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ME 323 Final Lecture – April 2012
Additional Topics
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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
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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.
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Example Given: a cantilever beam in bending P x L Find: approximate expression for vertical displacement v(x). Solution Assume, for example,
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P x Boundary conditions require that so and and
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and where v is defined positive downward. Thus, and for minimum P (using “Rayleigh-Ritz method”)
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Cylindrical Pressure Vessels
Thin-Walled Cylinders (ME 318 Lab S3) (closed cylinder) Thick-Walled Cylinders (See pp , Budynas) (closed cylinder)
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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.
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Differential Equations of Equilibrium
(See p. 86, Budynas)
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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
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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
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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 ….
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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 …
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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
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All the best with your studying
and your final exams!
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