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ANALISA STRUKTUR METODE MATRIKS RANGKA BIDANG (PLANE TRUSS)

ANALISA STRUKTUR METODE MATRIKS RANGKA BIDANG (PLANE TRUSS). Structural Analysis. Classical Methods. Matrix Methods. Vitrual Work. Stiffness by Definition. Force Method. Direct Stiffness. Slope Deflection. Trusses. Moment-Area. Beams. STRUKTUR RANGKA.........???.

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ANALISA STRUKTUR METODE MATRIKS RANGKA BIDANG (PLANE TRUSS)

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  1. ANALISA STRUKTUR METODE MATRIKSRANGKA BIDANG(PLANE TRUSS)

  2. Structural Analysis Classical Methods Matrix Methods Vitrual Work Stiffness by Definition Force Method Direct Stiffness Slope Deflection Trusses Moment-Area Beams

  3. STRUKTUR RANGKA.........??? • Composed of slender, lightweight members • All loading occurs on joints • No moments or rotations in the joints • Axial Force Members Tension (+) or Compression (-)

  4. A Simple Comparison Stiffness by Definition • 2 Degrees of Freedom Direct Stiffness • 6 Degrees of Freedom • DOFs 3,4,5,6 = 0 • Unknown Reactions (to be solved) included in Loading Matrix 6 5 2 1 4 3 Remember.. More DOFs = More Equations

  5. Hubungan antara “Gaya” dan “Deformasi”

  6. Persamaan matriks hub. “gaya” dan “ deformasi”

  7. nodal displacemen, terdiri dari ; ui ; vi ; uj ; vj atau vektor displacemen • nodal gaya, terdiri dari ; fi ; gi ; fj ; gj atau vektor gaya

  8. Matriks Kekakuan elemen pada sistem koordinat Lokal dimana : A = luas penampang elemen L = panjang elemen E = modulus elastis bahan

  9. y v x Transformasi Koordinat ui = Ui cos θ + Vi sin θ + 0 + 0 vi = -Ui sin θ + Vi cos θ + 0 + 0 uj = 0 + 0 + Uj cos θ + Vj sin θ vj = 0 + 0 - Uj sin θ + Vj cos θ X, Y ; sistem koordinat global x, y; sistem koordinat lokal

  10. Analog dengan diatas ; fi = Fi cos θ + Gi sin θ + 0 + 0 gi = -Fi sin θ + Gi cos θ + 0 + 0 fj = 0 + 0 + Fj cos θ + Gj sin θ gj = 0 + 0 - Fj sin θ + Gj cos θ

  11. Hubungan gaya dan displacemen pada elemen ; "Pre-multiplied" dengan [T]-1 dan mengingat [T]T = [T]-1 sehingga ;

  12. Matriks kekakuan elemen pada sistem koordinat global ; simetris dimana : c = cos  s = sin 

  13. OVER-ALL STIFFNESS MATRIX ....untuk 1 elemen dengan node i dan j ......untuk rangka bidang dengan node sejumlah n

  14. BOUNDARY CONDITIONS {Fe}= “Prescribed external force vector” {Fr} = “Un-known reaction vector” {Uu} = “Un-known displacement vector” {Uk} = “Known displacement vector “given by boundary condition. [Kij] = “Sub-matrix of overall stiffness matrix”.

  15. Stiffness by Definition vs Direct Stiffness = X Uunknown Fknown K = X K completed Uknown Funknown Zero Unless Settlement Occurs Reactions

  16. UN-KNOWN DISPLACEMENT & REACTION SEHINGGA DIPEROLEH ;

  17. MEMBERS FORCES dimana ;

  18. 2 4,00 m 1 3 3,00 m 5 ton 10 ton

  19. V2 V1 V3 G2 G1 G3 U2 U1 U3 F2 F1 F3 Nodal Force Nodal Displacemen U1 V1 U2 = 0 V2 = 0 U3 = 0 V3 = 0 F1 = -5 ton G1 = -10 ton F2 G2 F3 G3

  20. K2 K2 K2 K2 F} = [K] U} U1 V1 U2 V2 U3 V3 K1 F1 U1 U1 G1 V1 V1 F2 U2 U2 K3 G2 V2 V2 F3 U3 U3 G3 V3 V3

  21. F} = [K] U} U1 V1 U2 V2 U3 V3 Kover-all=K1+K2+K3 - 5 U1 U1 - 10 V1 V1 F2 0 U2 G2 0 V2 F3 0 U3 G3 0 V3

  22. F} = [K] U} U1 V1 U2 V2 U3 V3 - 5 U1 U1 - 10 V1 V1 F2 0 U2 G2 0 V2 F3 0 U3 G3 0 V3

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