^ = V{^-isin20} + H o ^{i.cos2^-cos^,-|} 



^-==r o 7^{f-cos^ + icos2^} + H o r 3 {f<^-2.sin</, + ism20} 



Arched Ribs of Uniform Section. 385 



0-1 sin 2<£}+: 

 + M o r 2 {l — cos0}, 



} r 3 {f— COS0 + 1 C O 



+ M o r 2 (0-sin0), 

 ^=F o r2{l-cos0}+H o r 2 (0-sin0) + M o r0. 



These values enable all questions to be solved concerning a 

 circular rib loaded at detached points only. For example, let it 

 be required to determine the bending moment at a given point 

 of one of the rings of a chain composed of circular links. 

 First, suppose the links not to be strengthened by the addition 

 of a cross bar; then, supposing measured from the point 



W 



of contact of two links, H = 0, F = — — , where W is the 



weight which the chain carries ; so that we have only to deter- 

 mine M , which is done from the equation 



d\J W 



_ = __ r2( l_ cos<W + Mo r0 = O, 



where = 7r, because U must be estimated through half the ring 

 up to the point of contact of the other ring in contact with it. 



The same result must manifestly be obtained if 0= — ; in either 



case „ Wr 



M °=v 



The stress on any section is therefore given by 



W 



F = F o .cos0 + H o . sin0= — — . cos 0, 



W 



H = H o .cos0— F . sin 0= -^-.sin0, 



M = M o + F o .r.sin0 + H o r(l-cos0), 



Wr W 

 = -g- r i sin © 



it 2 r 



Tltr 2 — 7r.sin0 



If the links had each of them a central bar placed at right angles 

 to the line of action of W, then H would be undetermined, and 



