

Bending of Thin Shells. 497 



quantities, viz.: — 



T x = a tension across A D parallel to A ; 



M 2 = a tangential shearing-stress along AD; 



N 2 = a normal shearing-stress parallel to OC ; 



G 2 = a flexural couple from C to A, whose axis is parallel 



to AD; 

 Hi= a torsional couple from B to C, whose axis is parallel 



to OA. 



Similarly the stresses which act across the section B D 

 consist of: — 



T 2 = a tension across B D parallel to B ; 



Mi= a tangential shearing-stress along B D; 



N x = a normal shearing-stress parallel to C ; 



Gi = a flexural couple from B to C, whose axis is parallel 



to BD; 

 H 2 = a torsional couple from C to A, whose axis is parallel 



to OB. 



By resolving the stresses and bodily forces (such as gravity 

 and the like), which act upon the element, parallel to the axes 

 A, OB, and C, and by taking moments about these lines, 

 we obtain the six equations of equilibrium of the element ; 

 but as these six equations connect ten unknown quantities, 

 namely the ten stresses which act across the sides of the 

 element, they are insufficient for the solution of the problem. 



2. In the case of a bell, or of a railway bridge which is 

 thrown into a state of oscillation by a passing train, the 

 displacements are all small quantities, and under these cir- 

 cumstances the ten sectional stresses can be expressed in 

 terms of the displacements of a point on the middle surface 

 and their differential coefficients ; and owing to the fact that 

 these displacements are small, we may neglect their squares 

 and products when determining the stresses, and their cubes 

 &c. when determining the potential energy due to strain. 

 There is, however, another class of problems of considerable 

 importance in which the deformation is finite instead of 

 infinitesimal ; and to such problems the theory of thin shells 

 is inapplicable. 



3. An ordinary clock-spring is one of the most familiar 

 examples of the finite bending of a thin plate or shell. Such 

 springs consist of a naturally curved steel strip whose thick- 

 ness is somewhere about one thirtieth of an inch, and whose 

 breadth is from an eighth to a quarter of an inch according 

 to the size of the clock ; and when the clock is wound up 

 an amount of bending takes place which it would be unsafe 

 to treat as infinitesimal. The hair-spring of a watch also 



