Introduction FEA.pptx

Introduction to
Finite Element Analysis
By
Dr. N.Vasiraja, M.Tech., Ph.D
Asst. Prof.(Sr.G)/Mech. Engg.
Mepco Schlenk Engg. College, Sivakasi

17-Jan-23 Introduction to Finite Element Analysis 2
FLOW OF PRESENTATION
1. History and Applications of Finite Element Analysis
2. Theory of Elasticity
3. Finite Element Equation of Bar element
4. Finite Element Equation of Truss element
5. Finite Element Equation of Beam element
6. Tutorial related to
1. Bar element
2. Beam element
7. Finite element simulation using ANSYS 15.0
1. Bar element
2. Truss element
3. Beam element
8. Summary

Method for Solving Engineering Problems

Numerical Methods
1. Variational method (Applying Minimum potential energy
Principle)
1. Rayleigh Ritz Method
2. Weighted residual method
1. Point collocation method
2. Subdomain method
3. Least square method
4. Galerkins’ method
3. Finite Difference method
4. Finite element Method
1. By manual
2. By coding using C++, Matlab
3. By FEA packages like ANSYS, NASTRAN, COMSOL, FEAST
etc

Historical Background
 Hrenikoff, 1941 – “frame work method”
 Courant, 1943 – “piecewise polynomial interpolation”
 Turner, 1956 – derived stiffness matrice for truss, beam, etc
 Clough, 1960 – coined the term “finite element”

Advantages of Finite Element Analysis
 Models Bodies of Complex Shape
 Can Handle General Loading/Boundary Conditions
 Models Bodies Composed of Composite and Multiphase
Materials
 Model is Easily Refined for Improved Accuracy by Varying Element
 Size and Type (Approximation Scheme)
 Time Dependent and Dynamic Effects Can Be Included
 Can Handle a Variety Nonlinear Effects Including Material
Behavior, Large Deformations, Boundary Conditions, Etc.

Steps involved in the finite element analysis
(i) Select suitable field variables and the elements.
(ii) Discritise the domain into small.
(iii) Select interpolation functions.
(iv) Find the element properties.
(v) Assemble element properties to get global properties.
(vi) Impose the boundary conditions.
(vii) Solve the system equations to get the nodal unknowns.
(viii) Make the additional calculations to get the required values.

Type of Problems
Equilibrium problems
• Time independent problems, BVPs.
Eigen value problems
• Steady state problems whose solution often requires the
determination of natural frequencies and modes of vibration of
solid and fluids.
• These are special class of BVPs where solution exists for only
certain ‘particular’ or ‘characteristic’ value of the parameter.
Propagation problems
• Time dependent problems, IVPs.

Quiz-1
Who have coined the name Finite Element?
Ans: Clough, 1960

Applications of Finite Element Methods
Structural & Stress Analysis
 Thermal Analysis
 Dynamic Analysis
 Acoustic Analysis
 Electro-Magnetic Analysis
 Manufacturing Processes
 Fluid Dynamics

Application of FEA in Bridges

Applications of FEA in Structural Analysis

Applications of FEA in Aerospace and Automobile

Other applications

Basic Term related to FEA
(i) The basic unknown and field variables which are encountered
in the engineering problems are displacement in solid
mechanics, velocity in fluid mechanics, electric and magnetic
potential in electrical engineering and temperature in heat
flow.
(ii) Finite number by dividing region in to small parts called
elements.
(iii) The field variables specified at points are called nodes.
(iv) DOF – are field variables
(v) Boundary condition
(i) Essential Boundary conditions ( Displacement and slope)
(ii) Natural boundary conditions ( Force and moment)

Basic Concept of the Finite Element Method
Exact Analytical Solution
x
T
Approximate Piecewise Linear Solution
x
T
One-Dimensional Temperature Distribution

Quiz 2
The total number of DOF is six. justify
Ans: Translation along x, y, z direction
Rotation along x, y, z direction

