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# Matrix Structural Analysis, Steel Frame Analysis in SAP2000

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### Matrix Structural Analysis, Steel Frame Analysis in SAP2000

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