Tie-Rods with Lateral Loads. 271 



In this paper I shall first give the general solution of such 

 problems ; and it will include the old problem of want of 

 straightness of the strut and inaccuracy of loading. 



I take as the axis of x the straight line through the centres 

 of area of the two ends of a strut. The origin is at the point 

 of bisection of this line. The length of strut is 21. Systems 

 of forces act at the ends such that their resultants are two 

 equal and opposite pushing forces F (or pulling forces FjJ in 

 the axis of x together with certain couples. Thus if a strut is 

 hinged at the ends, and if the resultant forces do not act 

 exactly through the centres of area, but at points whose dis- 

 tances are A (measured parallel to the plane in which bending 

 is most likely to occur) from the centre of area, I take it that 

 there are couples at the ends, of the amounts FA or — F t A. 

 Again, if a strut is fixed at an end, I take it that there is an 

 end couple — M or — M l whose amount will be determined by 

 given conditions of lateral loading and end push or pull. In 

 this case M and M 1 will include such terms as FA. I can 

 assume that the strut when unloaded is not quite prismatic, 

 the ordinate of its centre line being if. It is easy to make 

 the treatment even more general by taking the initial shape a 

 tortuous curve, and by considering bending as taking place in 

 various planes ; but for nearly all practical purposes it is suf- 

 ficient to deal with one plane only. 



I suppose the strut to have a lateral loading such that if it 

 were a beam supported merely at the ends, the bending- 

 moment would be $(#)*. 



It can readily be shown that the total bending-moment at 

 any section of a strut or tie-rod, if — M and — M x are the 

 couples at its ends, is 



Fy + #*)-i(Mi + Mi}'+*M«^M!;. . . (3) 



or M — M 



-% + ^W-|(M s + M,)+^-°^.. . . (4) 



* If the strut is loaded uniformly laterally with a total load W, 



(£(,:•) = | W(7 2 -.r 2 ). 



If the strut has a single load at the middle there is a discontinuity in the 

 function at the middle ; but we may put it 



<£(*) = iW (7 -VP), 



the minus sign being taken whether x is positive or negative. There are 

 easy graphical methods, well known to all practical engineers, for rinding 

 4>(z) for any system of lateral loading however complicated. 



U2 



