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Vachana Pitama Dr.P.G.Halakatti College of Engineering & Technology, Bijapur-03
DEPARTMENT OF MECHANICAL ENGINEERING
Page 1 of 89
Vachana Pitamaha Dr. P.G. Halakatti College of Engineering & Technology,
Bijapur 586 103
Department of Mechanical Engineering
Semester VI
Course Title: Finite element method (15ME61)
2017-2018
COURSE FILE
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Vachana Pitama Dr.P.G.Halakatti College of Engineering & Technology, Bijapur-03
DEPARTMENT OF MECHANICAL ENGINEERING
Page 2 of 89
Program Educational Objectives (PEOs)
The educational objectives of the Mechanical Engineering Program are to prepare our graduates to:
1. Establish a successful career in Mechanical Engineering or related fields in Industry and
other organizations where an engineering approach to problem solving is highly valued.
2. Develop the ability among the students to synthesize the data and technical concepts for
applications to the product design.
3. Contribute significantly in a multidisciplinary work environment with high ethical standards
and with understanding of the role of engineering in economy and the environment.
4. Excel in graduate study and research, reaching advanced degrees in engineering and related
disciplines.
5. Achieve success in professional development through life-long learning.
Program outcomes (POs)
a. an ability to apply knowledge of mathematics, science, and Mechanical Engineering b. an ability to design and conduct experiments, as well as to analyze and interpret data c. an ability to design a mechanical system, mechanical component, or process to meet desired needs within realistic constraints such as economic, environmental, social, political, ethical, health and safety, manufacturability, and sustainability d. an ability to function on multidisciplinary teams e. an ability to identify, formulate, and solve mechanical engineering problems f. an understanding of professional and ethical responsibility g. an ability to communicate effectively h. the broad education necessary to understand the impact of mechanical engineering solutions in a global, economic, environmental, and societal context i. a recognition of the need for, and an ability to engage in life-long learning, j. a knowledge of contemporary issues k. an ability to use the techniques, skills, and modern mechanical engineering tools necessary for engineering practice.
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Vachana Pitama Dr.P.G.Halakatti College of Engineering & Technology, Bijapur-03
DEPARTMENT OF MECHANICAL ENGINEERING
Page 3 of 89
COURSE PLAN
Semester: VI Year: 2017-18
Subject: Finite Element Method Subject Code: 15ME61
Total No. of Lecture Hours: 54 I A Marks : 20
Exam Marks: 80 Exam Hours: 03
Lesson plan prepared by : Prof. S.S.Chappar
Prof. R.K.Kanakaraddi
Date:5/1/2018
COURSE CONTENT
Module I
Introduction to Finite Element Method:
General description of the finite element method. Engineering applications of finite element method.
Boundary conditions: homogeneous and nonhomogeneous for structural, heat transfer and fluid flow
finite element formulation. Convergence criteria, Discretisation process, Types of elements: 1D, 2D and
3D, Node numbering, Location of nodes. Strain displacement relations, Stress strain relations, Plain stress
and Plain strain conditions, temperature effects. Interpolation models: Simplex, complex and multiplex elements, linear interpolation polynomials in
terms of global coordinates 1D, 2D, 3D Simplex Elements. 12Hours
Module II
One-Dimensional Elements-Analysis of Bars and Trusses: Linear interpolation polynomials in terms
cubic elements in natural coordinates, , , Constant strain triangle, Four-Nodded Tetrahedral Element (TET
4), Eight-Nodded Hexahedral Element (HEXA 8), 2D isoparametric element, Lagrange interpolation
functions, Numerical integration: Gaussian quadrature one point, two point formulae, 2D integrals. Fore
terms: Body force, traction force and point loads,
Numerical Problems: Solution for displacement, stress and strain in 1D straight bars, stepped bars and
tapered bars using elimination approach and penalty approach, Analysis of trusses 12Hours
Module III
Beams and Shafts: Boundary conditions, Load vector, Hermite shape functions, Beam stiffness matrix
based on Euler-Bernoulli beam theory, Examples on cantilever beams, propped cantilever beams,
Numerical problems on simply supported, fixed straight and stepped beams using direct stiffness method
with concentrated and uniformly distributed load.
