By Jacob Fish, Ted Belytschko

ISBN-10: 0470035803

ISBN-13: 9780470035801

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

**Example text**

E3 ; e4 "1 "E ¼ u d " u2 ! ¼ " p1 " p2 ! ¼ "e1 "e2 ! ¼ 0 10=kð1Þ ! Partitioning above after two rows and columns gives kð3Þ þ kð4Þ þ kð5Þ Àkð4Þ ! ! 10 Àkð5Þ 0 Àkð4Þ d ¼ À F 0 kð1Þ þ kð2Þ þ kð4Þ kð1Þ Àkð2Þ Letting ke ¼ 1 for e ¼ 1 to 5 and solving above gives ! ! 1, consist of bar elements positioned at arbitrary angles in space joined by pin-like joints that cannot transmit moments. In order to analyze such general truss 2 Recommended for Structural Mechanics Track. 12 A two-dimensional truss element in the local coordinate system x0e 1 , y1 .

1 À1 ~ ð4Þ )K À1 1 Kð4Þ ¼ kð4Þ Element 5, I ¼ 3, J ¼ 2: 1 À1 ~ ð5Þ )K À1 1 Assembled system matrix: 2 kð1Þ þ kð3Þ 5 X 6 e 0 ~ ¼6 K¼ K 4 Àkð3Þ e¼1 Àkð1Þ 0 0 0 0 0 kð4Þ 0 Àkð4Þ 2 ! Kð5Þ ¼ kð5Þ 0 60 6 ¼4 0 0 0 0 6 0 kð5Þ ¼6 4 0 Àkð5Þ 0 0 3 0 0 7 7: Àkð4Þ 5 kð4Þ 3 0 07 7: 05 0 0 Àkð5Þ kð5Þ 0 Àkð3Þ Àkð5Þ kð3Þ þ kð4Þ þ kð5Þ Àkð4Þ 0 kð2Þ þ kð5Þ Àkð5Þ Àkð2Þ 27 3 Àkð1Þ 7 Àkð2Þ 7: ð4Þ 5 Àk kð1Þ þ kð2Þ þ kð4Þ Equations for the mechanical system: ------------- 2 3 2 3 r1 " # 6 7 " ! 6 7 ð2Þ ð5Þ ð5Þ ð2Þ 6 7 dE rE r 0 k þ k Àk Àk 6 27 6-------------------------------------------7 ¼ ¼6 7 6 Àkð3Þ 7 ð5Þ ð3Þ ð4Þ ð5Þ ð4Þ 405 f Àk k þk þk Àk F 4 5 dF kð1Þ þ kð3Þ 0 Àkð1Þ Àkð2Þ Àkð3Þ Àkð1Þ Àkð4Þ kð1Þ þ kð2Þ þ kð4Þ 0 where the solution matrix for mechanical, piping and electrical systems is dF ¼ u3 u4 !

Number the elements and nodes. Assemble the global stiffness and force matrix. Partition the system and solve for the nodal displacements. Compute the reaction forces. 1. 17 is: k¼ 5Etab ; ða þ bÞl where E is the Young’s modulus and t is the width of the bar (Hint: subdivide the bar with a square hole into 3 elements). 2. 18. Nodes A and B are fixed. A force equal to 10 N acts in the positive x-direction at node C. Coordinates of joints are given in meters. Young’s modulus is E ¼ 1011 Pa and the cross-sectional area for all bars are A ¼ 2 Á 10À2 m2 .

### A First Course in Finite Elements [With CDROM] by Jacob Fish, Ted Belytschko

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