[ 294 J 



L. On the Calculation of the Equilibrium and Stiffness of Frames. 

 By J. Clerk Maxwell, F.R.S., Professor of Natural Philo- 

 sophy in King's College, London*. 



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 weak- 

 ened 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 de- 

 rived from the principle of Conservation of Energy, and is 

 referred to in Lame's Leqons sur V Elasticity Lecon 7 me , as Cla- 

 peyron's Theorem ; but I have not yet seen any detailed applica- 

 tion 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 e 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 ten- 

 sions 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 equations 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 3s displacements of the s points ; 6 of these displace- 

 ments, however, are equivalent 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 equations 

 of extensions, and e equations of elasticity; so that the solution is 

 always determinate. 



The following method enables us to avoid unnecessary com- 

 plexity by treating separately all pieces which are additional to 

 those required for making the frame stiff, and by proving the 

 identity in form between the equations of forces and those of 

 extensions by means of the principle of work. 



On the Stiffness of Frames. 



Geometrical definition of a Frame. — A frame is a system of 

 lines connecting a number of points. 



* Communicated by the Author. 



