Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

1.15 Stress–Strain Relationships 35

Inthecasewherethebarisreturnedtoitsoriginallengthorifthebarhadnotbeenallowedtoexpand
atall,thetotalstrainiszero,andfromEq.(1.56),

σ=−EαT (1.57)

Equations (1.42) may now be modified to include the contribution of thermal strain. Therefore, by
comparingEq.(1.56),

εx=

1

E

[σx−ν(σy+σz)]+αT

εy=

1

E

[σy−ν(σx+σz)]+αT

εz=

1

E

[σz−ν(σx+σy)]+αT

⎫

⎪⎪

⎪⎪⎪

⎪⎬

⎪⎪

⎪⎪

⎪⎪

⎭

(1.58)

Equations(1.58)maybetransposedinthesamewayasEqs.(1.42)togivestress–strainrelationships
ratherthanstrain–stressrelationships:

σx=

νE

( 1 +ν)( 1 − 2 ν)

e+

E

( 1 +ν)

εx−

E

( 1 − 2 ν)

αT

σy=

νE

( 1 +ν)( 1 − 2 ν)

e+

E

( 1 +ν)

εy−

E

( 1 − 2 ν)

αT

σz=

νE

( 1 +ν)( 1 − 2 ν)

e+

E

( 1 +ν)

εz−

E

( 1 − 2 ν)

αT

⎫

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪⎭

(1.59)

Forthecaseofplanestressinwhichσz=0,theseequationsreduceto

σx=

E

( 1 −ν^2 )

(εx+νεy)−

E

( 1 −ν)

αT

σy=

E

( 1 −ν^2 )

(εy+νεx)−

E

( 1 −ν)

αT

⎫

⎪⎪

⎬

⎪⎪

⎭

(1.60)

Example 1.6
AcompositebaroflengthLhasacentralcoreofcopperlooselyinsertedinasleeveofsteel;theends
ofthesteelandcopperareattachedtoeachotherbyrigidplates.Ifthebarissubjectedtoatemperature
rise T,determinethestressinthesteelandcopperandtheextensionofthecompositebar.Thecopper
corehasaYoung’smodulusEc,across-sectionalareaAc,andacoefficientoflinearexpansionαc;the
correspondingvaluesforthesteelareEs,As,andαs.Assumethatαc>αs.

If the copper core and steel sleeve were allowed to expand freely, their final lengths would be
different, since they have different values of the coefficient of linear expansion. However, since they
arerigidlyattachedattheirends,onerestrainstheotherandanaxialstressisinducedineach.Suppose