Arched Ribs of Uniform Section. 383 



bending moment, still be in two transverse planes, whence the geo- 

 metry of the question shows that the elongation and shortening 

 of lines in the beam parallel to its axis will be proportional to 

 their distances from some line situate in a transverse section, and 

 that consequently the stress must vary uniformly ; and since, 

 moreover, the total amount of the stress is a couple, the line of 

 no stress or neutral axis must pass through the centre of gravity 

 of the section. 



Now, if the beam be curved, the same considerations of sym- 

 metry apply to show that particles originally in two transverse 

 sections will still be in two transverse sections ; but since those 

 transverse sections are not parallel in their original positions, the 

 geometry of the question is different, and consequently the stress 

 does not vary uniformly, nor does the neutral axis pass through 

 the centre of gravity of the section. By the method indicated, 

 the law of variation of the stress, and the deviation of the neu- 

 tral axis from the centre of gravity, might be determined, and 

 also the maximum stress on the section. But it is probable that 

 the introduction of terms depending on the curvature of the 

 beam would not add materially to the accuracy of the results ; 

 for in general not only does a bending couple act on the beam, 

 but also forces which cause a shear on most of its sections; and 

 the accuracy of the above conclusions is thereby disturbed to an 

 unknown though (from the values of the coefficients of direct 

 and transverse elasticity of the materials used in construction) 

 small extent ; and this error, in cases where the curvature is 

 moderate in proportion to the depth, may probably be as great 

 as the error produced by neglecting the curvature of the beam. 

 "We may take therefore 



M 2 H 2 \ 

 2EI + 2EAJ 



as a first approximation to the work done. In cases in which 

 the angle subtended by the rib is considerable, and in which, 

 moreover, the distribution of the load differs much from that 

 which is necessary for the equilibrium of a linear arch of like 

 form, then, the thrust being small, the work done is almost en- 

 tirely done by the bending moment, and a sufficient approxima- 

 tion is given by 



Reference will be subsequently made to cases in which these con- 

 ditions are not satisfied. 



Using this formula, if the value of M given above were sub- 

 stituted and the result integrated throughout the beam, the result 



-ft 



