Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

26 CHAPTER 1 Basic Elasticity

Fig.1.16

(a) Stress system on rectangular element; (b) distorted shape of element due to stress system in (a).

Anelementinatwo-dimensionalbodysubjectedtothecomplexstresssystemofFig.1.16(a)distorts
intotheshapeshowninFig.1.16(b).Inparticular,thetriangularelementECDsuffersdistortiontothe
shape E′C′D′with corresponding changes in the length FC and angle EFC. Suppose that the known
directandshearstrainsassociatedwiththegivenstresssystemareεx,εy,andγxy(theactualrelationships
willbeinvestigatedlater)andthatwewanttofindthedirectstrainεninadirectionnormaltotheplane
EDandtheshearstrainγproducedbytheshearstressactingontheplaneED.
Toafirstorderofapproximation,

C′D′=CD( 1 +εx)

C′E′=CE( 1 +εy)

E′D′=ED( 1 +εn+π/ 2 )

⎫

⎪⎬

⎪⎭

(1.29)

whereεn+π/ 2 isthedirectstraininthedirectionED.FromthegeometryofthetriangleE′C′D′inwhich
angleE′C′D′=π/ 2 −γxy,

(E′D′)^2 =(C′D′)^2 +(C′E′)^2 − 2 (C′D′)(C′E′)cos(π/ 2 −γxy)

orsubstitutingfromEqs.(1.29),

(ED)^2 ( 1 +εn+π/ 2 )^2 =(CD)^2 ( 1 +εx)^2 +(CE)^2 ( 1 +εy)^2

− 2 (CD)(CE)( 1 +εx)( 1 +εy)sinγxy

Notingthat(ED)^2 =(CD)^2 +(CE)^2 andneglectingsquaresandhigherpowersofsmallquantities,this
equationmayberewrittenas

2 (ED)^2 εn+π/ 2 = 2 (CD)^2 εx+ 2 (CE)^2 εy− 2 (CE)(CD)γxy

Dividingby2(ED)^2 gives

εn+π/ 2 =εxsin^2 θ+εycos^2 θ−cosθsinθγxy (1.30)