**INTRODUCTION**

Buckling is a classic mechanical problem, the concept of which was introduced by Euler (Euler, 1744). Over the years, buckling analysis has found wide application, including in civil engineering (Śledziewski & Górecki, 2020; Toledo et al., 2020), mechanical engineering (Czechowski et al., 2020; Kubit et al.,2019) or the maritime industry (Corigliano et al., 2019; Shen et al.,2020).Currently, there is a renewed interest in buckling, in particular in applications in composite structures (Rozylo et al., 2020; Schilling & Mittelstedt, 2020; Xu & Wu, 2008) or micro and nano technologies (Barretta et al., 2019; Chandra et al.,2020). A large number of industrial applications and scientific research (Li & Batra, 2013; Nistor et al., 2017) of buckling analysis have resulted in the development of buckling modeling methods.

In (Harvey & Cain, 2020) authors investigated column behavior if member imperfection and load eccentricity are simultaneously present. Authors analyzed pinned members assuming linearly elastic, slender, uniform, and inextensible columns. To compare the relative significance of member imperfections and load eccentricity on the deflected shape a linear analysis based on Euler-Bernoulli theory was performed. The obtained results were experimentally validated using additively manufactured specimen. Additive manufacturing ensured accurate seeding imperfections to control, specimens buckling characteristics. A series of initially imperfect specimens with eccentric load application points were tested, exhibiting imperfection amplification and cancellation. Authors stated that good agreement between the theoretical predictions and the experimental results was observed.

In (Zhu et al., 2017) author presented, an analytical study on the buckling problem of nonlocal Euler-Bernoulli beams using Eringen’s two-phase nonlocal integral model (Eringen, 2002). Authors deduced the exact characteristic equation for the buckling loads, by the reduction method. In addition, a simple and explicit expressions of the buckling loads for four-type boundary conditions was developed. Authors proved that adopted nonlocal integral model has a consistent softening, in contrast to the nonlocal differential model. Authors stated that in comparison with differential model and the pure nonlocal model,the integral model considered advantages of well-posedness and selfconsistency.In addition, the established analytical formulae may be useful in providing guidelines for designing structures with nonlocal effect, since they contain the nonlocal parameter explicitly.

In (Su et al., 2019) authors presented a finite prebuckling deformation (FPD)buckling theory to analyze the FPD buckling behaviors of beams with the coupling of bending, twist and stretch/compression. To verify the correctness of the proposed theory, it was compared with various analytical and numerical methods of modeling the transverse buckling of a three-point bending beam, lateral buckling of pure beam bending and Euler buckling. In result, it was stated than proposed FPD buckling theory for beams is able to give a good prediction,while the conventional buckling theory (Timoshenko & Gere, 2009) and conventional numerical method (Dassault-Systèmes, 2010) yield unacceptable results (in some cases with 70% error for a three-point-bending beam).

In (Nikolić & Šalinić, 2017) authors presented a method of buckling analysis of non-prismatic columns based on rigid element method. Authors derived a general form of the characteristic equation, which enabled to perform buckling analysis of columns with continuously varying, doubly symmetric cross-section and multiple-stepped columns under different boundary conditions. The proposed method was verified through numerical examples. Authors concluded that results obtained on the basis of presented method have a high rate of convergence to the other results from the literature. Summing up the presented review of buckling analysis methods, it can be stated that despite the fact that buckling is a classic problem of mechanics, researchers are constantly proposing new approaches to solve this problem, as a result of which they determine the values characterizing buckling. One such approach is proposed in this article. Presented approach deals with the problem determining specific buckling amplitude of straight and bent bar, where it is considered as a function of axial displacement of one end of the bar. The main novelty is that assumption that the length of a buckled bar at any instant of buckling is the same as that of a straight bar, regardless of the size of axial displacement of one end

of the bar. A formula for the value of axial displacement of one bar end or

buckling amplitude in the middle of bar length as a function of compressive force was derived – based on energy equations. The proposed method was validated for bars with different cross-section dimensions by comparing results obtained on its basis with finite element model results and experimental tests.

The structure of the article is as follows: in Section 2, the proposed method of determining axial displacement of one bar end or buckling amplitude in the middle of bar length presented. Next, the results obtained on the basis of presented method are compared to finite element model and experimental test results. In Section 4, a discussion of the results obtained is provided. Section 5 contains the final conclusions that summarize the most important achievements of the article.

