Mathematics. — “(Graphical determination of the moments of 
transition of an elastically supported, statically undeterminate 
beam.” *) I. By C. B. Brzeno. (Communicated by Prof. 
J. CARDINAAI). 
(Communicated at the meeting of November 24, 1917). 
1. Let a rectangular prismatic beam be charged by forces which 
ent its axis at right angles and which are parallel to one of the 
two other principal axes of its centre of gravity. 
Its support, which is thought to be elastical, be applied in a 
number of points of support A, B, C... at the same level in such 
a way that the reactions of support Ra, Rg, kc... 
1. are parallel to the lines of action of the charging forces. 
2. are proportional to the local descents y4,yz,yc... of the 
axis of the beam, so that eR4=y4,8Rg=—ye,yRc=—ye... 
It is required to define graphically the moments of transition in 
the beam. 
2. In order gradually to conquer the difficulties which arise 
during the solution of the problem, the case of the beam on three, 
four and five points of support will successively be dealt with and 
that on the supposition, that the fieldlengths of the beam as well as 
the coefficients of stiffness of the elastic supports are equal. This 
restricting supposition can be introduced, because it does not influence 
the general construction, as will appear later. 
When the case of the beam on five points of support has been 
treated, the general problem, which finds its analytical interpretation 
in the so called “theorem of five moments”, must have been solved 
at the same time. 
3. In fig. 1 for the beam ALC, supposed to be charged in 
the middle of each of its fields by a force of 1 ton, the lines 
') In the following treatise the reader is supposed to be thoroughly acquainted 
with the construction of the elastic link-polygon, which we owe to O. Monr. 
(See for this construction: O Mour, Abhandlungen aus dem Gebiete der technischen 
Mechanik, Ze Auflage S. 367; J. Kropper, Leerboek der toegepaste Mechanica, 
Deel III, p. 160). 
41 
Proceedings Royal Acad. Amsterdam. Vol. XXII. 
