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 2 EJ
P  min ,
кр 2 (5.11)
  l

where  – built rod length ratio ( Yasinsky’s ratio).
The value l  l  is called erected length. This conventional
зв
hinged prop rod length for which the critical force is critical force
for a given rod length l .
Coefficient  depends on three factors:
1) character of ligature that imposed on end and intermediate
cross-sections of the rod;
2) type of rod load external forces (concentrated or
distributed) and place of application;
3) nature of the change of cross section of the rod along its
length.
Consider the impact of the first factor is the influence of the
fixing rod conditions. Formula (5.11) can be obtained if we
consider the equation of the fourth order longitudinal bending
obtained by double differentiation ratio (5.3):

2
4
d y 2 d y
 k  0. (5.12)
dx 4 dx 2

The general integral of this equation is

y  A sin kx B coskx Cx D  . (5.13)
Constant A , , , B C D are determined from the boundary
conditions. For example, in the second case, the consolidation
(fig. 5.4, b):

y  0 ; y  0 ; y  0 ; y  . (5.14)
0
x 0 x 0 x l x l

Using these conditions, we obtain a system of homogeneous
equations
 B D  0,

 Ak C  0,
 (5.15)
 A sin kl B coskl Cl D   0,

 A sin kl B coskl  0.
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