 0.033 -0.021 0.60 -1.00 3 0.029 -0.007 0 0 4 0.011 0 0 0.67 Member Elongation m Force MN 1 -0.021 -0.52 2 -0.004 -0.14 3 -0.008 -0.19 4 0.011 0.36 5 0.030 -0.60 6 0.012 0.23 Note that the reactions at node 1 and 4 are identical in the two cases but other results are changed by the addition of one more diagonal member. 9 Concluding remarks. If the number of nodes is N and the number of constrained DOF is C then a the number of simultaneous equations in the unconstrained stiffness equation is 2N. b the number of simultaneous equations for the solution of unknown nodal displacements is 2N-C. 26 Truss Analysis Matrix Displacement Method by S. T. Mau In the present example both truss problems have five equations for the five unknown nodal displacements. These equations cannot be easily solved with hand calculation and should be solved by computer. Problem 3. The truss shown is made of members with properties E 70 GPa and 1 430 mm2. Use a computer to find support reactions member forces member .