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



291 



Fnmkland (1942) can be cited as giving a simple and 

 logical introduction to a structure's response to an impact 

 load. A slender l)eam may vibrate in a number of modes 

 which can be conveniently described by the number, ?i, 

 of nodal points of the elastic curve. All vibralion modes 

 are excited by an impact and the deflection y is expressed 

 by a summation of contributions from all modes. 



Let L designate the length of a beam and Lj the dis- 

 tance from one end to the point of application of an 

 impulse Pt. The deflection y of the beam at a distance 

 X from the end apparently can he represented as 



y = (const)/'r 2_^ ./ I ^, --, (( ) >f(t, n) 



(li) 



where n is the number of nodal points. The function of 

 x/L, Li/L, and /( is rather complicated. The function 

 {t, n) is essentially sinusoidal with the period depending 

 on n. The deflection, y, is represented as the summation 

 of all possible modes of vibiation for which the point of 

 impact application does not coincide with a nodal point. 

 The deflection, y, represented by such a sum of harmonic 

 functions, is certain to display a transient l)ehavior. 

 Lockwood Taylor (194()) demonstrated mathematically 

 that one of the manifestations of transients is a grad- 

 ual shortening of the apparent period of vil)ralion with 

 time. 



Further insight into the behavior of a slender beam, 

 indicated by equation ((>), can lie obtained by examining 

 a simple case reported by Frankland (1948, etjuation 39). 

 Frankland considered the undamped vibration of a 

 simply supported beam of length L and mass m per unit 

 length subjected to a load p per unit length, uniformly 

 distributed over the centrally located length a. The 

 load p was assumed to be sutldenly a])plied at the time 

 t = and suddenly re]ea.sed at the time t = r. The 

 time-dependent deflection y of the beam at any point .r 

 and at a time t > t has been found to be 



y(-r, t) 



4pTJ V^ 2 r . Xw . Xira . Nrx 



= — — > — sm — sm sm 



EIw' A^ N' I 2 2L L 



sm —^ sni iV-co„(< — T/2) 



(7) 



where N is the vibration mode, and aj„ the circular fre- 

 quency corresponding to it. .V is related to the number 

 n of nodal points in equation (6) by N = n — 1. 



The bending moment M acting at a .section of the beam 

 located at x is 



il/ 



bhj _ 4pL^ Sr^ 2 

 ' dx- TT^ A^ N' 



(8) 



where the dashes in square Ijrackets designate Ave 



trigonometric factors identical with those of equation (7). 



Equation (8) can be s mplifled by considering only the 



fundamental mode, N = I, and by evaluating the ampli- 



tude of the bending moment at mid-point, in which case 

 the first, third, and last trigonometric factors are ei|ual 

 to unity. Equation (8) can represent a central con- 

 centrated impact if the distance a is made sufticicnily 

 small, a <C L. The duration r also can be assumed lo be 

 small, T <S^ T, corresponding to a slamming impact on a 

 cargo ship. With these assumptions the sine of an angle 

 in second and fourth factors in .squai'e brackets can be 

 replaced by the angle, and ('(|uation (S), aftci- letting 

 CO = 'lirlT , is reduced to 



.1/„,ax = (4/ir}(paT)L/T (9) 



The ani].)litude of the dynamic bending moment for a 

 small duration of an imijact ;■ is shown to be proportional 

 to the momentum of the impulse, par, and inver.sely 

 proportional to the natural period of vibration T. For 

 a given small impact duration t. the ratio r T increases 

 for higher harmonics which have smaller T. These 

 harmonics are, therefore, easily excited by an impact 

 and significantly contril)ute to the bending moment, as 

 this was .shown by Frankland {l'.)4S) and Ochi (see Sec- 

 tion 5.52). 



With the foregoing background in mind, the stress 

 history shown in Fig. 40 will be examined, l-'irst it is 

 noted that at the instant of the bottom slam, 8.5 .sec, 

 there is no significant effect of the slam on the stresses 

 amidships. The first and very weak sagging stress 

 maximum is found 1 sec later, at 9.5 sec. The amplitude 

 of the vibratory stress is then oliserved to increase during 

 the next two oscillations and reaches the maximum at the 

 third peak at 11 sec. This is 2V2 sec after the initial 

 impact. This peak coincides with the maximum of 

 sagging stress in pitching osc'illation and m(.)re than 

 doubles this stress. The vibratory amplitude there- 

 after decays very slowly but is boosted by the second 

 slam at 14 sec on the time scale. This examjjle demon- 

 strates the fact that e\'aluation of the slanuning force 

 does not lead directly to the knowledge of ship stresses. 

 In order to evaluate these stresses, it is necessary to solve 

 the problem of the elastic response of a ship to an im- 

 pact, to evaluate the transients, and to represent strcs.ses 

 as functions of time. 



5.51 Free vibrations. An investigation of the free 

 vibration of ships can be considered as a prerequisite to 

 the subse((uent consideration of forced vibrations. Also 

 it has been shown in the preceding section that the 

 maximum bending stress in a ship usually occurs at some 

 time after a slam when the trending stress of free \'ibra- 

 tions is superposed on the wave-caused sagging bending 

 stress. The following quotation from McGoldrick, 

 et al (1953) can serve as an introduction to this suliject: 



"The beam-like nature of a ship's hull is self-evident 

 and has formed the basis for the ordinary strength cal- 

 culations universally used in design wherein the ship is 

 assumed supported on trochoidal waves which exert a 

 buoyant force per unit length which varies with distance 

 from the end but is considered constant in time, that is, 

 the analysis is carried out as a problem in statics. 



"The simple bending theory of beams has been used 