**METHODOLOGY OF RESEARCH**

**Euler bar buckling**

Bending a bar caused by exceeding a critical value of axial compressive force is called buckling. The value of this critical force was determined by Euler (Bedford & Liechti, 2020; Euler, 1744; Gere & Goodno, 2009;Timoshenko & Gere, 2009). He considered the equilibrium of a bent bar (Fig. 1).

Solving the differential equation, Euler derived a well-known formula for the value of critical force:

The bent axis of the bar is a sinusoid:

Amplitude A at the bar half-length can be written:

However, a specific value of amplitude A at such approach to buckling cannot be determined.

The article aims to determine the course of axial displacement δ of one bar end(or amplitude A of buckling, measured at half-length of the bar) as the function of axial compressive force P. A model of buckling due to axial displacements δ of one bar end is presented (Fig. 2). It can be compared to a bar placed in the closing jaws of a vice. Axial forces occur then as reaction forces in the supports(jaws).

It was assumed that the bent bar has a shape of a sinusoid with amplitude A, at a length between the supports shortened by displacement δ:

To determine the relationship between amplitude *A* and a preset axial

displacement δ, we need to know the formula for the length of the sinusoid. The sinusoid length for each value must be equal to the initial straight bar length .The length of the sinusoid cannot be determined by elementary functions. A precise formula for the length of the sinusoid:

where:

*E * is a complete elliptic integral of the second kind, e is an eccentricity of an ellipse.

Elliptic integrals were encountered while calculating the ellipse circumference,hence their name. The term refers to integrals that cannot be expressed by elementary functions. We can imagine a sinusoid as an expansion of an ellipse formed by an intersection of a cylinder by a plane at a certain angle to the axis and passing through the diameter of this cylinder (Fig. 3a) (Czechowski et al.,2020; Kubit et al., 2019).

Half of the ellipse circumference is the sinusoid length. An approximate formula for ellipse length *L*:

where:

*a, b* are a semi-axes of an ellipse.

It follows from Figure 3b that

where:

*a* is a smaller minor semi-axis, equal to the radius of the cylinder from which the ellipse is expanded,

*γ* is a dihedral angle between the ellipse plane and a plane perpendicular to the cylinder axis.

As the minor semi-axis a and the dihedral angle *γ* are constant, equation *y(x)* represents a sinusoid. It follows from Figure 3b that the minor semi-axis a of the ellipse equals the radius of the cylinder with circumference length 2·Lδ:

The major semi-axis *b* results from the initial length of the bar, equal to half the circumference of the ellipse

In further steps, the attention will be focused on the determination of the major axis *b* from the above equation. After simple transformations

we temporarily adopt a constant value D:

Then

We remove the brackets:

to get a square trinomial:

The discriminant of the trinomial:

The roots of the equation

After the rejection of the unreal solution, the only real solution is

Replacing *D* by the previous value, we get

After the reduction of the brackets:

The final form of the major semi-axis b expressed by the minor semi-axis *a*:

If *δ *= 0 then *b* = L0/π, i.e. the two semi-axes a and b are equal and the ellipse becomes a circle. The other root of the square trinomial would be b =/(9π) unrealistic, as the major semi-axis b would be smaller than the minor semi-axis a.

The sinusoid amplitude is expressed by the formula (Fig. 3b):

The ultimate equation of the sinusoid:

By knowing the sinusoid equation, we can determine the elastic energy US

accumulated in the bar (Shen et al., 2020):

Elastic energy in the bar caused by bending:

An increment of elastic energy Δ accumulated in the bar along with an

increment of displacement Δδ is equal to work of the external compressive force *P *over a distance of axial increment of displacement Δδ:

Hence the value of the compressive force P:

where the constant C:

The verifying calculations were made for the following theoretical data: L = 750 mm,E =MPa, J = 1250 mm³

The calculations of the previously discussed quantities a, b, A, , P were made,depending on the stepwise changing value of the axial displacement δ. At the start of the compression, the bar was perfectly straight. The results of the calculations are contained in Table 1.

The diagram of axial displacement δ as the function of axial compressive force P for this case is presented in Figure 4a.

This is a vertical line intersecting the horizontal axis (axis of force) at the critical force value according to Euler. The amplitude A graph at half-length of the bar perpendicular to the bar axis as the function of the axial compressive force P for this case is presented in Figure 4b.

It is also a vertical line intersecting the horizontal axis at the identical value of the critical force according to Euler. Analytically, the value of this asymptote can be easily expressed by the formula obtained by the substitution to force P of a small value of displacement equal to, e.g

The minor semi-axis:

The major semi-axis:

Amplitude A:

The above value is identical to the value of the critical force determined by the Euler formula. Therefore, a graph of axial displacement δ of one bar end was obtained as the function of axial compression force P as well as a graph of buckling amplitude A at bar half-length as the function of the axial compression force P, which was the objective of this article.

