630 PROCEBDIKGS OP SECTION H. 



Diffei'entiating the expression for this bending- moment, witk 

 respect to a, and, equating- to 0, we obtain a cubic equation in a for 

 that vakie of a for which the bending moment at the point of appH- 

 cation of W is a maximum. A graphical solution of this problem is 

 afforded by Fig. 8, where a few val-aes of bending moments have been 

 plotted, as ordinates, the abscissae being values of a. From this we 

 see that the bending moment is a maximum when W is about 0'218^ 

 from the end of the beam, the said maximum value being 0'1037 W/. 

 The bending moments at the top of the post, corresponding to dif- 

 ferent positions of the load, i.e., to different values of a, are always 

 less than the bending moments at the points of application of the 

 load. 



The force, R^ (Fig. 6), is equal to that part of the vertical 

 component of the tension in the inclined rod which holds the end of 

 the beam down : the remaining portion of the said vertical component 

 is equal to the actual pressure on the abutment^ viz. — 



-J — -• (J^ig-0- 



With reference to the force, R3 (Fig. 6), its A-alue is the same as 

 though there were no tension rods, and the beam were supported at 

 the right hand end and at the centre, and held down at the left hand 

 end. Thus, if X be the point of contrary flexure, XC may be re- 

 garded as a beam, supported at X and C, and loaded with an isolated 

 load, W; and R3 is equal to the suppoiting force at C. It is a 



part of the total pressure, (^^^'g- ^)» on the abutment : the 



remaining part of that total pressure being equal to the vertical 

 component of the tension in the inclined rod. The tensions in the 

 two parts of the tension member are, of course, equal. 



In a completely braced truss, with an isolated load at a panel 

 point, the whole of the load is transmitted to the abutments through 

 the medium of the members of the truss. In the trussed beair. cvm- 

 sidered, on the other hand, when the load is not directly over the 

 post, a part of it only ; is transmitted to the abutments through the 

 post and tension rods: the remainder is transmitted by the beani 

 acting as a beam — i.e., oft'ering resistance to bending. 



The distance, x, of the point of contrai*}^ flexure, X, from the 

 left hand end is found from the equation — 



li' X = E. 



■^(^-0 



If a vertical section be taken through X, cutting the tension rod in Y 

 {Kte Figs. 5 and 9), then the couple, whose moment is equal to the 

 direct compressive stress in the beam, multiplied by XY, is equal to- 

 the moment, about XY, of the external forces acting on the structure, 

 reckoned from either end. This is true of no other vertical section^, 

 liecause at all other sections, except XY, there are stresses in the 

 beam, due to bending, and these help in resisting the couple due to- 

 extei-nal forces. By "moment of external foi-ces'' is meant either 



