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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

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**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

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**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

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**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

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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

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**Example 8.3 Bending of a Beam by Uniform Transverse Loading**

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

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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

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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

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**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

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**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

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**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

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**Example 8.4 Beam with Sinusoidal Loading**

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

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**Example 8.4 Beam Problem Bending Stress**

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

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**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

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**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

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**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

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**Polar Coordinate Formulation Plane Elasticity Problem**

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

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**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

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**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

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**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

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**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

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**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

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**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

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**Example 8.7 Stress Results**

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

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**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

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**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

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**Stress Concentration Around Stress Free Elliptical Hole – Chapter 10**

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

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**Stress Concentration Around Stress Free Hole in Orthotropic Material – Chapter 11**

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

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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 Island

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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

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**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

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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

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**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

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**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

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**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

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**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

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**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

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**Half-Space Under Uniform Normal Loading Over –a x a**

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

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**Half-Space Under Uniform Normal Loading - Results**

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

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**Generalized Superposition Method Half-Space Loading Problems**

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

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**Photoelastic Contact Stress Fields**

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

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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

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**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

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**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

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**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

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**Curved Beam Under End Moments**

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

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**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

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**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

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**Disk Problem – Superposition of Stresses**

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

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**Disk Problem – Results x-axis (y = 0) y-axis (x = 0)**

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

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**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

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**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

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**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

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**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

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**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

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