16 



But in the case of columns the error made is of much 

 more importance. Here, in certain cases at any rate, the 

 error made in calculating the deflection of the column under 

 load by the approximate method is considerable, and when 

 the computed deflection is made the basis for determining the 

 stress in the material, as is done in most modern column 

 formulae, the result may be seriously in error. Accordingly 

 in this paper an attempt is made to compute the deflections 

 of columns without resort to the usual approximations. 



Case 1. — Pin-jointed Column with Axial Load. 

 This is the case that has been worked out by Burgess, 

 and his calculations will not be repeated here. 



If / denotes the moment of inertia of the cross 

 section of the column; E, the coefficient of elasticity; and 

 P, the compressive force along the axis, let 



EI 



= a 2 



P 

 Then if I denotes the length of the column, measured 

 around the curve, and h the deflection at the centre, it is 

 demonstrated that 



T 



--2a V /- h$ \=2aR 



[ l St"**) 



Jo 

 where K is the complete elliptic integral of the first kind with 



h 

 the modulus k— — . 

 2a 

 Burgess then proceeds by expansion in series to obtain an 

 expression for h in terms of I and a. From the available 

 tables of elliptic integrals, however, the value of h may be 

 obtained much more simply. These tables give correspond - 



h I 



ing values of k and K, that is of — and — , from which 



2a 2a 



again we may deduce the corresponding value of h/l. 



It will be seen that if there is no deflection of the column, 

 so that h = o, the integral gives l/a = it or P = E I 7r 2 /l, 

 which is Euler's value for the greatest value of P that will 

 not cause collapse of the column. Consequently in order to 

 bend the column 1/2 a must>>j7r. Once this value is ex- 

 ceeded, the tables show that the deflection increases rapidly 

 with increase of P. 



