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Unformatted text preview: L δ _nodes C 4 H x C General Method for Deriving an Element Stiffness Matrix step I: select suitable displacement function beam likely to be polynomial with one unknown coefficient for each (of four) degrees of freedom v x ⋅ v 1 say ... ( ) = C 1 + x C 2 + x 2 ⋅ C 3 + x 3 ⋅ C 4 dof = δ = v' 1 in matrix notation: C 1 v' 2 C 3 ( ) := H x v 2 C := C 2 ( ) := ( 1 x x 2 x 3 ) v x ( ) ⋅ C 5.3.6 C 4 v x ⋅ v x ⋅ ⋅ ( ) → C 1 + x C 2 + x 2 ⋅ C 3 + x 3 ⋅ C 4 d ( ) → C 2 + 2 x ⋅ C 3 + 3 x 2 ⋅ C 4 dx v x v ( ) ( ) → C 1 + x C 2 + x 2 ⋅ C 3 + x 3 ⋅ C 4 δ C 1 ( ) → C 1 + L C 2 + L 2 ⋅ C 3 + L 3 ⋅ C 4 ⋅ ⋅ δ x ( ) := d v x δ x ( ) → δ L dx ( ) C 2 + 2 x ⋅ C 3 + 3 x 2 ⋅ C 4 C 2 C 2 + 2 L ⋅ C 3 + 3 L 2 ⋅ C 4 ⋅ ⋅ ⋅ ⋅ in matrix form: C 1 for manipulation ( ) = 1 x x 2 x 3 ⋅ C 2 δ _over_C x δ x ( ) := 1 x x 2 x 3 1 2x 3x 2 C 3 1 2x 3x 2 5.3.7 C 4 step II: relate general displacements within element to its nodal displacement δ _over_C 1 ( ) → 1 L L 2 L 3 ( ) → 1 δ _over_C L 1 2 L 3 L 2 ⋅ ⋅ in single matrix form: v 1 v 1_p define A such that δ _nodes = A ⋅ C δ _nodes = v 2 v 2_p 1 1 form by stacking ( ) , δ _over_C( ) ) A → 1 1 node 1 with node 2 A := stack ( δ _over_C L 1 L L 2 L 3 A = 1 L L 2 L 3 ⋅ ⋅ ⋅ 1 2 L 3 L 2 1 L 3 L 2 L 4 δ _nodes = A ⋅ C => C := A − 1 ⋅ δ _nodes A − 1 → 1 ⋅ L 4 5.3.8b ⋅ L 4 − 3 ⋅...
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 Spring '03
 DavidBurke
 δ, Finite element method in structural mechanics, Wint, E⋅ I⋅

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