TRU.SSED BEAMS, ETC. 631 



y^ n — „) 



1 X, as obtained irom the left hand end in Fig. 9, or 



— ^ (/ — A') — W (n — x), as obtained from the right hand end in 



Fig 0, these expressions being obviously identical. 



Note. — If we transfer the origin to the right hand end, or, what 

 is the same thing, if we imagine the load in Fig-s. 5, 6, 7, and 9 to be 

 placed to the left of the post, then, writing (1 — a) for a in (111. ), we 



W H 



get R, = -,— (8/' — 4rr), which is a simpler expression than the 



other. 



B. — Uniformly Distributed Load. 



The load, per unit of length, may be denoted by xv. So far as 

 the beam is concerned, it is simply supported at each end and at the 

 middle, and loaded uniformly. {See Fig. 11.) The results are given 

 in many text books, e.g., in Lanza's "Applied Mechanics," 3rd 

 edition, pp. 567-8, we find, mutatis mutandis, that — 



E, .= ^ W /, R. = R, = ^^ W J. 



s ' 16 



Hence, bending moment at post= _ =0" 03125 /r/-*. Also, the 



<) 

 maximum bending moment between post and abutment = _'-;^ /r/- = 



0" 0176 irl'^, and occurs at ' / from the end of the beam. 



1() 



R 



The tensile stress in the rods = — ^ cosec and the direct com- 

 pressive stress in the beam = ^ 'Cot 0. The latter has to be com- 

 pounded with the sti-esses due to bending. 



The actual pressure on each abutment is, of course, — . A part 



3 



of this, viz., E3=Ri= ' /r/, is transmitted to each abutment by the 

 Itj 



1 5 



beam, acting as a beam : the remainder, viz., 1^2=", irl, is trans- 

 mitted to each abutment by the tension rods, i.e., the vertical 

 component of the tension in the rod is - ' .- irJ. 



Case II. — Trussed Beam wrrn Tavo Posts. 

 A. — Isolated Load, 

 (a) When the load is between the posts. (Fig. 12.) 

 As mentioned previously, on the supposition that the materials 

 are inextensible and incompressible, and that practically no defor- 

 mation takes place, one of the points, B and C, will be raised as 

 much as the other is depressed : also, the compressive stresses in the 

 posts are equal. The vertical components of the latter are denoted by 

 Ro- We have, then, five forces acting on the beam, two of which are 

 equal, viz., Ro. Only one force, W, is known. We, therefore, need 

 three equations to enable us to determine the three unknown forces. 



