124 - 2 - 



cannot, so to speak, get more out of the !-,ethod Ih^n one puts in. To illustrate this point consider 



the proDlen of a simply supported elubtic bean subjected to bending by an impulsive load. The 



exact solution for the duflection y in this case can be expressed as an infinite series of the 

 fol lowing f o rm 



y = S <!i, (t) 0„ (x) (3) 



each t;m of which is tho produce of a timo function and 3 space function tttc latter corresponding 

 to the sh.ipa 0^ tho beijT. whori viorating in j nomal mode. If we apply the -ipproximato method to 

 this probl*! Dy 'issuning the solution to be the sum of two terras only of the form 



y = F^ (t) ^^ (X) + F^ (t) v^'s (x) Ci) 



then the application of the method wuld give 



f, (t) = 4., (t) 



(t) = Citj (t) 



(5) 



Thus in this example the approximate method wuld give the exact contribution for the two 

 terms assumed but lAould give no information regarding the remaining terms in the exact solution. 

 The accuracy of the approximate solution thus depends on whether the assumed terms are those giving 

 the major contribution to deflection, stress or whatever quantity it is desired to estimate. 



For application to practical problems the above limitation is not so drastic as at first 

 sight might appear since appeal can usually be made to experimental results, such as final deflected 

 shape, when deciding what space distributions to assume. Similar recourse may sometimes be had to 

 kno*n exact solutions for simpler problems, for example the solution for a beem problem may suggest 

 the deflection form for a plate problem. 



Possible ai>i'li cations . 



The method is of general application to the behaviour under impulsive loading of structures 

 such as a ship's hull composed of plates supported on frarreworks. The example given in the 

 Appendix is a simple pnobl'3m of this type for elastic deflections. An important practical problan 

 to which the method is to be appli^^^d is that of a ribbed circular cylinder, simulating the pressure 

 hull of a submarine, under Impulsive loading with particular reference to the relative amounts of 

 plastic distortion given to the ribs -and the plating under different systems of loading, 



A plastic problem to which the method has already been applied is that of the relative 

 amounts of energy absorbed respectively in plate stretching and In pull ing-in at edges for tho 

 target plates in Box Model trials. 



A general problem to which the method might also be usefully applied is that of the effect 

 of bodily motion of a target in reducing damage sustained by the target when subjected to an underwater 

 explosion. In general the method is not expected to be of great applicability to hydrodynamical 

 problems due to the difficulty of assuming the space distribution of velocity, etc. In this 

 particular problem, however, use can possibly be mf.de of the known solutions for bodily motion of a 

 rigid sphere. 



In general the method is expected to be more useful for problems in v«iich the deformation is 

 essentially of vibrational typir than to problems in which the essential feature is the propagation 

 of waves. Certain features of wave motion can, however, be incorporated when applying the method, 

 for example the deflection of a beam under impulsive loading could be assumed as given by tvw 

 different space functions in different parts of the beam, the boundary between these being taken as 

 moving along the beam with a certain velocity; this velocity would then appear as one of the 

 dependent variables in the resulting equations. As a tentative suggestion it may also be possi.ble 

 to generalise the method somewhat by leaving in certain arbitrary functions of a wave type and, after 

 substitution in the integrand of equation (l), obtaining the equations by use of the calculus of 

 variations. 



In conclusion, it must be emphasized that the method is proposed not as a substitute for 

 exact methods but ^s a supplementary method which can be applied in cases where either the exact 

 solution is unobtainable or a formal solution is possible but presents excessive ccmputational 

 di f ficul ties. 



