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Bi,A)ia #tre## #it%ation

In this case the stress situation consists of two principal stresses 1σ , 2σ , and the strains1 are

gi#en by

=1ε ( )21

1υσ σ −

E

, =2ε ( )12

1υσ σ −

E

, and=3ε ( )

21

1σ σ υ +−

E

Tri,A)ia #tre## #it%ation

This is the case of three principal stresses 1σ , 2σ , 3σ , and the strains in the directions of the

principal stresses are then gi#en by

=1ε ( )[ ]321

1σ σ υ σ +−

E !1"

=2ε ( )[ ]312

1σ σ υ σ +−

E !2"

=3ε ( )[ ]213

1

σ σ υ σ +− E !3"

ENERGY PER UNIT -OLUME AT STRESS LOCATION

Tota #train ener"y U

The total strain energy is the strain energy caused by the three principal stresses 1σ , 2σ , 3σ .

It is gi#en by

=U 112

1ε σ

(22

2

1ε σ

(33

2

1ε σ

!)"

$ubstituting the three strains 321 ,, ε ε ε and in equations !1",!2" and !3" into equation !)"

yields

=U ( )[ ]313221

2

3

2

2

2

1 22

1σ σ σ σ σ σ υ σ σ σ ++−++

E !*"

Strain Ener"y !%e to Chan"e of -o%me (Hy!ro#tati' #tre##* ony

The stress that causes change of #olume only !hydrostatic stress" may be considered as the

a#erage of the three principal stresses avσ , and deri#ed from the expression

=avσ

3

321 σ σ σ ++ !+"

$ubstituting for the hydrostatic stress avσ , into equation !*" yields

=vU ( )[ ]22323

2

1avav

E σ υ σ − !"

1 'echanical -ngineering esign/ $higley, 0oseph, pg 12), 'craw ill, $e#enth -dition, 2)



=vU [ ]υ σ

212

3 2

− E

av 4 [ ] 2

2

213av

E σ

υ − !5"

$ubstituting the #alue of avσ from equation !+" into equation !5" yields

=vU [ ]

2321

32

213

++− σ σ σ υ

E

4 [ ]( )2321

267

213σ σ σ

υ ++

−

E

=vU [ ]

( ) 2321+

21σ σ σ

υ ++

−

E 4 [ ]

( )[ ]313221

2

3

2

2

2

1 2+

21σ σ σ σ σ σ σ σ σ

υ +++++

−

E

=vU ( )[ ]313221

2

3

2

2

2

1 2+


υ +++++

−

E !7"

This vU

is the strain energy per unit #olume caused by the uniform !hydrostatic" stress,which is part of the three principal stresses 1σ , 2σ , 3σ .

Di#tortion Ener"y at the o'ation of $rin'i$a #tre##e# 1σ . 2σ . 3σ

The distortion energy can then be obtained as the difference between the total strain energy atthe location of principal stresses, and the strain energy due to the hydrostatic portion of the

stresses at the same location. The distortion energy is then deri#ed from the expression

=d U U 8 vU

Where,=d U istortion energy in the element at the location of principal stresses 1σ . 2σ . 3σ

=U ( )[ ]313221

2

3

2

2

2

1 22

1σ σ σ σ σ σ υ σ σ σ ++−++

E !*"

=vU ( )[ ]313221

2

3

2

2

2

1 2+


υ +++++

−

E !7"

Therefore,

d U 4 ( )[ ]313221

2

3

2

2

2

1 22

1σ σ σ σ σ σ υ σ σ σ ++−++

E 8

( )[ ]313221

2

3

2

2

2

1 2+


υ +++++

−

E

d U 4 ( ) ( )[ ]313221

2

3

2

2

2

1 +3+

1σ σ σ σ σ σ υ σ σ σ ++−++

E 8

( ) ( ) ( )[ ]313221

2

3

2

2

2

1313221

2

3

2

2

2

1 262622+

1σ σ σ σ σ σ υ σ σ σ υ σ σ σ σ σ σ σ σ σ ++−++−+++++

E



d U

4

( ) ( ) ( )( ) ( ) ( )

++++++++−

++−++−++

313221

2

3

2

2

2

1313221

2

3

2

2

2

1313221

2

3

2

2

2

1

262622

+3

+

1

σ σ σ σ σ σ υ σ σ σ υ σ σ σ σ σ σ

σ σ σ σ σ σ σ σ σ υ σ σ σ

E

d U 4

( ) ( )

