Using the Airy Stress Function approach, it was shown that the plane
elasticity formulation with zero body forces reduces to a single governing
biharmonic equation. In Cartesian coordinates it is given by
and the stresses are related to the stress function by
We now explore solutions
to several specific
problems in both
Cartesian and Polar coordinate systems
Cartesian Coordinate Solutions
Using Polynomials
I
n
Cartesian
coordinates we choose Airy
stress function
solution of polynomial form
where
A
mn
are constant coefficients to be determined. This
method
produces
polynomial stress distributions,
and thus would
not
satisfy
general boundary
conditions. However,
we can modify such boundary
conditions
using Saint

Venant’s
principle and replace
a
non

polynomial condition
with a statically equivalent
loading.
T
his
formulation
is
most useful for problems with rectangular
domains, and is
commonly based
on the inverse solution concept where we
assume
a
polynomial
solution form and
then try
to find what
problem
it will solve.
Noted
that the three lowest order terms with
m + n
1
do not contribute to the
stresses and will therefore be dropped. It should be noted that second order terms
will produce a constant stress field, third

order terms will give a linear distribution of
stress, and so on for higher

order polynomials.
Terms
with
m + n
3
will automatically satisfy the biharmonic equation
for
any
choice of constants
A
mn
. However, for higher order
terms, constants
A
mn
will have to
be related in order to have
the polynomial
satisfy
the biharmonic equation.
Example 8.1 Uniaxial Tension of a Beam
B
oundary Conditions:
Since the boundary conditions specify constant
stresses on
all boundaries
,
try
a second

order
stress function of the form
The first boundary condition
implies
that
A
02
=
T
/2,
and all other boundary conditions are identically
satisfied. Therefore the stress field solution
is
given by
Displacement Field (Plane Stress)
Stress Field
. . . Rigid

Body Motion
“Fixity conditions”
needed
to
determine RBM
terms
Example 8.2 Pure Bending of a Beam
B
oundary Conditions:
Expecting a linear bending stress distribution,
try second

order
stress function of the form
M
oment boundary condition implies
that
A
03
=

M
/4
c
3
,
and all other boundary conditions
are identically satisfied.
Thus the
stress field
is
Stress Field
“Fixity conditions”
to determine RBM terms:
Displacement Field (Plane Stress)
Example 8.2 Pure Bending of a
Beam
Solution Comparison of Elasticity
with Elementary Mechanics of Materials
Elasticity Solution
Mechanics of Materials Solution
Uses
Euler

Bernoulli beam
theory to
find
bending stress and deflection of
beam centerline
T
wo
solutions
are
identical, with the exception of the
x

displacements
Example 8.3 Bending of a Beam by
Uniform Transverse
Loading
B
oundary Conditions:
Stress Field
BC’s
Example
8.3 Beam Problem
Stress Solution Comparison of Elasticity
with Elementary Mechanics of Materials
Elasticity Solution
Mechanics of Materials Solution
Shear
stresses are identical, while
normal
stresses are not
Example
8.3 Beam Problem
Normal Stress Comparisons of Elasticity
with Elementary Mechanics of Materials
M
aximum differences
between the two theories
exist
at
top
and
bottom of beam
, and
actual
difference in
stress
values is
w
/5. For
most
beam
problems
where
l
>>
c
, the bending stresses will be much greater than
w
, and thus the differences between elasticity
and
strength of materials will be relatively small.
Maximum
difference between the two theories is
w
and this occurs at the top of the beam. Again this
difference will be negligibly small for most beam
problems where
l
>>
c
. These results are generally true
for beam problems with other transverse
loadings.
x
–
Stress at x=0
y

