Relaxation methods applied to engineering problems I—The deflexion of beams under transverse loading

Abstract
I—According to the approximate theory of flexure, the transverse loading and deflexion of a girder initially straight are related by the equations ω = d 2 M / dx 2 , (1) B d 2 y / dx 2 = M , (2) whence we have on elimination of M ω = d 2 / dx 2 ( B d 2 y / dx 2 ). (3) In (l)-(3), ω, y and M relate to a section of the girder distant x from one end; M is the bending moment at this section, y the deflexion of the central line from its initial (unstrained) position, and ω the line intensity of the transverse loading. B ( = El ) denotes the flexural rigidity. 2—The solution of (3) entails four arbitrary constants, which can be used to satisfy two imposed conditions at each end of the girder.

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