Abstract
This chapter describes the determination of deflections of straight and curved beams as a result of bending and shearing action. To determine slope and deflection as a result of bending only, it is necessary to find the shape of the elastic curve of the neutral axis. This curve is dependent on the bending moment distribution along the length. If displacements of the neutral axis from the original unloaded position of the beam are v measured perpendicular to the x-axis and positive downwards, then the deflections of each end of the element are v and v + δν. However, for small displacements, tan θ =▪. An alternative method of obtaining deflections as a result of bending, other than by graphical solutions, is to consider the geometry of the deformed beam. However, the proportion of the deflection as a result of shear is very small compared with that as a result of bending when the cross section is small in relation to the length. The deflections of beams and frames can also be determined by indirect methods using the Principle of Virtual Work or, alternatively, the First Theorem of Complementary Energy. The equivalence of elastic strain energy stored and external work done can be used in the case of a single concentrated load to determine the deflection under the load.
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