CVEN4304-TutorialSet5-IsoparametricElements

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THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING CVEN 4304: STRUCTURAL ANALYSIS AND FINITE ELEMENTS CVEN9820 COMPUTATIONAL STRUCTURAL MECHANICS Tutorial Set 5: Isoparametric Element Determine the 3 by 3 stiffness matrix if nodes of the element in figure are uniformly spaced. Let A and E be constant and perform integrations explicitly. Figure 1 Omit the internal node in Figure 1, so that the element becomes a bar with end nodes only. Using natural coordinates , derive the 2 by 2 stiffness matrix of a uniform element. In Figure 1, let 4 / 2 3 L x x and 4 / 3 1 2 L x x . If only node 3 has axial displacement ( 0 2 1 u u , 0 3 u ), what is the strain x at each node. Evaluate [J] and J for each of the four elements shown. Also determine the ratio of element area to the area of a square two units on a side. How is this ratio related to J, and why?

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Unformatted text preview: area to the area of a square two units on a side. How is this ratio related to J, and why? Figure 2 Repeat Question 1, but use two-point Gauss quadrature rather than explicit integration. A quadrilateral element is shown in the figure with the coordinates in millimeters. Let E =210 GPa and v =0.25 and assume plane stress condition. Use a spreadsheet (for example, MS Excel), MATLAB, MAPLE or MATHEMATICA 1. To evaluate the element stiffness matrix using 1-by-1 Gaussian quadrature. 2. To evaluate the element stiffness matrix using 2-by-2 Gaussian quadrature. 3. To determine the stresses at the local coordinates (0,0) when the displacements are given as u 1 =0, v 1 =0; u 2 =0.01mm, v 2 =0.005mm; u 3 =0.015mm, v 3 =0.02mm; u 4 =0, v 4 =0. Figure 3...
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• Three '10
• HneryYip
• Numerical Analysis, Ratio, Chemical element, Finite element method in structural mechanics, SCHOOL OF CIVIL, COMPUTATIONAL STRUCTURAL MECHANICS

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