290 



PRACTICAL MATHEMATICS 



If x be given any value between and 30 ft. the value of the 

 bending moment at any section can be calculated : 



Now 



M is a maximum when 



That is, when 

 or 



dx 60 



m 



dx 



x 2 = 300 



a; = 10V3 = 17-32 ft. 

 [_ = 57-73 ft. tons 



152. Let P be any section of a beam carrying some system of 

 loads. The only force action at P is the action between the par- 

 ticles ; while at that section there is acting a vertical external 

 force which is the equilibrant of all the forces acting on that part 

 of the beam to the right, or to the left, of the section ; this force 

 is spoken of as the " Shearing Force " at the section. Now the 

 portion of the beam to the right of the section is in equilibrium 

 against vertical translation, but it would have the tendency to 

 rotate, and the magnitude and direction of this rotation is deter- 

 mined by the bending moment at the section. At the section 

 there are elastic forces induced by the external loading, and these 

 forces produce at that section a couple whose magnitude is equal 

 to that of the bending moment, but the direction is opposite. 

 These elastic forces consist of pulls decreasing uniformly to zero 

 and thrusts increasing uniformly from zero. Hence the algebraic 

 sum of the moments of all these pulls and thrusts, taken with 

 reference to some fixed axis of the section, must balance the 

 algebraic sum of the moments of the external forces acting on 

 that portion of the beam to the right, or to the left of the 

 section ; or, 



Moment of resistance of the section = bending moment at the 

 section. 



The axis about which the moment of resistance is taken 

 is called the neutral axis of the section and is that axis, or that 

 fibre of the beam which is in an unstrained condition. Thus, in 

 brief, at a given section, the shearing force is the equilibrant of 

 all the external forces acting on that portion of the beam to the 

 right, or to the left of that section. The bending moment is the 



