TwoDimensional Problem Solution Using Airy Stress Function approach plane elasticity formulation with zero body forces reduces to a single governing biharmonic equation In Cartesian coordinates it is given by ID: 198287
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Slide1
Chapter 8 Two-Dimensional Problem Solution
Using Airy Stress Function approach, 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
Elasticity
Theory, Applications and NumericsM.H. Sadd , University of Rhode IslandSlide2
Cartesian Coordinate Solutions Using PolynomialsIn Cartesian coordinates we choose Airy stress function solution of polynomial formMethod produces polynomial stress distributions, and thus would not satisfy general boundary conditions. However, using Saint-Venant’s principle we can replace a non-polynomial condition with a statically equivalent polynomial loading. T
his formulation is most useful for problems with rectangular domains, and is commonly based on inverse solution concept where we assume a polynomial solution form and then try to find what problem it will solve. Notice that the three lowest order terms with
m + n 1 do not contribute to the stresses and will therefore be dropped. 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 biharmonic equation for any choice of constants Amn. However, for higher order terms, constants Amn
will have to be related in order to have polynomial satisfy biharmonic equation.
Elasticity
Theory, Applications and
Numerics
M.H. Sadd
, University of Rhode IslandSlide3
Example 8.1 Uniaxial Tension of a Beam Boundary 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
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide4
Example 8.2 Pure Bending of a BeamBoundary Conditions: Expecting a linear bending stress distribution, try second-order stress function of the form
M
oment boundary condition implies
that
A
03
= -M/4c
3, and all other boundary conditions are identically satisfied. Thus the stress field isStress Field
“Fixity conditions”
to determine RBM terms:
Displacement Field (Plane Stress)
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide5
Example 8.2 Pure Bending of a BeamSolution 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
Elasticity
Theory, Applications and NumericsM.H. Sadd ,
University of Rhode IslandSlide6
Example 8.3 Bending of a Beam by Uniform Transverse LoadingBoundary Conditions:
Stress Field
BC’s
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide7
Example 8.3 Beam ProblemStress Solution Comparison of Elasticity with Elementary Mechanics of Materials Elasticity Solution
Mechanics of Materials Solution
Shear
stresses are identical, while
normal
stresses are not
Elasticity
Theory, Applications and Numerics
M.H.
Sadd ,
University of Rhode IslandSlide8
Example 8.3 Beam ProblemNormal Stress Comparisons of Elasticity with Elementary Mechanics of Materials
Maximum differences between two theories exist at top and bottom of beam, difference in stress is w/5. For
most beam problems (l >> c), bending stresses will be much greater than w, and differences between elasticity and strength of materials will be relatively small.Maximum difference between two theories is w
and occurs at top of 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
Elasticity
Theory, Applications and NumericsM.H.
Sadd ,
University of Rhode IslandSlide9
Example 8.3 Beam ProblemNormal Stress Distribution on Beam Ends
End stress distribution
does not vanish and is nonlinear but gives zero resultant force.
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide10
Example 8.3 Beam Problem
Choosing
Fixity Conditions
Strength of Materials:
Good match for
beams where
l
>>
c
Displacement Field (Plane Stress)
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide11
Cartesian Coordinate Solutions Using Fourier MethodsFourier methods provides a more general solution scheme for biharmonic equation. Such techniques generally use separation of variables along with Fourier series or Fourier integrals.
Choosing
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide12
Example 8.4 Beam with Sinusoidal Loading
B
oundary Conditions:
Stress Field
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide13
Example 8.4 Beam Problem
Bending Stress
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide14
Example 8.4 Beam Problem
For the case
l
>>
c
Strength
of
Materials
Displacement Field (Plane Stress)
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide15
Example 8.5 Rectangular Domain with Arbitrary Boundary Loading Boundary Conditions
Must use series representation for Airy stress function to handle general
boundary
loading.
Using Fourier series theory to handle general
boundary
conditions, generates a doubly infinite set of equations to solve for unknown constants in stress function form. See text for details
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide16
Polar Coordinate FormulationAiry Stress Function Approach = (r,θ)
R
S
x
y
r
Airy Representation
Biharmonic Governing Equation
Traction Boundary Conditions
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide17
Polar Coordinate FormulationPlane Elasticity Problem
Strain-Displacement
Hooke’s Law
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide18
General Solutions in Polar CoordinatesMichell Solution
Choosing the case where
b
=
in
,
n = integer gives the general Michell solution
Will use various terms from this general solution to solve several plane problems in polar coordinates
Elasticity
Theory, Applications and Numerics
M.H.
Sadd
,
University of Rhode IslandSlide19
Axisymmetric Solutions
Stress Function Approach:
=(
r
)
Navier Equation Approach: u=ur(r)er(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
8-15
Underlined terms represent
rigid-body motion
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide20
Example 8.6 Thick-Walled Cylinder Under Uniform Boundary Pressure
B
oundary Conditions General AxisymmetricStress Solution
Using
Strain Displacement Relations and
Hooke’s
Law for plane strain gives the
radial
displacement
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide21
Example 8.6 Cylinder Problem ResultsInternal Pressure Only
r
1/r2 = 0.5
r/r
2
r /p
θ /pDimensionless Stress
Dimensionless Distance,
r/
r2Thin-Walled Tube Case:
Matches with Strength
of Materials Theory
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide22
Special Cases of Example 8-6Pressurized Hole in an Infinite Medium
Stress Free Hole in an Infinite Medium Under Equal Biaxial Loading at Infinity
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide23
Example 8.7 Infinite Medium with a Stress Free Hole Under Uniform Far Field Loading
B
oundary Conditions
Try Stress Function
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide24
Example 8.7 Stress Results
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide25
Superposition of Example 8.7Biaxial Loading CasesT1
T2T1T2
Equal Biaxial Tension CaseT1 = T2 = TTension/Compression CaseT1 = T , T2 = -T
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide26
Review Stress Concentration FactorsAround Stress Free Holes
K = 2
K = 3K = 4=
Elasticity
Theory, Applications and Numerics
M.H. Sadd
, University of Rhode IslandSlide27
Stress Concentration Around Stress Free Elliptical Hole – Chapter 10
Maximum Stress Field
Elasticity
Theory, Applications and
Numerics
M.H. Sadd ,
University of Rhode IslandSlide28
Stress Concentration Around Stress Free Hole in Orthotropic Material – Chapter 11
Elasticity
Theory, Applications and Numerics
M.H.
