116 KANSAS Academy of science. 



3IAXIMUM BENDING MOMENTS FOR MOVING LOADS IN A 

 DEAW BEAM. 



BY E. C. MUBPHY. 



This paper is part of an investigation which the writer has been making of 

 "Maximum bending moments in structures carrying moving loads." Other papers 

 by the writer on this general subject may be found in vol. 6, Mathematical 3Iessenger, 

 and vol. 1, Kajisas University Quarterly. 



The bending moment at any assumed point or section of a structure varies as 

 the load passes over the structure. For some position of the load the moment at 

 this point is greater than that for any other portion of the load, but the bending 

 moment at some other point of the structure may be greater than at the assumed 

 point; hence, to find the greatest bending moment which can ever be produced in 

 a given structure by a given load, we must find the position of the load when it pro- 

 duces this maximum moment,* and the point or section of the structure where the 

 maximum moment occurs. 



Two kinds of moving loads are considered — a concentrated load and a uniformly 

 distributed load. 



Case I — A Conoenteated Moving Load. 



Fig. 1 shows a draw beam sustaining its own weight and a concentrated moving 

 load, P; the end spans are of equal length. 



' X 



I Z. 



-^ B 



A 



Let P=:the concentrated load. 

 1^21^ + 1^. 



q = the weight of the beam per linear foot. 

 w = ql = the weight of the beam. 

 Rj and Rj =the abutment reactions. 

 X = the distance from left end of beam to load, P. 

 z = the distance from left end of beam to section considered. 

 m'i, M'2, M3 =the moment in the 1st, 2d and 3d spans respectively, when P 



is in the Ist span. 

 Ml, M2, M3 =:the moment in the 1st, 2d and 3d spans respectively, when P 



is in the 2d span. 

 m'i", M'2", M'3" = the moment in the 1st, 2d and 3d spans respectively, when 



P is in the 3d span. 



(M)max. =the maximum moment. 



Considering thewhole beam as a free body, and taking moments about A, we have — 



wl , , 



RJ^+-R.^[l^+l,)-Px-j = (1) 



From S (vertical forces) = o we have — 



Ri-\-ll2—w — P = o (2) 



'Hereafter In this paper the term moment Is used In phjce of bending moment. 



