CALCULATION OF THE TRANSVERSE STRENGTH OF SUBMARINES 



BY MARBEC'S METHOD. 



By Professor William Hovgaard, Member. 



[Read at the twenty-ninth general meeting of the Society of Naval Architects and Marine Engineers, held 



in New York, November 17 and 18, 1921.] 



The French naval constructor Marbec in 1905 contributed an essay on "Dilatation des 

 Tuyautages" to Memorial du Genie Maritime, in which he dealt with the strength of closed 

 elastic rings of uniform section and of relatively small transverse dimensions by a method 

 based on the fundamental work of M. Maurice Levy. In Bulletin de I'Association Technique 

 Maritime for 1908, in a paper entitled "Theorie de L'fiquilibre d'une Lame filastique," he 

 developed the method further and applied it not only to a simple closed ring but also to arches. 

 The method was for the first time, I believe, explained in English by Marbec himself in 

 1911, when he read a paper before the Institution of Naval Architects, "Notes on the Col- 

 lapsing of Curved Beams and Curved Elastic Strips" ; and in 1921 Mr. W. R. G. Whiting 

 published the results of an application of the method to elliptical forms in a paper, "The 

 Strength of Submarine Vessels," read before the same institution. In Germany Marbec's 

 method was explained and applied to arches by Dr. Rudolph Mayer in Zeitschrift fiir Mathe- 

 maiik und Physik, 1913. 



I shall endeavor to explain the method as briefly as possible, without following all the 

 interesting side issues of M. Marbec, including only the steps necessary to an understand- 

 ing of the method and its intelligent application to problems occurring in the design of closed 

 frame rings, and more especially of frames in submarines, in cases where the resultant effec- 

 tive forces can be assumed to be equivalent to a tmiform pressure all around the circumfer- 

 ence. The proof here given is direct and simple and is essentially that given by Dr. Mayer. 

 The method of application I have developed into a form which I believe is suitable for 

 practical purposes and is exemplified in the accompanying plates. 



Marbec's method is based ultimately on the same fundamental formula as that used in 

 my work on "Structural Design of Warships" in the treatment of the closed frame ring : 



.0 = f, (■) 



where aB \s the change in curvature. Marbec also makes use of the same equations of con- 

 dition based on the principle of continuity. This must necessarily be so, but in determining 

 the unknown bending moments and reactions Marbec follows a different path and by an ele- 

 gant graphical method arrives at a solution which exhibits the results in such a form that it 

 is easy to visualize the general distribution of the moments and reactions. The method ap- 

 pears to be simpler in application than any other, but, as I have not had an opportunity to 

 compare it thoroughly with other methods, I am not certain that it is in all cases and for all 

 purposes superior. 



The reason why Marbec's method has not become better known than it is today, ten 



