Simplified Elasticity Formulation, Elasticity Problems, 2 and 3 Dimensional Problems, Plane strain Problems, Plane Strain Problems, Airy Stress, Function Method, Polar Coordinate Formulation, Stress function

1. Formulation of Two-Dimensional
Elasticity Problems
Prof. Samirsinh Parmar
Asst. Professor, Dept. of Civil Engg.
Dharmasinh Desai University, Nadiad, Gujarat, INDIA
Mail: samirddu@gmail.com
1

2. Simplified Elasticity Formulations
Displacement Formulation
Eliminate the stresses and strains
from the general system of equations.
This generates a system of three
equations for the three unknown
displacement components.
Stress Formulation
Eliminate the displacements and
strains from the general system of
equations. This generates a system of
six equations and for the six unknown
stress components.
The General System of Elasticity Field Equations
of 15 Equations for 15 Unknowns Is Very Difficult
to Solve for Most Meaningful Problems, and So
Modified Formulations Have Been Developed.
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
2

3. Solution to Elasticity Problems
F(z)
G(x,y)
z
x
y
Even Using Displacement and Stress Formulations
Three-Dimensional Problems Are Difficult to Solve!
So Most Solutions Are Developed for Two-Dimensional Problems
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
3

4. Two and Three Dimensional Problems
x
y
z
x
y
z
Three-Dimensional Two-Dimensional
x
y
z
Spherical Cavity
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
4

5. Two-Dimensional Formulation
x
y
z
R
x
y
z
R
2h
Plane Strain Plane Stress
0
,
)
,
(
,
)
,
( 

 w
y
x
v
v
y
x
u
u
0
)
,
(
)
,
(
)
,
(















yz
xz
z
xy
xy
y
y
x
x
y
x
y
x
y
x
<< other dimensions
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
5

6. Examples of Plane Strain Problems
x
y
z
x
y
z
P
Long Cylinders
Under Uniform Loading
Semi-Infinite Regions
Under Uniform Loadings
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
6

7. Examples of Plane Stress Problems
Thin Plate With
Central Hole
Circular Plate Under
Edge Loadings
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
7

8. Plane Strain Formulation
0
2
1
,
,























yz
xz
z
xy
y
x
e
e
e
x
v
y
u
e
y
v
e
x
u
e
Strain-Displacement
0
,
2
)
(
)
(
2
)
(
2
)
(




























yz
xz
xy
xy
y
x
y
x
z
y
y
x
y
x
y
x
x
e
e
e
e
e
e
e
e
e
Hooke’s Law
0
,
)
,
(
,
)
,
( 

 w
y
x
v
v
y
x
u
u
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
8

9. Plane Strain Formulation
0
)
(
0
)
(
2
2














































y
x
F
y
v
x
u
y
v
F
y
v
x
u
x
u
Displacement Formulation
0
0


















y
y
xy
x
xy
x
F
y
x
F
y
x





















y
F
x
F y
x
y
x
1
1
)
(
2
Stress Formulation
R
So
Si
S = Si + So
x
y
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
9

10. Plane Strain Example
0
)
(
)
(
0
2
)
(
1
)
1
(
2
2
)
2
1
)(
1
(
)
2
1
)(
1
(
2
)
(
:
0
,
)
1
(
,
1
:
0
,
)
1
(
,
1
,
2
2
2


































































































yz
xz
xy
o
y
x
y
x
z
y
y
x
y
o
o
o
x
y
x
x
z
zx
yz
xy
o
y
o
x
o
o
e
e
e
e
e
E
E
E
E
e
e
e
e
e
e
e
E
y
v
e
E
x
u
e
w
y
E
v
x
E
u
Stresses
Strains
Stresses
and
Strains
the
Determine
nts
Displaceme
Following
the
Given
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
10

11. Plane Stress Formulation
Hooke’s Law
0
,
1
)
(
1
)
(
)
(
1
)
(
1


























yz
xz
xy
xy
y
x
y
x
z
x
y
y
y
x
x
e
e
E
e
e
e
E
e
E
e
E
e
Strain-Displacement
0
2
1
0
2
1
2
1
,
,



















































x
w
z
u
e
y
w
z
v
e
x
v
y
u
e
z
w
e
y
v
e
x
u
e
xz
yz
xy
z
y
x
0
,
)
,
(
,
)
,
(
,
)
,
( 













 yz
xz
z
xy
xy
y
y
x
x y
x
y
x
y
x
Note plane stress theory normally neglects some of the
strain-displacement and compatibility equations.
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
11

12. Plane Stress Formulation
R
So
Si
S = Si + So
x
y
Displacement Formulation
0
)
1
(
2
0
)
1
(
2
2
2












































y
x
F
y
v
x
u
y
E
v
F
y
v
x
u
x
E
u
0
0


















y
y
xy
x
xy
x
F
y
x
F
y
x
Stress Formulation





















y
F
x
F y
x
y
x )
1
(
)
(
2
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
12

