Fig. 11. 



BRIDGES — BEAMS. 137 



in order to change its shape, as in Fig. 11, the 

 side A B must become longer than C D, and he 

 supposed that all the fibres above C D were 

 extended by the action of the weight, and that 

 the tensile strength of the material was alone 

 called into action. 



It is evident, upon reflection, however, that 

 if the material is at all compressible, that the 

 fibres along C D, in the giving way, will be compressed. Mariotte 

 first suggested this, but very vaguely. James Bernoulli afterwards 

 examined the subject, and pointed out the fact clearly, and indicated 

 the position of the neutral axis. 



If in the Figs. 10 and 11 the upper fibres are extended, and the lower 

 ones compressed, there will evidently be a line along which the parti- 

 cles will suffer neither extension or compression ; and this line is 

 called the neutral axis. 



If the material is able to resist compression and extension equally 

 well, the neutral axis will be in the middle. If it is readily extended, 

 and resists compression, the neutral exis will be near to the compressed 

 side, and vice versa. As before stated, the beam will bend before it 

 breaks, and the amount of this bending is important, partly because 

 in many structures great stiffness is necessary, and we should know 

 how to attain it, and partly because it is found that any bending after 

 a certain amount, is injurious to the beam, although the weight applied 

 may not have been sufficient to break it at the time. 



The distance that the point of the beam sinks below the horizontal 

 line is called the deflection, and it can only be determined b} r experi- 

 ment upon the different materials, although we may deduce the general 

 laws which govern it. 



The formula by which the law of deflection is expressed, is as follows: 



Where D is the deflection, W t\\e weight, I the length of the beam, 

 b the breadth, and d the depth, c is a constant, determined by ex- 

 periment. 



That the deflection should be directly as the weight, that is, that if 

 we double the weight we will double the deflection, need hardly be 

 demonstrated. 



That the deflection is as the cube of the length is not quite so 

 obvious. We must remember that the effect of any force or weight 

 does not depend simply upon its amount, but also upon the distance 

 of the point of application from the fixed point, upon its leverage, or, 

 as it is properly called in mechanics, its moment. Now, when we in- 

 crease the length of the beam, the weight remaining the same, we in- 

 crease the moment of the weight, and therefore its deflecting power; 

 the length, therefore, comes into the expression in that way, once. 



Again, as the extension of the upper side is due to the increased 

 distance between the particles with any particular strain, if there are 

 more particles there will be greater extension, and so I comes again 

 into the expression. Lastly, the angle of the deflection being the 



