BEAM ANALYSIS USING THE STIFFNESS METHOD ! Development: The Slope-Deflection Equations ! Stiffness Matrix ! General Procedures ! Internal Hinges ! Temperature Effects ! Force & Displacement Transformation ! Skew Roller Support 1 Slope – Deflection Equations i P j k w Cj settlement = ∆j i P j M w ij Mji θi θ ψ j 2 • Degrees of Freedom M θΑ A B 1 DOF: θΑ L P θΑ B θ θ A C 2 DOF: Α , Β θΒ 3 • Stiffness Definition k kAA 1 BA A B L 4EI k = AA L 2EI k = BA L 4 k kAB BB A 1 B L 4EI k = BB L 2EI k = AB L 5 • Fixed-End Forces Fixed-End Forces: Loads P PL L/2 L/2 PL 8 8 L P P 2 2 w wL2 wL2 12 12 L wL wL 2 2 6 • General Case i P j k w Cj settlement = ∆j i P j M w ij Mji θi ψ θj 7 P i w j Mij Mji θi L settlement = ∆ θ j ψ j 4EI 2EI 2EI 4EI θi + θ j = M M = θ + θ L L ij ji L i L j θj θi + F (M ij) F ∆ (M ji)∆ settlement = ∆ + j P w F (MF ) (M ij)Load ji Load 4EI 2EI F F 2EI 4EI F F M = ( )θ + ( )θ + (M ij ) + (M ij ) , M = ( )θ + ( )θ + (M ji ) + (M ji ) ij L i L j ∆ Load ji L i L j ∆ Load 8 • Equilibrium Equations i P j k w Cj C Mji j Mjk Mji Mjk j + ΣM j = 0 : − M ji − M jk + C j = 0 9 • Stiffness Coefficients Mij i j Mji L θj θi 4EI kii = 2EI L k ji = ×θi L 1 + 2EI kij = 4EI L k = ×θ j jj L 1 10 • Matrix Formulation 4EI 2EI F M = ( )θ + ( )θ + (M ij ) ij L i L j 2EI 4EI F M = ( )θ + ( )θ + (M ji ) ji L i L j F M ij (4EI / L) (2EI / L) θiI M ij = + F M (2EI / L) (4EI / L) θ ji j M ji kii kij []k = k ji k jj Stiffness Matrix 11 P i w j Mij Mji θi [M ] = [K][θ ]+[FEM ] L θ ([M ]−[FEM ]) = [K][θ ] ψ j ∆j [θ ] = [K]−1[M ]−[FEM ] Mij Mji θj θi Fixed-end moment + Stiffness matrix matrix F (M ij) F ∆ (M ji)∆ [D] = [K]-1([Q] - [FEM]) + Displacement Force matrix F P F (M ij)Load w (M ji)Load matrix 12 • Stiffness Coefficients Derivation M Mi θi j Real beam i j L M i + M j M i + M j L L L/3 M j L M j 2EI EI Conjugate beam M i EI M i L 2EI θι M L L M L 2L From(1)and (2); + ΣM ' = 0 : − ( i )( ) + ( j )( ) = 0 i 2EI 3 2EI 3 4EI M i = ( )θi M i = 2M j − − − (1) L 2EI M i L M j L M j = ( )θi + ↑ ΣFy = 0 : θi − ( ) + ( ) = 0 − − − (2) L 2EI 2EI 13 • Derivation of Fixed-End Moment Point load P Real beam Conjugate beam A B ABL M M M EI EI M EI