Reactions at the Points of Support of Continuous Beams. 269 



variable moment of inertia by obtaining the theorem of three 

 moments in a slightly different form, the necessary summations for 

 each span being performed graphically, whence on substitution in 

 the original equations the bending moments can be obtained. The 

 author has reverted to the problem of finding the reactions at the 

 points of support and has based his method on a principle, definitely 

 stated by Bresse,* and applied by him to the case of a uniform con- 

 tinuous beam of two equal spans. 



The author claims that the method given below affords an easy 

 and accurate solution for continuous beam problems, and especially 

 those in which the moment of inertia is variable. It also permits of 

 the variations in the stresses due to alterations in the levels of the 

 supports being investigated. 



The principle may be reproduced as far as is necessary as 

 follows : — 



The displacement of any point by reason of the deformation of the 

 beam is the resultant of the displacements which icould be produced if 

 one supposed all the external known forces to act separately and one after 

 the other. 



This being so, the continuous beam may be considered as a simple 

 beam supported at each end and under the action of the given 

 loading acting vertically downwards, and also under the action of 

 the supporting forces at the intermediate piers acting vertically 

 upwards. 



If the neutral fibre of the beam in the unloaded condition is 

 assumed to be a straight, line, then the result of the action of these 

 two distinct systems of loading is to make the final deflection of the 

 neutral fibre at each of the intermediate points of support equal to 

 zero. 



If the beam consist of n spans there will be n — 1 intermediate 

 supports, the upward pressure of each of which acting by itself 

 would produce a definite deflection of the beam at any point: the 

 sum of the separate deflections produced by these pressures at any 

 one point of support must equal the deflection produced there by the 

 given loading, and as each of the constituent deflections can be 

 expressed in terms of the unknown concentrated load causing it, 

 there will, therefore, be (n— 1) equations each containing the reac- 

 tions at each intermediate point of support, and as there are (n — 1) 

 reactions these equations are sufficient to determine their values. 



Let A Ax A 2 . . . . A n be the points of support. 



A A„ = L. 



A Ax = 7 1? A A 2 = Z 2j A A 3 = l 3 



* ' Cours de Mecaniqtie Appliqnee,' vol. 1, p. 137. 



