Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.4 Combined Bending and In-Plane Loading of a Thin Rectangular Plate 239

Thetotalforceinthezdirectionisfoundfromthesummationoftheseexpressionsandis

Nx

∂^2 w

∂x^2

δxδy+

∂Nx

∂x

∂w

∂x

δxδy+Ny

∂^2 w

∂y^2

δxδy+

∂Ny

∂y

∂w

∂y

δxδy

+

∂Nxy

∂x

∂w

∂y

δxδy+ 2 Nxy

∂^2 w

∂x∂y

δxδy+

∂Nxy

∂y

∂w

∂x

δxδy

inwhichNyxisequaltoandisreplacedbyNxy.UsingEqs.(7.31)and(7.32),wereducethisexpressionto

(

Nx

∂^2 w

∂x^2

+Ny

∂^2 w

∂y^2

+ 2 Nxy

∂^2 w

∂x∂y

)

δxδy

Sincethein-planeforcesdonotproducemomentsalongtheedgesoftheelement,Eqs.(7.17)and
(7.18)remainunaffected.Further,Eq.(7.16)maybemodifiedsimplybytheadditionofthepreceding
verticalcomponentofthein-planeloadstoqδxδy.Therefore,thegoverningdifferentialequationfora
thinplatesupportingtransverseandin-planeloadsis,fromEq.(7.20),

∂^4 w

∂x^4

+ 2

∂^4 w

∂x^2 ∂y^2

+

∂^4 w

∂y^4

=

1

D

(

q+Nx

∂^2 w

∂x^2

+Ny

∂^2 w

∂y^2

+ 2 Nxy

∂^2 w

∂x∂y

)

(7.33)

Example 7.3
DeterminethedeflectedformofthethinrectangularplateofExample7.1if,inadditiontoauniformly
distributedtransverseloadofintensityq 0 ,itsupportsanin-planetensileforceNxperunitlength.

TheuniformtransverseloadmaybeexpressedasaFourierseries(seeEq.(7.28)andExample7.1);
thatis,

q=

16 q 0

π^2

∑∞

m=1,3,5

∑∞

n=1,3,5

1

mn

sin

mπx

a

sin

nπy

b

Equation(7.33)thenbecomes,onsubstitutingforq,

∂^4 w

∂x^4

+ 2

∂^4 w

∂x^2 ∂y^2

+

∂^4 w

∂y^4

−

Nx

D

∂^2 w

∂x^2

=

16 q 0

π^2 D

∑∞

m=1,3,5

∑∞

n=1,3,5

1

mn

sin

mπx

a

sin

nπy

b

(i)

Theappropriateboundaryconditionsare

w=

∂^2 w

∂x^2

=0atx=0anda

w=

∂^2 w

∂y^2

=0aty=0andb