Uploaded bySajjad Ahmad
Matrix Structural Analysis, Steel Frame Analysis in SAP2000
The document discusses the analysis of a frame structure. It provides details on the geometry of the frame, determining the degrees of freedom, formulation of load vectors, calculation of local and global stiffness matrices, and global assembly of the structure with a band width of 24. The equilibrium equations for the structure can then be formed from the assembled global stiffness matrix and load vector.
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Matrix Structural Analysis, Steel Frame Analysis in SAP2000
- 2. By SAJJAD AHMAD (2009-NUST-MSPhD-Str-06)
- 3. Geometry of frame DOFs Loadvector Stiffness matrices of members Assemblage and Band width Equilibrium equation for joint “F”
- 4.
- 5. 4m 2m 2m 2m 2m 2m 2m 2m 4m
- 6. PART (A) DEGREE OFFREEDOM (DOFs)
- 7. 8 17 26 29 7 30 16 25 28 9 18 27 5 14 23 13 4 22 6 15 24 2 11 20 3 1 10 19 12 21
- 8. DOFs Total DOFs are30 Known DOFs at joints “A, D and G” (by support conditions) 1, 2, 10, 11, 12, 19, 20, 21 = Zero Unknown DOFs (30 – 8 = 22)
- 9.
- 10. LOAD VECTOR P = F – FEF P is load vector F is nodal force FEF is the force observed at nodes due to application of loading between nodal points
- 11. FORMULATION OF LOADVECTORS Frame members 1, 2, 3, 4, 7, 8, 9 and 12 have no loading between nods so As FEF is null matrix for local axis so for global FEF will be
- 12. FORMULATION OF LOADVECTORS For member 5 and 6 FEF will be given by following
- 13. FORMULATION OF LOADVECTORS For member 5 and 6 For transforming it in to global, transformation matrix will be used
- 14. FORMULATION OF LOADVECTORS For member 5 and 6 θ = 0 degree FEF will be given by Which is
- 15. FORMULATION OF LOADVECTORS For member 10 and 11 FEF will be given by
- 16. FORMULATION OF LOADVECTORS As θ = 0 degree so for global FEF will be same
- 17. STRUCTURE LOAD VECTOR By combining all nodal forces and subtracting all FEFs we get total structural load vector
- 18. PART (C) STIFFNESS MATRIXOF MEMBERS
- 19. LOCAL STIFFNESS MATRIX Localstiffness matrix is given by following matrix
- 20. LOCAL STIFFNESS MATRIX INPUT CALCULATIONS Member 1 AE/L 5000 A (m2) 0.01 12EI/L3 5625 E (t/m2) 2E6 6EI/L2 11250 I (m4) 0.015 4EI/L 30000 L (m) 4 2EI/L 15000 FOR MEMBER 01 5000 0 0 -5000 0 0 0 5625 11250 0 -5625 11250 0 11250 30000 0 -11250 15000 -5000 0 0 5000 0 0 0 -5625 -11250 0 5625 -11250 0 11250 15000 0 -11250 30000
- 21. LOCAL STIFFNESS MATRIX INPUT CALCULATIONS Member 2 AE/L 2773.54 A (m2) 0.01 12EI/L3 960.099 E (t/m2) 2000000 6EI/L2 3461.637 I (m4) 0.015 4EI/L 16641.24 L (m) 7.211 2EI/L 8320.621 FOR MEMBER 02 2773.54 0 0 -2773.54 0 0 0 960.099 3461.637 0 -960.099 3461.637 0 3461.637 16641.24 0 -3461.64 8320.621 -2773.54 0 0 2773.54 0 0 0 -960.099 -3461.64 0 960.099 -3461.64 0 3461.637 8320.621 0 -3461.64 16641.24
- 22. LOCAL STIFFNESS MATRIX INPUT CALCULATIONS Member 5 AE/L 3333.333 A (m2) 0.01 12EI/L3 1666.667 E (t/m2) 2000000 6EI/L2 5000 I (m4) 0.015 4EI/L 20000 L (m) 6 2EI/L 10000 FOR MEMBER 05 3333.333 0 0 -3333.33 0 0 0 1666.667 5000 0 -1666.67 5000 0 5000 20000 0 -5000 10000 -3333.33 0 0 3333.333 0 0 0 -1666.67 -5000 0 1666.667 -5000 0 5000 10000 0 -5000 20000
- 23.
