**1. Introduction**

The loss in stability of elements designed for engineering structures remains of great importance and topicality for various applications in civil engineering (stadium roofs, bridges), shipbuilding, and aerospace construction (aircraft). These items are usually made of steel or alloys with high strength characteristics. Depending on the type of the element (slender column, beam, plate, or shell) and on the type of loading, there are different analytical models and numerical models used for the analysis of their stability.

The first problems regarding elastic instability were approached and solved by L. Euler [1,2] in the middle of the 18th century, over 200 years ago. Nowadays, the main problems regarding the theory of elastic stability for different types of elements (columns, beams, frames, rings, curved bars, arches, thin plates, and thin shells), were synthesized by S. P. Timoshenko and J. M. Gere in a reference book in the scientific literature [2]. The loss in stability of structural elements takes place under the action of compressive loads [3,4] or bending (lateral buckling of beams) [2,5,6]. The optimized designs of such elements involve

**Citation:** Botis, M.F.; Cerbu, C. Design Solutions for Slender Bars with Variable Cross-Sections to Increase the Critical Buckling Force. *Materials* **2022**, *15*, 6094. https:// doi.org/10.3390/ma15176094

Academic Editor: Krzysztof Schabowicz

Received: 15 July 2022 Accepted: 30 August 2022 Published: 2 September 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

modifying geometry in order to increase the critical buckling force and ensure a low weight. In the case of steel structures, just decreasing the volume can lead to weight reduction. In the context of the research approached in this article, the control of the variation in the cross-section along the axis of a slender bar subjected to compression is of great importance for increasing the critical buckling force and, last but not the least, for reducing the weight. The problem of designing the shapes of bars concerning the cross-section variation also becomes stringent for bars subjected to compression in tensegrity structures [7] used for roofs in modern civil buildings.

The questions are as follows: which geometric parameter (dimensions, area, or the second moment of inertia) of a cross-section should be considered, and which mathematical function models the continuous variation in the section along the axis of a bar to significantly increase the critical buckling force while keeping the length of the bar and boundary conditions unchanged? What is the best design solution for the shape of a bar: stepwise or continuous variation in the cross-section along the bar axis? Another issue raised is how much the ratio between the critical buckling force and volume of a bar is affected considering the variation in the cross-section along the axis of the bar.

In the last years, many researchers around the world have investigated the stability loss in columns having different boundary conditions and non-uniform cross-sections under the action of axially distributed force [3,4,8]. S. P. Timoshenko and J. M. Gere were among the first researchers who presented a theoretical approach of buckling for bars with changes in the cross-sections without considering an axial distribution of compressive force in the second edition of their book [2] (the first edition of this book was published in 1961).

Eisenberger M. [3] found an exact solution for the buckling loads of columns with polynomial variation in the bending stiffness of the cross-sections under an axial load with a polynomial distribution along the bar axis by considering the determinant of the stiffness matrix to be equal to zero at the stability loss. Considering the Euler–Bernoulli beam theory to model a column with a variable moment of inertia *I*(*x*) (linear or parabolic variation) in the rectangular cross-section along its axis, which is subjected to distributed axial force, Darbandi et al. [8] computed buckling loads, taking into account the Wentzel–Kramers– Brillouin method of singular perturbation. Just the buckling loads and corresponding mode shapes for the rectangular variable cross-sections of columns were reported in that research, and those results were compared with the results given by Eisenberger [3].

Coskun and Atay [9] used a variational integration method to compute the normalized critical buckling load for Euler's columns with variable cross-sections with different boundary conditions, considering that the flexural stiffness varied by exponential function or by power function (linear, quadratic, and cubic variation). Their results matched very well with the exact solutions, but the paper did not customize the solutions for different shapes of cross-sections (for example, rectangular, circular, and annular cross-sections). Regarding computation methods, Ma et al. [10] computed critical buckling force considering various higher-order shear deformation beam theories based on Engesser's hypothesis and Haringx's hypothesis in comparison with Euler's theory in order to show the effects of warping shape. That research showed that buckling loads were not influenced by the theory used in the case of very slender columns or in the case of those having high shear rigidity.

Taking into account the achievements of the research mentioned above, we focus on finding an analysis method that allows the accurate calculation of the critical buckling force both for a bar with a stepwise variable cross-section and for a bar with a continuous variable cross-section. The analysis model must be flexible in terms of easy adaptation for any function used for variation in the second moment of inertia of the cross-section along the axis. This is the reason why the research focuses on the use of a finite element method for the numerical analysis of a slender bar subjected to compression in order to compute the critical buckling force. However, only a program made with specific software for calculation (such as MATLAB) can ensure the necessary flexibility, not a commercial one used for finite element analysis. Another problem arises concerning the validation method. In this context, it is necessary to present the state-of-the-art methods in the literature regarding the numerical methods used to analyze the stability of columns subjected to compression.

Using a finite difference method, Soltani and Sistani [11] also investigated a stability analysis for columns having variable flexural stiffnesses subjected to variable axial force. In that research, a finite difference method was applied in the case of column having a rectangular cross-section or an I-shaped cross-section whose dimensions were variable along the axis of the column in order to compute the critical buckling load. On the other hand, in nanomechanics, there are specific methods to analyze the buckling of nanobeams resting on elastic substrate media [12].

