Tutorial Exercise 4: Finite Element Method 1. Consider a bar made from a linear elastic material with Young’s modulus E, cross sectional area A, and length L. The bar is loaded at one end by a force of magnitude F. The other end is fixed. Treat the problem as one-‐dimensional and solve using two linear finite elements. Compare the finite element solution with the finite volume solution from Tutorial 4 and the exact theoretical solution. 2. Construct the actual (not the parent) shape functions for a 2-‐D quadrilateral element with 4 nodes, and check that the shape functions satisfy all the necessary conditions. Assume shape functions of the following form: 𝑁! 𝑥, 𝑦 = 𝑎! + 𝑏! 𝑥 + 𝑐! 𝑦 + 𝑑! 𝑥𝑦. y= 2 x = 4 Figure 1. 3. Consider a steel disc rotating with an angular velocity of 100 rad/sec (See Figure 2). By considering quadrilateral element (2 x 4 m) illustrated in Figure 2, calculate the body force on the element using: (a) Single point numerical Gauss quadrature; (b) 2 × 2 point numerical quadrature; (c) Analytic Integration. ω = 100 rad/sec mm y = 4 y x x = -‐1 x = 1 Figure 2
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