1954MNRAS.114..210S THE BENDINGOFASQUAREPLATEONSPHERICALFORMER at thesuggestionofDrE.H.Linfoot,deformationaphotographicplate certain assumptionsweremade,andsimilararemadewhereappli- this papersquareplatesareconsidered.Whendealingwiththecircularplate in aSchmidtcamera.Inthatpaperonlycircularplateswereconsidered. tioned inthesectiononboundaryconditions.Thesolutionforsquareplate cable inthispaper.Themodificationsappropriateforasquareplatearemen- transverse pressure. is, ofcourse,morecomplicatedandmuchthesimplificationduetosymmetry is lost.Theproblemis,however,showntobeequivalentinthemathematical sense totheproblemofbendingaclampedsquareplatebyuniform bent bymeansofaframesothatitisincontact atallpointsofitsedges,witha matically inFig.1.Rectangular axesNx,Ny,Nsaretakensuchthat Nx,Ny spherical former.Thestrainedandunstrained positionsareshowndiagram- © Royal Astronomical Society • Provided by theNASA Astrophysics Data System WITH APPLICATIONSTOTHEPHOTOGRAPHICPLATEOF Introduction.—In apreviouspaper*oneofthepresentwritersinvestigated, Displacements andstrains.—Theplatewhichis flatintheunstrainedstateis the presentwritersobtainednecessarycorrectionsappropriatetoa circular plate.Inthispaperthecorrectionsforasquareplateareobtained. for inmakingmeasurementsafterexposure.Inapreviouspaperoneof Schmidt camera,strainsareproducedintheplatewhichmustbeallowed Fig. I.—Diagrammaticrepresentationofthestrainedand unstrained statesoftheplate. When aflatphotographicplateisbentonsphericalformer,asinthe IV. M.ShepherdandD.Radenkovib * W.M.Shepherd,M.N.,113, 450,1953. (Communicated byE.H.Linfoot) A SCHMIDTCAMERA (Received 1954February22) Summary 1954MNRAS.114..210S x No. 2,1954Thebendingofasquareplateonsphericalformer211 of contactNtheunstrainedplateandformer.Thedisplacements the displacementwofapointS(#,3/,o)isgivenbyordinatez equations* (i). so thatthedisplacementsaresmallandwillberelatedtostrainsby point (x,y,z)inthemiddlesurfaceofplatearew,v,wparalleltoaxes Nx Ny>Nzandthecorrespondingstrainsinplaneofplateares,o^. (x,yyz) ontheformer. are intheplaneofunstrainedplateandNzisinwardnormalatpoint 0 where Ristheradiusofcurvatureformer. the platevanishes.Forreasonsgivenin previouspaperzzcanherebe be likewisenegligible.Herewemaywrite coefficient offrictionisremarkablylarge,the surfaceshearstressesxz,yzwill are foundtosatisfythecompatibilitycondition yx assumptions thatyzvanishonthefacesof theplate,andmeanzzacross neglected incomparisonwiththestresses planeoftheplate,andunless Airy stressfunctionx(>y)fromthebody-stress equations,followingtheusual the meanstresses'xkyyy,ïyinplaneof platecanbederivedintermsofan may bewrittenintermsoftheoriginalposition oftheelement.Itfollowsthat small theequationsofequilibriumanelement oftheplateinitsownplane © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 2, The platesubtendsonlyasmallangleatthecentreofcurvatureformer If smallquantitiesoforderhigherthan{xfR)areneglectedinthestrains, Equations (i)nowtaketheform To therequireddegreeofaccuracywisgivenby On eliminatingthetwodisplacementsuandvfromequations(3)strains The differentialequationforthestressfunction.—Since thedisplacementsare * Cf.S.Timoshenko,Plates and Shellspp.304-5,1940. y xv =—5, — 2 oy~ 2 d*» ,_^ =(A dy dxdxdyR^ Sx+ Sy '~ dx2\dx)’ dy+2\dy) or Sy=+ Sx=+ ty 2 dx 2 xy 2 2 + 2 w =(x+y)¡2R du i/dw\ dy d+dx* du dvdzv dv i/dw\ y x +2 yy=^' dy^~ dxR' du dvxy du i dv I 2 d X R, R, y\2 X vy= — dxdy ° X -TO (5) (I) (3) (2) 1954MNRAS.114..210S 212 The straincomponentsin(3)areindependentofzsothatthemeans stress-strain relationsforthestressesinplaneofplateare stress function where £*,
1954Mnras.114..210S the Bending of a Square Plate
