Question

Q2 (a) (0) Explain what is meant by interpolation in the Finite Element Method and why it is used (3 marks) What is a shape function? (3 marks) PLEASE TURN OVER 16363,16367 Page 2 of 3 0.2 (a) (Continued) (iii) For an isoparametric element, explain the relationship between shape functions, the geometry of the element and the shape the loaded element will deform to. (3 marks) (iv) Describe the relationship between structural equilibrium and the minimum potential energy state. (3 marks) (b) wwwwww F FIGURE Q2b: 3 Springs (0) Derive an expression for the Total Potential Energy of the structure shown above in Q2b. (5 marks) (ii) Apply the minimum potential energy method to derive the governing equilibrium equations of the structure in terms of the joint displacements. (3 marks) (c) A vertical uniform bar of length L and uniform cross section A is held at one end. The bar has Young's Modulus E and density p. It is subject to gravity g in the vertical direction. It is modelled using a 3 node 10 bar element with the shape function 2r? [N] 2x x Determine the consistent load vector for the element. (4 marks) (d) Describe the processing steps required to carry out a practical Finite Element Analysis of an engineering component using commercial software such as ANSYS. 6 marks)

Answer 1

(i) In finite element method , the geometry is split into many discrete elements with a number of nodepoints for solving the analysis. Interpolation is used to derive a simple function from all the data points , so that function passes through all the points. So ,interpolation functions are used to derive equations for different sets of nodes to solve the analysis easily.

(ii) Shape functions are the polynomial functions(approximations) that are chosen to interpolate the field variables in the domain of the element ( such as Triangle, box, Tetrahedral) .They establish the relationship between the displacement at any point in the element and the nodal displacement with respect to it . Generally, low order polynomials are used as shape functions.