INSTABILITY IN OPEN STRUCTURES. 



323 



we get the equalities 



— BO. CO y~.-CO.AO 



COB 



AOC 



cC 



AO.BO 

 BOA 



that is to say, the three external pressures applied at the points A, B, C 

 balance each other just as if they had been applied directly to the point 0. 



When computing the internal strains caused by given external pressures, 

 the area ABC occurs in every case as a division ; if, then, the three points 

 were in one straight line, that is, if the area ABC were zero, the internal strains 

 would become infinitely great, unless the applied pressures were all in the 

 same line with them. Here we have the first and very well known example 

 of instability in construction. 



If the point O be removed to a very great distance, the directions 

 aA, b~B, cC of the external pressures become parallel as in fig. 2. The 

 intermediate pressure, in this figure MB, must be op- 

 posed to the direction of the others, its intensity being 

 the sum of those at A and C. 



The relation of the strain on AC to the external 

 pressure at B is then given by the formula 

 __ — AX.CX 



ac-6b. AC-BX ; 



so that if B were shifted along the line BX nearer to X, Fig. 2. 



the strain on AC would be augmented in the inverse ratio of the new to the 



former BX ; but the pressures «A, bB, cC, would still remain proportional to the 



lines XC, CA, AX. Were B brought actually to X the strains would become 



infinite. 



It is much to be regretted that, in lesson books on mechanics, the beginner 

 is taught the properties of this impossible straight lever, without a hint of 

 caution in regard to it. The strains on the arms, even that upon the fulcrum, 

 are left out of view. In this way hazy notions are engendered ; the load at A 

 is said to balance that at C, although both be pressing in one direction. 



The relative positions of four points, as A, B, C, D, fig. 3, are in general 

 fixed by the lengths of the six lines AB, CD, AC, BD, AD, 

 BC joining them two and two ; these form the boundaries 

 of a solid, called in Greek tetrahedron, which may get the 

 English name fournib, shorter and quite as descriptive ; 

 the potters call it crowfoot : it is the simplest of flat-faced 

 solids. 



As in the triangle pressures applied at the corners can 

 balance each other only when their directions meet in one Fig. 3. 



point ; so, reasoning by what is called analogy, we might infer that, of four 



