_ 19 - 503 



A.icj M otion d-ue to load suddenly attlicd and then maintained constant . 



If the loaJ is of the form: 



"(t) = 0, t < 



= = , t > C 



(86) 



then equation (68) Dpcomes: 



ph Aj W ^. A^ T^ W = P^ (87) 



anJ the initial conditions rest (69) w= ootain as solution: 



w = i w^ (i - Cos a t) (88) 



2P^ 



h'o 



(89) 



while a is jiven by (7u) js in tht orccedin^ section. 

 Then S and its d.'rivitives are givsn By: 



5 = "2 "l «^ = -5 ^m '1 - ^"^"^ ^'^ ('°' 



S = i a S^ :in a t (l - Cos a t) (91) 



S = i a^ S (Cvis c^ t - Cj5 ^ a t) (92) 



5 = 1 A w 2 ^ ^ "l (93) 



^m 2 1 (h A T 2 



1 



The motion mjst cease when 5 = w = C =nd a t = tt ?nd change to state of rest as considered 

 in tne svction a. 11(5) . 



Further, when c t = tt, we hc'.ve from (38) ana (89) and (90) that: 



S = W = 



2 P T (94) 



1., T W ^ ,. T W = L-J > P 



11 1 1 m J 1 



since T^ ~ T. Dy jur assumoticn (66). »s the load F(t) remains constant for a t >-tt, equation (61) 

 will thus Se satisfied indefinitely and no further motion will occur. Physically this means that 

 for a t ^TT the deformed plate remains at rest in statical equilibrium with the load P., the resulting 

 tension in tlate and at edj.s being too small to produce ^ny yielding. 



Since thoro is no further motion, it follows that W and S as given by (89) and (9?) are 



' m m ^ 



the final maximum v-i^lues of W ano 5, as their n.-tation anticipated. 



In order to decide the types of motion during O^a t ■$ 77, the curve for S from (9l) is 

 shewn in Figure 5. As will be seen from equation (?3), the slope S of the S curve increases 

 steadily from at t = up to .'. maximum value 9 a S /I6 at cos a t = i and thence decreases 

 steadily to a negative v.lus -.t a t = 77. 



If we new define w, as the bcrderline v>lue of W corrsspcnding to the maximum S = Y, 

 2 m ^ ^ ' 



then, using (U6) : 



32 Y 32 A^ 



^ -^ i (95) 



9 a"^ Aj^ 9 u^' 



Thence 