Sadd , University of Rhode IslandSlide29
2-D Thermoelastic Stress Concentration Problem Uniform Heat Flow Around Stress Free Insulated 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 = 6.5in/in/oF (11.7m/m/oC), qa/k = 100oF (37.7oC), the maximum stress becomes 19.5ksi (88.2MPa)
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide30
Nonhomogeneous Stress Concentration Around Stress Free Hole in a Plane Under Uniform Biaxial Loading with Radial Gradation of Young’s Modulus – Chapter 14
Elasticity
Theory, Applications and Numerics
M.H.
Sadd , University of Rhode IslandSlide31
Three Dimensional Stress Concentration Problem – Chapter 13 Normal Stress on the x,y-plane (z = 0)
Two Dimensional Case:
(
r
,
/2)/
S
Three Dimensional Case:
z
(
r
,0)/
S
,
= 0.3
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide32
Wedge Domain Problems
Use general
stress function
solution to
include terms that are bounded at
origin
and give uniform stresses on the boundariesQuarter Plane Example ( = 0 and =
/2)
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide33
Half-Space ExamplesUniform Normal Stress Over x 0
Try Airy Stress Function
Boundary Conditions
Use BC’s To Determine Stress Solution
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide34
Half-Space Under Concentrated Surface Force System (Flamant Problem)Try Airy Stress Function Boundary Conditions
Use BC’s To Determine Stress Solution
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide35
Flamant Solution Stress ResultsNormal Force Case
or in Cartesian components
y
=
a
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd , University of Rhode IslandSlide36
Flamant Solution Displacement ResultsNormal Force Case
On Free Surface
y
= 0
Note unpleasant feature of 2-D model that displacements become unbounded as
r
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide37
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
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide38
Half-Space Under Uniform Normal Loading Over –a x a
dY
=
pdx
=
prd
/sin
Elasticity Theory, Applications and NumericsM.H. Sadd
,
University of Rhode IslandSlide39
Half-Space Under Uniform Normal Loading - Results
max
- Contours
Elasticity
Theory, Applications and
NumericsM.H. Sadd , University of Rhode IslandSlide40
Generalized Superposition MethodHalf-Space Loading Problems
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd , University of Rhode IslandSlide41
Photoelastic Contact Stress Fields
Elasticity
Theory, Applications and NumericsM.H. Sadd
, University of Rhode IslandSlide42
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
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide43
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 theoryA terms give symmetric stress fields – Opening or Mode I behaviorB terms give antisymmetric stress fields – Shearing or Mode II behavior
Elasticity
Theory, Applications and
NumericsM.H. Sadd , University of Rhode IslandSlide44
Crack Problem ResultsContours 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
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide45
Mode III Crack Problem – Exercise 8-41
Anti-Plane Strain Case
Stresses Again
z
- Stress Contours
Elasticity
Theory, Applications and
NumericsM.H.
Sadd
,
University of Rhode IslandSlide46
Curved Beam Under End Moments
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide47
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
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide48
Disk Under Diametrical Compression+
P
P
D
=
+
Flamant Solution (1)
Flamant Solution (2)
Radial Tension Solution (3)
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide49
Disk Problem – Superposition of Stresses
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide50
Disk Problem – Results
x
-axis (
y
= 0) y-axis (x = 0)
Elasticity
Theory, Applications and Numerics
M.H.
Sadd
, University of Rhode IslandSlide51
Applications to Granular Media ModelingContact Load Transfer Between Idealized Grains
(Courtesy of URI Dynamic Photomechanics Lab)
P
P
P
P
Four-Contact Grain
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode IslandSlide52
Contact Between Two Elastic SolidsCreates Complicated Nonlinear Boundary Condition:Boundary Condition Changing With Deformation; i.e. w and pc
Depend on Deformation, Load, Elastic Moduli, Interfacial Friction Characteristics
w
p
c
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode Island
Generates:
Contact
Area (
w
)
Interface Tractions
(
p
c
)
Local Stresses in Each BodySlide53
2-D Elastic Half-Space Subjected to a Rigid Indenter
x
y
Rigid Indenter
a
a
Local stresses and deformation determined from
Flamant
solution
See Section 8.4.9 and Exercise 8.38
Elasticity
Theory, Applications and
Numerics
M.H.
Sadd
,
University of Rhode Island
Frictionless Case (
t
= 0
)Slide54
2-D Elastic Half-Space Subjected Frictionless Flat Rigid Indenter
Elasticity Theory, Applications and
NumericsM.H. Sadd
, University of Rhode Island
x
y
Rigid Indenter
a
a
P
Max Shear
Stress Contours
Solution
Unbounded Stresses at Edges of IndenterSlide55
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
2-D Elastic Half-Space Subjected
Frictionless Cylindrical Rigid Indenter
Max Shear
Stress Contours
Solution