- 21 - 505 



corresponding in shaoe to the S curve, and the tanijent AC (Figure 5) at 6 = 6 is then given by: 



{\- ZosQ ) * [0 -0 ] (Cos 0„ - Cos 2 ) (105) 







This tangent was then drawn Dy calculating from (105) two points on it, namely, the point 

 where it cuts Y = ana tne point wncre it cuts either = 77 or t? = 77/2 tne latter beinj used for 

 the steeper tjngcnti to avoid ths; pcints Deing too *tdely soaced. (Tne usi of th? graphical 

 property -.f tjns,£ncy is such would be relatively inaccurate). wnere this tangent cut the curve 

 (lO«) th-'n gave a first approximation t'' (say). 



Now, if we put if = 0' . + e in (lOO), expand in powers cf e and neglect pcwers higher than 

 the f i rst, we obtain: 



e (Cos 0^ - Cos 20^- Cos 0- ^ + Cos 0' ^) (106) 



= Sin(9'j (1 - Cos^'^) - Sin0^ (l - Cos 0^) - {0\ -0^)(Cos0^- Cos 2(9.) 



Using therefore the qraphic.il value cf C", in (1O6), we jbtained e and thence a corrected 

 value of 6 , for given 5 . Since £ was in all cases small, no further correction by successive 

 Substitution in (1O6) was found necessary. 



The results for S' /S as a function of w /W are given in Table 2 but are not plotted as 



m rr. m 2 



such in Figure 2 since the somewhat remarkable fact was noted that this curve was virtually 



indistinguishable from the curve S' /3, versus W /w, for the instantaneous impulse over the range 

 ^ m rn m 1 ' 



plotted. MatheiT.atically there does not seem any obvious connection between the set of equations 



(85) and the set (lOO), (lOi) and (102) while physically no explanation can be offered. However, 



this does not mean that the relative ar"a of oull-in is the same in the two cases for 3 given central 

 deflection, since from (80) and (95): 



,, 2 _ 32 u. 2 

 '2 - T •*! 



(107) 



In order to bring out this difference, tne ratio S' /S for the present case has been 



mm 



plotted in Figure 2 against ths same abscissa W /W as for the instantaneous impulse. 

 4.24 Sume r\ cat valves of parameter s. 



For numerical application to Box Model results, we note that the qualitative variation of 

 S' /S with *< /W in Figure 2 does not depend on the numerical values of the parameters A etc., 

 but that W depends on the constant K(say) defined by: 



2 



(108) 



4a b A 



A A 

 2 "3 



whence from (80) and (95) we have: 



2 



T - T w ' W ' 



Tjj «ab 32 i»ab 



The constant K depends on the s;< parameters defined in (9) and (13) but not on A , and we 

 must therefore make specific assumptions regaroing the forms of the displacement functions of 

 f J ^2 f, of equations (l), (2) o.nd (3). We assume them to be of the simple type: 



I - '1 f^'^' = ^ '-; ^ (^' ("0) 



I - '2 '"•>' = ^2 (3) s (|> (111) 



^ = '3 (x.y) = F^ (|) F^ (^) (112) 



It will 