Saraçao ˘glu and Uzun [13] showed critical buckling loads obtained with Ansys 19.0 software for certain columns having square or circular cross-sections that were variable along the axis of the column (linear variation combined with a portion having a constant crosssection). Szmidla et al. [14] investigated and showed results concerning a stability analysis for steel columns consisting of portions having inhomogeneous cross-sections (composed cross-sections). Regarding columns with inhomogeneous cross-sections, Li et al. [15] made a parametric optimization of composite columns against buckling.

For columns consisting of certain portions having constant cross-sections, Maalawi [16] approached the optimization of buckling calculation in order to obtain design variables (area of the cross-section, length of each portion, radius of gyration) for the maximization of the critical buckling load and for certain input data (number of portions, boundary conditions, cross-section type).

A localized differential quadrature method was used by Yilmaz et al. [17] in order to compute non-dimensional critical buckling loads for non-uniform columns with continuous elastic restraint and different boundary conditions.

Using a discretized Hencky bar-chain model and a parallel genetic algorithm, Ruocco et al. [4] provided in 2017 an optimization method against buckling for columns with non-uniform variation in the cross-section subjected to both distributed and concentrated compressive forces. In this case, the geometrical variation in the cross-section was not given as input data. The paper [4] reported the optimal variation in the circular cross-section in non-dimensional coordinates and its corresponding non-dimensional normalized buckling load by keeping the same length and volume of the column with a uniform cross-section.

In the literature [18], optimization methods were also presented for columns having thin-walled, open cross-sections in order to obtain the maximum critical buckling by considering the constraint that the volume of the column remained constant.

As mentioned before, a retrospective analysis of the literature highlights the fact that there is a lack of research on design methods and solutions to increase the critical buckling force of slender bars with annular cross-sections subjected to compression by continuous variation in the dimensions of the cross-sections along the axis of the bars. Furthermore, no comparative studies have been reported regarding the critical buckling force for slender bars with continuous annular cross-sections compared to bars with stepwise variable cross-sections.

The present research aims to make some contributions to the scientific literature regarding design solutions for slender bars with variable cross-sections along the bar axis. The research provides a numerical model with finite elements validated by an analytical model, which is used to compute the critical buckling forces both for bars with stepwise variation in the cross-section and for bars with continuous variation in the cross-section along the bar axis. The MATLAB program used in this research based on the finite element model presented allows easy adaptation for any variation function considered for the moment of inertia. Considering variable annular cross-sections, the MATLAB program calculates additionally the inner and outer diameters that define the geometry of the bar. Moreover, a comparison is shown of the critical buckling forces for different particular cases considered for variation in the cross-section. In this way, the comparative analysis leads to finding some functions for variation in the second moment of inertia of the cross-sections, which ensure a major increase in the critical buckling force.

In this context, the main purpose of the research is to present a numerical model with finite elements used for a comparative analysis of increasing the critical buckling force for two types of slender bars subjected to compression: (i) bars with stepwise variation in the cross-section and (ii) bars having annular cross-sections with continuous variation in the moment of inertia along the bar axis. The results are also compared with the values of the critical buckling forces corresponding to bars having constant cross-sections along the bar axis. The case of bars that are pin-connected at one end and simple-supported at the other end is taken into account, considering the wide range of applications of these kinds of bars in engineering construction (stadium roofs, bridges, aircraft, and so on).

The main objectives of the research are: (i) the generalization of an analytical model for the calculation of the critical buckling force for a slender bar with stepwise variation in the cross-section, as well as a bar consisting of three portions; (ii) a numerical analysis using the MATLAB program, which uses modeling with finite elements for a slender bar with a stepwise variable cross-section along the axis of the bar in order to obtain the critical buckling force; (iii) the validation of the finite element model (FEM), with the results obtained with an analytical model for a bar with stepwise variation in the crosssection; (iv) an adaptation of the MATLAB program for a numerical analysis of a bar with a continuous variable cross-section; and (v) a comparison of the results regarding the critical buckling force for slender bars with stepwise variation and with continuous variation in the cross-sections. It is considered that the dimensions of a cross-section vary along the bar axis as the second moment of inertia *I*(*x*) varies from *I* to 4*I*. For a slender bar with continuous variation in the cross-section, the following types of variation are considered for the second moment of inertia *I*(*x*) along the bar axis: parabolic, sinusoidal, triangular, and trapezoidal. The numerical simulation with FEM is made using a computer program made with MATLAB R2014a software.

All the results are interpreted in terms of the normalized critical buckling force with respect to the critical force for stability loss corresponding to the bars having constant cross-sections along the axes of the bars. It is assumed that all the bars involved in this study lose their stability in the elastic field.

Finally, this research reports accurate results on increasing the normalized critical buckling force for slender bars with stepwise or continuous variations in the cross-sections involved in this study. In this way, this research shows design solutions regarding the mathematical functions for variation in the second moment of inertia of a cross-section along the bar axis in order to increase the critical buckling force for slender bars subjected to compressive loads.
