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IX. The Equations of Equilibrium of an Elastic Plate under 

 Normal Pressure. By John Prescott, ALA., D.Sc, 

 Lecturer in Mathematics in the Faculty of Technology of 

 Manchester University *. 



THE earliest writers who had any success with the 

 problem o£ the bending of an elastic plate followed 

 the analogy of the beam so faithfully that they missed 

 altogether one very important difference between a plate and 

 a beam or rod. They were so obsessed with the idea that 

 only the curvatures of the plate mattered that they ignored 

 the effect of the stretching of the middle surface that must 

 accompany any bending of the surface. It is a very well- 

 known fact that a plane area cannot be made to fit on the 

 surface of a sphere, for example, without a stretching or 

 contraction of some of its parts. Yet this strain has nearly 

 always been neglected by writers on the bending of thin 

 plates. It is true that there are many cases in which it is 

 justifiable to neglect these stretchings and shortenings, but 

 there are so many practical problems in which it is not 

 justifiable that no theory of thin plates is satisfactory which 

 does not take them into account, ur else point out the 

 limitations of the restricted theory. 



The usual Poisson-Kirchhoff equations need no alteration 

 when the maximum deflexion of any particle of the middle 

 surface is small in comparison with the thickness of the 

 plate, this deflexion being measured from a plane which 

 touches the middle surface at any convenient point, or from 

 any developable surface which touches the middle surface at 

 any point and nearly coincides with it elsewhere. But when 

 this deflexion is of the same order as the thickness, then 

 Poisson's equations may be very much in error. In such 

 cases the mean tensions over sections perpendicular to the 

 middle surface play as great a part in supporting a pressure 

 on the plate as do those stresses of which the usual theory 

 takes account. In fact, to make Poisson's theory correct 

 there must be quantities of three different orders of magni- 

 tude. In ascending order these quantities are 



(1) the maximum deflexion perpendicular to the unstrained 



middle surface or perpendicular to a developable 

 surface ; 



(2) the thickness of the plate; 



(3) the lateral dimensions of the plate. 



* Communicated by the Author. 

 Phil. Mag. S. 6. Vol. 43. No. 253. Jan. 1922. H 