13. Correspondence Between Plane Problems
Plane Strain Plane Stress
0
)
(
0
)
(
2
2














































y
x
F
y
v
x
u
y
v
F
y
v
x
u
x
u
0
0


















y
y
xy
x
xy
x
F
y
x
F
y
x





















y
F
x
F y
x
y
x
1
1
)
(
2
0
)
1
(
2
0
)
1
(
2
2
2












































y
x
F
y
v
x
u
y
E
v
F
y
v
x
u
x
E
u
0
0


















y
y
xy
x
xy
x
F
y
x
F
y
x





















y
F
x
F y
x
y
x )
1
(
)
(
2
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
13

14. Elastic Moduli Transformation Relations for Conversion
Between Plane Stress and Plane Strain Problems
2
1 

E



1
2
)
1
(
)
2
1
(




E



1
E v
Plane Stress to Plane Strain
Plane Strain to Plane Stress
Plane Strain Plane Stress
Therefore the solution to one plane problem also yields the solution
to the other plane problem through this simple transformation
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
14

15. Airy Stress Function Method
Plane Problems with No Body Forces
0
0
















y
x
y
x
y
xy
xy
x
0
)
(
2




 y
x
Stress Formulation
y
x
x
y
xy
y
x

















2
2
2
2
2
,
,
Airy Representation
0
2 4
4
4
2
2
4
4
4
















y
y
x
x
Biharmonic Governing Equation
(Single Equation with Single Unknown)
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
15

16. Polar Coordinate Formulation

































r
u
r
u
u
r
e
u
u
r
e
r
u
e
r
r
r
r
r
1
2
1
1
,
Strain-Displacement
0
,
2
)
(
)
(
2
)
(
2
)
(





































rz
z
r
r
r
r
z
r
r
r
r
e
e
e
e
e
e
e
e
e
Strain
Plane
0
,
1
)
(
1
)
(
)
(
1
)
(
1


































rz
z
r
r
r
r
z
r
r
r
e
e
E
e
e
e
E
e
E
e
E
e
Stress
Plane
Hooke’s Law
0
2
1
0
)
(
1
r
r
































F
r
r
r
F
r
r
r
r
r
r
r
Equilibrium Equations
































r
r
r
r
r
r
r
r
1
1
1
2
2
2
2
2
Airy Representation
x1
x2

r



dr
r
r
r






r
r








 
 d
dr
r
r
r




 







 
 d
r
r
r
F

F
dr
rd
d
0
1
1
1
1
2
2
2
2
2
2
2
2
2
2
4







































r
r
r
r
r
r
r
r
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
16

17. Solutions to Plane Problems
Cartesian Coordinates
y
x
x
y
xy
y
x

















2
2
2
2
2
,
,
Airy Representation
0
2 4
4
4
2
2
4
4
4
















y
y
x
x
Biharmonic Governing Equation
)
,
(
,
)
,
( y
x
f
T
y
x
f
T y
y
x
x 

Traction Boundary Conditions
R
S
x
y

18. Solutions to Plane Problems
Polar Coordinates
R
S
)
,
(
,
)
,
( 


 
 r
f
T
r
f
T r
r
Traction Boundary Conditions
Airy Representation





























 

r
r
r
r
r
r
r
r
1
,
,
1
1
2
2
2
2
2
Biharmonic Governing Equation
0
1
1
1
1
2
2
2
2
2
2
2
2
2
2
4







































r
r
r
r
r
r
r
r
x
y

r


19. Cartesian Coordinate Solutions
Using Polynomial Stress Functions
0
2 4
4
2
2
4
4
4













y
y
x
x









 




2
02
11
2
20
01
10
00
0 0
)
,
( y
A
xy
A
x
A
y
A
x
A
A
y
x
A
y
x
m n
n
m
mn
y
x
x
y
xy
y
x

















2
2
2
2
2
,
,
terms do not contribute to the stresses and are therefore dropped
1

n
m
terms will automatically satisfy the biharmonic equation
3

 n
m
terms require constants Amn to be related in order to satisfy biharmonic equation
3

n
m
Solution method limited to problems where boundary traction conditions
can be represented by polynomials or where more complicated boundary
conditions can be replaced by a statically equivalent loading
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
19

20. Stress Function Example
)
(
6
,
0
)
(
6
:
)
2
3
(
:
3
2
2
2
3
2
2
2
3
y
d
y
d
F
y
x
x
y
d
x
d
F
y
y
d
xy
d
F
xy
y
x



























Stresses
the
Determine
Function
Stress
Following
the
Consider
Appears to Solve the Beam Problem:
x
y
d
F
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
20

21. Thank You
Two Dimentional Elasticity
Problems- SPP, DoCL, DDU, Nadiad.
21