( ) ( )

+++++−

−++−++

2

3

2

2

2

1313221

313221

2

3

2

2

2

1

622

22

+

1

σ σ σ υ σ σ σ σ σ σ

σ σ σ σ σ σ υ σ σ σ

E

d U 4 ( )( ) ( )( )[ ]2222+

1313221

2

3

2

2

2

1 +++−+++ υ σ σ σ σ σ σ υ σ σ σ

E

d U 4

( )

( ) ( )[ ]313221

2

3

2

2

2

1+

221

σ σ σ σ σ σ σ σ σ

υ

++−++

+

E

d U 4 ( ) ( ) ( )[ ]313221

23

22

21

3


υ ++−++

+

E !1"

9ut

( )3132212

32

221 σ σ σ σ σ σ σ σ σ ++−++ 4

( ) ( ) ( )

2

231

232

221 σ σ σ σ σ σ −+−+−

Therefore

d U 4 ( ) ( ) ( ) ( )( )[ ]2

31

2

32

2

21362

1σ σ σ σ σ σ

υ −+−+−+

E !11"

d U 4 ( )

( ) ( ) ( )( )[ ]2

31

2

32

2

21+

1σ σ σ σ σ σ

υ −+−+−

+

E !12"

THE CASE OF SIMPLE TENSION TEST /HEN YIELDING OCCURS



:or the simple tension test specimen, the three principal stresses when yielding occurs are

1σ 4 yS , 2σ 4, 3σ 4

$ubstituting for the principal stresses in equation !12" yields

d U 4 ( ) ( ) ( ) ( )( )[ ]222

+

1−+−+−

+ y y S S

E

υ

d U 4 ( ) [ ]2

2+

1 yS

E

υ +!13"

THE CASE OF THREE DIMENSIONAL STRESS /HEN YIELDING OCCURS

The distortion energy theory of failure states

When Yie!in" occurs in any material, the !i#tortion #train ener"y $er %nit &o%me at the

point of failure equals or exceeds the !i#tortion #train ener"y $er %nit &o%me when

yie!in" occurs in the ten#ion te#t #$e'imen.

This can be restated that when yielding occurs in any situation

d U 4 ( )

( ) ( ) ( )( )[ ]2

31

2

32

2

21+

1σ σ σ σ σ σ

υ −+−+−

+

E !12"

E0UALS

d U 4 ( ) [ ]2

2+

1 yS

E

υ +!13"

( ) ( ) ( )231

2

32

2

21 σ σ σ σ σ σ −+−+− 4 2

2 yS

( ) ( ) ( )

−+−+−

2

2

31

2

32

2

21 σ σ σ σ σ σ

4 yS !1)"

E0UI-ALENT (-on,Mi#e#* STRESS

The expression on the left hand side of equation !1)" is therefore considered as the

e1%i&aent #tre## eσ , which causes failure by yielding. The equi#alent stress is then gi#en

by

eσ

4

( ) ( ) ( )

−+−+−

2

2

31

2

32

2


!1*"

The equi#alent stress eσ is also referred to as -on Mi#e# stress.

DESIGN E0UATION BASED ON THE DISTORTION ENERGY THEORY

This is deri#ed by ad;usting the yield strength of the material in simple tension with an

appropriate factor of safety .. s f The design equation then becomes



eσ 4 ( ) ( ) ( )

−+−+−

2

2

31

2

32

2


4.. s f

S y!1+"

APPLICATION OF THE DESIGN E0UATION

The principal stresses 1σ . 2σ . 3σ are first determined by stress analysis. $uch analysis

describes the principal stresses as a function of the oa! carried, and the "eometry and

!imen#ion# of the machine or structural element.

The equi#alent stress in the design equation is then expressed in terms of the !imen#ion# of

the machine or structural element, while the right hand side is the ten#ie yie! #tren"th of

the material.

The fa'tor of #afety is simply a number chosen by the designer. The factor of safety together

with the strength of the material, gi#es the wor<ing2 !design, allowable" stress expected in the

machine part. The solution to the design equation then gi#es the minim%m !imen#ion# required to a#oid fai%re of the element by yie!in".

2 Wor<ing $tress, page *2,andboo<, 'etals -ngineering =esign, >merican $ociety of 'echanical

-ngineers !>$'-"

beyond the syllabus f f

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