176 STRENGTH OF MATERIALS 



than elsewhere, and this will cause the pressure line, or linear an-h, 

 to move down somewhat, thus taking a position nearer to the mil Id It- 

 third than is required by the principle of least resistance, if applied 

 to the arch as a rigid body. 



The above brief account of Moseley's principle of least resistance 

 and Culmann's modification of it are given chiefly for their historical 

 interest and the importance formerly attached to them. The modern 

 theory of the arch is based upon the principle of least work, and is 

 therefore rigorously correct from the standpoint of the mathematical 

 theory of elasticity. 



143. Application of the principle of least work. Although Ho. 

 law is not rigorously true for such materials as stone, cement, and 

 concrete, the best approximation to actual results is obtained by 

 assuming that the materials of which the arch is composed con: 

 to Hooke's law, and then basing the theory of the arch <-n the general 

 theorems of the strength of materials. On this assumption the ; 

 tion of the linear arch can be determined by means of Casti^l; 

 theorem, which states that for stable equilibrium the work of d< 

 mation must be a minimum (Articles 78 and 80). 



Consider a section of the arch perpendicular to the center line of 

 the arch ring, or, in general, normal to the in trades. Let F denote 

 the area of the section, R the resultant pressure on tbe section, c tin- 

 distance of the point of application of R from the center of gravity 

 of the section, and ds an infinitesimal element of the center line. 

 Then the work of deformation will consist of two parts, that due 

 to the axial thrust R, and that due to a moment M = Re. Since the 



R 



direct stress per unit of area of the section is > the unit deformation 



R ^ 



due to the stress is > where E denotes Young's modulus; and 



hence the work of deformation due to R is - R I 1 or 



J \ FJtJ 'J 1 

 From Article 72, Chapter IV, the work of deformation due to the 



J/ 2 



bending moment Mis - - Therefore the work of deformation </!/' 

 L 1 



for a portion of the arch included between two cross sections at a 



distance ds apart is 



