Two-Dimensional Elasticity
2D
3D
elem. only
2D
Plane Stress
Plane Strain
Plane Stress:
A state of stress in which we have a very thin elastic
body and no load in the direction parallel to the
thickness exist.
Chapter 23
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K. Haghighi
' s along t ar
SOME ASPECTS OF CONTINUUM
MECHANICS FOR SOLIDS
Continuum Concept:
Material is continuously distributed
throughout its volume and completely
fills the space it occupies.
Continuum Mechanics:
Studying the behavior of solids , liquids
and gases.
Continuum
FINITE ELEMENT METHOD (FEM / FEA)
INTRODUCTION
The FEM is a numerical procedure for solving
Boundary Value Problems (BVP and structural &
s)
solid mechanics problems in engineering.
The method had its birth in the aerospace industry
in the early 1950s a
ERROR ESTIMATION AND ADAPTIVITY
FOR BOUNDARY VALUE PROBLEMS
Governing Differential Equation
Dx
2
x
2
+ Dy
2
y
2
G + Q = 0
(1)
Error Estimation
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K. Haghighi
with boundary conditions:
= 0 (x, y )
on 1
Dx
cos + D y
sin
x
y
= Mb + S
on 2
(2)
(3)
E
CONVECTIVE BOUNDARY VALUE PROBLEMS
2
2
Dx
u v G + Q = 0
+ Dy
x
y
y 2
x 2
u and v are the velocity components in the x and y
directions, is a coefficient depending on . If is
temperature is the product of the density and heat
capacity. u and v are not v
ELEMENT MATRICES
Higher order elements have curved boundaries
Integration is a challenge
Integrate in natural coordinate system!
Need numerical integration!
Chapter 27
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G. Subbarayan
Changing the Variables of Integration
One-dimensional integrals
x
Two-Dimensional Elasticity
2D
3D
elem. only
2D
Plane Stress
Plane Strain
Plane Stress:
A state of stress in which we have a very thin elastic
body and no load in the direction parallel to the
thickness exist.
Chapter 23
Page 1
K. Haghighi
' s along t ar
Eigenvalue Problems
and Eigenvalues of a Matrix
e.v. Problems: special class of BVP; elasticity,
vibration.
In any e.v. problem, the information of interest are
the eigenvalues for the system.
In a vibration prob., these e.v.s are natural
frequencies of
Theory of Elasticity
Solid Mechanics applications
Elasticity prob., plates & shells, buckling and
stability prob., vibrations, plasticity, elastoplasticity, viscoelasticity. . . (procedure is the same)
Theory, elem. matrices, 2 - D elasticity, axisym.
e
2-D Elements
Local Coordinate System:
-independent of orientation?
s
t
t
y
y
x
s
x
Chapter 6
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K. Haghighi
-range
1 D s 0 to L
2 D s, t ?
2b
2a
0 to 2b t
0 to 2a s
Natural Coordinate System:
A local system that permits the specification of a point
wi
ELEMENT SHAPE FUNCTIONS
Quadratic elements
Shape functions of one and two-dimensional
elements
Chapter 26
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G. Subbarayan
Local Node Numbers
When number of nodes in element increases, need
alternative notation for nodes
I,j no longer convenient
Elem