382 Mr. J. H. Cotterill on the Equilibrium of 



and 



— =F.p = Kp + F .p.cos(£ + H .p.sin<£; 



.\M=l K/w?0 + F o .l p.cos^d(f) + EA /3.sin<£d<£ + M . 



Jo Jo Jo 



Thus M, H, and F are expressed in terms of known functions of <£, 

 and three undetermined constants which, if no distributed forces 

 act on the rib, are the shear, thrust, and bending moment on 

 the initial section. In these expressions shear is reckoned posi- 

 tive when it tends to draw away from the centre of curvature 

 that portion of the rib which is toivards the algebraical increase 

 of <)), thrust when it tends to move the same portion towards the 

 increase of <j>, and bending moment when tending to twist the 

 same portion in the direction of <£. 



Now it may be that something is known about the stress on 

 one or more sections, and in that case one or more of the con- 

 stants will be definite ; but in general it is impossible to deter- 

 mine them without a knowledge of the physical constitution of 

 the rib, because their values entirely depend on that constitu- 

 tion. But if the nature of the material be known, so that the 

 work done in it by the action of the forces can be estimated, then 

 will the actual values of the constants be such as to make that 

 work the least possible, and the differential coefficients of that 

 work with respect to the constants will furnish the means of find- 

 ing their values. I proceed, therefore, to find the general values 

 of those coefficients, supposing that the rib is composed of 

 homogeneous material strained within the limits of perfect elas- 

 ticity. 



Before doing so, however, I may remark that it is frequently 

 convenient to use rectangular equations of equilibrium ; the rib 

 is to be divided into plates by planes parallel to the axis, and 

 differential equations obtained expressing the equilibrium of one 

 of those plates*. 



2. When a straight beam is subjected to a bending moment, the 

 strain being supposed within the limits of perfect elasticity, then 

 it is known that the stress at different points of any transverse 

 section varies uniformly, vanishing at points situated in a straight 

 line, through the centre of gravity of the section, parallel to the 

 axis of the applied couple. 



In effect, considerations of symmetry show that any particles 

 originally in two transverse planes will, after application of the 



* It is worth remarking that qp= -jt {pp} is the equation of a linear 

 arch. 



