[From the Philosophical Magazine, Vol. XXVII.] 



* XXVI. On the Calculation of the Equilibrium and Stiffness of Frames. 



THE theory of the equilibrium and deflections of frameworks subjected to 

 the action of forces is sometimes considered as more complicated than it really 

 is, especially in cases in which the framework is not simply stiff, but is 

 strengthened (or weakened as it may be) by additional connecting pieces. 



I have therefore stated a general method of solving all such questions in 

 the least complicated manner. The method is derived from the principle of 

 Conservation of Energy, and is referred to in Lamp's Lemons sur VElasticite, 

 Lecon 7*, as Clapeyron's Theorem ; but I have not yet seen any detailed 

 application of it. 



If such questions were attempted, especially in cases of three dimensions, 

 by the regular method of equations of forces, every point would have three 

 equations to determine its equilibrium, so as to give 3s equations between 

 < unknown quantities, if s be the number of points and e the number of 

 connexions. There are, however, six equations of equilibrium of the system 

 which must be fulfilled necessarily by the forces, on account of the equality 

 of action and reaction in each piece. Hence if 



e = 3s-6, 



the effect of any external force will be definite in producing tensions or pressures 

 in the different pieces ; but if e > 3s 6, these forces will be indeterminate. 

 This indeterminateness is got rid of by the introduction of a system of e equa- 

 tions of elasticity connecting the force in each piece with the change in its 

 length. In order, however, to know the changes of length, we require to assume 

 3* displacements of the s points ; 6 of these displacements, however, are equiva- 

 lent to the motion of a rigid body so that we have 3s 6 displacements of 

 points, e extensions and e forces to determine from 3s 6 equations of forces, e 



* [Owing to an oversight this paper is out of its proper place ; it should have been immediately 

 before the memoir on "The Electro-magnetic Field." (No. XXV.)] 



