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P  0x y    2 EJ z ; P  0x z    2 EJ y ,
кр 2 кр 2
 z  l   l y

where J and J – main moments of inertia of the cross section
y z
of the rod.

5.4 Critical stress. The limits of applicability of the
Euler’s formula

In a compressed rod critical stress arising under critical forces:
P кр  2 EJ
   min .
кр 2
F F   l
Taking into account that the ratio J F is the square of the
min
minimum radius of inertia i , we obtain
min
 2 E
  .
кр 2
 l i 
min
We introduce the dimensionless quantity  , called rod flexibility
and is equal to the length of the summary to a minimum radius of
inertia of the cross section:
l 
  . (5.17)
i
min
We see that the rod flexibility is its generalized geometric
characteristics. The higher it is, the worse the rod resists
longitudinal bending.
Now, the critical stress is determined by the formula
 2 E
  . (5.18)
кр 2

The relationship between the critical stress and flexibility can be
represented as a hyperbolic curve – Euler’s hyperbola (fig. 5.6).
Since Euler's formula is derived under the assumption that the
critical stress does not exceed the boundary of proportionality,
there are certain limits of its applicability, which are defined by
the inequality

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