Steady Load Failure Steady Load Failure TheoriesTheories

Lecture 5Lecture 5

Engineering 473Engineering 473Machine DesignMachine Design

Steady Load Failure TheoriesSteady Load Failure Theories

� Maximum-Normal-Stress� Maximum-Normal-Strain� Maximum-Shear-Stress� Distortion-Energy

� Shear-Energy� Von Mises-Hencky� Octahedral-Shear-Stress

� Internal-Friction� Fracture Mechanics

DuctileMaterials

BrittleMaterials

UniaxialStress/Strain

Field

MultiaxialStress/Strain

Field

Many theories have been put forth � some agree reasonably well with test data, some do not.

The MaximumThe Maximum--NormalNormal--Stress TheoryStress Theory

Postulate: Failure occurs when one of the three principal stresses equals the strength.

321 σσσ >>stresses principal

are σ and σ σ 32,1,

Failure occurs when either

c3

t1

Sσ

Sσ

−=

= Tension

CompressionnCompressioin Strength S

Tensionin Strength S

c

t

≡≡

MaximumMaximum--NormalNormal--Stress Failure Stress Failure SurfaceSurface

(Biaxial Condition)(Biaxial Condition)

1σ

2σ

tS

tS

cS-

cS-

According to the Maximum-Normal-Stress Theory, as long as stress state falls within the box, the material will not fail.

locus of failure states

MaximumMaximum--NormalNormal--Stress Failure Stress Failure SurfaceSurface

(Three(Three--dimensional Case)dimensional Case)

1σ

2σ

3σ

tS

cS-

According to the Maximum-Normal-Stress Theory, as long as stress state falls within the box, the material will not fail.

~

~~

The MaximumThe Maximum--NormalNormal--Strain Strain TheoryTheory

(Saint(Saint--Venant’s Venant’s Theory)Theory)

Postulate: Yielding occurs when the largest of the three principal strains becomes equal to the strain corresponding to the yield strength.

( )( )( ) y2133

y3122

y3211

SσσνσEε

SσσνσEε

SσσνσEε

±=+−=

±=+−=

±=+−=

Ratio sPoisson'νModulus sYoung'E

≡≡

MaximumMaximum--NormalNormal--Strain TheoryStrain Theory(Biaxial Condition)(Biaxial Condition)

1σ

2σ

yS

yS

yS-

yS-y12

y21

Sνσσ

Sνσσ

±=−

±=−

As long as the stress state falls within the polygon, the material will not yield.

locus of failure states

MaximumMaximum--ShearShear--Stress TheoryStress Theory((Tresca Tresca Criterion)Criterion)

Postulate: Yielding begins whenever the maximum shear stress in a part becomes equal to the maximum shear stress in a tension test specimen that begins to yield.

1σ2σ3σ σ

τmax1/3 ττ =

y1 Sσ =

32 σ,σ σ

τyτ

Stress State in PartStress State in Part Tensile Test SpecimenTensile Test Specimen

1/2τ2/3τ

321 σσσ >>

MaximumMaximum--ShearShear--Stress TheoryStress Theory(Continued)(Continued)

ys 0.5SS =y1 Sσ =

32 σ,σ σ

τsmax Sτ =

Tensile Test Specimen

The shear yield strength is equal to one-half of the tension yield strength.

MaximumMaximum--ShearShear--Stress TheoryStress Theory(Continued)(Continued)

Stress State in PartStress State in Part

2σσττ

2σστ

2σστ

31max1/3

322/3

211/2

−==

−=

−=

1σ2σ3σ σ

τmax1/3 ττ =

1/2τ2/3τ

321 σσσ >>

MaximumMaximum--ShearShear--Stress TheoryStress Theory(Continued)(Continued)

2S

S ys = From Mohr�s circle for a

tensile test specimen

2σσττ 31

max1/3−== From Mohr�s circle for a three-

dimensional stress state.

31y σσS −=

MaximumMaximum--ShearShear--Stress TheoryStress Theory(Hydrostatic Effect)(Hydrostatic Effect)

( )3211h

hd3

hd2

hd1

σσσ31I3

1σ

σσσ

σσσ

σσσ

3

2

1

++==

+=

+=

+=

Principal stresses will alwayshave a hydrostatic component (equal pressure)

2σστ

2σστ

2σστ

d3

d1

1/3

d3

d2

2/3

d2

d1

1/2

−=

−=

−=

The maximum shear stresses are independent of

the hydrostatic stress.

d => deviatoric componenth => hydrostatic

MaximumMaximum--ShearShear--Stress TheoryStress Theory(Hydrostatic Effect (Hydrostatic Effect –– Continued)Continued)

stress. chydrostati theof magintude theof

regardless yielding no is thereand ,0Then τ

σσσ If

max

d3

d2

d1

=

==

The Maximum-Shear-Stress Theory postulates that yielding is independent of a hydrostatic stress.

Hydrostatic Stress StateHydrostatic Stress State

MaximumMaximum--ShearShear--Stress TheoryStress Theory(Biaxial Representation of the Yield Surface)(Biaxial Representation of the Yield Surface)

31y

32y

21y

σσS

σσSσσS

−=±

−=±

−=±

Yielding will occur if any of the following

criteria are met.

For biaxial case(plane stress)

0σ3 =

1y

2y

21y

σS

σSσσS

=±

=±

−=±

In general, all three conditions must be checked.

MaximumMaximum--ShearShear--Stress TheoryStress Theory(Biaxial Representation of the Yield Surface)(Biaxial Representation of the Yield Surface)

For biaxial case(plane stress)

0σ3 =

1y

2y

21y

σS

σSσσS

=±

=±

−=±1σ

2σ

yS

yS

yS-

yS-

III

IIIIV

Note that in the I and III quadrants the Maximum-Shear-Stress Theory and Maximum-Normal-Stress Theory are the same for the biaxial case.

locus of failure states

MaximumMaximum--ShearShear--Stress TheoryStress Theory(Three(Three--dimensional Representation of the Yield Surface)dimensional Representation of the Yield Surface)

Hamrock, Fig. 6.9

failure surface

AssignmentAssignmentFailure Theories, Read Section 5-9.

(a) Find the bending and transverse shear stress at points A and B in the figure. (b) Find the maximum normal stress and maximum shear stress at both points. (c) For a yield point of 50,000 psi, find the factor of safety based on the maximum normal stress theory and the maximum shear stress theory.