LOADS ACTING ON A SHIP AND THE ELASTIC RESPONSE OF A SHIP 



257 



with writers on ship motions unci siiij) bending moments. 

 Quoting from a recent paper by Radoslavljevic (1957a): 

 ". . . the amplitudes of forced oscillations taking into 

 account Smith Effect can be over 50% smaller in heav- 

 ing and over 30% smaller in pitching, in relation to the 

 same amplitudes when the distribution of buoyancy in 

 the disturbed sea is hydrostatic." 



W. E. Smith included the effect of orbital water 

 velocities in the wave, but neglected the wave celerity 

 and ship motions. Except for the adjustment in the 

 effective weight of water, the calculations are still based 

 on statics. The definition "static calculation" is often 

 used, therefore, in the contemporary technical literature^ 

 in reference to calculations which include the Smith 

 effect. 



T. C. Read (3-1890) called attention to ihe fact that the 

 effective ship weight is also modified bj' heaving accelera- 

 tions. It is increased in wave troughs and decreased on 

 wave crests. Read sIiowcmI that there is a corresponding 

 increase of the sagging moment from 8.4 to 24 per cent 

 and decrease of hogging moment from to 7.7 per cent. 

 These figures refer to two examples given by Read. 

 The larger figures correspond to a larger bending moment 

 and a fine vessel and the smaller figures to a full vessel 

 and a smaller bending moment. The excess of sagging 

 over hogging bending moment was later confirmed 

 qualitatively by sea observations on the MS Smi Fran- 

 cisco (Schnadel, 3-1936) and SS Ocean Vulcan (3-Adm. 

 Ship Weld. Comm., 1953, 1954). It also was confirmed 

 by model tests in towing tanks (Sato, 1951; Ochi, 1956, 

 1957; Lewis and Dalzell, 1958). 



1.3 Dynamics of Ship Motions. The significance and 

 magnitude of Smith and Head effects were further demon- 

 strated by Alexander (1911) and Robb (1918) without 

 introducing new concepts. Kriloff (3-1896, 1898a and 

 b), Horn (1910), Hazen and Nims (3-1940) and Bull 

 (3-Adm. Ship. Weld. Comm., 1953) considered shi]5 

 motions and inertial forces more completely. The dif- 

 ferential eciuations of a ship's hea\'iiig and pitching were 

 formed and in\-estigated. The common weak points in 

 this activity were the failure to consider the coupling 

 between heaving and pitching motions, the failure to 

 consider the modifications of the wave pressure gradient 

 (Smith effect) by interference of a ship's hull, and the 

 inadequacy of the available hydrodynamic data. At- 

 tention was concentrated on the dynamics of ship motion 

 while the necessary hydrodynamic development was 

 neglected. 



An advanced theory of coupled ship motions was de- 

 veloped by Haskind (3-1947), but it did not extend to 

 the calculation of bending moments and was not com- 

 plete in regard to the forces caused by waves. Recently, 

 Hanaoka (3-1957a, b) extended a similar theory and pre- 

 sented a sample of bending moment calculations. It 

 was pointed out in Chapter 2 that such advanced mathe- 

 matical methods are valuable in guiding simpler ap- 

 proaches. However, they cannot be used in engineering 



^ For example, Admiralty Ship Welding Committee Reports 

 Nos. 8 and 12 (1953, 11)54), and Hanaoka (1957). 



problems because of excessive mathematical complexity 

 and because their application is limited to certain ideal- 

 ized ship forms. In the present monograph, therefore, 

 attention will be concentrated on a simpler engineering 

 approach. 



A solution of the coupled ditferential ecjuations of 

 pitching and heaving motions was developed in a simple 

 form in the course of the past few years (Korvin-Krou- 

 kovsky and Lewis, 3-1955). This was followed by an 

 evaluation of hydrodynamic-force coefficients which 

 included the interference effects between a ship and 

 waves (Korvin-Kroukovsky and Jacobs, 3-1957). The 

 material has now been collected for the formulation of a 

 simple rational theory of bending moments. The word 

 "rational" is used here to indicate a theory formulated 

 on the basis of and compatible with the physical laws of 

 nature and free of empirical assumptions. The word 

 rational should not be confused with exact. The process 

 of application of physical laws to an engineering problem 

 necessarily involves \'arious approximations. The.se 

 approximations, however, are compatible with dynamic 

 and hydrodynamic concepts and are not based on em- 

 piricism. 



1.4 Rate of Load Application to a Ship's Structure. 

 In the iliseu.'^si(jn of bending moments it is necessary to 

 distinguish between slowly and rapidly varying hy- 

 drodynamic loads. The slowly varying loads are im- 

 posed by waves which have a period of encounter from 

 5 to 15 sec in the case of a normal cargo ship. These 

 periods are from 8 to 25 times longer than the period of 

 the two-node hull vibration. A ship's structure responds 

 to such loads as though they were static loads. At any 

 instant the stress is equal to the bending moment divided 

 by the effective .section modulus and a ship's elastic 

 properties need not be considered. The historical out- 

 line, presented earlier, refers to bending moments 

 caused by such slowly varying loads and this subject will 

 be discussed in the next three sections of this chapter. 



Slamming is a well-known example of rapidly applied 

 loads which cause vibration of a ship's structure. Less 

 well known is the nature of wave shocks felt in fast ships 

 and ships with excessive bow flare. The.se loads are 

 applied with such rapitlity that stresses cannot be ascer- 

 tained by static considerations and it is necessary to con- 

 sider a ship's elastic properties. These loads will be dis- 

 cussed later following the exposition of the slowly acting 

 loads. 



2 Rational Theory of Bending Moments 



The weight-distribution curve and the bending moment 

 acting on a ship floating in still water are readily evalu- 

 ated by conventional methods and will not lie considered 

 here. 



With increasing speed of a ship a certain sinkage occurs. 

 This sinkage wall affect the bending moment because the 

 water-suction forces are not distributed in proportion to 

 displacement. Furthermore the formation of a long 

 wave is observed at the sides of a fast ship. This wave 