Common Types of Elements
One-Dimensional Elements
Line
Rods, Beams, Trusses, Frames
Two-Dimensional Elements
Triangular, Quadrilateral
Plates, Shells, 2-D Continua
Three-Dimensional Elements
Tetrahedral, Rectangular Prism (Brick)
3-D Continua

Discretization Examples
One-Dimensional
Frame Elements
Two-Dimensional
Triangular Elements
Three-Dimensional
Brick Elements

Stresses on a typical element

Equation of Equilibrium

Strain Displacement Equation

Generalized Hooks law

Hooke’s Law for Different Types of Materials
Anisotropic Material
The most general stress–strain relationship is given as follows for a three-dimensional
body in a 1–2–3 orthogonal Cartesian coordinate system:
• Where 6×6 [D] matrix is called stiffness matrix
• It can be shown that the 36 constants in above equation actually reduce to 21 constants
due to the symmetry of the stiffness matrix [D]. That is called anisotropic materials.

Orthotropic Material (Orthogonally Anisotropic)/Specially Orthotropic
If a material has three mutually perpendicular planes of material symmetry, it is an
orthotropic materials , then the stiffness matrix is given by
The number of constant is reduced to 9

Quiz 3
What is Poissons’ ratio?
Ans: Lateral strain/Linear strain

Development of Finite Element Equation
• The Finite Element Equation Must Incorporate the
Appropriate Physics of the Problem
• The Appropriate Physics Comes from Either Strength of
Materials or Theory of Elasticity
• FEM Equations are Commonly Developed Using
•Direct Method
•Variational-Virtual Work Method
•Virtual work & Principle of minimum potential
energy
•Weighted Residual Method- Galerkin Method

Finite Element Equation of Bar Element
𝐴𝐸
𝐿
−1 1
−1 1
𝑢1
𝑢2
=
𝜌𝑔𝐴𝐿
2
𝜌𝑔𝐴𝐿
2
+
𝑞0𝐿
2
𝑞0𝐿
2
+
𝑃1
𝑃2
• One dimensional bar element has two variables
• when there is no distributed load q0=0
• When there is no self weight ρg=0
•Size of the global stiffness matrix = number of nodes ×1

Finite Element Equation of Truss Element
𝐴𝐸
𝐿
𝑙2
𝑙𝑚
𝑙𝑚 𝑚2
−𝑙2
−𝑙𝑚
−𝑙𝑚 −𝑚2
−𝑙2 −𝑙𝑚
−𝑙𝑚 −𝑚2
𝑙2 𝑙𝑚
𝑙𝑚 𝑚2
𝑢1
𝑢2
𝑢3
𝑢4
=
𝑃1
𝑃2
𝑃3
𝑃4
• One dimensional truss element has four variables
• There is no self weight
•Where
𝑙 =
𝑥2 − 𝑥1
𝐿
= cosθ m =
𝑦2 − 𝑦1
𝐿
= 𝑠𝑖𝑛θ

Finite Element Equation of Beam Element
• One dimensional beam element has four variables
• when there is no distributed load p=0

Tutorial 1- Bar element
Consider a bar as shown in figure. An axial load of 200kN is applied at
point P. Take A1=2400 mm2; E1=70×109 N/m2; A2=600 mm2;
E1=200×109 N/m2. Calculate the nodal displacement at point P and
stresses in the each section.

Tutorial 2- Beam Element
For the beam and loading shown in figure, compute the slope at the
hinged support points. Take E=200 GPa and I=5×10-6 m4.

Tutorial 3- Truss element
Consider a three bar truss as shown in figure. It is given that Young’s
modulus E= 2x105 N/mm2. Calculate the following:
(i) Nodal displacements.
(ii)Stress in each member.
(iii)Reaction at the support.
Take,
area of element (1) = 2000 mm2

Introduction FEA.pptx

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