ML M 2EI M ML EI 2EI P PL2 PL PL2 16EI 4EI 16EI ML ML 2PL2 PL + ↑ ΣF = 0 : − − + = 0, M = y 2EI 2EI 16EI 8 14 P PL PL 8 L 8 P P P/2 2 2 P/2 PL/8 -PL/8 -PL/8 - -PL/8 -PL/16 - -PL/16 -PL/8 − PL − PL PL PL PL/4 + + = + 16 16 4 8 15 Uniform load w Real beam Conjugate beam A B ABL M M M EI EI M EI ML M 2EI M ML EI 2EI 2 wL3 wL wL3 w 24EI 8EI 24EI ML ML 2wL3 wL2 + ↑ ΣF = 0 : − − + = 0, M = y 2EI 2EI 24EI 12 16 Settlements M M Mi = Mj Real beam j Conjugate beam EI L A B M + M ∆ ∆ i j M L M i + M j M EI L M ML EI ML 2EI 2EI M M EI ∆ ML L ML 2L + ΣM = 0 : − ∆ − ( )( ) + ( )( ) = 0, B 2EI 3 2EI 3 6EI∆ M = L2 17 C • Typical Problem B P P1 w 2 A C B L1 L2 2 wL 2 PL P PL w wL 12 8 8 12 L L 0 4EI 2EI P1L1 M AB = θ A + θ B + 0 + L1 L1 8 0 2EI 4EI P1L1 M BA = θ A + θ B + 0 − L1 L1 8 0 2 4EI 2EI P2 L2 wL2 M BC = θ B + θC + 0 + + L2 L2 8 12 0 2 2EI 4EI − P2 L2 wL2 M CB = θ B + θC + 0 + − L2 L2 8 12 18 C B P P1 w 2 A C B L1 L2 C MBA B MBC B 2EI 4EI P1L1 M BA = θ A + θ B + 0 − L1 L1 8 2 4EI 2EI P2 L2 wL2 M BC = θ B + θC + 0 + + L2 L2 8 12 + ΣM B = 0 : CB − M BA − M BC = 0 → Solve for θ B 19 C B P P1 w 2 M M BA AB C A MCB M B BC L1 L2 Substitute θB in MAB, MBA, MBC, MCB 0 4EI 2EI P1L1 M AB = θ A + θ B + 0 + L1 L1 8 0 2EI 4EI P1L1 M BA = θ A + θ B + 0 − L1 L1 8 0 2 4EI 2EI P2 L2 wL2 M BC = θ B + θC + 0 + + L2 L2 8 12 0 2 2EI 4EI − P2 L2 wL2 M CB = θ B + θC + 0 + − L2 L2 8 12 20 C B P P1 w 2 MBA MAB MCB A MBC C Ay B L1 L2 Cy By = ByL + ByR B C P P 1 M 2 B BA MCB MAB A MBC A ByR C y ByL y L1 L2 21 Stiffness Matrix • Node and Member Identification • Global and Member Coordinates • Degrees of Freedom •Known degrees of freedom D4, D5, D6, D7, D8 and D9 • Unknown degrees of freedom D1, D2 and D3 6 9 5 3 8 2EI 2 EI 2 1 4 1 21 3 7 22 Beam-Member Stiffness Matrix i j 1 4 3 6 E, I, A, L 2 5 k41 k14 AE/L = k k11 = AE/L AE/L AE/L 44 d1 = 1 d4 = 1 12 3 45 6 1 AE/L - AE/L 2 0 0 [k] = 3 0 0 4 -AE/L AE/L 5 0 0 6 0 0 23 i j 1 4 3 6 6EI/L2 = k E, I, A, L 32 2 2 5 6EI/L = k65 2 2 k62 = 6EI/L 6EI/L = k35 d2 = 1 d5 = 1 3 3 12EI/L = k52 12EI/L = k25 k = 12EI/L3 3 22 12EI/L = k55 12 3 45 6 1 AE/L 0 - AE/L 0 2 0 12EI/L3 0 - 12EI/L3 [k] = 3 0 6EI/L2 0 - 6EI/L2 4 -AE/L0 AE/L 0 5 0 -12EI/L3 0 12EI/L3 6 0 6EI/L2 0 - 6EI/L2 24 i j 1 4 3 6 E, I, A, L 2 5 k33 = 4EI/L = k 2EI/L = k63 = k 4EI/L 66 d3 = 1 2EI/L 36 d6 = 1 k = 2 2 2 2 23 6EI/L 6EI/L = k53 k26 = 6EI/L 6EI/L = k56 12 3 45 6 1 AE/L 0 0 - AE/L 0 0 2 0 12EI/L3 6EI/L2 0 - 12EI/L3 6EI/L2 [k] = 3 0 6EI/L2 4EI/L 0 - 6EI/L2 2EI/L 4 -AE/L0 0 AE/L 0 0 5 0 -12EI/L3 -6EI/L2 0 12EI/L3 -6EI/L2 6 0 6EI/L2 2EI/L 0 - 6EI/L2 4EI/L 25 • Member Equilibrium Equations i j Fxi F M xj i Mj E, I, A, L F = yi Fyj AE/L AE/L AE/L AE/L x δi x δj 1 1 + + 6EI/L2 6EI/L2 6EI/L2 6EI/L2 1 1 x ∆i x ∆j 3 3 3 12EI/L 12EI/L + 12EI/L + 12EI/L3 4EI/L 1 2EI/L 2EI/L 4EI/L x θi x θj 1 + 6EI/L2 6EI/L2 6EI/L2 2 FF 6EI/L F yi FFF FFF yj FFF FF xi xj M ii FF M jj 26 F Fxi = (AE / L)δ i + (0)∆i (0)θi + (−AE / L)δ j + (0)∆j + (0)θ j + Fxi 3 2 3 2 F Fyi = (0)δ i + (12EI / L )∆i (6EI / L )θi (0)δ j (−12EI / L )∆j (6EI / L )θ j Fyi 2 2 F Mxi = (0)δ i (6EI / L )∆i (4EI / L)θi (0)δ j (−6EI / L )∆j (2EI / L)θ j Mi F Fxj = (−AE / L)δ i (0)∆i (0)θi (AE / L)δ j (0)∆j (0)θ j Fxi 3 2 2 F Fyj = (0)δ i (−12EI / L )∆i (−6EI / L )θi (0)δ j (0)∆j (−6EI / L )θ j Fyj 2 2 F Mj = (0)δ i (6EI / L )∆i (2EI / L)θi (0)δ j (−6EI / L )∆j (4EI / L)θ j Mj F Fxi AE/L 0 0 − AE/L 0 0 δ i Fxi 3 2 3 2 F F 0 12EI/L 6EI/L 0 −12EI/L 6EI/L ∆ F yj i yi 2 2 F Mi 0 6EI/L 4EI/L 0 − 6EI/L 2EI/L θ i Mi = + F δ F xj − AE/L 0 0 AE/L 0 0 j Fxj F 0 −12EI/L3 − 6EI/L2 0 12EI/L3 − 6EI/L2 ∆ F F yj j yi M 2 2 θ F j 0 6EI/L 2EI/L 0 − 6EI/L 4EI/L j M j Stiffness matrix Fixed-end force matrix [q] = [k][d] + [qF] End-force matrix Displacement matrix 27 6x6 Stiffness Matrix δi ∆i θi δj ∆j θj Ni AE/L 0 0 − AE/L 0 0 V 3 2 3 2 i 0 12EI/L 6EI/L 0 −12EI/L 6EI/L 2 2 Mi 0 6EI/L 4EI/L 0 − 6EI/L 2EI/L []k 6×6 = Nj − AE/L 0 0 AE/L 0 0 3 2 3 2 Vj 0 −12EI/L − 6EI/L 0 12EI/L − 6EI/L 2 2 Mj 0 6EI/L 2EI/L 0 − 6EI/L 4EI/L 4x4 Stiffness Matrix ∆i θi ∆j θj 3 2 3 2 Vi 12EI/L 6EI/L −12EI/L 6EI/L M 6EI/L2 4EI/L − 6EI/L2 2EI/L []k = i 4×4 3 2 3 2 Vj −12EI/L − 6EI/L 12EI/L − 6EI/L 2 2 Mj 6EI/L 2EI/L − 6EI/L 4EI/L 28 2x2 Stiffness Matrix θi θj Mi 4EI / L 2EI / L []k 2×2 = Mj 2EI / L 4EI / L Comment: - When use 4x4 stiffness matrix, specify settlement.