Torsion of Shafts: Finite element formulation of shafts, determination of stress and twists in circular
shafts. 08 Hours
Module IV
Heat Transfer: Basic equations of heat transfer: Energy balance equation, Rate equation: conduction,
convection, radiation, energy generated in solid, energy stored in solid, 1D finite element formulation
using vibrational method, Problems with temperature gradient and heat fluxes, heat transfer in composite
sections, straight fins. Fluid Flow: Flow through a porous medium, Flow through pipes of uniform and
stepped sections. 12 Hours
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Vachana Pitama Dr.P.G.Halakatti College of Engineering & Technology, Bijapur-03
DEPARTMENT OF MECHANICAL ENGINEERING
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Module V
Axi-symmetric Solid Elements: Derivation of stiffness matrix of axisymmetric bodies with triangular elements, Numerical solution of axisymmetric triangular element(s) subjected to point loads. Dynamic Considerations: Formulation for point mass, Consistent element mass matrix of one
dimensional bar element, truss element, lumped mass matrix of bar element, truss element. 10 Hours
Text Books:
1. Logan, D. L., A first course in the finite element method,6th Edition, Cengage Learning, 2016.
2. Rao, S. S., Finite element method in engineering, 5th Edition, Pergaman Int. Library of
Science, 2010.
3. Chandrupatla T. R., Finite Elements in engineering, 2nd Edition, PHI, 2013.
Reference Books:
- McGraw -Hill International Edition.Bathe K. J. Finite Elements Procedures, PHI. - 4th Edition, Wiley & Sons, 2003.
Prerequisites:
Elementary mathematics, Mechanics of materials.
Course Description:
VTU Belgaum.
Basically the subject deals with
Basics of finite element methods Detailed procedure in finite element analysis Applications of finite element method
Course outcomes:
Upon successful completion of this course you should be able to: 1. Understand the concepts behind formulation methods in FEM.
2. Identify the application and characteristics of FEA elements such as bars, beams, plane and
iso-parametric elements.
3. Develop element characteristic equation and generation of global equation.
4. Able to apply suitable boundary conditions to a global equation for bars, trusses, beams, circular shafts, heat transfer, and fluid flow, axi symmetric and dynamic problems and solve them displacements, stress and strains induced Relevance of the course: The primary function of design engineer is to give size and shape to components of machine elements. While doing so he has to check its safety by finding the stress
distribution in that element. If the geometry, material properties and loading conditions are
B.L.D.E.A͛s
Vachana Pitama Dr.P.G.Halakatti College of Engineering & Technology, Bijapur-03
DEPARTMENT OF MECHANICAL ENGINEERING
Page 5 of 89
simple the formulae of mechanics of materials can be used to analyze the stress. If these are complicated, in the absence of exact methods he has to go for approximate methods like finite element methods. As the computational capabilities of modern computers are very high, the use of FEM is vital in industries. So it is essential to know the details of finite element analysis.
Application Areas:
Stress analysis Fluid flow analysis Heat transfer Computational fluid dynamics
Unit wise plan
Course Title / Code: Finite Element Methods (15ME61) Module: 1 Introduction to Finite Element Method & Interpolation models
Planned hours: 12
Learning Objectives:
At the end of the Unit, the student should be able to;
1. Explain the basics of theory of elasticity.
2. Discuss the need of FEA
3. Understand steps in FEA
4. Apply potential energy and virtual work methods to formulate.
5. Discuss the method of discretization.
6. Explain types and size of elements
7. Discuss interpolation models for different applications.
Lesson Plan:
Lesson
No. Topics covered Teaching
Method
POs attained COs attained
Text/Reference
Book/Chapter
No. L1
General description of the finite
element method. Engineering applications of finite element method
Chalk &
Board
a,b,e,k
1 T1,T2,T3,R1
L2 Boundary conditions: homogeneous
and nonhomogeneous for structural,
Chalk &
Board
1 T1,T2,T3,R1
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Vachana Pitama Dr.P.G.Halakatti College of Engineering & Technology, Bijapur-03
DEPARTMENT OF MECHANICAL ENGINEERING
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heat transfer and fluid flow problems.
L3 Chalk &
Board
1 T1,T2,T3,R1
L4 Displacement method of finite
element formulation.
