5. An innovative way to solve
engineering problems
FEA: A matrix method
through use of computer
6. Any Engineering Problem
Analytical Methods Experimental Methods Numerical Methods
• Classical method
• Infinite elements
• Assumptions
• Solution of
differential equations
• Exact solution
• Simple linear
problems
• Matrix method
• Finite elements (no
Assumptions considered)
• Real life situation
• Solution of algebraic
equations
• Approximate solution
• Real life complicated
problems
• Actual Measurements
• Time Consuming &
need Exp. Set up
• Results can not be
believed blindly &
requires verification
1
7. CLASSICAL METHOD vs. FEM
FEM
1. Exact equations but
approximate solutions
2. Solutions for all problems
3. Linear algebraic equations
4. Finite degrees of freedom
5. Can handle all types of
material properties.
6. Can handle all types of
Nonlinearities
CLASSICAL
1. Exact equations and exact
solutions
2. Solutions for few standard
cases only
3. Partial differential
equations
4. Infinite degrees of freedom
5. Good for homogeneous and
isotropic materials
6. Cannot handle nonlinear
problems
8. FEM is superior to the classical methods only
for the problems involving a number of
complexities which cannot be handled by
classical methods without making drastic
assumptions.
However for all regular, standard and simple
problems, the solutions by classical methods
are the best solutions.
The finite element knowledge makes a
good engineer better
9. Need for Studying FEM?
•Any FEM software tool used as a black box
•Any input to the black box results in an output.
•Garbage in and garbage out
•Interpretation of results
•Debugging the errors during the analysis
•No knowledge of FEA may produce more
dangerous results.
10. Why it is called Finite elements ?
• The domain or region is discretized into finite
number of elements.
11. Simple Approach of FEM Concept
•Basic concept of discretization of a Domain by Finite
elements.
•Assume that you don’t know how to calculate area of the
circle but knew the formula for the area of the Triangle.
Area ‘unknown’ )
h
b
(
2
1
Area
b
h
15. Various Element Types
• Divide the body into a systems of finite
elements with nodes and the appropriate
element type
• Element Types:
– One-dimensional Element
(Bar/Spring/Truss/Beam)
– Two-dimensional Element
(Plates/Shells)
– Three-dimensional Element
16.
17.
18. Common Types of Elements
One-Dimensional Elements
Line
Rods, Beams, Trusses,
Frames
Two-Dimensional Elements
Triangular, Quadrilateral
Plates, Shells
Three-Dimensional Elements
Tetrahedral, Rectangular Prism
(Brick)
33. Discontinuity of Load
• Concentrated loads and sudden
change in the intensity of
uniformly distributed loads are
the sources of discontinuity of
loads.
• A node or a line of nodes should
be there to model the structure.
34.
35.
36. Discontinuity of Boundary Conditions
• If the boundary condition for a structure
suddenly change we have to discretize such
that there is node or a line of nodes.
38. Refining Mesh
• To get better results, the finite element mesh should
be refined in the following situations
(a) To approximate curved boundary of the structure
(b) At the places of high stress gradients.
39. Use of Symmetry
• Wherever there is symmetry in the problem,
it should be made use.
• By doing so lot of computer memory
requirement is reduced.
40.
41. Element Aspect Ratio
• The shape of the element also affects the
accuracy of analysis.
• It is defined as the ratio of largest to smallest
size in an element.
• It is applied to 2D and 3D elements.
• For good accuracy or better results, the aspect
ratio should be as close to unity as possible.
ension
Smaller
ensio
er
L
Ratio
Apspect
dim
dim
arg
45. General Procedure for FEA
Select a suitable field variable.
Discretization or meshing into a number of
elements.
Selection of shape/interpolation functions.
Development of element equations.
e e e
k q f
F = kq for a spring
This is a matrix
equation
This is a
algebraic
equation
46. Assembly of the element equations to
produce a global system of equations.
Imposition of the boundary conditions. (In
this step, the assembled system of equations
is modified by applying the prescribed
boundary conditions)
K Q F
47. Solution of equations. (In this step, modified
global system of equations is solved and
primary variables at the nodes are obtained)
Additional computations (if desired). (In
this step, various secondary quantities are
computed from the obtained solution. For
example, stresses and strains are computed
from the obtained nodal displacements)
1
Q K F
49. What is Finite Element Analysis ?
•Finite Element Analysis is
a computer simulation
technique used in
engineering analysis.
•It is a way to simulate
loading conditions on a
design and determine its
response to those loading
conditions.
51. General Steps of Any FEA Software
• Set the type of analysis
• Create CAD model
• Assign the material
• Define the element type
• Divide the geometry into nodes and elements (meshing)
• Element equations created in the background
• Assemble equations created in the background
• Apply loads and boundary conditions
• Modified equations are framed
• Solution and Interpretation of results
52.
53. Stages of Any FEA Software
..General scenario..
Preprocessing
Analysis/Solution
Postprocessing
Step 1
Step 2
Step 3
57. Preprocessing
• Define the geometric domain of the problem.
• Define the element type(s) to be used.
• Define the material properties of the elements.
• Define the geometric properties of the
elements (length, area, and the like).
• Define the element connectivity (meshing)
• Define the boundary conditions.
• Define the loadings.
58. Processing/Solution/Analysis
• The user asks the software to calculate
values of a set of parameters as per
requirement
• Assembles the governing algebraic
equations in matrix form and computes the
unknown values of the primary field
variable(s).
• The computed values are then used by back
substitution to compute reaction forces,
element stresses, and heat flow, etc.
59. Postprocessing
• Calculates stresses/strains
• Check equilibrium.
• Calculate factors of safety.
• Plot deformed structural shape.
• Animate dynamic model behavior.
• Produce color-coded temperature plots.
