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In fig. 6.15 y and  – deflection and rotation angle of the
A A
cross-section at a point A . Using the geometric meaning of the first
y
derivative of the function   x , we can write
tg   x  y   x ,

and in small deformations
  x  y   x . (6.30)
To determine the beam deformation the Hooke's law under
bending is used
1 M   x
k   z . (6.31)
 EJ
z
Now we establish the correlation between the curvature k and
y
bending   x , having considered the point A and A lying at a
1
distance ds from each other (fig. 6.16). The tangent to the line of
deflections at the point A forms an angle  with the horizontal.

At the point A corresponding angle is equal to   d , and d
1
the angle between normal OA and OA . Express the length of the
1
arc by a central angle
ds   d ,
hence the curvature is

Figure 6.15 Figure 6.16

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