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ME 323 Final Lecture – April 2011

Additional Topics

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The Principle of Stationary Potential Energy(A Different Form of Castigliano’s 1st Theorem)

Define the “potential energy” as

U

where U is the strain energy, and

ii uF

whereFi = an applied force;ui = displacement in direction of Fi at point of application of Fi

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and, for “stationary” (which means minimizing ),

So

nn2211 uFuFuFU

iii

FuU

0u

ii u

UF

or

i.e., Castigliano’s First Theorem. The 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,2

321 xcxccv

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2321 xcxccv

Boundary conditions require that

1c0)0(v

xc2cv 32

2c0)0(v

x

P

so 23xcv

and 3c2EIvEIM

and dxEIc2dxEIc2EI21

dxEI2

MU 2

3

2

3

2

3c2v

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dxEIc2dxEI2

MU 2

3

2

and 23Lx LPcvP

where v is defined positive downward. Thus,

L

0

23

2

3 LPcdxEIc2

and for minimum (using “Rayleigh-Ritz method”)

L

0

23

3

PLdxEIc40c

EI4PL

c3 2xEI4

PLv

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Cylindrical Pressure Vessels

Thin-Walled Cylinders (ME 318 Lab S3)

tpr t2

prz (closed cylinder)

Thick-Walled Cylinders (See pp. 350-352, Budynas)

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1 r

CC 1z C (closed cylinder) 2

21r r

CC

2i

2o

2oo

2ii

1 rr

rprpC

2i

2o

io

2

oi2 rr

pprrC

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Spinning Disks

(See p. 355, Budynas)

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2

oi2o

2i

2r r

r

rrrr

83

22

oi2o

2i

2 r3

31r

rrrr

83

where is Poisson’s ratio, is the density of the material, and is the angular velocity.

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- 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:

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- 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;

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- 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.

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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

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Good luck with your studying.

Please continue to check your e-mail and the course website for announcements regarding the final exam.

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