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**1. Introduction:**

1.1 Introductory Concepts: Introduction to FEM, Historical Background, General FEM procedure, Applications of FEM in various fields Advantages and disadvantages of FEM

1.2 Mathematical Modelling of field problems in engineering, Governing equations, Differential equations in different fields

1.3 Approximate solution of differential equations, Weighted residual techniques, Boundary value problems

**2. FEA Procedure:**

2.1 Discrete and Continuous Models, Weighted Residual Methods - Ritz Technique- Basic Concepts of the, Finite Element Method

2.2 Definitions of various terms used in FEM like element, order of the element, internal and external node/s, degree of freedom, primary and secondary variables, boundary conditions.

2.3 Minimization of a functional, Principle of minimum total potential, Piecewise Rayleigh-Ritz method, Formulation of 'stiffness matrix', transformation and assembly concepts

**3. One Dimensional Problems:**

3.1 One dimensional second order equations - discretization-element types - linear and higher order elements -derivation of shape functions and stiffness matrices and force vectors

3.2 Assembly of Matrices- solution of problems in one dimensional structural analysis, heat transfer and fluid flow (stepped and taper bars, fluid network, spring-Cart Systems)

3.3 Analysis of Plane trusses, Analysis of Beams

3.4 Solution of one dimensional structural and thermal problems using FE Software, Selection of suitable element type, modelling, meshing, boundary condition, convergence of solution, result analysis, case studies

**4. Two Dimensional Finite Element Formulations:**

4.1 Introduction, three node triangular element, four node rectangular element, four node quadrilateral element, eight node quadrilateral element

4.2 Natural coordinates and coordinates transformations: serendipity and Lagrange’s methods for deriving shape functions for triangular and quadrilateral element

4.3 Sub parametric, Isoparametric, super parametric elements, Compatibility, Patch test, Convergence criterion, sources of errors

**5. Two Dimensional Vector Variable Problems:**

5.1 Equations of elasticity - Plane stress, plane strain and axisymmetric problems

5.2 Jacobian matrix, stress analysis of CST and four node Quadratic element

**6. Finite Element Formulation of Dynamics and Numerical Techniques:**

6.1 Applications to free vibration problems of rod and beam, Lumped and consistent mass matrices

6.2 Solutions techniques to Dynamic problems, longitudinal vibration frequencies and mode shapes, Fourth order beam equation, transverse deflections and natural frequencies of beams