1 ME 323 Final Lecture – April 2011 Additional Topics

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

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

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

5 Boundary conditions require that x P so and

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

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

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

9 -free body diagrams; -equilibrium equations -fundamental equations of “The Chart”; -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; Topics we’ve studied in stress analysis (or “mechanics of materials”) in GE 213, ME 313, ME 323 include:

10 -shearing stresses in pins; -torsion of circular members; -torsion of nonlinear materials; -torsion of tapered shafts; -torsion of noncircular open sections; -torsion of noncircular closed sections; -shearing stresses due to transverse loading in beams;

11 -strain gauge data analysis; -temperature effects; -buckling of columns; -statically indeterminate problems; -failure criteria (von Mises, Tresca, max. normal stress, Mohr-Coulomb); and -problems requiring energy methods; -more.

12 Additional topics in your future careers may include: -plate and shell analyses; -curved beam analysis; -contact problems; -finite element methods; -other computer methods; -life-long reference to textbooks, handbooks, data-bases and the internet

13 Good luck with your studying. Please continue to check your and the course website for announcements regarding the final exam.

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