Note on Continuous Beams. 307 



taken 33,000,000 sq. ft. tons, or the elastic modulus 1,440,000 

 tons per square foot, and the moment of inertia 3300/144 ft. 4 

 instead of 3300 ft. 4 * 



The main features of our method are due to Culmann ; in 

 some points, however, we differ in our treatment from what 

 we believe to be the practice of Continental engineers : thus, 

 we get rid of the variability of the flexural rigidity ab initio, 

 while some of them at least appear to allow for it by dis- 

 placements of the " verticals " and a series of discontinuous 

 changes in the position of the pole of the second vector 

 polygon f. 



(2) If the flexural rigidity of a girder changes very 

 slightly, we still have the relation 



flexural rigidity x j curvature J = Ending-moment 



approximately true. 



Now if hi -J be the change in curvature after strain, M be 



the bending-moment at any cross-section, EA& 2 the flexural 

 rigidity at that cross-section, and EA & 2 that at any other 

 selected section, the above relation may be written : — 



EAFxofi) = M, 



or, again : — 



2 



EA V*S(J)=^-M = M'. 



In this last form the result is identical with that for a 

 girder of constant flexural rigidity and bending-moment M' 

 instead of the actual moment M. Accordingly in our method 

 we start by altering all the known bending-moments in the 

 ratio of A & 2 /A£-, and the unknown bending-moments when 

 found will then have to be multiplied by A# 2 /A # 2 to determine 

 their actual values. If this reduction of bending-moments be 

 gone through ah initio, then the girder may be treated as if 

 it had a constant instead of a variable flexural rigidity. Of 

 course the change back to the actual bending-moments must 

 be made before the reactions are found. 



We shall now indicate briefly the graphical stages requisite 



* This value was suggested by Mr. Wilson in a reply to a letter from 

 one of us pointing out the impossible value of the rigidity assumed in his 

 paper. 



t See Culmann: Graphische Statik, 1875, S. 576; W. Ritter: Die 

 elastische Linie, Zurich, 1883, S. 18; M. Levy : La Statique Graphique, II e . 

 Partie ; Eddy : New Constructions in Graphical Statics, 1876, chapter ii. 

 for a variety of constructions on this and similar points. 