- 24. GLOBAL STIFFNESS MATRIX INPUT Member 1 θ (Degree) 90 cos θ 0 sin θ 1 A (m2) 0.01 E (t/m2) 2000000 I (m4) 0.015 L (m) 4 FOR MEMBER 1 5625 0 -11250 -5625 0 -11250 0 5000 0 0 -5000 0 -11250 0 30000 11250 0 15000 -5625 0 11250 5625 0 11250 0 5000 0 0 5000 0 -11250 0 15000 11250 0 30000
- 25. GLOBAL STIFFNESS MATRIX INPUT Member 2 θ (Degree) 33.69 cos θ 0.83205 sin θ 0.55469 A (m2) 0.01 E (t/m2) 2000000 I (m4) 0.015 L (m) 7.211 FOR MEMBER 2 2215.546 836.9573 -1920.14 -2215.55 -836.957 -1920.14 836.9573 1518.049 2880.255 -836.957 -1518.05 2880.255 -1920.14 2880.255 16641.24 1920.135 -2880.25 8320.621 -2215.55 -836.957 1920.135 2215.546 836.9573 1920.135 -836.957 1518.049 -2880.25 836.9573 1518.049 -2880.25 -1920.14 2880.255 8320.621 1920.135 -2880.25 16641.24
- 26. GLOBAL STIFFNESS MATRIX INPUT Member 5 θ (Degree) 0 cos θ 1 sin θ 0 A (m2) 0.01 E (t/m2) 2000000 I (m4) 0.015 L (m) 6 FOR MEMBER 5 3333.333 0 0 -3333.33 0 0 0 1666.667 5000 0 -1666.67 5000 0 5000 20000 0 -5000 10000 -3333.33 0 0 3333.333 0 0 0 1666.667 -5000 0 1666.667 -5000 0 5000 10000 0 -5000 20000
- 27. PART (D) GLOBAL ASSEMBLAGEAND BAND SEMI WIDTH
- 28. GLOBAL ASSEMBLAGE DOFs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 7841 837 -13170 -5625 0 -11250 0 0 0 0 0 0 -2216 -837 -1920 2 837 6518 2880 0 -5000 0 0 0 0 0 0 0 -837 -1518 2880 3 -13170 2880 19641 11250 0 15000 0 0 0 0 0 0 1920 -2880 8321 4 -5625 0 11250 14583 0 0 -5625 0 -11250 0 0 0 -3333 0 0 5 0 5000 0 0 11667 5000 0 -5000 0 0 0 0 0 -1667 5000 6 -11250 0 15000 0 5000 80000 11250 0 15000 0 0 0 0 -5000 10000 7 0 0 0 -5625 0 11250 8958 0 11250 0 0 0 0 0 0 8 0 0 0 0 5000 0 0 6667 5000 0 0 0 0 0 0 9 0 0 0 -11250 0 15000 11250 5000 50000 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 5625 0 -11250 -5625 0 -11250 11 0 0 0 0 0 0 0 0 0 0 5000 0 0 -5000 0 12 0 0 0 0 0 0 0 0 0 -11250 0 30000 11250 0 15000 13 -2216 -837 1920 -3333 0 0 0 0 0 -5625 0 11250 20132 837 1920 14 -837 1518 -2880 0 1667 -5000 0 0 0 0 5000 0 837 14851 -2880 15 -1920 2880 8321 0 5000 10000 0 0 0 -11250 0 15000 1920 -2880 116641 16 0 0 0 0 0 0 -3333 0 0 0 0 0 -5625 0 11250 17 0 0 0 0 0 0 0 1667 -5000 0 0 0 0 5000 0 18 0 0 0 0 0 0 0 5000 10000 0 0 0 -11250 0 15000 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 0 0 0 0 -3333 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 1667 -5000 24 0 0 0 0 0 0 0 0 0 0 0 0 0 5000 10000
