Basics of finite element method 19.04.2018

BASICS TO FINITE ELEMENT
METHOD
MOHD ZAMEERUDDIN
MOHD SALEEMUDDIN
ASSISTANT PROFESSOR
DEPARTMENT OF CIVIL ENGINEERING
MGM’s COLLEGE OF ENGINEERING,
NANDED (m. s)
Email: zameerstd1@hotmail.com
Mobile: +919822913231

2
Numerical Methods
FORMULATION OF
STRUCTURAL ANALYSIS
•EQUILLIBRIUM
•CONSTITUTIVE
•COMPATIBILITY
•ANALYTICAL
•NUMERICAL
ANALYTICAL
•Closed Form Solutions.
•Applicable to;
Simple geometry
Boundary conditions
Loading and material
properties
NUMERICAL METHODS
•Complex in nature.
•Approximate solution of differential
equations
Finite difference method
Finite volume method
Boundary element method
Mesh less Method
© MZS Engineering Technologies

3
Let us go through finite difference method
It involves replacing the governing differential equations and
boundary condition by static algebraic equations.
Let us consider beam bending
problem
2
12
2
1121
2
2
1
4
4
)2(
)(()(
)()()(
0
lim)(
)()(
)(;
h
www
h
wwww
h
ww
dx
d
dx
wd
h
ww
x
xfxxf
xdx
wd
xfxxfw
xfw
EI
q
dx
wd
iii
iiiiii
ii








 














Displacement function

4
Let us go through finite difference method
4
1112
4
4
3
123
3
3
)464(
)33(
h
wwwww
dx
wd
h
wwww
dx
wd
iiiii
iiii






Thus the equation for deflection can be written in finite
difference method as;
4
2112 )464( h
EI
q
wwwww iiiii  
Finite difference equation at node i

COMPARISONS
5
 FDM gives values at
node points only. For
other points interpolation
is required
 FDM needs larger
number of nodes to get
good results
 Fairly complicated
problems can be handled
by FDM
 FEM gives values at any point
including node points
 FEM needs fewer nodes to get
good results
 All type of complicated
problems can be handled by
FEM
Finite Difference Method
(FDM)
Finite Element Method
(FEM)

COMPARISONS
6
 FDM makes stair type of
approximation to slopping
and curve boundaries.
 FDM makes point wise
approximation that is
satisfy continuity at node
points only. Along the
sides continuity are not
ensured.
 FEM can consider sloping and
curved boundaries exactly.
 FEM makes piece wise
approximation that is satisfy
continuity at node as well as
along sides/edges of element
(FDM)
(FEM)

COMPARISONS
7
 It is less efficient and
more approximate
 Not applicable for
nonlinearity of domain
 Difficult to apply for
unusual boundary and
loading condition
 It is difficult to write
general purpose
computer codes in FDM
 It is more effective and
approximate
 Applicable for nonlinearity of
domain
 Can be applied for any
boundary and loading
conditions
 It is easier to write general
purpose computer codes for
FEM formulations.
(FDM)
(FEM)

8
Finite Element Method- Basics
The most distinctive feature of finite element method that
separates it from others is the division of a given domain into a set
of simple sub domains, called finite elements
These elements are connected through number of joints
which are called “Nodes” © MZS Engineering Technologies

9
These elements may be 1D elements, 2D elements, Axi-
symmetric elements or 3D elements
One Dimensional Elements
These elements are suitable for the analysis of one dimensional
problem and may also called as line elements.

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Two Dimensional Elements
1. Plane stress
2. Plane strain
3. Plate Problem
Constant strain triangle (CST) or
Linear displacement triangle
 
)3,2,1(
),(;,,1


i
uyxUyx e
i
e
i
e
i
e































3
2
1
33
22
11
3
1
1
1
1
2
c
c
c
yx
yx
yx
u
u
u

11
Six noded Linear strain triangle or
Quadratic displacement triangle
Ten noded triangular elements or
Cubic displacement triangle

12
Cubic strain triangle Quartic strain triangle

13
4 node rectangular element
Lagrange family rectangular element

14
Serendity family of rectangular
elements
Quadrilateral elements
Based on isoparametric
parameters;
Same function used to define
geometrics and nodal unknowns

15
Quadrilateral elements generated using triangular element

16
Curved two dimensional elements

17
Axi-symmetric Elements
•These are also known as ring type elements.
•These elements are useful for the analysis of axi-symmetric
problems such as analysis of cylindrical storage tanks, shafts,
rocket nozzles.
•Axi-symmetric elements can be constructed from one or two
dimensional elements.
•One dimensional axi-symmetric element is a conical frustum and,
•Two dimensional axi-symmetric element is a ring with a
triangular or quadrilateral cross section.

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Axi-symmetric Elements
One dimensional
Two dimensional

19
Three dimensional Elements
Tetrahedron
Brick element
(rectangular prism)

20
Three dimensional Elements
Arbitrary hexahedron
3D-quadratic element

21
Nodes
Nodes are the selected finite points at which basic unknowns
(displacements in elasticity problems) are to be determined in the
finite element analysis.
The basic unknowns at any point inside the element are
determined by using approximating/interpolation/shape functions
in terms of the nodal values of the element.
There are two types of nodes;
a) External nodes
b) Internal nodes

22
External Nodes
External nodes are those which occur on the edges/ surface of the
elements and they may be common to two or more elements
These nodes may be further classified as
(i) Primary nodes and
(ii) Secondary nodes.
Primary nodes occur at the ends of one dimensional elements or at
the corners in the two or three dimensional elements.
Secondary nodes occur along the side of an element but not at
corners

23
External Nodes
Internal nodes are the one which occur inside an element. They are specific
to the element selected i.e. there will not be any other element connecting
to this node. Such nodes are selected to satisfy the requirement of
geometric isotropy while choosing interpolation functions

24
Nodal Unknowns
•Displacements for stress analysis
•Temperatures for heat flow problems
•Potentials for fluid flow or in magnetic field
In truss problem continuity of displacement are satisfied and no
change in slopes at any nodal point known as zeroth continuity
problem (C0).
In case of beam s and plates both continuity of displacement and
slope continuity has to be satisfied (first derivative of
displacement) known as first order continuity problem (C1)

25
Nodal Unknowns
In exact plate bending analysis second order continuity has to be ensured
hence known as second order continuity problem (C2)
Cr continuity problem = rth derivatives of the basic unknowns

26
Degrees of Freedom
Finite number of displacements is the number of degrees of
freedom of the structure
u, v, w and θx,y, θz represents displacement and rotation respectively

27
Degrees of Freedom

28
Degrees of Freedom

29
Degrees of Freedom

30
Coordinate systems
1. Global coordinates
2. Local coordinates and
3. Natural coordinates

31
Global coordinates
The coordinate system used to define the points in the entire
structure is called global coordinate system.
Global Cartesian coordinate system

32
Local coordinates
For the convenience of deriving element properties, in FEM
many times for each element a separate coordinate system is
used.

33
Local coordinates
A natural coordinate system is a coordinate system which permits
the specification of a point within the element by a set of
dimensionless numbers, whose magnitude never exceeds unity
One dimensional element

34
IDEALIZATION OF A CONTINNUM

Basics of finite element method 19.04.2018

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Basics of finite element method 19.04.2018