CHAP. I.] SUPPLEMENT TO OHAP. XIV. 275 



SUPPLEMENT TO CHAPTER XIV. 



DEMONSTRATION OF ANALYTICAL FORMULA GIVEN IN TEXT. 



In order to complete our discussion of the braced arch, we shall now 

 give the analytical development of the formulae of which we have made 

 use in the preceding chapter. We do this the more readily, as in no book 

 of easy access to the student are these formulae made out. In the work of 

 Winkler, already referred to in the text, will be found a very thorough 

 discussion of the subject. We shall confine ourselves at present to the 

 case of a single concentrated load.* 



CHAPTER I. 



GENERAL CONSIDERATIONS AND FORMULA! FOR FLEXURE. 



1. Fundamental Equations. The resultant of all the forces act-' 

 ing upon a curved piece in a common plane may be decomposed into a 

 force normal to the piece N, and into a compressive or tensile force in the 

 direction of the axis or of the tangent to the axis G ; and this latter force, 

 if taking effect above or below the axis, acts to bend the piece, and gives 

 rise to a moment M as well as to a compressive or tensile force G. These 

 forces cause corresponding strains. Thus, if P is the tangential strain per 

 unit of area d a, then 



* (1) 



while, if is the distance of any fibre from the axis, 



M . , (2) 



(a) Coefficient of elasticity. 



Let the length of a piece be *, its area of cross-section A, and, as above 



a 



the force acting upon this area G. Then ^ will be the force per unit of 

 area. Let the displacement [elongation or compression] produced by this 

 f orce _ foe A g- the sign A indicating and reading "elongation." Now 



A. 



* A full and complete discussion of the whole subject will be found in " Die 

 Lehre von der Elasticitat und Festigkeit," by Prof. Winkler, Prag, 1867, from 

 which the following is in the main extracted. 



