1. Application Solutions of Plane Elasticity
Professor M. H. Sadd

2. Solutions to Plane Problems Cartesian Coordinates
Airy Representation
Biharmonic Governing Equation R S
Traction Boundary Conditions x y

3. Uniaxial Tension of a Beamy T 2l 2c

4. Note Integrated Boundary Conditions
Pure Bending of a Beam x y M 2l 2c
Note Integrated Boundary Conditions

5. Bending of a Beam by Uniform Transverse Loadingx y w 2c 2l wl
x/w - Elasticity
x/w - Strength of Materials
l/c = 2
l/c = 4
l/c = 3
Dimensionless Distance, y/c

6. Bending of a Beam by Uniform Transverse Loadingx y w 2c 2l wl
Note that according to theory of elasticity, plane sections do not remain plane
For long beams l >>c, elasticity and strength of materials deflections will be approximately the same

7. Cantilever Beam Problemx y N P L 2c
Stress Field
Displacement Field

8. Cantilever Tapered Beamx y L p  A B
Stress Field
x = L
x = L

9. Solutions to Plane Problems Polar Coordinates
Airy Representation
Biharmonic Governing Equation R S
Traction Boundary Conditions x y  r 

10. General Solutions in Polar Coordinates

11. Thick-Walled Cylinder Under Uniform Boundary Pressure
r1/r2 = 0.5
r/r2
r /p
 /p
Dimensionless Distance, r/r2
Internal Pressure Case

12. Stress Free Hole in an Infinite Medium Under Uniform Uniaxial Loading at Infinity
r/a

13. Stress Concentrations for Other Loading Cases
Biaxial Loading T
Biaxial Loading T
Unaxial Loading
K=3
K=2
K=4

14. Stress Concentration Around Elliptical Holex y b a
()max/S
Circular Case (K=3)

15. Half-Space Under Concentrated Surface Force System (Flamant Problem)x y Y X r  C
Normal Loading Case (X=0)
Dimensionless Distance, x/a
y/(Y/a)
xy/(Y/a)
y = a

16. Notch-Crack Problems Contours of Maximum Shear Stress y r    x
 = 2 -  r  x
Contours of Maximum Shear Stress

17. Two-Dimensional FEA Code MATLAB PDE Toolbox
- Simple Application Package For Two-Dimensional Analysis Initiated by Typing “pdetool” in Main MATLAB Window
Includes a Graphical User Interface (GUI) to: Select Problem Type Select Material Constants Draw Geometry Input Boundary Conditions Mesh Domain Under Study Solve Problem Output Selected Results

18. FEA Notch-Crack Problem
(vonMises Stress Contours)

19. Curved Beam Problem P  r a b  = /2 b/a = 4 a/P
Dimensionless Distance, r/a
a/P
Theory of Elasticity Strength of Materials
 = /2 b/a = 4

20. Disk Under Diametrical Compression= P
Flamant Solution (1) + +
Flamant Solution (2)
Radial Tension Solution (3)

21. Disk Under Diametrical Compression+ + = P 2 y x 1 r1 r2

22. Disk Results Theoretical, Experimental, Numerical
Theoretical Contours of Maximum Shear Stress
Finite Element Model (Distributed Loading)
Photoelastic Contours
(Courtesy of Dynamic Photomechanics
Laboratory, University of Rhode Island)