HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY UNIVERSITY OF MECHANICS Finite Element Methods Prepared by Prof. M.C. Nguyen, Dept of Mechatronics 1 UNIT - 1 Introduction to FEM: Stiffness equations for a axial bar element in local co-ordinates using Potential Energy approach and Virtual energy principle – Finite element analysis of uniform, stepped and tapered bars subjected to mechanical and thermal loads – Assembly of Global stiffness matrix and load vector – Quadratic shape functions – properties of stiffness matrix 2 Axially Loaded Bar Review: Stress: Stress: Strain: Strain: Deformation: Deformation: 3 Axially Loaded Bar Review: Stress: Strain: Deformation: 4 Axially Loaded Bar – Governing Equations and Boundary Conditions Differential Equation d  du  EA(x)  f (x)  0   dx  dx  0  x L Boundary Condition Types prescribed displacement (essential BC) •prescribed force/derivative of displacement (natural BC) 5 Axially Loaded Bar –Boundary Conditions Examples fixed end simple support free end 6 Potential Energy Elastic Potential Energy (PE) Spring case Unstretched spring PE  0 Stretched bar 1 2 PE  kx 2 x Axially loaded bar undeformed: deformed: Elastic body PE  PE  0 L 1 PE   Adx 20 1 T σ εdv  2V 7 Potential Energy Work Potential (WE) WP    u  fdx  P  u B f: distributed force over a line P: point force u: displacement 0 Total Potential Energy Adx  u  fdx  P  u Principle of Minimum Potential Energy For conservative systems, of all the kinematically admissible displacement fields, those corresponding to equilibrium extremize the total potential energy. If the extremum condition is a minimum, the equilibrium state is stable. 8 Potential Energy + Rayleigh-Ritz Approach Example: Step 1: assume a displacement field u   aii x  i  1 to n i  is shape function / basis function n is the order of approximation Step 2: calculate total potential energy 9 Potential Energy +