

173 



STRENGTH OF MATERIALS 



of the center of pressure from the center of gravity of the section. 

 For a first approximation, therefore, the first term may be disregard.-, I 

 in comparison with the second, and hence the problem of makinu W 

 a minimum reduces to that of making the integral J ds as small 



as possible. 



To effect a still further reduction, suppose that R is resolved into 

 vertical and horizontal components so that the vertical component 

 shall pass through the center of gravity G of the section . Pig, 131), 

 and let z denote the perpendicular distance of the hori/ontal oom- 



ponent P k from d. Then M - /',: and 



/M a C / '" ~~ 



ds becomes I -TJT^ 



or, since P k is constant for all sections, 

 this may be written Pj / 



Ordinarily the thickness of the nn-li 

 ring varies, being least at the crown 

 and greatest at the abutments. In this 

 case let l r denote tin- thn-km-ss of the 

 crown, and suppose that the law of variation in thickness is su-h 

 that the thickness b at any other point is given by the expression 



where dx is the horizontal projection of ds. Under this 

 the expression Pi t * becomes 



Therefore the problem of making W a minimum is now reduced to 

 that of making the integral | s?dx as small as possible. 



This latter expression, however, consists of only positive terms, 

 and reduces to zero for the center line of the arch. From tl 

 follows that if an equilibrium polygon is drawn for tin- stem 



of loads, and then the center line of the arch is so chosen as to coin- 

 cide with this equilibrium polygon, the true linear arch can 

 but little from this center line. 



