113 



inhere A « 8a / 3 Is the area of the plate. The mean work done per unit volume o' the material is 



(3») 

 RectangMlar plate dxshed by a non-uniform distribution of pressure. 



Wp = 5.76 P(f)^/a^ 



The only simple case in which the displacement o' a rectangular memDrane by non-uniform 

 pressure can 6e calculated is when the pressure Is proportional to cos -^ cos —^ . The 

 equation of equilibrium Is 



Po "^ IT 



This is satisfied by taking z ' <^. cos ^ 



and 



p = Pt 





(«o) 



(«2) 



The mean value of cos -j^ cos ^g^ over the rectangle is -j = O.uos.so that for this distribution 

 of pressure " 



1/h = 0.«0« or h /z 







2.46 (»3) 



Contours of the displacements are shown in Figure 2. 



Dishing of thin plastic plates under suddenly applied pressures , 



Wonnal modes . 



The foregoing study of the dishing of a plsstic plate under static pressui^ leads to the 

 conclusion that it may, without much error, be treated as a membrane with uniform tension in all, 

 directions. If this approximation is made to a plastic plate distorted by a very large load 

 applied for a very short time, the motion of the plate can be identical with that of a membrane 

 during the first quarter of its period. The condition that this particular solution will apply 

 is that the plate Is given initi.illy a distribution of velocity corresponding with the velocity 

 of a membrane which is executing simple harmonic oscillations. If this condition is satisfied, 

 the timer required for the dishing of the plate is j of the period of the vibration 

 of the corresponding membrane. Tne membrane of course would return towards its 

 equilibrium position after the first ^ period but the plastic plate stays in the position cf maxifr 

 distortion. The longest period cf a circular membrane jf radius a is 2.6la/c where c^ = P/p, 

 p is Its density and Pt the -stress per unit length; t is the thickness. The period of a 

 rectangular membrane of sides 2a and 2b is Hab/c / a ♦ b . 



Thus T = 0.65 a/c for a circular plate 



J — 5 f 



and T = ab/c / a + b for a rectangular plate 



(«*) 



It will be noticed that c depends only on the stress per unit thickness of the membrane. 

 Thus T is the same for all thicknesses of plate. For steel plates of density 7.8 with 

 20 tons/square inch yield, c^ = 20 x 1.5U x 10^/7.8, so that 



and 



c = 1.99 X 10 cm. /second = 654 feet/second for 20 ton steel 

 c = 2.44 X 10 cm. /second = 800 feet/second for 30 ton steel 



(45) 



For the 6 feet x 4 feet plates used in the investigation 



T = 3 X 2/654 /p+2^ => 

 T = 2.08 X lO"-' seconds 



2.55 X 10 -^ seconds for 20 ton yield 

 for 3 ton yield 



. (46) 



Uniform ., 



