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XXII. On the Flexure of Continuous Girders. By Mansfield 

 M err i man, C.E., Instructor in Civil Engineering in the Shef- 

 field Scientific Sclwol, New Haven, Conn., U. S. A.* 



IN the ordinary text-books on Mechanics the theory of the 

 flexure of girders is developed for particular cases. First 

 a beam fastened at one end and supporting a weight at the other 

 is considered, and the equation of the elastic line for that case is 

 applied to girders of one span with free ends. Then, by methods 

 tedious and particular, beams with fixed ends are treated -, and 

 occasionally may be found investigations upon girders of two 

 spans for special cases of loading, such as a uniform load over 

 the whole of the girder, or a single load in the middle of one of 

 the spans. The impression is conveyed that the investigation 

 of continuous girders is too difficult to be undertaken except when 

 the load in each span is uniformly distributed over its entire 

 length. The student who is interested in this branch of mathe- 

 matical analysis and who seeks to extend his investigations, finds 

 but little to assist his progress, and is apt to drop the subject in 

 despair, particularly as he finds authorities hinting that the theory 

 is too complicated for development \. 



I propose in this article to demonstrate and present a few new 

 and simple formulas which shall include the whole theory of the 

 moments, shearing-forces, and reactions for girders of any num- 

 bers of spans, equal or unequal, subject to any assignable loads. 

 The remarkable theorem of three moments for concentrated 

 weights will serve as a basis for my investigation ; and as I am 

 unaware of any work in English which presents a proof of that 

 theorem, 1 shall first give an abridged demonstration of it. The 

 formulae that I shall deduce from it will be found to be perfectly 

 general, and yet in a form easily applied to any particular case, 

 as the examples which follow them will illustrate. Armed with 

 these algebraic expressions, which may be written upon a sheet 

 of common note-paper, the student may attack and solve every 

 problem relating to the flexure of girders over level supports. 



The following are the conditions which enable the theory of 

 continuous girders to be mathematically investigated : — (1) the 

 extensions and compressions of different fibres are proportional 

 to their distances from the neutral axis ; (2) the deflection is 

 small compared with the length of the beam ; (3) the moment 

 of inertia is constant. In the following pages the supports will 

 also be considered upon the same level. 



The curve which a beam assumes under the action of its own 

 weight and the loads that it supports is known as the elastic 



* Communicated by the Author. 



t For example, Humber, 'Strains in Girders,' art. 3\. 



N2 



