Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.3 Plates Subjected to a Distributed Transverse Load 229

Takingmomentsaboutthexaxis

Mxyδy−

(

Mxy+

∂Mxy

∂x

δx

)

δy−Myδx+

(

My+

∂My

∂y

δy

)

δx

−

(

Qy+

∂Qy

∂y

δy

)

δxδy+Qx

δy^2

2

−

(

Qx+

∂Qx

∂x

δx

)

δy^2

2

−qδx

δy^2

2

= 0

Simplifyingthisequationandneglectingsmallquantitiesofahigherorderthanthoseretainedgive

∂Mxy

∂x

−

∂My

∂y

+Qy= 0 (7.17)

Similarly,takingmomentsabouttheyaxis,wehave

∂Mxy

∂y

−

∂Mx

∂x

+Qx= 0 (7.18)

SubstitutinginEq.(7.16)forQxandQyfromEqs.(7.18)and(7.17),weobtain

∂^2 Mx

∂x^2

−

∂^2 Mxy

∂x∂y

+

∂^2 My

∂y^2

−

∂^2 Mxy

∂x∂y

=−q

or

∂^2 Mx

∂x^2

− 2

∂^2 Mxy

∂x∂y

+

∂^2 My

∂y^2

=−q (7.19)

ReplacingMx,Mxy,andMyinEq.(7.19)fromEqs.(7.7),(7.14),and(7.8)gives

∂^4 w

∂x^4

+ 2

∂^4 w

∂x^2 ∂y^2

+

∂^4 w

∂y^4

=

q

D

(7.20)

Thisequationmayalsobewrittenas
(
∂^2
∂x^2

+

∂^2

∂y^2

)(

∂^2 w

∂x^2

+

∂^2 w

∂y^2

)

=

q

D

or
(
∂^2
∂x^2

+

∂^2

∂y^2

) 2

w=

q

D

Theoperator(∂^2 /∂x^2 +∂^2 /∂y^2 )isthewell-knownLaplaceoperatorintwodimensionsandissometimes
writtenas∇^2 .Thus,

(∇^2 )^2 w=

q
D
Generally, the transverse distributed loadqis a function ofxandyso that the determination of
thedeflectedformoftheplatereducestoobtainingasolutionofEq.(7.20),whichsatisfiestheknown
boundaryconditionsoftheproblem.ThebendingandtwistingmomentsfollowfromEqs.(7.7),(7.8),