13 



the deflections of columns under axial and 

 Eccentric Loading. 



By R. W. Chapman, M.A., B.C.E. 



[Read May 9, 1918.] 



In a paper (*) published in The Physical Review for March, 

 1917, Mr. R. W. Burgess has demonstrated that the ordinary 

 method of solution, for the problem of determining the deflec- 

 tion of a column under load, leads to results that are very 

 considerably in error when applied to the case of a long, thin 

 column, pin-jointed at the ends, and subjected to compression 

 along its axis. In all such problems the solution is obtained 

 by the integration of a certain differential equation. The 

 exact solution is, as a rule, somewhat difficult; but a very 

 simple solution can generally be obtained by neglecting the 



dy 

 term involving the square of — , and as this is supposed to 



\ d x 



give results that are quite sufficiently accurate for all practical 

 purposes, the usual solution of the problem is obtained in this 

 way. This is the method universally adopted in all the text 

 books on Engineering and Physics. It comes, therefore, 

 somewhat as a shock to learn from Mr. Burgess that this 

 universal method of calculation, when applied to determine 

 the deflection of a pin-jointed column under axial load, leads 

 to results that are in error by as much as 100 per cent. One 

 naturally thinks of all the ' other problems, fundamental in 

 engineering practice, in which the computations of practical 

 engineers are all based upon formulae derived by precisely 

 the same approximate method of solution, and one wonders 

 whether all this practice is fundamentally defective. Of 

 such, for instance, are all computations on continuous girders 

 and all calculations of beam deflections. 



As an illustration we will consider the case of the 

 deflection of a beam supported at each end and loaded with a 

 concentrated load IF at the centre. 



Let I denote the span, and take the origin of co-ordinates 

 at the centre of the beam. 



(i) "The Comparison of a. certain Case of the Elastic Curve 

 with its Approximation." 



