26 HENDLNG. 



and the maximum bending moment, M B , at the centre of 

 the beam, is 



MB = E -^l-= E *- 1 



= W ' l (XX.) 



4 



The immediate effect of hanging a weight W on any 

 beam of elastic material will be a downward deflection, as 

 shown in the figure, the upper surface becoming concave, 

 and the lower convex. 



In part of the figure (2) the beam is shown before 

 any load has been put upon it, and therefore in a perfectly 

 straight condition. The beam is shown in its side 

 view by the rectangle A B C D. If, now, the load W is 

 hung upon it, a deflection will take place, with the 

 result above described. If the beam had been composed 

 of a number of layers, or laminae, the effect of the deflec- 

 tion and curvature would have been to simply alter the 

 shape of each separate layer without either lengthening or 

 shortening it. With a uniform elastic beam this is, how- 

 ever, not the case. The beam, as a whole, is curved, its 

 upper layers are shortened, and its lower layers extended. 

 That this is so can easily be shown by making a V shaped 

 notch on the upper surface G (Fig. 11) ; under the deflec- 

 tion caused by the load this notch will close up and become 

 a parallel slit. Thus the upper parts of the beam are 

 shortened in the bending, and this shortening must be due 

 to a compressive stress. Just in the same way it may be 

 shown that the lower parts are thrown into tension by 

 making a saw cut in the lower surface ; this opens under 

 the stress, and becomes a V shaped notch, similar to the 

 one that was made in the upper side of the beam ; in fact, 

 the state of things is exactly reversed. It will be further 

 seen that the width of opening at the lower side is greater 

 at the surface and decreases towards zero near the centre, 

 thus showing that the tensile stress is greater at the outer 

 surface on the under side of the beam and zero at the 

 centre ; the stress, both the tensile on one side and 

 the compressive on the other, is proportional to its distance 

 from the centre. The part of the beam towards the centre 

 where the stresses each diminish to zero is called the 

 "neutral surface," and the intersection of this with a 

 vertical plane the neutral axis. 



