The shape of the optimal rod determined in the work meets the condition of mass conservation in relation to the reference rod. At the same time, this rod shows a significant increase in resistance to axial force. In the examples presented, this increase was 80% and 117%, respectively, for rods with slenderness of 125 and 175. A practical benefit from the use of compression rods of the proposed shapes is clearly visible. The example presented in this publication shows how great the utility in the structural mechanics can be, resulting from the applications of complex analysis (complex numbers). This approach to many problems can find its solutions, while they are lacking in the real numbers domains. What is more, although these are operations on complex numbers, these solutions have often their real representations, as the numerical example shows. There are too few applications of complex numbers in the technique and science, therefore it is obvious that the use of complex analysis should have an increasing range. One of the first people to use complex numbers was Girolamo Cardano. Cardano, using complex numbers, was solving cubic equations, unsolvable to his times – as the famous Franciscan and professor of mathematics Luca Pacioli put it in his paper Summa de arithmetica, geometria, proportioni et proportionalita (1494). It is worth mentioning that history has given Cardano priority in the use of complex numbers, but most probably they were discovered by another professor of mathematics – Scipione del Ferro (cf. [1]). We can see, that already then, they were definitely important (complex numbers).

The paper deals with a shape optimisation procedure of steel, compressed bars. Circular hollow sections (CHS) of variable cross sections and variable wall thickness are taken into account. The proposed procedure for designing of steel rods exhibiting maximum compression resistance is effective and possible to use in engineering practice. The advantage of the proposed shape of the bar is that it allows to increase the value of its load carrying capacity, i.e. it ensures the transfer of a higher value of compressive force than similar, solid struts of the same mass and length. The extent of the increase in the load capacity relative to the load capacity of the reference solid, cylindrical bar depends on the slenderness of the reference bar and ranges from 60% to 170%. Due to this very beneficial fact, it can be used wherever it is required to maintain a certain stiffness and an increased value of compressive force is desired, as well as in constructions where it is necessary to reduce weight while maintaining the adopted mechanical parameters, e.g. values of load bearing capacity. Final results achieved in the research were presented in the form of the flow chart allowing to design the compressed columns of optimum shape.

Brilliant in its simplicity, Southwell’s method of experimentally determining the critical force on the basis of the so-called Southwell diagram has been of interest to structural stability specialists since its presentation in 1932. In its original version, the correctness of the Southwell method was proved for compressed bars with a constant cross section and showing the loss of stability only within the elastic range while maintaining the constant value of Young's modulus. In numerous works on the method, proposals for its modification have emerged ensuring the extension of the scope of applications with problems other than the classic buckling of a compressed bar. There has also been an attempt to justify the correctness of Southwell’s method in the case of a bar with a variable cross section. This paper presents a strict solution of the differential equation of bar stability with a parabolic variable moment of inertia of the cross section and a parabolic form of initial geometric imperfections. The solution arrived at with the use of the Mathematica™ system was the basis for constructing the Southwell diagram and justifying the correctness of the method for this particular case of a bar. To confirm the correctness of Southwell’s method also the numerical approach exploiting FEM was adopted. Two different cases of column’s shape variabilities and different boundary conditions were considered.

Experimental determination of the critical force of a compression bar made of linear material does not pose major problems. The widely used Southwell method is also applicable to bars with cross-sections varying in length. The purpose of the research undertaken was to demonstrate that this method can also be successfully applied to bars made of elastic materials exhibiting nonlinear σ(ε) characteristics. Using analytical relationships for a rod satisfying the assumptions of Euler-Bernoulli theory, F(δ) relationships were derived for a rod with an initial arc imperfection, and Southwell diagrams were constructed on this basis. Nonlinear equilibrium paths F(δ) were also determined numerically using the COSMOS/M program for this purpose. For the pre-assumed different nonlinear characteristics, full confirmation of the validity of the application of the Southwell method for determining the critical force was obtained.

The paper presents an analytical framework for the classification of buckling problems of compressed rods with variable bending stiffness. The governing Euler–Bernoulli stability equation with spatially varying coeffi-cients is transformed into a Sturm–Liouville form and further reduced to a Schrödinger-type equation using the Liouville transformation. This formulation establishes a direct correspondence between the bending stiffness distribution and the associated spectral problem, allowing a systematic mapping of stiffness profiles to classes of differential equations. Depending on the resulting Liouville potential, the eigenvalue problems can be ex-pressed in terms of classical special functions such as Airy, Bessel, Hermite, or error-function-based solutions. The proposed approach provides a unified classification of variable-stiffness buckling problems within a single operator framework and organizes known analytical results as special cases of a general spectral formulation. The study focuses on analytical structure and classification rather than on optimization procedures, and it does not assume or solve a shape optimization problem.

The article presents a stochastic approach to the shape optimization of compressed rods, taking into account random deviations of Young’s modulus and the second moment of area. The parameters are described as stationary random fields with a specified correlation length, and their influence on the critical buckling load is determined using a first-order perturbation expansion. The obtained expressions for the expected value and variance of the critical load made it possible to assess the sensitivity of the structure to local stiffness disturbances. Subsequently, a probabilistic optimization problem was formulated within a family of Gaussian-type profiles, with the aim of minimizing mass while maintaining the required level of reliability. Numerical analysis shows that accounting for uncertainties makes it possible to obtain a rod approximately 12 % lighter than the reference rod, without reducing load-carrying capacity. The presented approach demonstrates that modeling