Fine element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
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Finite_Element_Method analysis and theory of yeild line .ppt
1. G.H. RAISONI COLLEGE OF ENGINEERING &
MANAGEMENT, PUNE.
(An Autonomous Institute Affiliated to SPPU)
Department of Civil Engineering
TAE – 2
Name : Atif Malik
Roll no : 05
Reg. no : 23ASTR2101007
Subject : FINITE ELEMENT METHOD
PPT Topic : FINITE ELEMENT METHOD ANALYSIS
Year : F.Y. SEM - II
Course : MTECH STRUCTURAL ENGINEERING
2. Fundamental Concept of FEM
• A continuous field of a certain domain having infinite degrees of
freedom is approximated by a set of piecewise continuous models
with a number of finite regions called elements. The number of
unknowns defined as nodes are determined using a given relationship
i.e.{F}=[K]*{d}.
3. Fundamental Concept of FEM
Domain
x
x
Domain with degrees of freedom
•Red line-Continuous
field over the entire
domain.
•Blue line-Finite number
of linear
approximations with
the finite number of
elements
x
1
2
3 4
5 6
x
Subdomain e
Domain divided with subdomains
with degrees of freedom
4. General Steps:
1) Discretize the domain
a) Divide domain into finite elements using appropriate
element types (1-D, 2-D, 3-D, or Axisymmetric)
2) Select a Displacement Function
a) Define a function within each element using the
nodal values
3) Define the Strain/Displacement and
Stress/strain Relationships
4) Derive the Element Stiffness Matrix and
Equations
a)Derive the equations within each element
5. General Steps:
5) Assemble the Element Equations to Obtain the
Global or Total Equations and Introduce
Boundary Conditions
a)Add element equations by method of superposition to
obtain global equation
6) Solve for the Unknown Degrees of Freedom (i.e
primary unknowns)
7) Solve for the Element Strains and Stresses
8) Interpret the Results
6. Applications:
•Stress Analysis
• Truss and frame analysis
• Stress concentration
•Buckling
•Vibration analysis
•Heat transfer
•Fluid flow
7. Advantages of FEM :
•Model irregularly shaped bodies
•Compute General load conditions
•Model bodies composed of different materials
•Solve unlimited numbers and kinds of boundary
conditions
•Able to use different element sizes in places where
loads or stresses are concentrated
•Handle non-linear behavior using linear
approximations
•Reduce System Cost
8. FEM Packages
•Large Commercial Programs
• Designed to solve many types of problems
• Can be upgraded fairly easily
• Initial Cost is high
• Less efficient
•Special-purpose programs
• Relatively short, low development costs
• Additions can be made quickly
• Efficient in solving their specific types of problems
• Can’t solve different types of problems
10. Note on Stiffness matrix
For a 1-D bar, the stiffness matrix is derived from the
stress/strain relationship in Hooke’s law and the definitions
of stress and strain.
σx = Eεx. ; σx = P/A ; εx = du/dx = (d2x – d1x)/L
By substitution: -f1x = EA (d2x – d1x)
L
f1x = EA (d1x – d2x)
L
Similarly for f2x: f2x = EA (d2x – d1x)
L
Combining into matrix form, the stiffness matrix is defined
as
[k] = EA
L
1 -1
-1 1
f2
f1
d1x d2x
L
A
E
11. Note on the displacement function
•For a given set of nodes there exists a function that approximates
the displacement at any position along the bar.
•This function, called the displacement function, is derived from
Pascal’s Triangle.
•A new constant is introduced into the function for every node in
the discretized domain.
u(x) = a1 + a2x + a3x2 + …
For 1-D
12. Note on the displacement function
If a 1-D bar is broken into 2 elements, the displacement function
would be u(x) = a1 + a2x + a3x2.
Putting it into matrix notation: u(x) = [1 x x2] a1
a2
a3
By knowing the distances to the nodes
and the displacements at those nodes, the
equation becomes: u1
u2
u3
=
1 0 0
1 x2 x2
2
1 x3 x3
2
a1
a2
a3
, where
x1 = 0, x2 and x3 are the distances to the nodes and u1, u2, and u3 are the
displacements.
The coefficients are found by solving the equation.
13. Example
A A
L L
E 2E
P
Determine displacements of materials a and b if
the load P is applied to the end of the bar given
the above information.
a b
14. SOLUTION:
1) Discretize the domain with appropriate elements.
Element a
1 2
Element b
2 3
u1 u2
u2 u3
1 2 3
u1 u2 u3
f3 = P
f1
f1 f3 = P
f21 f22
15. SOLUTION:
2) Select a displacement function
There will be new term for each element, and the terms are
1 2 3
u1 u2 u3
u(x) = a1 + a2x + a3x2
derived from Pascal’s triangle.
16. SOLUTION:
3) Define stress/displacement and stress/strain relationships
σx = Eεx εx = du/dx
4) Derive the element stiffness matrix and element equations
{F} = [k]{d} [k] = stiffness matrix
[k] = EA
L
1 -1
-1 1
{F} = Force
{d} = displacement
f1
f21
= L
1 -1
-1 1
EA u1
u2
f22
f3
= L
2 -2
-2 2
EA u2
u3
,
a b
17. SOLUTION:
5) Construct Global equation and introduce boundary
conditions and known variables.
f1
f21+f22
f3
=
L
1 -1 0
-1 3 -2
0 -2 2
EA u1
u2
u3
Global Equation
B.C.: (x =0) u1 = 0
Known variables: f3 = P and f2 = f21+f22 = 0
18. SOLUTION:
6) Solve for unknowns.
f1
0
P
=
L
1 -1 0
-1 3 -2
0 -2 2
EA 0
u2
u3
f1 = -EAu2
L
0 = EA(3u2-2u3)
L
P = EA(2u3-2u2)
L
u2 = PL
EA
u3 = 3PL
2EA f1 = -P
19. SOLUTION:
7) Solve for the element strains and stresses.
εa = P = u2
EA L
σa = Eεa = P
A
εb = 3P = u3
2EA L
σa = 2E εb = P
3 A
8) Interpret the Results
•After solving for the displacements, the coefficients of the
displacement function can be determined.