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Figure 6.6

The static aspect. In fig. 6.6, b a cut off part of the beam is
shown. Considering its equilibrium we recall (see. p.1.6) that the
bending moment M   x is the algebraic sum of the moments of
z
elementary normal forces  dF :
x
M   x   y dF .
z x (6.4)
F
Similarly M   x   z dF  0 ,
y x (6.5)
F

N   x   x  dF  0.
x (6.6)
F
The geometric aspect. Considering the geometric hypotheses of
Bernoulli the equation of displacements can be set up. Select the
element of the beam with the length dx that is the form of
deformation - fig. 6.7, and in deformation - fig. 6.7 b. Since the
upper fibers of the beam are reduced, and the lower become
longer, it is natural that among them there is a neutral layer - the
fiber which length does not change.

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