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XXIX. Note on Beams fixed at the Ends. 

 By W. E. Aykton, F.R.S., and John Pekry, F.R.S* 



I. A HORIZONTAL beam with vertical loads, fixed at 

 -£*- the ends, being given, to find the bending-moment 

 everywhere : this is a problem the solution of which has not 

 hitherto been put in an elementary form. A knowledge of 

 the bending-moment everywhere leads, of course, to a com- 

 plete knowledge of the strength and stiffness of the beam. 

 The problem is quite soluble by a method which is obvious to 

 any one who has worked out the theorem of three moments ; 

 but even for advanced students the work is tedious ; and prac- 

 tically the answers known for the two cases — (1) when a 

 uniform beam is loaded merely in the middle, (2) when a 

 uniform beam is loaded uniformly — are regarded as roughly 

 applicable in all cases which occur in practice. 



If M is the bending-moment at a section, I the moment of 

 inertia of the section about its neutral line, and E Young's 



M 

 modulus of elasticity for the material, then ^ is the curvature 



M 



of the beam. If 00 / is a short length of the beam, ^j 00' 



is the angle which the originally parallel sections at and 0' 



now make with one another. Hence, if we divide the beam 



into a great number of parts, and if M is taken in the middle 



of each division, the moment of inertia there being I, the sum 



M 

 of all such terms as ^j. 00 / (if 00' is one of the elementary 



lengths) gives the angle between the two end sections of the 



beam. 



M 

 This principle, that 2=nvf . 00' gives the angle between the 



two end sections of the beam, is, of course, well known. It 

 is the basis of the theory of flat spiral springs. As applicable 

 to beams, it may be put in the words of Professor Cotterill 

 (< Applied Mechanics,' § 169, p. 334) :— " ... The angle (t) 

 between two tangents to the deflection-curve of a beam is pro- 

 portional to the area of the curve of bending moments [had he 

 been speaking of non -uniform beams, Mr. Cotterill would have 



M 



said the area of the curve whose ordinate is ^y] intercepted 



between two ordinates at the points considered." It is also 

 the basis of Prof. Fuller's beautifully simple method of dealing 



* Communicated by the Physical Society : read June 11, 1887. 