Chalk &
Board
1 T1,T2,T3,R1
L5 Convergence criteria, Discretisation
process,
Chalk &
Board
1 T1,T2,T3,R1
L6 Types of elements: 1D, 2D and 3D, Chalk &
Board
2 T1,T2,T3,R1
L7 Node numbering, Location of nodes. Chalk &
Board
2
L8 Strain displacement relations, Stress
strain relations,
Chalk &
Board
2 T1,T2,T3,R1
L9 Plain stress and Plain strain
conditions, temperature effects
Chalk &
Board
2 T1,T2,T3,R1
L10 Simplex, complex and multiplex
elements,
Chalk &
Board
2 T2,T3,R1 L11
Linear interpolation polynomials in
terms of global coordinates 1D, 2D,
3D Simplex Elements
Chalk &
Board
2 T2,T3,R1
L12 Simple numericals Chalk &
Board 2 T2,T3,R1
Assignment Questions COs attained
Explain the steps in FEA 1
Discuss the applications of FEA 1
Explain essential and natural boundary conditions with examples 1 Derive the expression for total potential energy for one dimensional bar subjected to an axial force. 1 Obtain the equilibrium equation of the system shown in figure using principle of minimum potential energy 1 K1 K2 K3 F
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Vachana Pitama Dr.P.G.Halakatti College of Engineering & Technology, Bijapur-03
DEPARTMENT OF MECHANICAL ENGINEERING
Page 7 of 89
Find the stress at x=0 and displacement at the midpoint of the rod as shown in figure. Use Rayligh Ritz method 1 1 Compute the value of central deflection for the beam shown in figure, considering trigonometric functions and Rayligh Ritz method. 1 Obtain the stress strain relations in Plane stress problem 1 Explain descretization process and different types of elements with sketches.[element library] 2 Explain simplex, complex and multiplex elements with sketches 2
Explain node location scheme 2
L/2 L/2
P E, I
1 unit
1 unit
2 units X
Take E=1 unit
A=1 unit
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Vachana Pitama Dr.P.G.Halakatti College of Engineering & Technology, Bijapur-03
DEPARTMENT OF MECHANICAL ENGINEERING
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Course Title / Code: Finite Element Methods (15ME61) Module: II One-Dimensional Elements-Analysis of Bars and Trusses Planned hours: 12
Learning Objectives:
At the end of the Unit, the student should be able to;
1.Derive shape functions for different elements.
2. Determine displacement, stress, strain and reactions in different bars.
3. Determine displacement and stress in different strusses.
Lesson Plan:
Lesson.
No. Topics covered Teaching
Method
POs
Attained
COs
Attained
Reference
book/Chapter no. L13
Linear interpolation polynomials
1D, 2D elements.
Chalk and
Board
a,b,e,k
2,3 T2,T3,R1
L14
Higher order interpolation
functions for 1D quadratic and cubic elements in natural coordinates
Chalk and
Board 2,3 T2,T3,1
L15
Constant strain triangle, Four-
Nodded Tetrahedral Element
(TET 4),
Chalk and
Board 2,3 T2,T3,R1
L16
Eight-Nodded Hexahedral
Element (HEXA 8), 2D
isoparametric element
Chalk and
Board 2,3 T2,T3,R1
L17 Lagrange interpolation functions Chalk and
Board 2,3 T2,T3,R1
L18
Numerical integration: Gaussian
quadrature one point, two point formulae, 2D integrals
Chalk and
Board 2,3 T2,T3,R1
L19
Force terms: Body force, traction
force and point loads,
Chalk and
Board 4 T2,T3,R1
L20 Solution for displacement, stress
and strain in 1D straight bars
Chalk and
Board 4 T2,T3,R1
L21 Problems on stepped bars bars. Chalk and
Board 4 T2,T3,R1
L22 Problems on tapered bars. Chalk and
Board 4 T2,T3,R1
L23 Problems on truss. Chalk and
Board 4 T2,T3,R1
L24 Problems on truss. Chalk and
Board 4 T2,T3,R1
Assignment Questions COs
attained
Explain linear, quadratic and cubic models 2
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Vachana Pitama Dr.P.G.Halakatti College of Engineering & Technology, Bijapur-03
DEPARTMENT OF MECHANICAL ENGINEERING
Page 9 of 89
Define shape function. What are the properties of shape function? 2,3 Derive the shape function for bar element in global coordinate system 2,3 Derive the shape function for bar element in natural coordinate system. 2,3 Derive the shape function for quadratic bar element. 2,3 Derive the shape function for quadratic bar element. 2,3
Derive the shape function for CST element 2,3
Derive the shape function for four noded tetrahedral element 2,3 Derive the shape function for eight noded hexahedral element 2,3 Find the values of following integrals using 2point and 3point Gauss quadrature methods a)
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