7.3 Plates Subjected to a Distributed Transverse Load 231
7.3.2 The Built-In Edge
Iftheedgex=0isbuilt-inorfirmlyclampedsothatitcanneitherrotatenordeflect,then,inaddition
tow,theslopeofthemiddleplaneoftheplatenormaltothisedgemustbezero.Thatis,
(w)x= 0 = 0
(
∂w
∂x
)
x= 0
= (^0) (7.24)
7.3.3 The Free Edge
Alongafreeedgetherearenobendingmoments,twistingmoments,orverticalshearingforces,sothat
ifx=0isthefreeedge,then
(Mx)x= 0 = 0 (Mxy)x= 0 = 0 (Qx)x= 0 = 0
giving,inthisinstance,threeboundaryconditions.However,Kirchhoff(1850)showedthatonlytwo
boundaryconditionsarenecessarytoobtainasolutionofEq.(7.20),andthatthereductionisobtained
byreplacingthetworequirementsofzerotwistingmomentandzeroshearforcebyasingleequivalent
condition.ThomsonandTait(1883)gaveaphysicalexplanationofhowthisreductionmaybeeffected.
TheypointedoutthatthehorizontalforcesystemequilibratingthetwistingmomentMxymaybereplaced
alongtheedgeoftheplatebyaverticalforcesystem.
Consider two adjacent elements,δy 1 andδy 2 , along the edge of the thin plate of Fig. 7.11. The
twisting momentMxyδy 1 on the elementδy 1 may be replaced byforces Mxya distanceδy 1 apart.
Note thatMxy, being a twisting moment per unit length, has the dimensions of force. The twisting
momentontheadjacentelementδy 2 is[Mxy+(∂Mxy/∂y)δy]δy 2 .Again,thismaybereplacedbyforces
Mxy+(∂Mxy/∂y)δy.Atthecommonsurfaceofthetwoadjacentelements,thereisnowaresultantforce
(∂Mxy/∂y)δyor a vertical force per unit length of∂Mxy/∂y. For the sign convention forQxshown in
Fig.7.9,wehaveastaticallyequivalentverticalforceperunitlengthof(Qx−∂Mxy/∂y).Theseparate
Fig.7.11
Equivalent vertical force system.