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By Tai Hun Kwon

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5 Bandwidth The global stiffness matrix is usually a banded and partially-filled (sparse) matrix. The nodal numbering system will determine the bandwidth. Bandwidth is also closely related with the storage of computer memory. “How to store the matrix” is one of issues in conjunction with the solution scheme. f. for each node. © 2005 by T. H. Kwon 33 Minimization of bandwidth Bandwidth depends on the nodal numbering system. bandwidth with an appropriate nodal numbering. f.. f. before proceeding the assembly procedure.

Kwon 54 Assembly procedure is a matter of summation of scalar quantities yielding δΠ = ∑ δΠ e 2 e = δy i (K ij y j − Fi ) = 0 for any δy i which provide us with the following global matrix equation: [K ]{y} = {F } {F } being the work equivalent nodal force3. Consider specifically a linear element for simplicity. y e ( x) = N 1 ( x) y1e + N 2 ( x) y 2e N 1 (ξ) = 1 − ξ 1 N 2 (ξ) = ξ with ξ≡ ∴ 1 ξ xl called a normalized coordinate. le dN i dN i dξ 1 dN i = = dx dξ dx l e dξ N2(ξ) N1(ξ) where the shape functions can be found as ξ=0 dN 1 1 =− , dx le ⇒ ξ=1 dN 2 1 = dx le 1 − 1⎤ l e ⎣− 1 1 ⎥⎦ [K ]e = T ⎡⎢ { f }e = ∫0 le ⎧N ⎫ wl ⎧1⎫ w⎨ 1 ⎬dx = e ⎨ ⎬ 2 ⎩1⎭ ⎩N 2 ⎭ 3 In this particular case, three elements yields the global matrix equation as follows: 2 The assembly procedure is in fact the same as before, identifying the global nodal number and add the element stiffness matrix coefficient to the corresponding row and column in the global stiffness matrix.

F. {x c } are sufficient to prevent rigid body motion. Then the unknown {xu } can be obtained by {xu } = [K11 ]−1 {{Fc } − [K12 ]{xc }} Then the unknown load can be determined as {Fu } = [K 21 ]{xu } + [K 22 ]{xc } Instead of following the procedure of rearranging and partitioning matrix described above, one can find several other methods of introducing boundary conditions, which will be discussed next. 7 Methods of introducing boundary conditions We have a global matrix equation [K ]{x} = {F } © 2005 by T.

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