74. Shape/Interpolation Functions
In FEA, the main aim is to find the
field variables at nodal points. The
value of the field variable at any point
inside the element is a function of
values at nodal points of the element.
This function which relates the field
variable at any point within the
element to the field variables at nodal
points is called shape function.
75. Why Polynomial Shape Functions?
They are easy to handle
mathematically i.e. differentiation and
integration of polynomials is easy.
Using polynomial, any function can
be approximated reasonably well. If a
function is highly nonlinear we may
have to approximate with higher order
polynomial.
76. Important FE Equations
: e e e
Element Eqn K q F
:
T
e
v
Element Stiffness Matrix K B D B dv
Body force Surface force
:
T T
e
v s
Element LoadVector F N f dv N T ds P
Point load
1 1
:
n n
e e e
e e
Global Eqn K q F
1 1
n n
n n
K Q F
77. Prerequisites to Study FEA
Working knowledge on Matrix Algebra
Basic Elementary Knowledge on SOM
and HT
78. APPLICATIONS OF FEA
Analysis of Bar/Spring
Analysis of Beam
Analysis of Truss
Bar: Any structural member under axial load
Bar: Any member either under tension or compression
It means it may either elongate or contract
79. Beam: It is a structural member under transverse load
80. Shape Functions
2 1
1
1 2
1 1 2 2
1
1 2
2
1
q q
u x q x
l
x x
q q
l l
N q N q
q
N N
q
1 2
1
x x
N and N
l l
2 1
1 2
u x N q
Analysis of Bar
81. Properties of Shape Functions
At any point x,
At node 1, x = 0;
At node 2, x = l;
1 2 1
N N
1 2
1, 0
N N
1 2
0, 1
N N
82.
du d dN
Strain N q q B q
dx dx dx
1 2
, 1
1
1 1
dN dN d x d x
Where B
dx dx dx l dx l
Straindisplacement matrix
l
Stress E D B q
2 1
1 2
2 1
1 2
2 1
1 1 1 2
For one -dimensional problems
u x N q
B q
D B q
83. Element Stiffness Matrix of a Uniform Bar
0
0
2 1 2
0
2 1
2 2
1
1 1
1 1
1
1
1 1
1
1 1
1 1
l
T T
e
v
l
l
k B D B dv B E B Adx
AE dx
l l
AE
dx
l
AE
l
84. Element Stiffness Matrix of a Tapered Bar
1 1
1 1
e m
A E
K
l
1 2
, Mean or average area
2
m
A A
Where A
85. Element and Global Equations
2 2 2 1
2 1
1 1
2 2
:
1 1
1 1
e e e
Element Equation k q F
q F
AE
q F
l
1 1
:
n n n n
Global Equation K Q F
86. Example 1: Determine the nodal displacements at node 2,
stresses in each material and support reactions in the bar due to
applied force P = 400 kN. Given:
A1 = 2400 mm2, A2 = 1200 mm2, l1 = 300 mm, l 2 = 400 mm
E1 = 0.7 × 105 N/mm2, E2 = 2 × 105 N/mm2
87. Example 2: Determine the nodal displacement, element
stresses and support reactions of the axially loaded bar as
shown in Figure. Take
E = 200 GPa and P = 30 kN
88. ANALYSIS OF BEAMS
• Vertical displacement v (Translational degree of freedom)
• Slope, dv
dx
(Rotational degree of freedom)
1 1 2 2
T
q v v
1 2
1 2
and
dv dv
dx dx
89. Shape Functions
2 3 2 3
1 2
2 3 2
2 3 2 3
3 4
2 3 2
3 2 2
1 ,
3 2
,
x x x x
N N x
l l l l
x x x x
N N
l l l l
The displacement at any point within the element is
interpolated from the four nodal displacements .
1 1 2 1 3 2 4 2
v x N v N N v N
1
1
1 2 3 4
2
2
v
N N N N
v
4 1
1 4
N q
90. Element Stiffness Matrix
2 2
3
2 2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
e
l l
l l l l
EI
k
l l
l
l l l l
91. Element Load Vector
1
1
1 1 2 2
2
2
1
2
3
4
T
e
F
M
F F M F M
F
M
92.
1
2
1
2
2
2
2
1
12
2
2
3
12
4
o
o
e
o
o
q l
F
q l
M
F
q l
F
q l
M
1
2
1
2
2
2
3
20
1
1
2
30
7 3
20 4
1
20
o
o
e
o
o
q l
F
q l
M
F
F
q l
M
q l
For a downward UDL, For a downward UVL,
93. Element and Global Equations
4 4 4 1
4 1
1 1
2 2
1 1
3
2 2
2 2
2 2
:
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
e e e
Element Equation k q F
v F
l l
M
l l l l
EI
v F
l l
l
M
l l l l
1 1
:
n n n n
Global Equation K Q F
94. Example 1:A beam of length 10 m, fixed at both ends carries
a 20 kN concentrated load at the centre of the span. By taking
the modulus of elasticity of material as 200 GPa and moment
of inertia as 24 × 10–6 m4, determine the slope and deflection
under load.
95. Example 2: Determine the rotations at the supports.
Given E = 200 GPa and I = 4 × 106 mm4.
96. Example 3: Find the slopes at nodes the beam shown in
Figure by finite element method and determine the end
reactions. Also determine the deflections at mid spans given
E = 2 × 105 N/mm2 and I = 5 × 106 mm4
97. Introduction to FEA
• A computing technique to obtain approximate solutions
to boundary value problems.
• Uses a numerical method called FEM
• Involves a CAD model design that is loaded and
analyzed for specific results
• Simulates the loading conditions of a design and
determines the design response in those conditions
• A better FEA knowledge helps in building more accurate
models
