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8-Dec-10 Prof. Samy J. Ebeid ١

MANF231

Theories of Elastic Failure

Engineering MechanicsIII

(Stress Analysis)

Part F3

8-Dec-10 Prof. Samy J. Ebeid ٢

Theories of Elastic Failure

If a ductile material is

subjected to a gradually

increasing axial tensile load,

the part will begin to have

permanent deformations

when the load reaches Py.

In machine components no

permanent deformations are

usually allowed and thus the

yield point indicates elastic

failure. When the tensile test

piece reaches the yield point,

we are faced with certain

simultaneous quantities:

P

P

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8-Dec-10 Prof. Samy J. Ebeid ٣

Theories of Elastic Failure

P

P

σσσσmax = σσσσy

σσσσavg = σσσσmax/2

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8-Dec-10 Prof. Samy J. Ebeid ٤

Theories of Elastic Failure

In machine components no permanentdeformations are usually allowed andthus the yield point indicates elasticfailure. When the tensile test piecereaches the yield point, we are faced

with certain simultaneous quantities:

1. The principal normal stress

2. The maximum shear

stress

will reach ττττ yield= σy/2

3. The strain εreaches the

strain at yield

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8-Dec-10 Prof. Samy J. Ebeid ٥

Theories of Elastic Failure

All these quantities happen at the same

time in the tensile test specimen and it is impossible to state

which of these

From the design point

of view it is important

to assume which of

these quantities when

reached,

8-Dec-10 Prof. Samy J. Ebeid ٦

Elastic failure occurs when the maximum principal normal stress reaches the tensile elastic limit = σyield.

The Maximum Principal Stress Theory

Rankine’s Theory

σmax

If n is the factor of safety, the design of

any mechanical element according to thistheory is governed by:

For Brittle Materials

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8-Dec-10 Prof. Samy J. Ebeid ٧

Elastic failure begins when the maximum shear stress on some inclined plane through the considered point

reaches the value of ττττ max.

If n is the factor of safety, the design of

any mechanical element according tothis theory is governed by:

For Ductile Materials

The Maximum Shear Stress Theory

Coulomb’s Theory

ττττ max

8-Dec-10 Prof. Samy J. Ebeid ٨

Transmission shafts are usually subjected to:

Theories of Elastic Failure as applied to the

Design of Transmission Shafts

1.

Rankine’s Theory

Coulomb’s Theory

σmax =

ττττ max =

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8-Dec-10 Prof. Samy J. Ebeid ٩

Elastic failure occurs when the maximum strain in the element

considered reaches the maximum strain at the yield point.

E= Young’s modulus

µµµµ = Poisson’s ratio

For Brittle Materials

The Maximum Strain Theory

εεεε max =

The maximum strain

σσσσmax

εεεε y =Strain at yield

8-Dec-10 Prof. Samy J. Ebeid ١٠

For Brittle Materials

The Maximum Strain Theory

εεεε max =

σσσσmax

σσσσmin

Element before deformation

εεεε y = σy/EStrain at yield

σ max

n = factor of safety.

σ max