Stress
Example 8.3
Beam Problem
Normal Stress Distribution on Beam Ends
End stress distribution
does not
vanish and is nonlinear but gives
zero resultant force
.
Example 8.3
Beam Problem
Choosing
Fixity Conditions
Strength of Materials:
Good match for
beams where
l
>>
c
Displacement Field (Plane Stress)
Cartesian Coordinate Solutions
Using
Fourier Methods
A more general solution scheme for the biharmonic equation may be
found
using
Fourier methods
. Such techniques generally use
separation of
variables
along with
Fourier series
or
Fourier integrals
.
Choosing
Example 8.4 Beam
with
Sinusoidal Loading
B
oundary Conditions:
Stress Field
Example 8.4 Beam
Problem
Bending Stress
Example 8.4 Beam
Problem
For the case
l
>>
c
Strength
of
Materials
Displacement Field (Plane Stress)
Example
8.5 Rectangular Domain with
Arbitrary Boundary Loading
B
oundary Conditions
Must use series representation for Airy stress
function to handle general
boundary
loading.
Use Fourier series theory to handle general
boundary
conditions, and this generates a
doubly infinite set of equations to solve for
unknown constants in stress function form.
See text for details
Polar Coordinate
Formulation
Airy Stress Function Approach
㴠
(
r,
θ
)
R
S
x
y
r
Airy Representation
Biharmonic Governing Equation
Traction Boundary Conditions
Polar Coordinate
Formulation
Plane Elasticity Problem
Strain

Displacement
Hooke’s Law
General Solutions in Polar
Coordinates
Michell
Solution
Choosing the case where
b
=
in
,
n
= integer gives the general
Michell
solution
We will use various
terms from this general
solution to solve
several plane problems
in polar coordinates
Axisymmetric Solutions
Stress Function Approach:
=
(
r
)
Navier
Equation Approach:
u=
u
r
(
r
)
e
r
(Plane Stress or Plane Strain)
Displacements

Plane Stress Case
Gives Stress Forms
•
a
3
term leads to multivalued behavior, and is not found following the
displacement formulation approach
•
Could
also have an axisymmetric elasticity problem using
=
a
4
which gives
r
=
=
0 and
r
=
a
4
/
r
0, see Exercise
8

14
Underlined terms represent
rigid

body motion
Example 8.6 Thick

Walled Cylinder
Under Uniform Boundary
Pressure
B
oundary Conditions
General Axisymmetric
Stress Solution
Using
Strain Displacement
Relations and
Hooke’s
Law
for plane strain gives the
radial
displacement
Example 8.6
Cylinder Problem Results
Internal Pressure Only
r
1
/r
2
= 0.5
r/r
2
r
/p
θ
/p
Dimensionless Stress
Dimensionless Distance,
r
/
r
2
Thin

Walled Tube Case:
Matches with Strength
of Materials Theory
Special Cases of Example 8

6
Pressurized Hole in an Infinite Medium
Stress Free Hole in an Infinite Medium
Under Equal Biaxial Loading at Infinity
Example 8.7 Infinite Medium with a Stress
Free Hole Under Uniform Far Field Loading
B
oundary Conditions
Try Stress Function
Example 8.7
Stress Results
Superposition of Example 8.7
Biaxial Loading Cases
T
1
T
2
T
1
T
2
Equal Biaxial Tension Case
T
1
= T
2
= T
Tension/Compression Case
T
1
= T , T
2
=

T
Review Stress Concentration Factors
Around Stress Free Holes
K = 2
K = 3
K = 4
=
Stress Concentration Around
Stress Free Elliptical Hole
–
Chapter 10
Maximum Stress Field
Stress Concentration Around
Stress
Free
Hole in Orthotropic
Material
–
Chapter
11
2

