On the Calculation of the Equilibrium and Stiffness of Frames. 295 



A stiff frame is one in which the distance between any two 

 points cannot be altered without altering the length of one or 

 more of the connecting lines of the frame. 



A frame of s points in space requires in general 3s — 6 con- 

 necting lines to render it stiff. In those cases in which stiffness 

 can be produced with a smaller number of lines, certain condi- 

 tions must be fulfilled, rendering the case one of a maximum or 

 minimum value of one or more of its lines. The stiffness of 

 such frames is of an inferior order, as a small disturbing force 

 may produce a displacement infinite in comparison with itself. 



A frame of s points in a plane requires in general 2s — 3 con- 

 necting lines to render it stiff. 



A frame of s points in a line requires 5 — 1 connecting lines. 



A frame may be either simply stiff, or it may be self-strained 

 by the introduction of additional connecting lines having ten- 

 sions or pressures along them. 



In a frame which is simply stiff, the forces in each connect- 

 ing line arising from the application of a force of pressure or 

 tension between any two points of the frame may be calculated 

 either by equations of forces, or by drawing diagrams of forces 

 according to known methods. 



In general, the lines of connexion in one part of the frame 

 may be affected by the action of this force, while those in other 

 parts of the frame may not be so affected. 



Elasticity and Extensibility of a connecting piece. 



Let e be the extension produced in a piece by tension-unity 

 acting in it, then e may be called its extensibility. Its elasticity, 



that is, the force required to produce extension-unity, will be -. 



"We shall suppose that the effect of pressure in producing com- 

 pression of the piece is equal to that of tension in producing 

 extension, and we shall use e indifferently for extensibility and 

 compressibility. 



Work done against Elasticity. 



Since the extension is proportional to the force, the whole 

 work done will be the product of the extension and the mean 

 value of the force ; or if x is the extension and F the force, 



x — eY, 



work = \ ¥x= \ eF 2 = \ i x*. 

 & £ Z e 



TVhen the piece is inextensible, or e = 0, then all the work ap- 

 plied at one end is transmitted to the other, and the frame may 



