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Stiffness Method

Stiffness Method. Lecture No. : 13. المحاضرة الثالثة عشر. Analysis of Grids Using Direct stiffness method. Preliminary example :. Calculate the deformations of the shown space frame where E = 1200 kN/cm 2 , G = 500 kN/cm 2 and the sections are shown in figure. 100 kN. A. 200 kN. B.

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Stiffness Method

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  1. Stiffness Method Lecture No. : 13 المحاضرة الثالثة عشر

  2. Analysis of Grids Using Direct stiffness method

  3. Preliminary example : Calculate the deformations of the shown space frame where E = 1200 kN/cm2, G = 500 kN/cm2 and the sections are shown in figure 100 kN A 200 kN B C 20x60 cm 30x80 cm 8 m 10 m

  4. y z x Sections Properties 20x60 cm y x z 30x80 cm A =1200 cm2 Iz =360,000 cm4 Iy =40,000 cm4 Ix =126,435 cm4 A =2400 cm2 Iz =1,280,000 cm4 Iy =180,000 cm4 Ix =550,180 cm4

  5. z x y Modeling d6 C A d3 d4 d5 d2 d1

  6. Stiffness matrix k12 k15 k11 k13 k14 k16 k22 k25 k21 k23 k24 k26 k32 k35 k31 k33 k34 k36 K = k42 k45 k41 k43 k44 k46 k52 k55 k51 k53 k54 k56 k62 k65 k61 k63 k64 k66

  7. z x y First column in Stiffness matrix A C d1=1 B

  8. E A L 12x1200X180,000 10003 12EIy 6 EIy z L2 L3 x y y y d1=1 x z x d1=1 z 1200X1200 800 1,800 2.592 6x1200X180,000 10002 1,296

  9. z d6 x y d3 d4 d5 d2 d1 1,800 2.6 k11 1,802.6 1,296 k21 0 k31 0 = k41 0 k51 0 k61 -1,296

  10. First column in Stiffness matrix k11 1,802.6 k21 0 k31 0 = k41 0 k51 0 k61 -1,296

  11. z x y Second column in Stiffness matrix A d2=1 C B

  12. E A y L z x 1200X2400 12x1200X40,000 1000 8003 12EIy 6 EIy z L2 L3 6x1200X40,000 x y 8002 450 y d2=1 d2=1 x z 2,880 1.125

  13. z d6 x y d3 d4 d5 d2 d1 450 2,880 1.1 k12 0 k22 2,881.1 k32 0 = k42 0 k52 0 k62 450

  14. Second column in Stiffness matrix k12 0 k22 2,881.1 k32 0 = k42 0 k52 0 k62 450

  15. z x y Third column in Stiffness matrix A d3=1 C B

  16. y z x 12EIz 6 EIz 12EIz 6 EIz L3 L2 L3 L2 d3=1 d3=1 y x z 9,216 4,050 10.1 18.4

  17. z d6 x y d3 d4 d5 d2 d1 4,050 9,216 10.1 k13 0 k23 0 18.4 k33 28.5 = k43 9,216 k53 - 4,050 k63 0

  18. Third column in Stiffness matrix k13 0 k23 0 k33 28.5 = k43 9,216 k53 - 4,050 k63 0

  19. z x y Fourth column in Stiffness matrix A C d4=1 B

  20. y z x 500X126,435 800 4 EIz 6 EIz z L L2 GIx x L y y x 6,144,000 d4=1 d4=1 z 9,216 79,022

  21. z d6 x y d3 d4 d5 d2 d1 79,022 6,144,000 k14 0 k24 0 9,216 k34 9,216 = k44 6,223,022 k54 0 k64 0

  22. Fourth column in Stiffness matrix k14 0 k24 0 k34 9,216 = k44 6,223,022 k54 0 k64 0

  23. z x y Fifth column in Stiffness matrix A d5=1 C B

  24. y z x 4 EIz 6 EIz z L2 L GIx x L y y x d5=1 z d5=1 2,160,000 4,050 275,090

  25. z d6 x y d3 d4 d5 d2 d1 2,160,000 4,050 k15 0 275,090 k25 0 k35 - 4,050 = k45 0 k55 2,435,090 k65 0

