METHODOLOGY FOR THE COMPUTATION OF
CRITICAL BUCKLING FORCE AT STEEL TUBES WITH
were installed on the machine (1) and the speci-men (4) was fixed by bolt (3).
Figure 1. Shape of Steel Tube – broken half
Figure 2. Experimental Measurement
A fact, that tubes were deformed in the plane which is perpendicular on the rotation plane, was noticed from these measurements, [4].
3. Mathematical Modeling
The governing differential equation for the buckling of axial loaded column with variable cross-section can be derived from the equilibrium of moment as:
( )
22( )
22( ) 0
2
2
+ =
w x
dx N d x dx w x d dx EI
d
(1)where EI(x) is the flexural rigidity and N is the axial force. This equation will be solved numerically by Finite Difference Method. The values of moment of inertia are exactly computed with the step of 1 mm.
For a both side fixing, we require to meet the boundary conditions:
0 ) ( ) 0 ( ) ( ) 0
( = w l = w ′ = w ′ l =
w
(2)Rayleigh energy method is based on the formula (3), which uses the rule of minimum potential energy [5]:
( )
∫ ( )
∫ ′
=
ll
crit
EJ dx x w
dx x w N
0 2 0
2
(3)
The second variant of Rayleigh method is:
∫
∫
′
′′
=
ll
crit
dx x w
EJdx x w N
0 2 0
2
) (
) (
(4)
where the function (5) is used.
( )
−
⋅
= l
x x w
w 2 π
cos
2
01
(5)In practice, there are also used the other methods like Ritz or Vianello method, but they are identical with the second variant of Rayleigh method. Also, Euler method with a recommended buckling coefficient β = 0.65 will be used.
4. Results and Discussion
For each variant of diameter and slenderness, the values of critical buckling force were computed and compared with average measured values. A comparison of these values is shown in Fig. 3 and Fig. 4.
Figure 3. Comparison of experimental and computed values of critical buckling forces for diameter of 12 mm Legend:
DE – N Differential Equation (numerically) RM – 0.5 Rayleigh Method using the buckling
coefficient β = 0.5
RM – 0.866 Rayleigh Method using the buckling coefficient β ≈ 0.866
RV – 0.65 Euler Method using the buckling coefficient β = 0.65
Methodology for the Computation of Critical Buckling Force at Steel Tubes with Flattened Ends 108
Figure 4. Comparison of experimental and computed values of critical buckling forces for diameter of 14 mm
The computation realized by differential equation provides non-stabile values therefore some simplifi-cations have to be carried out. In comparison with experiment, these values are more than 3 times overestimated in dependence on diameter and slen-derness.
Using the Rayleigh energy method and its first variant, differences between tubes with flattened ends and without flattened ends are negligible. The buckling coefficient has the value β ≈ 0.866 by editing on the form of Euler equation. These values are the most closest to the experiment; however they are significantly overestimated in some cases.
We will consider the hypothesis that flattening does not have any effect on the value of critical buckling force, which was verified by the other experiments performed in past.
Using the second variant of Rayleigh energy method, the buckling coefficient has the value of 0.5, which is in conformity with Euler method. Also, this value appears using the Ritz and Vianello method [6]. The computation is not carried out in elastic area and so the computation with this coef-ficient is not correct; however it will be considered in the computation, because at the first time dimensioning perhaps we do not have knowledge of low slenderness value. These values are also more than three times overestimated. Considering the recommended value of buckling coefficient 0.65, the values of critical buckling force are more than two times overestimated.
5. Conclusion
The aim of this contribution was to find out the value of critical buckling force for the column with flattened ends by defined methods. For a comparison of the values, the experiment was performed. We can find out that original assumed pinned joining is not correct and the column is deformed as clamped. Owing to this, the computation was carried out for clamped column.
At the computation, Rayleigh energy method for both variants, Euler method for β = 0.65 and differential equation were used. It was verified, that the flattening does not have any effect on the value of critical buckling force and so there was considered a constant moment of inertia in cases of β = 0.5 and β = 0.65 without a big deviation. The results are not considered as adequate in compa-rison with the experiment. The most closest is Rayleigh energy method; however these values are significantly higher than measured.
Complications are caused by flattened part, because in the certain time, an axial force achie-ves the value which does not cause the exceeding of limit state in uniform part of column, but yield stress in a place of flattening is exceeded. The computation does not contain this aspect in the formulas. Deformation of column appears earlier than the theory considers with. We recommend to ignore this area of flattening and put the pinned joining to the place between the flattened and tran-sition area. In past, some computations were car-ried out, where the computed values are under-estimated, because the added bending moment was neglected. This moment is relatively small therefore differences between the experiment and theory are also small and the mathematical model is suitable for the computation.
6. Acknowledgment
The article has been written within the research grant project: KEGA no. 019TU Z-4/2015 “The innovation of forms and methods within the educational process in the field of agricultural and forest technology”.
7. References
[1] C. M. Wang, C. Y. Wang and J. N. Reddy, Exact Solutions for Buckling of Structural Members, CRC Press LLC, Florida, 2005.
[2] R. M. Jones, Buckling of Bars, Plates, and Shells, Bull Ridge Publishing, Blacksburg Virginia, 2006.
[3] W. F. Chen and E. M. Lui, Handbook of Structural Engineering, CRC Press, Florida, 2005.
[4] S. Kotšmíd, M. Minárik and P. Beňo, „Strata stability prúta definovaného tvaru pri rôznych hodnotách štíhlostí“, Trendy lesníckej a environmentálnej techniky a jej aplikácie vo výrobnom procese, 2014, 6 pages.
[5] F. Trebuňa and F. Šimčák, Odolnosť prvkov mechanických sústav, Technická univerzita v Košiciach, Košice, 2004.
[6] S. N. Patnaik, and D. A. Hopkins, Strength of Materials: A Unified Theory for the 21st Century, Butterworth – Heinemann, 2003.
S. Kotšmíd, P. Beňo, D. Kozak, G. Królczyk 109