ME 323 Final Lecture – April 2012

1. ME 323 Final Lecture – April 2012 Additional Topics

2. The Principle of Stationary Potential Energy (A Different Form of Castigliano’s 1st Theorem) Define the “potential energy” P as where U is the strain energy, and where Fi = an applied force; ui = displacement in direction of Fi at point of application of Fi

3. So and, for “stationary” P (which means minimizing P ), or i.e., Castigliano’s First Theorem. The P function is used extensively in the finite element method.

4. x L Example Given: a cantilever beam in bending P Find: approximate expression for vertical displacement v(x). Solution Assume, for example,

5. P x Boundary conditions require that so and and

6. and where v is defined positive downward. Thus, and for minimum P (using “Rayleigh-Ritz method”)

7. (closed cylinder) (closed cylinder) Cylindrical Pressure Vessels Thin-Walled Cylinders(ME 318 Lab S3) Thick-Walled Cylinders (See pp. 350-352, Budynas)

8. Spinning Disks (See p. 355, Budynas) where n is Poisson’s ratio, r is the density of the material, and w is the angular velocity.

9. Differential Equations of Equilibrium (See p. 86, Budynas)

10. Strain-Displacement Relations (See p. 24, Budynas) u is the x-component of displacement v is the y-component of displacement w is the z-component of displacement

11. Topics Covered in GE 213, ME 313, and ME 323 • Axial loading of rods • Bearing stresses in bolt holes and pin holes • Symmetric Bending of Beams • Unsymmetric Bending of Beams • Bending of composites and nonlinear materials • Torsion of circular members • Torsion of noncircular open sections • Torsion of noncircular closed sections • Shearing stresses in pins • Shearing stresses due to transverse loading in beams • Shear center

12. Topics Covered in GE 213, ME 313, and ME 323 • Strain gauge analysis • Temperature effects • Statically indeterminate problems • Energy methods for impact analysis • Energy methods for deflection analysis • General 3-D stress states • Failure criteria • Tensor mathematics • Generalized Hooke’s law • Resultant-stress relations • And more ….

13. Topics for Future Study • Restrained warping in noncircular torsion • Curved beams • Plate theory • Shell theory • Contact problems • Buckling and instability • Fracture mechanics • Nonlinear problems • And more …

14. Sources of Information • Senior engineers and co-workers • Textbooks and Handbooks • Technical literature – journals, conference proceedings • The internet (with judgment) • “Roark’s Formulas for Stress and Strain” • Your own intellectural resources

15. All the best with your studying and your final exams!