- 29. GLOBAL ASSEMBLAGE 0 -5000 10000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5625 0 -11250 0 0 0 -3333 0 0 0 0 0 0 0 0 0 -5000 0 0 0 0 0 -1667 5000 0 0 0 0 0 0 11250 0 15000 0 0 0 0 -5000 10000 0 0 0 0 0 0 12292 0 11250 0 0 0 0 0 0 -3333 0 0 0 0 0 0 8333 0 0 0 0 0 0 0 0 -1667 5000 0 0 0 11250 0 70000 0 0 0 0 0 0 0 -5000 10000 0 0 0 0 0 0 5625 0 -11250 -5625 0 -11250 0 0 0 0 0 0 0 0 0 0 5000 0 0 -5000 0 0 0 0 0 0 0 0 0 0 -11250 0 30000 11250 0 15000 0 0 0 0 0 0 0 0 0 -5625 0 11250 14583 0 0 -5625 0 -11250 0 0 0 0 0 0 0 5000 0 0 11667 -5000 0 -5000 0 0 0 0 0 0 0 -11250 0 15000 0 -5000 80000 11250 0 15000 0 0 0 -3333 0 0 0 0 0 -5625 0 11250 18958 0 11250 -10000 0 0 0 1667 -5000 0 0 0 0 5000 0 0 51667 40000 0 -45000 45000 0 5000 10000 0 0 0 -11250 0 15000 11250 40000 110000 0 -45000 30000 0 0 0 0 0 0 0 0 0 -10000 0 0 10000 0 0 0 0 0 0 0 0 0 0 0 0 45000 -45000 0 45000 -45000 0 0 0 0 0 0 0 0 0 0 45000 30000 0 -45000 60000
- 30. DEFLECTED FRAME FROMSAP2000
- 32. COMPARISON BETWEEN DEFORMATIONBY HAND CALCULATONS AND SAP2000 HAND CALCULATION SAP2000 ERROR % ERROR 1 0 1 0 1 0 0 2 0 2 0 2 0 0 3 0.000192 3 0.000640 3 0.000448 0.699258328 4 0.001125 4 0.001250 4 0.000125 0.100006278 5 -0.000532 5 -0.000482 5 0.000051 -0.105428095 6 -0.000186 6 -0.002980 6 -0.002794 0.937441803 7 0.001125 7 0.009250 7 0.008125 0.878379833 8 -0.000639 8 -0.000723 8 -0.000084 0.115717957 9 -0.000253 9 -0.003810 9 -0.003557 0.933711891 10 0 10 0 10 0 0 11 0 11 0 11 0 0 12 0 12 0 12 0 0 13 0.001125 13 0.001150 13 0.000025 0.021746083 14 -0.000937 14 -0.001220 14 -0.000283 0.232283224 15 -0.000017 15 0.000874 15 0.000891 1.018878576 16 0.001125 16 0.009090 16 0.007965 0.87623831 17 -0.001161 17 -0.001770 17 -0.000609 0.343998416 18 0.000050 18 0.000010 18 -0.000040 -3.859000675 19 0 19 0 19 0 0 20 0 20 0 20 0 0 21 0 21 0 21 0 0 22 0.001125 22 0.001120 22 -0.000005 -0.004453084 23 -0.000166 23 -0.000815 23 -0.000649 0.796117634 24 0.000412 24 0.001940 24 0.001528 0.787753348 25 0.001125 25 0.009090 25 0.007965 0.876237562 26 -0.000061 26 -0.001360 26 -0.001299 0.954949649 27 0.000412 27 0.004100 27 0.003688 0.899523869 28 0.001125 28 0.009030 28 0.007905 0.87541522 29 0.000808 29 -0.020180 29 -0.020988 1.040056451 30 0.000446 30 0.011570 30 0.011124 0.961432181
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- 37.