34 CHAPTER 1 Basic Elasticity
Fig.1.17
Mohr’s circle of strain for Example 1.5.
arelocated.ThecenterCofthecircleistheintersectionofQ 1 Q 2 andtheOεaxis.Thecircleisthen
drawnwithradiusCQ 1 ,andthepointsB(εI)andA(εII)arelocated.Finally,angleQ 1 CB= 2 θandangle
Q 1 CA= 2 θ+π.
1.15.1 Temperature Effects
The stress–strain relationships of Eqs. (1.43) through (1.47) apply to a body or structural member
at a constant uniform temperature. A temperature rise (or fall) generally results in an expansion (or
contraction)ofthebodyorstructuralmembersothatthereisachangeinsize—thatis,astrain.
Consider a bar of uniform section, of original lengthLo, and suppose that it is subjected to a
temperaturechange Talongitslength; Tcanbearise(+ve)orfall(−ve).Ifthecoefficientoflinear
expansionofthematerialofthebarisα,thefinallengthofthebaris,fromelementaryphysics,
L=Lo( 1 +α T)sothatthestrain,ε,isgivenby
ε=L−Lo
Lo=α T (1.55)Suppose now that a compressive axial force is applied to each end of the bar such that the bar
returnstoitsoriginallength.Themechanicalstrainproducedbytheaxialforceisthereforejustlarge
enoughtooffsetthethermalstrainduetothetemperaturechangemakingthetotalstrainzero.Ingeneral
terms,thetotalstrain,ε,isthesumofthemechanicalandthermalstrains.Therefore,fromEqs.(1.40)
and(1.55),
ε=σ
E+α T (1.56)