D
Thermoelastic
Stress
Concentration
Problem Uniform Heat Flow Around
Stress
Free Insulation
Hole
–
Chapter
12
Stress
Field
Maximum
compressive stress on
hot
side of
hole
Maximum
tensile stress on
cold
side
Steel
Plate
:
E
= 30Mpsi (200GPa) and
= 6.5
in/in/
o
F
(
11.7
m/m/
o
C
),
qa
/k
= 100
o
F (37.7
o
C), the maximum stress becomes
19.5ksi (88.2MPa)
Nonhomogeneous Stress Concentration
Around
Stress
Free Hole in a Plane Under Uniform Biaxial Loading
with Radial Gradation of Young’s
Modulus
–
Chapter 14
Three Dimensional Stress Concentration
Problem
–
Chapter 13
Normal Stress
on the
x,y

plane (
z
= 0)
0
0.5
1
1.5
2
2.5
3
3.5
1
2
3
4
5
Normalized Stress in Loading Direction
Dimensionless Distance, r/a
Two Dimensional Case:
(
r
,
/2)/
S
Three Dimensional Case:
z
(
r
,0)/
S
,
= 0.3
Wedge
Domain
Problems
Use general
stress function
solution to
include
terms that are bounded at
origin
and give
uniform stresses on the
boundaries
Quarter Plane
Example
(
= 0 and
=
/2)
Half

Space
Examples
Uniform
Normal Stress Over
x
0
Try Airy Stress Function
Boundary Conditions
Use BC’s To Determine Stress Solution
Half

Space Under Concentrated Surface
Force System (
Flamant
Problem)
Try Airy Stress Function
Boundary Conditions
Use BC’s To Determine Stress Solution
Flamant
Solution Stress Results
Normal Force Case
o
r in Cartesian
components
y
=
a
Flamant
Solution
Displacement Results
Normal Force Case
On Free Surface
y
= 0
Note unpleasant feature of 2

D model that
displacements become unbounded as
r
Comparison of
Flamant
Results with
3

D Theory

Boussinesq’s
Problem
Cartesian Solution
Cylindrical Solution
Free
Surface Displacements
Corresponding 2

D Results
3

D Solution eliminates the
unbounded far

field behavior
Half

Space Under Uniform Normal
Loading Over
–
a
x
a
dY
=
pdx
=
prd
/sin
Half

Space Under Uniform Normal
Loading

Results
max

Contours
Generalized Superposition Method
Half

Space Loading
Problems
Photoelastic
Contact Stress Fields
Notch/Crack Problem
Boundary
Conditions:
At Crack
Tip
r
0:
Try Stress Function:
Finite Displacements and Singular
Stresses at
Crack Tip
1<
<2
=
3/2
Notch/Crack
Problem Results
Transform to
Variable
•
Note special singular behavior of stress field O(1/
r
)
•
A
and
B
coefficients are related to
stress intensity factors
and are useful in fracture
mechanics
theory
•
A
terms give symmetric stress fields
–
Opening or Mode I behavior
•
B
terms give
antisymmetric
stress fields
–
Shearing or Mode II
behavior
Crack
Problem
Results
Contours of Maximum Shear Stress
Mode I (Maximum shear stress contours)
Mode II (Maximum shear stress contours)
Experimental
Photoelastic
Isochromatics
Courtesy of URI Dynamic
Photomechanics
Laboratory
Mode III Crack Problem
–
Exercise 8

32
Anti

Plane Strain Case
Stresses Again
z

Stress Contours
Curved Beam Under End Moments
Curved
Cantilever Beam
P
a
b
r
Dimensionless Distance,
r
/
a
Dimensionless Stress
,
a
/
P
Theory of Elasticity
Strength of Materials
=
/2
b/a
= 4
Disk Under Diametrical Compression
+
P
P
D
=
+
Flamant Solution (1)
Flamant Solution (2)
Radial Tension Solution (3)
Disk Problem
–
Superposition of Stresses
Disk Problem
–
Results
x

axis (
y
= 0)
y

axis
(
x
= 0)
Applications to Granular Media
Modeling
Contact Load Transfer Between Idealized Grains
(Courtesy of URI Dynamic Photomechanics Lab)
P
P
P
P
Four

Contact Grain
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