Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

1.16 Experimental Measurement of Surface Strains 37

1.16 ExperimentalMeasurementofSurfaceStrains...................................................

Stresses at a point on the surface of a piece of material may be determined by measuring the strains
atthepoint,usuallybyelectricalresistancestraingaugesarrangedintheformofarosette,asshown
inFig.1.18.SupposethatεIandεIIaretheprincipalstrainsatthepoint,thenifεa,εb,andεcarethe
measured strains in the directionsθ,(θ+α), and (θ+α+β)toεI, we have, from the general direct
strainrelationshipofEq.(1.31),

εa=εIcos^2 θ+εIIsin^2 θ (1.61)

whereεxbecomesεI,εybecomesεII,andγxyiszero,sincethexandydirectionshavebecomeprincipal
directions.RewritingEq.(1.61),wehave

εa=εI

(

1 +cos2θ

2

)

+εII

(

1 −cos2θ

2

)

or

εa=^12 (εI+εII)+^12 (εI−εII)cos2θ (1.62)

Similarly,

εb=^12 (εI+εII)+^12 (εI−εII)cos2(θ+α) (1.63)

and

εc=^12 (εI+εII)+^12 (εI−εII)cos2(θ+α+β) (1.64)

Therefore,ifεa,εb,andεcaremeasuredingivendirections—thatis,givenanglesαandβ—thenεI,εII,
andθaretheonlyunknownsinEqs.(1.62)to(1.64).

Fig.1.18

Strain gauge rosette.