Answer Any One Question From The Following Two Questions In Section B B1 A Two Node One Dimensional Linear Finite Elem 1 (58.94 KiB) Viewed 28 times
Answer Any One Question From The Following Two Questions In Section B B1 A Two Node One Dimensional Linear Finite Elem 2 (39.67 KiB) Viewed 28 times
Answer any ONE QUESTION from the following TWO questions in Section B. B1. A two-node one dimensional linear finite element with its physical (x-y) co-ordinate system and natural ) co-ordinate system is shown in Figure 1(a) and 1(b), respectively. The element has a uniform cross-sectional area A and the Young's modulus of its material is E. (a) Using any logical method of your choice, develop expressions for interpolation functions in the natural space associated with displacements di and q at node i and j, respectively. [10 Marks] (b) In an analysis using the above element, it is found that the displacement at node i is q = +0.003mm and that at node jis 4 =-0.005mm. Using the interpolation functions you have developed in part (a) above, calculate the displacement vector u, at generic point P on the element [6 marks) (c) The general expression of stiffness matrix for a solid continuum finite element is given by [k] [D]" dV where the various terms have their usual meaning. Using the above relationship, develop an explicit expression of [k] for the element shown in Figure 1. [9 marks) Total [25 marks 0,0 0,0 X; - 20mm -X, - 24mm 4 - 36mm (1) Figure 1
SECTION B: STANDARD FORMULA SHEET Useful Expressions for Two Nodded linear Truss elements k 4,E, 1 1 1 11 f. 1.21 61 212 S. A.B.1.1 2 1 9 1. 4, ( ) 1 E 1 Useful Expressions for Two Nodded Beam finite element: Convention used: downward force +ve, clockwise moment +ve. 12 12 61 41 EI k 61 12 61, 21 61 41 Symmetric 3 7 4 4 1 3 21 3 w 1. NO' 20 3 21 3 1. 4 1 1 Sampling points and Weights for Gaussian Quadrature: No. of Points, n IN 2 3 Location, 0.0 +0.5773502692 +0.7745966692 0 +0.8611363116 +0.3399810436 Weights, W. 2.0 1.0 0.5555555556 0.8888888889 0.3478548451 0.6521451549 4
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