9.5. Empirical formula for determining the critical stresses
If, as it very often happenes in practice, the slenderness ratio of the bar is lesser than the pointed values, Euler’s formula will be inapplicable, because the critical stresses surpass the proportional limit and Hooke’s law will lose the action.
There are approximate theoretical methods for determining the critical forces under the loss of stability in the inelastic range but their consideration isn’t included into the present course of studies.
In this case the following empirical Tetmajer-Jasinsky formula based on the numerous experiment is used:
(9.11)
where a and b are the coefficients depending on the material.
For steel
Ст.
3 under slenderness ratio
the coefficients a
and
b
can
be taken equal: a=310
MPa,
b=1,14
MPa.
Under slenderness ratio λ<40 the bars can be calculated on the strength without considering the danger from elastic buckling.
9.6. The practical formula for the stability analysis
It is comfortable to have one formula under any bar slenderness ratio then two formulas (Euler and Jasinsky) as each of them is good for a certain range of the slenderness ratio.
This practical formula being applied widely in the construction design has the form
(9.12)
where
is the allowable working stress in compression;
is the reduction coefficient for the allowable working stress (or a
coefficient
of buckling).
The value
depends on the material and the slenderness ratio. Its values are
given in tabl. 9.1; A
is the bar cross section area.
The value
can be considered as the allowable stress in the stability analysis,
i.e.
(9.13)
The allowable working stress has the form:
(9.14)
where
is the limit stress taken as the yield stress for plastic materials
or the ultimate strength for the brittle ones.
The
relation between the coefficient
the critical stress
the limit stress
of the safety factor coefficients n
and stability
can
be derived as follows:
from which
(9.15)
Using the formula (9.14) we get
(9.16)
To select the section the formula is led to the following form:
(9.17)
The value
has to be set as the slenderness ratio
is unknown because the section area A
is unknown but the slenderness ratio depends on it. Then one has to
determine i min
and λ values from tabl. 9.1 It is recommended to take φ 1
=
0.5. And then we find the values A,
l min.
If we
receive a large difference between the values
and
,
the calculation must be repeated setting a new value
and so on as long as the difference between the successive values do not exceed 4-6%.
For bars which sections have considerable weakness (for example, from holes) besides the stability design, the strength analysis must be done by the formula
(9.18)
where
is the working area (net) of the bar section.
In the
stability design the total section area
is taken. In some cases (for example, in the machine-building
construction element design) the values of the safety stability
coefficients
provided for composing the tables of the coefficients
are insufficient. In this case the design must be done by using the
demanded coefficient
and Euler’s
or Jasinsky’s
formula. It must be done the same in the design of the bar stability
from materials which do not have coefficients in the table
Tabl. 9.1.
Slenderness ratio
|
for |
||||
Steels Ст. 1, Ст. 2, Ст. 3, Ст. 4 |
Steel Ст. 5 |
Steels of increased quality
|
Cast-iron |
Tree |
|
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 |
1,00 0,99 0,96 0,94 0,92 0,89 0,86 0,81 0,75 0,69 0,60 0,52 0,45 0,40 0,36 0,32 0,29 0,26 0,23 0,21 0,19 |
1,00 0,98 0,95 0,92 0,89 0,86 0,82 0,76 0,70 0,62 0,51 0,43 0,37 0,33 0,29 0,26 0,24 0,21 0,19 0,17 0,16 |
1,00 0,97 0,95 0,91 0,87 0,83 0,79 0,72 0,65 0,55 0,43 0,35 0,30 0,26 0,23 0,21 0,19 0,17 0,15 0,14 0,13 |
1,00 0,97 0,91 0,81 0,69 0,57 0,44 0,34 0,26 0,20 0,16 - - - - - - - - - - |
1,00 0,99 0,97 0,93 0,87 0,80 0,71 0,60 0,48 0,38 0,31 0,25 0,22 0,18 0,16 0,14 0,12 0,11 0,10 0,09 0,08 |