  26. Fifth column in Stiffness matrix k15 0 k25 0 k35 - 4,050 = k45 0 k55 2,435,090 k65 0

  27. z x y Sixth column in Stiffness matrix d6=1 A C B

  28. y z x 4 EIy 4 EIy 6 EIy 6 EIy z L L2 L L2 x y y d6=1 d6=1 x z 450 1,296 240,000 864,000

  29. z d6 x y d3 d4 d5 d2 d1 450 240,000 1,296 k16 -1,296 864,000 k26 450 k36 0 = k46 0 k56 0 k66 1,104,000

  30. Sixth column in Stiffness matrix k16 -1,296 k26 450 k36 0 = k46 0 k56 0 k66 1,104,000

  31. k11 1,802.6 k21 0 k31 0 = k41 0 k13 0 k51 0 k23 0 k61 -1,296 k33 28.5 = k14 k12 k43 k15 0 0 9,216 k16 0 -1,296 k24 k22 k53 2,881.1 0 k25 k26 - 4,050 0 450 k34 k32 k63 9,216 0 k35 - 4,050 k36 0 0 = = = = k44 k42 6,223,022 0 k45 0 k46 0 k54 k52 k55 0 0 k56 2,435,090 0 k64 k62 k65 k66 0 450 0 1,104,000 31

  32. 1,802.6 0 0 0 -1,296 0 0 2,881.1 0 450 0 0 0 0 28.5 0 9,216 - 4,050 K = 0 0 9,216 0 6,223,022 0 0 0 - 4,050 0 0 2,435,090 -1,296 450 0 1,104,000 0 0

  33. Force vector 100 kN A 200 kN B C 20x60 cm 30x80 cm 8 m 10 m

  34. 100 kN 100 200 kN 250 50 100 250 100 50 100 Fixed End Reaction (FER)

  35. 100 kN 100 200 kN 250 50 100 250 100 50 100 Fixed End Action (FEA)

  36. d6 d3 d4 d5 d2 d1 100 kN 100 200 kN 250 50 100 250 50 F1 0 100 F2 0 F3 - 150 100 = F4 - 250 F5 100 F6 0

  37. Stiffness Equation F = K D d1 0 1,802.6 0 0 0 -1,296 0 d2 0 0 2,881.1 0 450 0 0 d3 - 150 0 0 28.5 0 9,216 - 4,050 = d4 0 0 9,216 0 - 25000 6,223,022 0 0 d5 0 - 4,050 0 10000 0 2,435,090 -1,296 d6 450 0 1,104,000 0 0 0

  38. Stiffness Equation D = K-1 F -1 1,802.6 0 0 0 -1,296 0 d1 0 0 2,881.1 0 450 0 0 d2 0 0 0 28.5 0 9,216 - 4,050 d3 - 150 0 = 0 9,216 0 6,223,022 0 d4 - 25000 0 0 - 4,050 0 0 2,435,090 d5 10000 -1,296 450 0 1,104,000 0 0 d6 0

  39. Deformations d1 0 d2 0 d3 - 11.8714 = d4 0.013564 d5 - 0.015638 d6 0

  40. 100 kN A 200 kN B C 20x60 cm 30x80 cm 8 m 10 m

  41. Grids

  42. d6 d5 d2 d3 d1 d4 d1 d3 d2 Grids

  43. z y x Degrees of Freedom d1 d3 d2

  44. Example 1: Draw all diagrams for the shown grid where E = 1200 kN/cm2, G = 500 kN/cm2 and the sections are shown in figure 100 kN A 200 kN B C 20x60 cm 30x80 cm 8 m 10 m

  45. y z x Sections Properties 20x60 cm y x z 30x80 cm Iz =360,000 cm4 Ix =126,435 cm4 Iz =1,280,000 cm4 Ix =550,180 cm4

  46. z x y Modeling C A d1 d3 d2

  47. Stiffness matrix k12 k11 k13 K k22 k21 k23 = k32 k31 k33

  48. z x y First column in Stiffness matrix A d1=1 C B

  49. y z x 12EIz 6 EIz 12EIz 6 EIz L3 L2 L3 L2 d1=1 d1=1 y x z 9,216 4,050 10.1 18.4

  50. z x y 4,050 9,216 10.1 d1 18.4 28.5 d3 k11 d2 9,216 k21 = - 4,050 k31

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