lokli Lateral and Non-Axial Loads. 205 



to the lateral system acting alone. For the present purpose 

 let us call this the primary deflexion diagram. If we now 

 multiply the ordinates of this primary deflexion diagram by 

 the end load P, we shall obtain a curve which will give the 

 bending moment that would be produced if the ultimate 

 curve of the strut under both lateral and end systems 

 corresponded exactly to the shape o£ the primary deflexion 

 diagram. A second approximation to the correct curve may 

 be obtained by treating this additional or secondary bending 

 moment curve in exactly the same manner as we previously 

 treated the bending moment curve due to the lateral system 

 acting alone. The secondary deflexion so obtained is usually 

 unimportant but produces bending moments of small amount 

 which may be added to those already found. By continuing 

 the process we should finally arrive at a curve which would 

 represent, within the limit of accuracy possible in graphical 

 work, the exact shape of the bended strut. As the end-long 

 load increases in magnitude and approaches the value given 

 by Euler's formula for the crippling of a strut having no 

 lateral loading, the effects of the secondary and higher 

 deflexion curves become of great importance. For practical 

 purposes of design, however, it will usually be found 

 sufficiently accurate to take the total bending moment as 

 being that due to the lateral loads plus that moment 

 caused by the end loads acting through distances cor- 

 responding to the ordinates of the primary deflexion 

 diagram. The degrees of accuracy may be judged 

 by comparing the maximum ordinate of the lateral load 

 bending moment diagram with that due to the primary 

 deflexion diagram. It will usually be found that the latter 

 is not more than one tenth the former, from which it is 

 justifiable to assume that the maximum ordinate of the 

 bending moment diagram due to the secondary deflexion 

 curve would again be less than one tenth that of the primary 

 deflexion diagram bending moment curve. The error 

 involved by neglecting secondary deflexion curves would 

 therefore be less than one per cent. 



If the end load acts non-axially then the initial bending 

 moment curve will be increased by an amount equal to the 

 product of the end load and the initial eccentricity, and it 

 will be necessary therefore to add a constant ordinate equal 

 to this to the bending moment curve for the lateral system 

 before drawing the primary deflexion curve. After this the 

 work may be proceeded with exactly as before. 



In the case of tie-rods under lateral loadings the bending 

 moment diagrams obtained from the primary, etc. deflexion 



