. TRUSSED BEAMS, ETC. 629 



deflection at the centre W would cause, if it acted alone, and the ends 

 (if the beam were supported, would be equal and opposite to the 

 deflection which R2 "^vould cause at the same point, if it acted alone 

 and the ends of the beam were held down. Here we can make use of 

 the formulse for deflections of beams, as found in text books, and the 

 text book chosen for reference is that of Lanza (" Applied 

 Mechanics "), whose symbols will be adopted in what follows. At the 

 bottom of p. 298 of the third edition of the volume mentioned is 

 o-iven the value of the deflection, v, which a load, W, acting at a. 

 distance, a, from one end of a beam, will produce at a point, distant 

 r, from the said end, x being less than a. The equation is that 

 niarked -'(2)", viz.— 



' = "^TeT " + 67E-I ^'"' - -' - "'^ ''■ 

 This may be put in the simpler form — 



Owing to the convention as to signs, adopted by Lanza, v is 



positive when measured upwards; hence this expression is negative. 



We shall, for the sake of simplicity, give it its positive value and 



write — ■ 



W.r (1 -o) ^ , 



v= — C2(f/ — X- — (r) (a I. 



6/EI ^ ^ 



Fig. 7 explains the meaning of all the symbols, except E and L 

 E is the modulus of elasticity of the material in the beam; I is the 

 moment of inertia of the cross-section of the beam, about the neutral 

 axis. 



We have, then, for the deflection, ' \V, which W would produce 

 at the centre of the beam, putting for x in equation (u) — 



12 El V ^ / 



Similarly, putting ^ for both x and a, we get, for the deflection 

 which R2 would produce at the centre of the beam — 



48 EI 



which is a well-kno\vn expression. 



On the assumption of inextensibility and incompressibility, 

 above-mentioned — 



.-.E. = i:^^(2«/-«-^). (in.) 



This, then, is our third equation. Combining it with equation 

 (II.), we get Rj. 



The bending moment, at the point of application of W, can then 

 be found. 



