Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

1.13 Principal Strains 27

ThestrainεninthedirectionnormaltotheplaneEDisfoundbyreplacingtheangleθinEq.(1.30)by
θ−π/2.Hence,

εn=εxcos^2 θ+εysin^2 θ+

γxy

2

sin2θ (1.31)

TurningourattentionnowtothetriangleC′F′E′,wehave

(C′E′)^2 =(C′F′)^2 +(F′E′)^2 − 2 (C′F′)(F′E′)cos(π/ 2 −γ) (1.32)

inwhich

C′E′=CE( 1 +εy)

C′F′=CF( 1 +εn)

F′E′=FE( 1 +εn+π/ 2 )

SubstitutingforC′E′,C′F′,andF′E′inEq.(1.32)andwritingcos(π/ 2 −γ)=sinγ,wefind

(CE)^2 ( 1 +εy)^2 =(CF)^2 ( 1 +εn)^2 +(FE)^2 ( 1 +εn+π/ 2 )^2

− 2 (CF)(FE)( 1 +εn)( 1 +εn+π/ 2 )sinγ

(1.33)

Allthestrainsareassumedtobesmallsothattheirsquaresandhigherpowersmaybeignored.Further,
sinγ≈γandEq.(1.33)becomes

(CE)^2 ( 1 + 2 εy)=(CF)^2 ( 1 + 2 εn)+(FE)^2 ( 1 + 2 εn+π/ 2 )− 2 (CF)(FE)γ

FromFig.1.16(a),(CE)^2 =(CF)^2 +(FE)^2 andtheprecedingequationsimplifiesto

2 (CE)^2 εy= 2 (CF)^2 εn+ 2 (FE)^2 εn+π/ 2 − 2 (CF)(FE)γ

Dividingby2(CE)^2 andtransposing,

γ=

εnsin^2 θ+εn+π/ 2 cos^2 θ−εy

sinθcosθ

Substitutionofεn+π/ 2 andεnfromEqs.(1.30)and(1.31)yields

γ

2

=

(εx−εy)

2

sin2θ−

γxy

2

cos2θ (1.34)

1.13 PrincipalStrains......................................................................................

IfwecompareEqs.(1.31)and(1.34)withEqs.(1.8)and(1.9),weobservethattheymaybeobtainedfrom
Eqs.(1.8)and(1.9)byreplacingσnbyεn,σxbyεx,σybyεy,τxybyγxy/2,andτbyγ/2.Therefore,for
eachdeductionmadefromEqs.(1.8)and(1.9)concerningσnandτ,thereisacorrespondingdeduction
fromEqs.(1.31)and(1.34)regardingεnandγ/2.