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XXXII. Elastic Stability of Long Beams under Transverse 

 Forces. By A. G. Ml Michell, M.C.E. Melh* 



I. TT is a matter of common experience that a thin flat bar 

 A or blade of elastic material, subjected to bending in 

 its own plane, may be unstable in the plane form and fail by 

 combined lateral displacement and twist. It will appear 

 below that the instability of such a beam is due to its want 

 of torsional, rather than of flexural, rigidity, so that the same 

 kind of instability as occurs in a thin blade may affect beams 

 of other forms. The mode of instability will depend upon 

 the proportions of the beam, but if the length is so great 

 compared with the other dimensions that the ordinary 

 equations for the bending and twisting of a linear rod are 

 applicable, a criterion of instability may be obtained from 

 them. 



In engineering construction beams which fail under ex- 

 cessive loads by buckling or lateral displacement are very 

 frequently employed. In many commonly occurring cases 

 the assumptions made below as to the relative dimensions of 

 the beam would be justifiable, and lead to results at least 

 sufficiently accurate for purposes of practical design. In 

 other cases the results might be used as bases for empirical 

 formulae. 



Taking the case of a cylindrical beam subject to forces 

 applied in the axial plane of greatest bending rigidity, denote 

 by fti the rigidity in this plane, by /3 2 that in the perpen- 

 dicular axial plane, and by 7 the torsional rigidity. Let B 1? 

 B 2 be the bending moments, at any point of the beam, in the 

 planes of fti, /3 3 respectively, and Gr the twisting couple ; also 

 let (j>i, <£ 2 be the angular displacements of the axis in the 

 planes of fti and /3 2 , and 6 the total twist from the origin. 



Then 



*$ = B *> w 



d9 „ 



tds =G 



are the ordinary approximate equations. It will be supposed 

 throughout that yjfii can be treated as a small fraction. /3 2 

 may be either of the same order as, or much smaller than /3 1? 

 but is supposed not to be small in comparison with 7. 



* Communicated by the Author. 



