264 



CALCULATION OF THE TRANSVERSE STRENGTH 



occurs, as illustrated in Figs. 5 and 6, where the ends of a circular arch are respectively fixed 

 or connected by pin joints to an absolutely rigid structure. In such case we may imagine the 

 circle of the neutral axis to be continued through the rigid mass of the foundation. Even 

 if the flexibility is quite unevenly distributed in the arches, and the center of gravity does 

 not lie in the vertical through the center of the circle of the neutral axis, the center of the 

 nodal circle O will still coincide with the center of that circle and there will be no bending in 

 the arch. 



It is important to note that the condition for the truth of these statements is that the 

 extensibility and compressibility of the ring all around the circle is negligible compared with 

 the flexibility. If, for instance, the two points "k and v in Figs. 5 and 6 approach each other 

 under the effects of the pressure, the above theorems do not hold good. 



Consider next a noncircular arch fixed at both ends to a foundation of absolute rigidity 

 (Fig. 7). Here, again, the fundamental theorem is applicable, simply assigning to each part 

 its flexibility. The rigid part Xt does not enter into the calculations for G or for the nodal 

 circle, since its flexibility is zero. 



If a closed frame ring is hinged at one point A (Fig. 8), the flexibility is infinite at 



FIG. 7. 



FIGr. 8. 



jAfeofer/ C/rtIa 



,f/e<tifal Axis 



that point and the center of gravity of the flexibility, G, coincides with it. The ordinates 

 of O are determined as usual. The nodal circle passes tlirough A where the bending mo- 

 ment is zero and, since 



Zp = o, we have z/ = a^ + b' 



The method is illustrated in Plates 1 to 11. Plate 1 shows the form and structural 

 details of the frame system of a submerged body designed for a special purpose and calcu- 

 lated to stand a pressure corresponding to a head of SO feet of water. The following plates, 

 which are believed to be self-explanatory, give the calculations. Since the frame is not 

 quite symmetrical about a horizontal axis, the example here given is much more involved 

 than where there are two axes of symmetry. 



Professor H. H. W. Keith, of the Massachusetts Institute of Technology, has prepared 

 Plate 1, and Mr. F. A. Magoun, of the same institute, has prepared the diagrams in the text 

 and performed the calculations of the plates. 



I 



