640 ■ PROCEEDINGS OF SECTION H. 



Comparing,; bridges of the same span, one with a single post 

 and the other .^th two posts, the advantage Has with the double-post 

 bridge, so far a^ bending stresses in the beam are concerned. This 

 is true, no matter what the inclination of the rods at the ends may 

 be, provided longitudinal extension and compression of the members 

 may be neglected. The reverse is the case with regard to the longi- 

 tudinal stresses in rods and beam : these are greater, under both 

 central and distributed loads, in the case of the double-post trussed 

 beam, if the inclination of the rods at the ends are the same in both 

 single and double post stinactures. 



Case III. — Trussed Beam, with Three Posts. 

 (^See Figures 3 and 3a, 27, 28, and 29.) 



The writer has not had time yet to fully investigate this case, 

 but hopes to do so before long. It would be desirable to ascertain the 

 best spacing of the posts, under an isolated load and under a 

 imiformly distributed load; also, to compare expressions for bending 

 moments, in this case, with those in the cases of single and double 

 post trusssed beams, and thus determine the advantages arising from 

 multiplying posts. In the meantime, a method of finding the stresses 

 may be indicated. 



Take the case Avhere the posts bisect the angles between 



adjacent sections of tension rod, and the rods make equal angles 



with each other ; then, by symmetry, the tensions in all sections of 



the rods must be equal, and the compressive stresses in the posts 



must be equal. Therefore, if R2 denotes the compressive stress 





 common to the three posts, R2 cos ;, will be the vertical component 



of the compressive stresses in the two outer posts. W and B being 

 known, we must form three equations to determine the unknown 

 quantities, Ri, R2, and R3. Summing up forces in opposite directions 

 and taking moments will give us two equations. For the third 

 equation, we have to express a relation among the deflections at the 

 tops of the posts, assuming inextensibility and incompressibility. 



This relation will be that existing at the joints, h, c, d, of an 

 inextensible chain, whose links have the lengths and slopes that the 

 sections of the tension rod have. (See Fig. 27.) 



If the loading be symmetrical, if c goes downwards, or upwards, 

 through a distance 8y, then h and d will rise or fall through a dis- 

 tance 8/?, and the relation between % and ?>h, under these circum- 

 stances, is found as follows, viz. — 



Referring to Fig. 27 — 



X, cos ^ -j- A2 cos ^ =^ ^ 



:. A, sin ^. 8 6* + A, sin (^. 8 c^ = . . (1) 

 ^ = A, sin B 

 y = A, sin -\- \^ sin c/) 

 .-. 87? = A, cos e. dO" ... (2) 



8y — 8// = A.^ cos <f>. ?> (f> ... (3) 



From (1), (2), and (3), we deduce 



sin <^ cos 



8h r= 8j/ 



sin («^ - 6) 



