## 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

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.

## Post a comment