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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” 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

6

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)

2

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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Differential Equations of Equilibrium

(See p. 86, Budynas)

0Bzyx xzxyxxx

0Bzyx yzyyyxy

0Bzyx zzzyzxz

xx x

yx

yxyx

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Strain-Displacement Relations

xu

xx

yv

yy

zw

zz

xv

yu

xy

yw

zv

yz

zu

xw

zx

u is the x-component of displacementv is the y-component of displacementw is the z-component of displacement

(See p. 24, Budynas)

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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!