260 



THEORY OF SEAKEEPING 



Table 2 Computed Shear Forces and Bending Moments for 5.71 -Ft Model 1723 Destroyer at 5 fps, in Waves of Model 



Length and 1/48 in Height 



Stations 1 

 Loads, Ih/ft 



Ship mass inertia 0.90 



Added mass inertia. ... 0.16 



Damping — 0.12 



Added displacement ... — 1 . 27 



Modified Smith effect. . 0.04 



Total load, lb/ft -0.29 



Shear, lb -0.09 



Bending moment, 



Ib-ft -0.02 



(For condition when wave crest is at Midsection) 



3 5 7 9 11 



13 



15 



17 



Phase lag of heaving after pitching, 55 deg 



Note: The Model was divided into 10 segments. There were 

 21 equidistant stations and the loads were computed at the odd 

 stations. 



19 



moment linearized analyses. Greater discrepancies be- 

 tween calculated and measiu'ed bending moments 

 appeared in the analy.sis of the T-2 tanker model in 

 higher waves (X//i = 20). These discrepancies can be 

 attributed to two primary causes; i.e., the uncertainties 

 in the damping data and the non-constancy of the hy- 

 drodynamic coefhcients of ecjuations of motions. The 

 examples analyzed by Korvin-Ivroukovsky and Jacobs 

 (3-1957) showed that the use of linearized eciuations of 

 motions leads to satisfactory results in case of ships of 

 usual form, but fails for such extreme forms as the sailing 

 yachts. A strong effect of nonlinearity on ship-motion 

 phases also was demonstrated by the experiments of 

 Akita and Ochi (3-1955) in case of an unusually shallow- 

 draft, flat-bottomed hull. 



A complete analysis of motions and bending moments 

 of ships of unusual form, or of normal ships in high waves, 

 requires a step-by-step integration of equations of 

 motions with time-dependent coefficients. Such inte- 

 grations were performed (on basis of uncoupled equa- 

 tions) by Horn (1910) and by Hazen and Nims (3-1940). 

 The results indicated that time variation of the bending 

 moment in harmonic waves distinctly deviates from a 

 sinu.soidal form. Horn emphasized that, because of this 

 deviation, it is necessary to calculate the bending mo- 

 ments for a series of small time intervals. 



The work of Korvin-Kroukovsky and Jacobs (3-1957) 

 and of Gerritsma (3-1957c and d) indicated that non-con- 

 stancy of coefficients apparently affected predicted 

 motions of ships of usual form only to a small degree. 

 The predominant discrepancies in this case were traced 

 to the error in estimating mean values of damping co- 

 efficients. This can be explained rationally by observing 

 that motions are obtained by double integration of the 

 forces acting on a ship. Variations of instantaneous 



forces are smoothed out in the process of this integration 

 and only the total work of the forces is of significance. 

 This observation will often permit an important short 

 cut in the nonlinear evaluation of bending moments: 

 relative ship-wave positions, velocities, accelerations 

 and phase relationships can be assumed to remain as in- 

 dicated by analytical solution of the coupled linearized 

 differential equations of motions. Subsequently, the 

 loading curve can be computed for isolated instances, 

 using the coefficients based on the instantaneous con- 

 ditions, (i.e., instantaneous waterline positions) at each 

 ship section. The analysis of this type, but extended to 

 include slamming vibrations, was attempted by Dalzell 

 (1959) with a reasonable success. This analysis will be 

 discussed in Section 5.55 in connection with slamming. 



3 Bending Moment— Observations on Ships at Sea 



The content of this section is limited to the quanti- 

 tative correlation of a ship's bending moment with the 

 wave size. The paper by G. Schnadel (1937/38) and 

 the reports of the Admiralty Ship Welding Committee 

 appear to be the only available sources of this informa- 

 tion. 



3.1 MS San Francisco. Schnadel's paper contains 

 the analysis of measurements taken aboard the MS 

 Ban Francisco over the period of a few seconds. During 

 this period, the ship passed through the hogging and 

 sagging conditions on an unusually large and steep wave. 

 The general description and data on the voyage of the 

 MS San Francisco will be found in Section 3-5. 12. ^ 



'• Reference to sections, equations, and figures in preceding 

 chapters will be designated by chapter nimiber and section, 

 etjuation, or figure number; reference above, for example, is to 

 chapter 3, section 5.12. 



