Basic Kinematic Assumption Curvature

47 Fig.12.7 shows portion of an originally straight beam which has been bent to the radius p by end couples M. support conditions, Fig. 12.1. It is assumed that plane cross-sections normal to the Figure 12.7: Deformation of a Beam under Pure Bending length of the unbent beam remain plane after the beam is bent.

48 Except for the neutral surface all other longitudinal fibers either lengthen or shorten, thereby creating a longitudinal strain ex. Considering a segment EF of length dx at a distance y from the neutral axis, its original length is

49 To evaluate this strain, we consider the deformed length E'F'

The strain is now determined from:

x EF dx or after simplification

where y is measured from the axis of rotation (neutral axis). Thus strains are proportional to the distance from the neutral axis.

50 p (Greek letter rho) is the radius of curvature. In some textbook, the curvature k (Greek letter kappa) is also used where k = - (12.15)

p thus,

5i It should be noted that Galileo (1564-1642) was the first one to have made a contribution to beam theory, yet he failed to make the right assumption for the planar cross section. This crucial assumption was made later on by Jacob Bernoulli (1654-1705), who did not make it quite right. Later Leonhard Euler (1707-1783) made significant contributions to the theory of beam deflection, and finally it was Navier (1785-1836) who clarified the issue of the kinematic hypothesis.

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