By Soubbaramayer, J. P. Boujot

ISBN-10: 3540139176

ISBN-13: 9783540139171

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**Example text**

50) is proved. The above proof is based on the fact that any two valid representations of the element must be energetically consistent, that is, the element must absorb the same amount of energy irrespective of the coordinate system in which it is described. One way of explaining this is that energy is a scalar, so it is independent of the alignment of the coordinate system. Scalar physical variables like pressure, temperature and energy do not depend on the coordinate system that is chosen. Furthermore, energy has to be independent of what generalized deformation modes are used to describe the deformation of the system.

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. The process is identical to what we did for bar elements. fþr¼ nel X LeT Fe by Equation ð2:38Þ LeT Ke de by Equation ð2:36Þ e¼1 ¼ nel X e¼1 ¼ nel X LeT Ke Le d by Equation ð2:37Þ: e¼1 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} K As indicated by the underscore, the assembled system matrix is given by K¼ nel X LeT Ke Le : ð2:39Þ e¼1 This system is obtained by a sequence of scatter and add operations, which corresponds to direct assembly.

C. d. Number the elements and nodes. Assemble the global stiffness and force matrix. Partition the system and solve for the nodal displacements. Compute the stresses and reactions. 3. 19. The Young’s modulus is E ¼ 1011 Pa, the cross-sectional area of the bar BC is 2 Â 10À2 m2 and that of BD and BF is 10À2 m2 . Note that point D is free to move in the x-direction. Coordinates of joints are given in meters. a. Construct the global stiffness matrix and load matrix. b. Partition the matrices and solve for the unknown displacements at point B and displacement in the x-direction at point D.

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