148 



STRENGTH OF MATERIALS 



which is the required formula for calculating the bending stress at 

 any point of a curved piece. 



125. Simplification of formula for unit stress. In formulas (96) 

 and (97), derived in the preceding article, the integrals involved 

 make the formulas difficult of application. The following geometrical 

 transformation, which is due to Resal,* greatly simplifies the formulas 

 and their application. 



The first step is a geometrical transformation of the boundary of 

 the given cross section. Consider a symmetrical cross section, for 

 example the circle shown in Fig. 103, and let OK be an axis of 



symmetry passing through the center 

 of curvature C of the section, and OZ 

 a gravity axis perpendicular to OY. 

 Now suppose radii drawn from C to 

 each point M in the boundary ..! tin- 

 cross section. From //. the point of 

 intersection of CM with the gravity 

 axis OZ, erect a perpendicular to OZ, 

 and from M draw a perpendicular 

 to OK Then these two perpendicu- 

 lars will intersect in a point of the 

 transformed boundary, as shown in 

 Fig. 103. 



It will now be proved (1) that tin- 

 distance of the center of gra\ 

 of the transformed section from the 



center of gravity of the original section is the value of d given by 

 formula (96), and (2) that the moment of inertia of the transformed 

 section is the integral which occurs in formula (97). 



In Fig. 103 the distance NM' is the ^coordinate of the point M '; 

 let it be denoted by z'. Then 



CN p + y 

 The distance d r of the center of gravity G of the transformed 



* Resistance des Materiaux, pp. 385 et seq, 



