232 CHAPTER 7 Bending of Thin Plates
conditions for a free edge of (Mxy)x= 0 =0 and (Qx)x= 0 =0 are therefore replaced by the equivalent
condition
(
Qx−
∂Mxy
∂y)
x= 0= 0
orintermsofdeflection
[
∂^3 w
∂x^3
+( 2 −ν)∂^3 w
∂x∂y^2]
x= 0= 0 (7.25)
Also,forthebendingmomentalongthefreeedgetobezero,
(Mx)x= 0 =(
∂^2 w
∂x^2+ν∂^2 w
∂y^2)
x= 0= 0 (7.26)
The replacement of the twisting momentMxyalong the edgesx=0andx=aof a thin plate by
a vertical force distribution results in leftover concentrated forces at the corners ofMxyas shown in
Fig. 7.11. By the same argument, there are concentrated forcesMyxproduced by the replacement of
thetwistingmomentMyx.SinceMxy=−Myx,thenresultantforces2Mxyactateachcornerasshown
andmustbeprovidedbyexternalsupportsifthecornersoftheplatearenottomove.Thedirections
of these forces are easily obtained if the deflected shape of the plate is known. For example, a thin
platesimplysupportedalongallfouredgesanduniformlyloadedhas∂w/∂xpositiveandnumerically
increasing, with increasingynear the cornerx=0,y=0. Hence,∂^2 w/∂x∂yis positive at this point,
andfromEq.(7.14),weseethatMxyispositiveandMyxnegative;theresultantforce2Mxyistherefore
downward. From symmetry, the force at each remaining corner is also 2Mxydownward so that the
tendencyisforthecornersoftheplatetorise.
Havingdiscussedvarioustypesofboundaryconditions,weshallproceedtoobtainthesolutionfor
therelativelysimplecaseofathinrectangularplateofdimensionsa×b,simplysupportedalongeach
ofitsfouredgesandcarryingadistributedloadq(x,y).Wehaveshownthatthedeflectedformofthe
platemustsatisfythedifferentialequation
∂^4 w
∂x^4+ 2
∂^4 w
∂x^2 ∂y^2+
∂^4 w
∂y^4=
q(x,y)
Dwiththeboundaryconditions
(w)x=0,a= 0(
∂^2 w
∂x^2)
x=0,a= 0
(w)y=0,b= 0(
∂^2 w
∂y^2)
x=0,b= 0
Navier(1820)showedthattheseconditionsaresatisfiedbyrepresentingthedeflectionwasaninfinite
trigonometricalorFourierseries
w=∑∞
m= 1∑∞
n= 1Amnsinmπx
asinnπy
b