712 Prof. A. Morley on 
alone produce is added that which would be caused by the 
axial load if the deflexions of the beam were due to the trans- 
verse loads only and were unaffected by the axial loads. 
This is the method usually given in the text-books for other 
than very short beams. 
(c) The bending moment resulting from the axial loads 
is estimated by means of the deflexions calculated from the 
axial and lateral loads jointly, and is added to that resulting 
from the lateral loads. This is the method adopted in the 
paper mentioned above ; but in the common case chosen for 
illustration an approximation is made in estimating the 
deflexion. 
Object of this Pajier. 
The present paper is mainly concerned with the simple but 
most important cases of uniformly distributed and single 
concentrated loads and simple conditions of end support ; its 
main object is to record the more exact solutions and to 
examine under what circumstances the simpler methods of 
calculation are approximately correct and to indicate the 
degree of approximation. 
Notation. 
The axis of x is taken through the centres of area of the 
two ends of the bar and the origin is midway between these 
points. The length of the strut or tie-rod in all cases is taken 
as I. The axial force is +P a thrust in the case of a strut, 
and —Pa pull in the case of a tie-rod. The radius of gyration 
of the area of cross section A about a central axis perpen- 
dicular to the plane of bending is k, and the moment of 
inertia, PA, of the area A about the same axis is I. Only 
sections symmetrical about this axis are considered and the 
depth of section is taken as d. The average intensity of 
P 
stress over the section is p = ± t* 
Since the curvature is always small within the limits of 
d 2 y 
elasticity, it is taken as -=^, and the bending moment, 
reckoned negative when it tends to bend the bar concave 
d 2 y 
towards its undeflected position, is equal to EI -y-|, where E 
is the modulus of direct elasticity. 
