Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

36 CHAPTER 1 Basic Elasticity

thatthisstressisσx.TheninEqs.(1.58),σx=σcorσsandσy=σz=0;thetotalstraininthecopperand
steelisthen,respectively,

εc=

σc

Ec

+αc T (i)

εs=

σs

Es

+αs T (ii)

Thetotalstraininthecopperandsteelisthesame,sincetheirendsarerigidlyattachedtoeachother.
Therefore,fromcompatibilityofdisplacement,

σc

Ec

+αc T=

σs

Es

+αs T (iii)

Thereisnoexternalaxialloadappliedtothebarsothat

σcAc+σsAs= 0

thatis, σs=−

Ac

As

σc (iv)

SubstitutingforσsinEq.(iii)gives

σc

(

1

Ec

+

Ac

AsEs

)

= T(αs−αc)

fromwhich σc=

T(αs−αc)AsEsEc

AsEs+AcEc

(v)

Also,αc>σssothatσcisnegativeandthereforecompressive.NowsubstitutingforσcinEq.(iv),

σs=−

T(αs−αc)AcEsEc

AsEs+AcEc

(vi)

whichispositiveandthereforetensileaswouldbeexpectedbyaphysicalappreciationofthesituation.
Finally,theextensionofthecompoundbar,δ,isfoundbysubstitutingforσcinEq.(i)orforσsin
Eq.(ii).Then,

δ= TL

(

αcAcEc+αsAsEs

AsEs+AcEc

)

(vii)