160 THE LONGITUDINAL STRENGTH OF RIGID AIRSHIPS. 
VI. APPLICABILITY OF THE THEORY OF BENDING TO AN ACTUAL AIRSHIP. 
We have hitherto dealt chiefly with airships of ideal section, where the shearing force 
at one frame initiates a system of P-forces in the adjacent frame conforming to the bend- 
ing theory. It appeared, as far as the investigation went, that in such a structure the theory 
applies at least as well as in an ordinary ship. Actually airship structures deviate some- 
what from the ideal form. The sections are not always of a regular polygonal form through- 
out, the longitudinals and the wires may not be of uniform strength, and the keel structure 
is much heavier than the top structure. It is desirable, therefore, to inquire into the funda- 
mental assumptions underlying the bending formula: 
i a Gn 
i y 33 
so as to ascertain whether or not an airship possesses features that seriously impair its 
validity. 
The fundamental assumptions in the established theory of bending of a beam are : 
1. That the beam is homogeneous, of uniform transverse section, and perfectly elastic. 
2. That sections plane before bending are plane after bending. 
3. That longitudinal layers of the beam expand and contract laterally in the same way 
as if they were disconnected from each other. 
It has been proved by theory and experience that these assumptions can be departed 
from to some extent without invalidating the theory. It is found that, if the material is 
homogeneous and perfectly elastic, the theory applies even though the transverse sections vary, 
provided they vary in a continuous manner. It applies to curved as well as to straight beams 
even although the curvature varies, provided it varies in a continuous manner. At sections 
where great discontinuity exists, either in section or curvature, the theory does not apply. 
The way in which the bending couple is applied does not sensibly influence the distribution 
of the stresses except locally in the nearest vicinity of the sections where the forces are ap- 
plied. The plane of the bending couples need not be a principal plane of the beam. 
Assuming that the wires are set up with an initial tension and that hence, in general, both 
shear and counterwires are active, we shall now examine in how far the assumptions stated 
above hold good in an airship. 
As regards the first assumption, the sections in an airship vary in a strictly continuous 
manner, more so than in ordinary ships, where the continuity is often broken by a forecastle, 
a poop and by deckhouses. Each section approaches in form to that of a regular many- 
sided polygon, while in a ship a great discontinuity in form occurs at the deck edges. In 
a general way rigid airships are also of a more homogeneous construction than ordinary 
ships provided the wiring is fairly uniform or proportional to the girders; they are also 
perfectly elastic. Now, in spite of the deviations from the fundamental assumptions in 
ordinary ships, the theory has been found to give a fairly close approximation to observed 
data in cases where measurements of strains and deflections have been made, and it is gen- 
erally accepted by naval architects as an established fact that the method is reliable as a 
means of comparison between ships of not too different type. It appears, therefore, that 
the requirements of the first assumption are complied with in a satisfactory manner. 
The second assumption probably holds good under pure bending, where the deflections 
are entirely due to elongation and compression of the longitudinals, since the deviations of 
these strains from the requirements of the theory are extremely small in relation to the 
