314 On the Buckling of Deep Beams. 



numerical values o£ which are quickly calculated for a 

 given value of c 2 by means of equations (65). I£ we write 



/(o') = l + a ? + *+..., 



then f(2), /(3), /(4), can be calculated and the results 

 plotted. The curve gives an approximate root of equation {QS). 

 The root is, in fact, very near 3. By this process and then 

 by successive approximations it was found that 



c 2 = 3-131. (69) 



Therefore 



that is 



wl z ^ 16 i/3-!31\/EnUK; 



W/ 2 = 28-31 y/En()K (70) 



The foregoing are the simplest cases. There are still 

 many more cases to be worked out, as, for example, the 

 beam with uniform load and clamped ends ; the beam with 

 a single load not equidistaut from the ends ; or again, the 

 cases of a load applied at the top or bottom of a section 

 instead of at the middle of a section. But most or all 

 of these new cases will lead to troublesome equations for the 

 critical loads, such equations as Case 5 led to, or worse. 

 Time and assiduity are, however, all that are necessary for the 

 solution of fresh cases. 



It is worth while to make one comparison with Euler's 

 strut formulae. 



The critical thrust R, applied at each end of a rod of 

 length I, for which the rod just fails when the ends are not 

 constrained in any way, is given by 



m 2 = 7t 2 ec. 



Thus R is proportional to the flexural rigidity of the beam 

 and to the inverse square of the length. 



The case of buckling that may be compared with the strut 

 is Case 4. Here 



PZ 2 = 16-94 •EClSi. 



Thus P is proportional to the geometric mean between 

 the flexural rigidity EC and the torsional rigidity Kn, and 

 to the inverse square of I. Also 



P 16-94 /S 1 _ A . /Kk 



r = VVbc = 1,704 Vfc 



That is, the ratio of the buckling load to the Euler thrust is 

 proportional to the square root of the ratio of the torsional 

 rigidity to the flexural rigidity. t 



