MECHANICAL PKINC2PLES. IOI 



The line through the cross-section of any beam where the 

 fibres are not strained is termed the " neutral axis" of the 

 beam. In the case of all vertical loads, this neutral axis exists 

 and passes through the centre of gravity of the beam cross- 

 section parallel to the top and bottom faces of the beam. 



Bending and Resisting Moments. The effect of any ver- 

 tical load, acting through the centre of gravity of the beam to 

 produce flexure, is the amount of the load sustained and the 

 point of application, or its leverage, as well the (t bending 

 moment" M at any cross-section of a beam, or the algebraic 

 sum of the vertical forces on the left or right of the section, 

 where the tendency of the forces is to cause motion by rota- 

 tion around that point. The maximum bending moment 

 occurs, of course, where the beam is most greatly strained. 

 Without demonstration, the bending moment of a beam, 

 uniformly loaded and supported at both ends, M = \Wl\ 

 where W = the total load and / its leverage. 



The resistance offered by the fibres and their arrangement 

 to the effects of the applied load is determined by the " re- 

 sisting moment," R, of the beam, and is found by obtain- 

 ing the algebraic sum of all the moments of the horizontal 

 stresses producing tension and compression of the fibres, act- 

 ing in opposite directions but parallel to each other. These 

 moments are determined, with respect to the neutral axis, by 

 adding together or summing up algebraically all the moments 

 of all the unit stresses acting upon all the elementary areas 

 of which the cross-section consists. 



When this value equals that of the applied weight when 

 multiplied by its leverage of action, called the "moment of 

 rupture," or J/, we have the equation, R = J/, indicating 

 equilibrium between the forces tending to cause rupture and 

 those which offer resistance to the former forces. 



Moment of Inertia. In the consideration and design of 

 beams, the effect of the shape or cross-section of the beam has 

 to be taken into account and is analyzed by the aid of a 



