# Chapter 8 Two-Dimensional Problem Solution

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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

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

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

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

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

Stress Field Boundary Conditions: BC’s Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

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

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

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

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

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

Stress Field Boundary Conditions: Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

Example 8.4 Beam Problem Bending Stress
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

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

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

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

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

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

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

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

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

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

Example 8.7 Infinite Medium with a Stress Free Hole Under Uniform Far Field Loading
Boundary Conditions Try Stress Function Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

Example 8.7 Stress Results
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

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

Review Stress Concentration Factors Around Stress Free Holes
K = 2 K = 3 = K = 4 Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

Stress Concentration Around Stress Free Elliptical Hole – Chapter 10
Maximum Stress Field Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

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

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

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

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

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

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

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

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

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

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

dY = pdx = prd /sin Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

max - Contours Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

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

Photoelastic Contact Stress Fields
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

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

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

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

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

Curved Beam Under End Moments
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

Curved Cantilever Beam
P 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

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

Disk Problem – Superposition of Stresses
Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

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

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

Contact Between Two Elastic Solids
pc 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

2-D Elastic Half-Space Subjected to a Rigid Indenter
x 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

2-D Elastic Half-Space Subjected Frictionless Flat Rigid Indenter
x 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

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