110 SUPPLEMENT TO CHAP. VH. [CHAP. HI. 



CHAPTER III. 



THEORY OF FLEXURE. 



9. Coefficient of Elasticity. Let us now take up the theory of 

 flexure, and see if it is not possible so to present the subject that, in the 

 light of the preceding principles, we may be able to solve all such prob- 

 lems as present themselves. 



If a weight P acts upon a piece of area of cross-section A, and elongates 

 or compresses it by a small amount I, we know from experiment that, 

 within certain limits, twice, three times, or four times that weight will 

 produce a displacement of 2 I, 8 Z, 4 1, etc. These limits are the limits of 

 elasticity. Within them practically, then, the displacement is directly as 

 the force. If we assume this law as strictly true for all values of the dis- 

 placement, and if we denote the original length by L, then, since the 



p 

 force per unit of area is -r, and since this unit force causes a displacement 



A, 



Z, in order to cause a displacement L equal to the original length, this 



L PL 



unit force must be -j- times as great, or equal to -r- y. This force we call 



I At 



the modulus or coefficient of elasticity. It is always denoted by E. Hence 



=' > 



The coefficient of elasticity, then, is the unit force which would elongate a 

 perfectly elastic body BY ITS OWN LENGTH. It is a theoretical force then ; 

 but as the law of perfect elasticity upon which its value is based is true 

 practically within certain limits, by experiments made within those limits, 

 knowing P, A, and L, and measuring I, we can find what the force would 

 have to be if the law were always true. Such experiments have been made, 

 and the values of B for different materials are to be found in any text- 

 book upon the strength of materials. 



From (6) we have for the unit force of displacement 



FEZ 

 A = IT W 



These expressions will be found useful as enabling us to replace often 

 expressions containing an unknown displacement by a definite or experi- 

 mentally known value. 



10. Moment of Inertia. This is also a convenient abbreviation, 

 and enables us to replace unknown expressions by a, in any given case, 

 perfectly determinate value. 



The moment of inertia, with respect to any axis, is the algebraic sum of the 



