Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.1 Pure Bending of Thin Plates 221

SubstitutingforεxandεyfromEqs.(7.1)into(7.2)andrearranginggives

σx=

Ez

1 −ν^2

(

1

ρx

+

ν

ρy

)

σy=

Ez

1 −ν^2

(

1

ρy

+

ν

ρx

)

⎫

⎪⎪

⎪⎬

⎪⎪

⎪⎭

(7.3)

Aswouldbeexpectedfromourassumptionofplanesectionsremainingplane,thedirectstressesvary
linearlyacrossthethicknessoftheplate,theirmagnitudesdependingonthecurvatures(i.e.,bending
moments) of the plate. The internal direct stress distribution on each vertical surface of the element
mustbeinequilibriumwiththeappliedbendingmoments.Thus,

Mxδy=

∫t/^2

−t/ 2

σxzδydz

and

Myδx=

∫t/^2

−t/ 2

σyzδxdz

SubstitutingforσxandσyfromEqs.(7.3)gives

Mx=

∫t/^2

−t/ 2

Ez^2

1 −ν^2

(

1

ρx

+

ν

ρy

)

dz

My=

∫t/^2

−t/ 2

Ez^2

1 −ν^2

(

1

ρy

+

ν

ρx

)

dz

Let

D=

∫t/^2

−t/ 2

Ez^2

1 −ν^2

dz=

Et^3

12 ( 1 −ν^2 )

(7.4)

Then,

Mx=D

(

1

ρx

+

ν

ρy

)

(7.5)

My=D

(

1

ρy

+

ν

ρx

)

(7.6)

inwhichDisknownastheflexuralrigidityoftheplate.