By Jacob Fish, Ted Belytschko

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**Extra resources for A First Course in Finite Elements [With CDROM]**

**Example text**

13 Two-element truss structure. 14 Local (element) and global coordinate systems. 13. The finite element model consists of two elements numbered 1 and 2 and three nodes. Step 2 deals with the formulation of each element starting with element 1. Element 1: Element 1 is numbered with global nodes 1 and 3. 14. Other relations are as follows: cos 90 ¼ 0; sin 2 0 0 6 AE 6 0 1 Kð1Þ ¼ 6 l 40 0 0 À1 ½1 90 ¼ 1; lð1Þ ¼ l; 3 0 0 ½1 0 À1 7 7 7 0 0 5 ½3 0 1 ½3 kð1Þ ¼ Að1Þ Eð1Þ AE ; ¼ l lð1Þ Element 2: Element 2 is numbered with global nodes 2 and 3.

35) in matrix form, we have ! e! 1 1 À1 ie1 e1 ¼ e : ð2:36Þ e i2 ee2 R À1 1 |ﬄ{zﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄ{zﬄ ﬄ} de Ke fe The continuity of the voltage at the nodes is enforced by de ¼ Le d: 1 Recommended for Science and Engineering Track. 10 A resistance element for a circuit and a hydraulic element for a piping network; the nodal flux is positive when it exits the domain of the element. 2. The system equation can then be obtained by enforcing the condition that the sum of the currents at any node is equal to any external sources of currents.

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. 15) are rearranged into larger augmented element stiffness matrices and zeros are added where these elements have no effect. The results are 2 3 2 32 3 0 u1 0 0 0 ð1Þ 4 F2 5 ¼ 4 0 kð1Þ Àkð1Þ 5 4 u2 5 ð1Þ 0 Àkð1Þ kð1Þ u3 F1 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄ{zﬄﬄ} |ﬄﬄﬄﬄ{zﬄﬄﬄﬄ} ð1Þ ð1Þ d ~ ~ K F or ~ ð1Þ d: ~ ð1Þ ¼ K F ð2:16Þ Note that we have added a row of zeros in row 1 corresponding to the force at node 1, as element 1 exerts no force on node 1, and a column of zeros in column 1, as the nodal displacement at node 1 does not affect element 1 directly.