1. 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 Numerics M.H. Sadd , University of Rhode Island

2. Cartesian Coordinate Solutions Using Polynomials
In Cartesian coordinates we choose Airy stress function solution of polynomial form
Method 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. This 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 Island

3. Example 8.1 Uniaxial Tension of a Beam
Stress Field
Displacement Field (Plane Stress)
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 A02 = T/2, and all other boundary conditions are identically satisfied. Therefore the stress field solution is given by
. . . Rigid-Body Motion
“Fixity conditions” needed to determine RBM terms
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

4. Example 8.2 Pure Bending of a Beam
Stress Field
Displacement Field (Plane Stress)
Boundary Conditions:
Expecting a linear bending stress distribution, try second-order stress function of the form
Moment boundary condition implies that
A03 = -M/4c3, and all other boundary conditions are identically satisfied. Thus the stress field is
“Fixity conditions” to determine RBM terms:
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

5. 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
Two solutions are identical, with the exception of the x-displacements
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

6. Example 8.3 Bending of a Beam by Uniform Transverse Loading
Stress Field
Boundary Conditions:
BC’s
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7. 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
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

8. Example 8.3 Beam Problem Normal Stress Comparisons of Elasticity with Elementary Mechanics of Materials
x – Stress at x=0
y - Stress
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.
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

9. 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.
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

10. Example 8.3 Beam Problem Displacement Field (Plane Stress)
Choosing Fixity Conditions
Strength of Materials:
Good match for beams where l >> c
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

11. Cartesian Coordinate Solutions Using Fourier Methods
Fourier 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
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12. Example 8.4 Beam with Sinusoidal Loading
Stress Field
Boundary Conditions:
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13. Example 8.4 Beam Problem Bending Stress
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14. Example 8.4 Beam Problem Displacement Field (Plane Stress)
For the case l >> c
Strength of Materials
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

15. Example 8.5 Rectangular Domain with Arbitrary Boundary Loading
Must use series representation for Airy stress function to handle general boundary loading.
Boundary Conditions
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 Island

16. Polar Coordinate Formulation Airy Stress Function Approach  = (r,θ)
Airy Representation
Biharmonic Governing Equation R S
Traction Boundary Conditions x y  r 
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

17. Polar Coordinate Formulation Plane Elasticity Problem
Strain-Displacement
Hooke’s Law
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

18. General Solutions in Polar Coordinates Michell 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 Island

19. Axisymmetric Solutions
Stress Function Approach: =(r)
Navier Equation Approach: u=ur(r)er
(Plane Stress or Plane Strain)
Gives Stress Forms
Displacements - Plane Stress Case
Underlined terms represent
rigid-body motion
a3 term leads to multivalued behavior, and is not found following the displacement formulation approach
Could also have an axisymmetric elasticity problem using  = a4 which gives r =  = 0 and r = a4/r  0, see Exercise 8-15
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

20. Example 8.6 Thick-Walled Cylinder Under Uniform Boundary Pressure
General Axisymmetric
Stress Solution
Boundary Conditions
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 Island

21. Example 8.6 Cylinder Problem Results Internal Pressure Only
r1/r2 = 0.5
r/r2
r /p
θ /p
Dimensionless Stress
Dimensionless Distance, r/r2
Thin-Walled Tube Case:
Matches with Strength
of Materials Theory
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

22. Special Cases of Example 8-6
Stress Free Hole in an Infinite Medium Under Equal Biaxial Loading at Infinity
Pressurized Hole in an Infinite Medium
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23. Example 8.7 Infinite Medium with a Stress Free Hole Under Uniform Far Field Loading
Boundary Conditions
Try Stress Function
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24. Example 8.7 Stress Results
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25. Superposition of Example 8.7 Biaxial Loading Cases
Tension/Compression Case
T1 = T , T2 = -T
Equal Biaxial Tension Case
T1 = T2 = T
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

26. Review Stress Concentration Factors Around Stress Free Holes
K = 2
K = 3 =
K = 4
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27. Stress Concentration Around Stress Free Elliptical Hole – Chapter 10
Maximum Stress Field
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

28. Stress Concentration Around Stress Free Hole in Orthotropic Material – Chapter 11
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

29. 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 = 6.5in/in/oF (11.7m/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 Island

30. 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 Island

31. 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 Island

32. 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)
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

33. Half-Space Examples Uniform Normal Stress Over x  0
Boundary Conditions
Try Airy Stress Function
Use BC’s To Determine Stress Solution
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

34. Half-Space Under Concentrated Surface Force System (Flamant Problem)
Boundary Conditions
Try Airy Stress Function
Use BC’s To Determine Stress Solution
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

35. Flamant Solution Stress Results Normal Force Case
or in Cartesian
components
y = a
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

36. Flamant Solution Displacement Results Normal Force Case
Note unpleasant feature of 2-D model that displacements become unbounded as r  
On Free Surface y = 0
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

37. Comparison of Flamant Results with 3-D Theory - Boussinesq’s Problem
Cartesian Solution
Free Surface Displacements
Cylindrical Solution
Corresponding 2-D Results
3-D Solution eliminates the unbounded far-field behavior
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

38. Half-Space Under Uniform Normal Loading Over –a  x  a
dY = pdx = prd /sin
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39. Half-Space Under Uniform Normal Loading - Results
max - Contours
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40. Generalized Superposition Method Half-Space Loading Problems
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41. Photoelastic Contact Stress Fields
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42. Notch/Crack Problem
Try Stress Function:
Boundary Conditions:
At Crack Tip r  0:
Finite Displacements and Singular Stresses at Crack Tip  1<  <2   = 3/2
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

43. 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
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

44. 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
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

45. Mode III Crack Problem – Exercise 8-41
Anti-Plane Strain Case
z - Stress Contours
Stresses Again
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

46. Curved Beam Under End Moments
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47. Curved Cantilever BeamP a b r 
Curved Cantilever Beam
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 Island

48. Disk Under Diametrical Compression= P
Flamant Solution (1) + +
Flamant Solution (2)
Radial Tension Solution (3)
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

49. Disk Problem – Superposition of Stresses
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50. Disk Problem – Results x-axis (y = 0) y-axis (x = 0)
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

51. Applications to Granular Media Modeling Contact Load Transfer Between Idealized Grains
(Courtesy of URI Dynamic Photomechanics Lab) P
Four-Contact Grain
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

52. Contact Between Two Elastic Solidspc
Generates:
Contact Area (w)
Interface Tractions (pc)
Local Stresses in Each Body
Creates Complicated Nonlinear Boundary Condition: Boundary Condition Changing With Deformation; i.e. w and pc Depend on Deformation, Load, Elastic Moduli, Interfacial Friction Characteristics
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

53. 2-D Elastic Half-Space Subjected to a Rigid Indenterx y
Rigid Indenter a
Local stresses and deformation determined from Flamant solution See Section and Exercise 8.38
Frictionless Case (t = 0)
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

54. 2-D Elastic Half-Space Subjected Frictionless Flat Rigid Indenterx y
Rigid Indenter a P
Solution
Unbounded Stresses at Edges of Indenter
Max Shear Stress Contours
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

55. 2-D Elastic Half-Space Subjected Frictionless Cylindrical Rigid Indenter
Solution
Max Shear Stress Contours
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island