notes_36_element_stif

notes_36_element_stif - L _nodes C 4 H x C General Method...

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Unformatted text preview: L _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 = = 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 C 1 ( ) C 1 + L C 2 + L 2 C 3 + L 3 C 4 x ( ) := d v x x ( ) 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 _over_C x 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 _over_C 1 ( ) 1 L L 2 L 3 ( ) 1 _over_C L 1 2 L 3 L 2 in single matrix form: v 1 v 1_p define A such that _nodes = A C _nodes = v 2 v 2_p 1 1 form by stacking ( ) , _over_C( ) ) A 1 1 node 1 with node 2 A := stack ( _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 _nodes = A C => C := A 1 _nodes A 1 1 L 4 5.3.8b L 4 3...
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notes_36_element_stif - L _nodes C 4 H x C General Method...

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