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Finite Element for Trusses in 2-D

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Finite Element Analysis for trusses in 2-D

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Finite Element for Trusses in 2-D

1. 1. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Introduction to the Finite Element Method Trusses in 2-D
2. 2. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Trusses • A truss is a set of bars that are connected at frictionless joints. • The Truss bars are generally oriented in the plain.
3. 3. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Trusses • Now, the problem lies in the transformation of the local displacements of the bar, which are always in the direction of the bar, to the global degrees of freedom that are generally oriented in the plain.
4. 4. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Equation of Motion                                            0 0 0000 0101 0000 0101 2 1 2 2 1 1 F F v u v u h EA
5. 5. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Transformation Matrix                 DOF dTransforme DOFLocal v u v u CosSin SinCos CosSin SinCos v u v u                                            2 2 1 1 2 2 1 1 00 00 00 00         DOF dTransforme DOFLocal T 
6. 6. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik The Equation of Motion Becomes • Substituting into the FEM: • Transforming the forces: • Finally:      FTK           FTTKT TT      FK 
7. 7. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Recall       TKTK T                                      CosSin SinCos CosSin SinCos T 00 00 00 00 Where:                  0000 0101 0000 0101 h EA K
8. 8. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Element Stiffness Matrix in Global Coordinates                                                                                      CosSin SinCos CosSin SinCos CosSin SinCos CosSin SinCos h EA K 00 00 00 00 0000 0101 0000 0101 00 00 00 00
9. 9. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Element Stiffness Matrix in Global Coordinates                                                                22 22 22 22 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 SinSinSinSin SinCosSinCos SinSinSinSin SinCosSinCos h EA K
10. 10. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Example: 4.6.1 pp. 196-201 • Use the finite element analysis to find the displacements of node C.
11. 11. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Element Equations                  0000 0101 0000 0101 1 L EA K                  1010 0000 1010 0000 2 L EA K                    3536.03536.03536.03536.0 3536.03536.03536.03536.0 3536.03536.03536.03536.0 3536.03536.03536.03536.0 3 L EA K
12. 12. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Assembly Procedure                              3536.13536.0103536.03536.0 3536.03536.0003536.03536.0 101000 000101 3536.03536.0003536.03536.0 3536.03536.0013536.03536.1 L EA K
13. 13. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Global Force Vector                                              P P F F F F F F F F F F F y x y x y x y x y x 2 2 2 1 1 3 3 2 2 1 1 Remember! NO distributed load is applied to a truss
14. 14. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Boundary Conditions 02211  VUVU Remove the corresponding rows and columns                     P P V U L EA 23536.13536.0 3536.03536.0 3 3 Continue! (as before)
15. 15. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Results EA PL V EA PL U 3 ,828.5 33  PFF PFPF yx yx 3,0 ,, 22 11  
16. 16. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Post-computation e e e e e A P A P 21                           e e ee e e u u L EA P P 2 1 2 1 11 11                                                            2 2 1 1 2 2 1 1 00 00 00 00 v u v u CosSin SinCos CosSin SinCos v u v u    
17. 17. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Post-computation A P A P 2, 3 ,0 )3()2()1(  
18. 18. #WikiCourses http://WikiCourses.WikiSpaces.com Trusses in 2-D Mohammad Tawfik Summary • In this lecture we learned how to apply the finite element modeling technique to bar problems with general orientation in a plain.