If we assume the bar is initially bent A(), which can be regarded as initial

displacement of the bar end by value (Fig. 5) (we omit elastic strains

caused by compressive stresses), then the elastic energy of the strain

caused by further movement of the bar end must be expressed by the difference of amplitudes A = A(δ) – A() (Fig. 5).

and the value of the compressive force P:

This case of buckling was calculated for the initial value of the bar end

displacement according to Table 2.

The results of the calculations for four initial displacements of the bar end according to Table 2 are presented in Figures 9 and 10.

**A simplified buckling model of a bent bar**

In this approach only buckling amplitudes are considered. This method of determining the critical force is commonly used in lab classes on buckling(Buczkowski & Banaszek, 2006). The amplitude increment in a bar A = A(δ) –A() initially bent at its half-length by a value A() is determined with sufficient accuracy by means of the approximate formula (Buczkowski & Banaszek,2006):

Hence

The graph of this approximate value of amplitude A = A(δ) – A() increment as the function of axial compressive force P is presented for A() = 0.59 mm in Figure 8a.

**Finite element model**

Simulation calculations were made using the **midas NFX **2018 R1 preprocessor(midas Information Technology Co. Ltd., Seongnam, Korea) (midas, 2011)computer program based on the finite element method. The modeled bar had dimensions as given in the previous section. The one-dimensional finite beam elements (CBEAM) were used. The finite element model was composed of 81nodes and 80 elements. The bar ends were modeled as hinged support, and one end had a preset axial displacement δ = −1mm (Fig. 6).

The non-linear static module (SOL 106) was used for the calculations. Further in this study the initial axial displacement δ was calculated according to Table 2.The simulation results are shown in Figures 8 to 10, denoted by FEM (Finite Element Method).

**Experimental tests**

Experimental buckling tests were conducted for a steel bar with dimensions as given in the previous section. The initial amplitudewas 0.59 mm. The value *P* of the axial force compressing the bar and amplitude *A* of buckling measured with a micrometer at bar half-length perpendicularly to the bar axis (Fig. 7).

The experiment results are given in Figure 8.

**DISCUSSION**

Summarizing the research presented in this article, the experimental studies show, that:

• In a wide range of straight bar end axial displacement δ (Fig. 4a) and amplitudes A (Fig. 4b) the compressive force P is constant and equal to Euler’s critical force (Euler, 1744; Timoshenko & Gere, 2009);

• The more the bar is bent at the initial compression stage, the more flattened are the graphs illustrating axial displacement δ (Fig. 10) and buckling amplitude A (Fig. 9);

• The asymptotic character of axial displacement δ of a straight bar end A(Fig. 10) and its amplitudes A (Fig. 9) as the function of axial compression forces is evident for compressive forces close to the value of Euler's critical force;The amplitude graphs differ slightly for the four methods: analytical,simulation, approximate and experimental (Fig. 8a);

• The analytical and simulation graphs of axial displacements δ of the bar end(Fig. 3a) are significantly different in the first phase of compression, but theyhave a joint asymptote;

• The conformity of amplitude A graphs (Fig. 9) as a function of axial compressive forces obtained by three methods confirms the correct

approach to the bar buckling problem by considering the axial displacementof one end of the bar.

**CONCLUSIONS**

Presented article deals with the problem of straight and bent bar buckling, where bar buckling is considered as a function of axial displacement of one end. Owing to the derived formula it was possible to determine a specific value of amplitude A, which is not possible using Euler method. Obtained results (buckling amplitude and axial displacement of bar end for bars with different cross-section dimensions.) show high agreement with FEM analysis and experimental results.

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**Abstract:**

A new approach has been taken to the problem of straight and bent bar

buckling, where bar buckling is considered as a function of axial displacement of one end. It was assumed that the length of a bar being buckled at any instant of buckling is the same as that of a straight bar, regardless of the size of axial displacement of one end of the bar. Based on energy equations, a formula was derived for the value of axial displacement of one bar end or buckling amplitude in the middle of bar length as a function of compressive force. The established relationships were confirmed by simulation tests using the finite element software **midas NFX** and by experimental tests.

**Keywords: **

buckling, Euler buckling, straight bar, bent bar, critical force View publication

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首次发布时间：2024-09-28

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