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M M
*
N   1 1 dF  1 1  ydF  1 1 S , (6.14)
1 
z
F * J z F * J z
M M
*
2 
N   2 2 dF  2 2  ydF  2 2 S , (6.15)
z
F * J z F * J z
T   bdx , (6.16)
where  ,  and M , M – normal stresses and
1 1 2 2 1 1 2 2
bending moments in accordance with cross-sections of a beam
*
1 1 and 2 2 ; F – area of the cut off cross-section of a beam
*
(the area that is shaded in fig. 6.9, b); S – static moment of the
z
area of a cut off part of the section relative to the neutral axis.
Noting the equation (6.16) we take into account Zhuravsky’s
hypothesis, which states that the tangential stresses are evenly
across the section width.
Substituting the expressions for the effort (6.14) - (6.16) in the
equilibrium equation (6.13), we obtain

S *
z M  M   bdx . (6.17)
J z 2 2 1 1
Considering the increase of the bending moment on the length
dx equals to dM  M  M , and taking into account the
z 2 2 1 1
equation (6.2), we obtain the formula Zhuravsky’s formula
Q S * z
y
  . (6.18)
bJ
z
Consider a rectangular cross-section (fig. 6.11, a). For it
b  const , J  bh 3 . Through a point A located at a distance
z 12
from the center of section gravity, draw a line that is parallel to the
neutral axis.

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