ON THE STRENGTH OF MATERIALS FOR IRON SHIPS. 



249 



Fig. 6. 



-N. 



used to express either of these moments. It must be observed, therefore, 

 that all the expressions hereafter given for M, the moment of rupture or 

 the moment of flexure, as the case may be, are equal to the moment of the 

 forces tending to break the beam. 



16. Let h equal the distance of the upper edge of the section of rupture 

 from the neutral axis passing through the centre of gravity of the section ; 

 7*, the distance of the lower edge from the neutral axis ; S= the resistance 

 of the material, per square inch, to compression at the upper edge ; Sj = the 

 resistance of the material, per square inch, to extension at the lower edge ; 

 and I == the moment of inertia of the section about its centre of gravity ; 

 then 



M =f'*o' OT |-Io (1) 



The value of I depends solely on the geometrical form of the transverse 

 section of the beam. 



17. The following values of I for different sections of material will be 

 found useful in calculating the transverse strength of a beam having a com- 

 plex section, such as that of an iron ship. 



(1) For a thin plate, AB, about any axis, 

 NCO, passing through its centre of gra- 

 vity C, 



I o = T VE7 2 sin 2 0, 



where K is the area of the section, and I sin 

 the projection of I, the length of the plate, 

 upon the vertical or upon a line perpendicular 

 to the axis. 



(2) For a hollow rectangle, 



where b= the breadth of the section, d= its depth, b l = the internal breadth, 

 and rf,= its internal depth. 



(3) For an ellipsis about any axis NO passing 

 through the centre C, 



WW 



where 2p=AB, the vertical depth of the section 

 when the major and minor axes are equal, p=r the 

 radius of the circle. 



(4) For a hollow circle, 



where d and cZ, are the external and internal 

 diameters respectively. 



When the plate is thin, I =£foP very nearly, which also approximately 

 expresses that of a hollow elliptical section, d being the vertical axis. 



(5) "When the depth of any surface is small as compared with its distance 

 from the axis, 



I=aa, 2 very nearly, 



where «= the area of the section, and a= the distance of its centre of gravity 

 from the axis about which the moment of inertia is taken. 



(6) The moment of inertia of a square is the same for all axes passing 



Fig. 7. 



