TRUSSED BEAMS, ETC. 637 



The two ordinary statical equations, formed by summing up 

 forces acting transversely to the beam and by taking moments, are 

 identical in this case, and of a simple character, provided we know 

 whether Rj and R3, which are equal, act upwards or downwards, and 

 this can readily be determined, as will be explained shortly. We 

 need, however, an additional equation, for there are two un- 

 known forces, viz., R3 and Rj (or R3). This equation is to express 

 that the beam, at the tops of the posts, is at the same level as at the 

 abutments; i.e., the downward deflection at B, which the distributed 

 load would cause, if it alone acted on the beam, the latter being 

 merely supported at the ends, is equal to the sum of the upward 

 deflections that R^, at B, and R.2, at C, would separately cause at B, 

 if the beam were acted on by these two forces, R2, and held down at 

 the ends. 



Let ^^xTT- be the deflection that the load id would cause at B. 



Let v-D be the rise that would be due to R2 at B, and y'-p the rise 

 due to Ro at C : then — 



^w^^'e + ^'r 



©Yj^r is obtained by putting l-^ for x in (B), viz. — 



wr 



The sum, Vj^ + v'^ will be expressed by the same symbols as 

 when the isolated load was considered, viz. — 



Equating the values of v^ and {v^ -{- v'-^), we obtain — 

 ^ ^ it> (/,3 — 211- + h 

 41, (SI— 4/,) 

 In a given case, knowing R2, we see, at once, whether Rj and R3 



act upwards or downwards, because, if R3 exceeds — , then Rj and 



and R3 must act- downwards, and vice versa. 



To tind where the posts should be placed, in order that Ri and 

 and R3 should vanish; in other words, to find where the supports 

 should be placed, under a uniformly loaded beam, in t)rder that the 

 beam should be at the same level at the top of the posts as at the 

 ends; or — expressing it in yet another way — to find, in a trussed 

 beam, where the posts should be placed m order that no portion of 

 the load should be transmitted to the abutments by the beam, acting 

 as a beam, and, at the same time, no part of the vertical component 

 of the tension in the rods be employed in holding the ends of the 

 beam down, but the said vertical component be equal to the actual 

 supporting force afforded bv the abutments, we must equate — 



E, to !i^ 

 2 



ie k"^JZ^JLl+Jl 

 21, (SI -41,) 

 from which we get — 



Z,3 -\-6U,-' - iil'l, -\-P=^0' 



