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# Matrix Structural Analysis, Steel Frame Analysis in SAP2000

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### Transcript

• 2. Geometry of frame DOFs Load vector Stiffness matrices of members Assemblage and Band width Equilibrium equation for joint “F”
• 3. FRAME GEOMETRY
• 4. 4m 2m 2m 2m 2m 2m 2m 2m 4m
• 5. PART (A) DEGREE OF FREEDOM (DOFs)
• 6. 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
• 7. DOFs Total DOFs are 30 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)
• 8. PART (B) LOAD VECTORS
• 9. 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
• 10. FORMULATION OF LOAD VECTORS 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
• 11. FORMULATION OF LOAD VECTORS For member 5 and 6 FEF will be given by following
• 12. FORMULATION OF LOAD VECTORS For member 5 and 6 For transforming it in to global, transformation matrix will be used
• 13. FORMULATION OF LOAD VECTORS For member 5 and 6 θ = 0 degree FEF will be given by Which is
• 14. FORMULATION OF LOAD VECTORS For member 10 and 11 FEF will be given by
• 15. FORMULATION OF LOAD VECTORS As θ = 0 degree so for global FEF will be same
• 16. STRUCTURE LOAD VECTOR By combining all nodal forces and subtracting all FEFs we get total structural load vector
• 17. PART (C) STIFFNESS MATRIX OF MEMBERS
• 18. LOCAL STIFFNESS MATRIX Local stiffness matrix is given by following matrix
• 19. 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
• 20. 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
• 21. 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
• 22. GLOBAL STIFFNESS MATRIX
• 23. 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
• 24. 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
• 25. 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
• 26. PART (D) GLOBAL ASSEMBLAGE AND BAND SEMI WIDTH
• 27. 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
• 28. 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
• 29. DEFLECTED FRAME FROM SAP2000
• 30. COMPARISON BETWEEN DEFORMATION BY 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
• 31. AXIAL FORCES
• 32. SHEAR FORCE DIAGRAM
• 33. BENDING MOMENT DIAGRAM