Loaded Shaft supported in Three Bearings. 



637 



slightly deflected, to that of the elastic forces in the shaft 

 tending to restore it to its undetected form. 



§ 2. General Case. 



In the general case the shaft is supposed to change in 

 diameter at various sections along its length, and to be of 

 uniform size between such sections. It is assumed to cany 

 at various places along it, a series of loads each of which is 

 balanced. The shaft ax ; s is assumed to be straight in its 

 unstrained condition, and the three bearings to lie in a straight 

 line. The bearings are assumed to act as free supports and 

 not to fix the direction of the shaft to any extent for the 

 small deflexions contemplated. The usual theories of elastic 

 bending will be supposed to hold right up to each section 

 change point, without introducing any end effect due to the 

 sudden change of section. 



The general problem of a number of loads on the shaft is 

 greatly simplified by the introduction of a principle originally 

 enunciated by Bresse. This principle enables the loads to be 

 considered in their effect on the deflexion of the shaft ore 

 at a time, and it may be stated as follows : — The displacement 

 of any point by reason of the deformation of the shaft is the 

 resultant of the displacements which would be produced if 

 one supposed all the external known forces to act separately 

 and one after the other. Thus the shaft may be considered 

 as supported at each end under the action of the given 

 loading acting downwards and also under the action of the 

 supporting force at the intermediate bearing acting upwards. 

 The result of the action of these two systems of forces is to 

 make the deflexion zero at the intermediate bearing. 



Fig. 1. 



It will be convenient to consider the shaft as carrying one 

 load and one couple in each of its two spans. Let tig. 1 

 represent the deflected centre line of the shaft. 



Take the intermediate bearing as origin, the spans being 

 OL and 017, and L and L' being the end bearings. Let ; 



