JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

Course Objectives

The objectives of this course is to impart knowledge of

• 1. About the fundamentals of domain discretization, interpolation, application of boundary conditions, assembly of global matrices, and solution of the resulting algebraic systems.
• 2. Understand the core concepts of variational and weighted residual methods in FEM.
• 3. Derive the element stiffness matrix for 1-D, 2-D and 3-D problems.
• 4. Formulate the simple structural problems in to finite elements.

Course Outcomes

At the end of this course, students will be able to

• 1. Build and analyse the FEA models for various engineering problems.
• 2. Identify the requirements and sources for analysis, design and evaluation.
• 3. Use the standard finite element software to solve the structural engineering problems.
• 4. Interpret the results obtained from FEA software, and arrive at the conclusions

UNIT - I

Introduction to FEM: Types of Problems – Types of Materials – Elastic / Inelastic situations – Types of forces: Body forces / Surface Traction / Point loads – Deformable bodies – Types of Deformations – Homogeneous / Non homogeneous Problems – Equations of equilibrium for elastic 2-D / 3-D continua - Equilibrium equations for 2-D / 3-D boundary elements – Boundary conditions – Strain-displacement relation for 2-D / 3-D – Stress-strain relation for 2-D / 3-D – Plane stress / Plane strain problems. Virtual Work Formulation: Application to problems of plane trusses with static indeterminacy not exceeding three.

UNIT - II

Variational Formulation: Approximate methods of Analysis- Weighted residual method - Rayleigh-Ritz Method -Strong form weak form -Variational principle - Stationarity Functional or Differential equation Finite element formulation for1-D problems: Minimum Potential Energy Approach, week form approach, introduction to natural coordinates -Finite element approximations in one dimension- Lagrangian approximation-Hermitian approximations, FE formulation for Axial bar, Euler Bernoulli beam -Numerical Examples Finite element formulation for 2-D problems: FE Approximation in 2-Dimension, Pascals triangle, Convergence criterion, Compatible and incompatible elements, FE Formulation for plane stress, plane strain and Axi-symmetrical problems, Shape functions for 2-Dimensional CST Element-4 noded quadrilateral element -Highier order triangular and rectangular elements- Consistent Nodal load vector -Numerical Examples

UNIT - III

Iso-parametric elements: Quadrilateral elements: FE Formulation for linear and quadratic isoparametric elements- Construction of shape functions using natural coordinates/Strain-displacement matrices/Load matrices for body force and surface traction/ Expressions for stiffness matrix, load matrices for 4-noded quadrilateral elements/ Gauss Quadrature of numerical integration / Problems with rectangular elements, kinematic indeterminacy not exceeding three- Determination of shape functions for 2nd order quadrilateral elements and for elements of with serendipity / Strain-displacement matrices / Load matrices for body force and surface traction.

UNIT - IV

Finite element formulation for 3 -D elements:

FE Formulation for Tetrahedral and Hexahedral elements: Volume coordinates, Strain-displacement matrix, stiffness matrix, load matrices due to body force and surface traction/ introduction to Hexahedron

• (brick) elements Galerkin’s Method of Weighted Residuals – Application to problems of mathematics / structural engineering, number of trial functions not exceeding two. Weak form of Trial Function - Application to problems of mathematics / structural engineering, number of elements limited to two - Strain-displacement relationship/stress-strain relationship / determination of stiffness matrix for 3-noded ring element and load matrices for body force and surface traction/ Problems with kinematic indeterminacy not exceeding three for 3-noded ring elements only

UNIT - V

Numerical examples: Simple 1-D model, 2-D model and a 3-D model/ analysis and post processing of the results using commercially available FEA software and available codes.

TEXT BOOKS:

• 1. Reddy, J. N, (1993). ―An Introduction to the Finite Element Methodǁ, McGraw Hill, New York.
• 2. Cook, R. D. (1981). ―Concepts and Application of Finite Element Analysisǁ, John Wiley and Sons.
• 3. Zienkiewicz, O. C. And Taylor, R. L, (1989). ―The Finite Element Methodǁ, Vol.1, McGraw Hill Company Limited, London.
• 4. Chandrupatla, T. R. And Belegundu, A. D, (2001). ―Introduction to Finite Elements in Engineeringǁ, Prentice Hall of India, New Delhi.
• 5. Seshu. P, (2003). ―Finite Element Analysisǁ, Prentice Hall of India Private Limited, New Delhi.
• 6. David V. Hutton, (2005). ―Fundamentals of Finite Element Analysisǁ, Tata McGraw-Hill Publishing Company Limited, New Delhi.
• 7. Bathe, K. J, (2006). ―Finite Element Proceduresǁ, Prentice Hall of India, New Delhi M. Tech in COMPUTER AIDED STRUCTURAL ENGINEERING I YEAR II – SEMESTER