Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

1.9 Strain 23

Now cosA′O′C′=cos(π/ 2 −γxz)=sinγxz,andasγxzis small, then cos A′O′C′=γxz. From the
trigonometricalrelationshipsforatriangle,

cosA′O′C′=

(O′A′)^2 +(O′C′)^2 −(A′C′)

2 (O′A′)(O′C′)

2

(1.19)

Wehavepreviouslyshown,inEq.(1.17),that

O′A′=δx

(

1 +

∂u

∂x

)

Similarly,

O′C′=δz

(

1 +

∂w

∂z

)

Butforsmalldisplacements,thederivativesofu,v,andwaresmallcomparedwithlsothat,asweare
concernedherewithactuallengthratherthanchangeinlength,wemayusetheapproximations

O′A′≈δx O′C′≈δz

Againtoafirstapproximation,

(A′C′)^2 =

(

δz−

∂w

∂x

δx

) 2

+

(

δx−

∂u

∂z

δz

) 2

SubstitutingforO′A′,O′C′,andA′C′inEq.(1.19),wehave

cosA′O′C′=

(δx^2 )+(δz)^2 −[δz−(∂w/∂x)δx]^2 −[δx−(∂u/∂z)δz]^2

2 δxδz

Expandingandneglectingfourth-orderpowersgive

cosA′O′C′=

2 (∂w/∂x)δxδz+ 2 (∂u/∂z)δxδz

2 δxδz

Similarly,

γxz=

∂w

∂x

+

∂u

∂z

γxy=

∂v

∂x

+

∂u

∂y

γyz=

∂w

∂y

+

∂v

∂z

⎫

⎪⎪

⎪⎪

⎪⎪

⎪⎬

⎪⎪

⎪⎪

⎪⎪

⎪⎭

(1.20)

ItmustbeemphasizedthatEqs.(1.18)and(1.20)arederivedontheassumptionthatthedisplacements
involved are small. Normally, these linearized equations are adequate for most types of structural
problem,butincaseswheredeflectionsarelarge—forexample,typesofsuspensioncable,andsoon—
thefull,nonlinear,largedeflectionequations,giveninmanybooksonelasticity,mustbeused.