158 CHAPTER 5 Energy Methods
Fig.5.32
Beam of Example 5.14.
Example 5.14
Determine the deflection of the tip of the cantilever in Fig. 5.32 with the temperature gradient
shown.
ApplyingaunitloadverticallydownwardatB,M 1 = 1 ×z.Alsothetemperaturetatasectionzis
t 0 (l−z)/l.SubstitutinginEq.(5.32)gives
(^) Te,B=
∫l
0
z
α
h
t 0
l
(l−z)dz (i)
IntegratingEq.(i)gives
(^) Te,B=
αt 0 l^2
6 h
(i.e.,downward)
References
[1] Charlton,T.M.,EnergyPrinciplesinAppliedStatics,Blackie,1959.
[2] Gregory,M.S.,IntroductiontoExtremumPrinciples,Butterworths,1969.
[3] Megson,T.H.G.,StructuralandStressAnalysis,2ndedition,Elsevier,2005.
Further Reading
Argyris,J.H.,andKelsey,S.,EnergyTheoremsandStructuralAnalysis,Butterworths,1960.
Hoff,N.J.,TheAnalysisofStructures,JohnWiley,1956.
Timoshenko,S.P.,andGere,J.M.,TheoryofElasticStability,McGraw-Hill,1961.
Problems..............................................................................................
P.5.1 FindthemagnitudeandthedirectionofthemovementofthejointCoftheplanepin-jointedframeloaded
asshowninFig.P.5.1.ThevalueofL/AEforeachmemberis1/20mm/N.
Ans. 5.24mmat14.7◦toleftofvertical.