Read e-book online A First Course in Finite Elements PDF

By Jacob Fish, Ted Belytschko

ISBN-10: 0470035803

ISBN-13: 9780470035801

This can be a nice booklet for introductory finite parts. the entire uncomplicated and primary stuff is there. Too undesirable, although, that it really is a nearly note for notice reproduction of the publication through Ottosen and Petersson (1992!). And, as is usually the case, the unique is simply that little bit higher - so minus one celebrity.

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Extra info for A First Course in Finite Elements

Example text

0 1 0 4 5 ¼ u2 ¼ Lð2Þ d; 1 0 0 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} u3 Lð2Þ ð2:20Þ or in general de ¼ Le d: ð2:21Þ The matrices Le are called thegather matrices. The name gather originates from the fact that these matrices gather the nodal displacements of each element from the global matrix. Note that these equations state that the element displacement at a node is the same as the corresponding global displacement, which is equivalent to enforcing compatibility. The matrices Le are Boolean matrices that consist strictly of ones and zeros.

Whenever this equation appears, it indicates assembly of the element matrices into the global matrix (for general meshes, the range of e will be 1 to nel ) . 19), we can see that ~ e ¼ LeT Ke Le : K ð2:26Þ So the stiffness matrix scatter corresponds to pre- and postmultiplications of Ke by LeT and Le , respectively. 25) gives the global equation 2 ð2Þ 32 3 2 3 " r1 u1 k Àkð2Þ 0 4 Àkð2Þ kð1Þ þ kð2Þ Àkð1Þ 54 u2 5 ¼ 4 f2 5: ð2:27Þ u3 f3 0 Àkð1Þ kð1Þ The above system of three equations can be solved for the three unknowns u2 , u3 and r1 as described in the next section.

7(a) and (b), respectively. 13)), in terms of the global nodal displacements of the element. 12) gives " ð1Þ # " # ! F1 kð1Þ Àkð1Þ u3 ¼ : ð2:14Þ ð1Þ u2 Àkð1Þ kð1Þ F2 Notice that we have replaced the nodal displacements by the global nodal displacements. This enforces compatibility as it ensures that the displacements of elements at common nodes are identical. 12) gives " ð2Þ F1 ð2Þ F2 # " ¼ kð2Þ Àkð2Þ Àkð2Þ kð2Þ # ! 13) because the matrices are not of the same size. 15) by adding zeros; we similarly augment the displacement matrices.

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A First Course in Finite Elements by Jacob Fish, Ted Belytschko

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