Closed-form solution for optimization of buckling column

Abstract The optimization problem for a column, loaded by compression forces is studied in this article. The direction of the applied forces coincides until buckling moment with the axis of the column. The critical values of buckling are equal among all competitive designs of the columns. The dimensional analysis eases the mathematical technique for the optimization problem. The dimensional analysis introduces two dimensionless factors, one for the total material volume and one for the total stiffness of the columns. With the method of dimensional analysis, the solution of the nonlinear algebraic equations for the Lagrange multiplier will be superfluous. The closed-form solutions for Sturm-Liouville and mixed types boundary conditions are derived. The solutions are expressed in terms of the higher transcendental functions. The principal results are the closed-form solution in terms of the hypergeometric and elliptic functions, the analysis of single- and bimodal regimes, and the exact bounds for the masses of the optimal columns. The isoperimetric inequality was formulated as the strict inequality sign, because the optimal solution could not be attained for any finite seting of the design parameter. The additional restriction on the minimal area of the cross-section regularizes the optimization problem and leads to the definite attainable shape of the optimal column.


Optimization for stability problem of a compressed rod
The stability of columns is the traditional problem of structural optimization. The history of this subject begins with the manuscript (Lagrange 1868). This volume reflects the original scripts of Lagrange, dated by the years 1770-1773. The optimal shape of a compressed column was derived by Clausen (1853). The formulation of the optimization problem without the additional restrictions to the cross-sections of the column is commonly referred to as the Lagrange problem. Nikolai (1955) introduced an additional constraint concerning the magnitude of permissible stress, preventing the vanishing of the thickness of this optimum column and unbounded increase of the compressive stress. The formulation of the optimization problem with the additional restrictions to the cross-sections of the column is referred to as the Nikolai problem. Since these prominent deliberations, the subject of the stability optimization of columns was reflected in a number of motivating works.
The design against instability is the problem of finding the shape of a compressed column that has minimum weight and can withstand a given load, without the loss of stability was studied with the application of analytical and numerical methods. Keller (1960) found the strongest hinged (simply supported) column with the maximum buckling load, which was 33.33% higher than that of the uniform column. (Tadjbakhsh and Keller 1962) extended this work to cover other end conditions, essentially both clamped ends. The analytical solution using the symmetry considerations for the buckling fundamental functions. The cross-sectional area reached zero values at some locations along the column's length. As it was demonstrated in the subsequent studies, the optimal solution of Tadjbakhsh and Keller (1962) remains correct in the cases of the Sturm type boundary conditions, but needs a revision for the mixed-type boundary conditions. Taylor (1967) presented a more direct and concise energy method than that developed by Tadjbakhsh and Keller. His approach was also restricted to the same quadratic relation, and applied to a simply supported column. The resulting suboptimal buckling load, however, was less than that found by Keller by about 5.5%. Prager and Taylor (1968) treated a variety of problems of optimal design of sandwich structures where a linear relationship between the second moment and area of cross-section was assumed. A simply supported column was optimized resulting in a parabolic wall thickness distribution with vanishing magnitudes at the ends. Afterward, Taylor and Liu (1968) applied Valentine (1937) mathematical procedure for variational problems to establish optimum shapes of sandwich cantilevered columns under an inequality constraint on the cross-sectional area. Their maximum buckling load reached a value 21.6% higher than that of the uniform cantilever. Strongest column was also addressed by Simitses et al. (1973) where a finite element displacement formulation was applied to elastically restrained columns subjected to a varying axial load. The attained optimization gain, under the same specified length and volume, for a cantilever divided into 20 equally spaced elements was about 32.5% with the penalty of reaching infinite compressive stress at the free end. Masur (1975) treated other types of columns built of covering plates, with the design variables taken to be only the locations of the plates along the column. The complementary energy formulation was applied to an iterative, sequential design procedure. This numerical method leads to the optimal solutions for the cases pinned-pinned and clamped-free boundary conditions. It was demonstrated that the optimal buckling design could be multi-modal in which the final optimum solution can have a double (bimodal) or triple (trimodal) eigenvalue with distinct eigenfunctions. Olhoff and Rasmussen (1977) considered both single and bimodal buckling optimization of clamped columns with geometrically similar cross sections. The authors realized that the first eigenvalue does not vary smoothly with design parameters at the points where its multiplicity exceeds one. Their resulting suboptimal odd-shaped columns with a complicated non-linear area distribution violate fabrication and production feasibility. Since the solution contains a point of zero area (moment of inertia), that design must contain a hinge and possibly cannot present the optimal design. There have followed a sequence of alternative designs which have come closer and closer in both shape and buckling load to the solution (Tadjbakhsh and Keller 1962). Haug and Rousselet (1981) noticed that simple eigenvalues are Gateaux differentiable with respect to design but repeated eigenvalues can only be expected to be directionally differentiable in general. The design sensitivity analysis of eigenvalue variations and explicit directional derivatives of repeated eigenvalues were present in Zolesio (1981). Kirmser and Hu (1983) discussed of the shape of the strongest fixed-fixed column of given weight, suggested that the exact buckling load "could be quite close to that found by Tadjbakhsh and Keller". Masur (1984) examined a problem of two-degree-of-freedom mass spring system which results in constraint surfaces which are not smooth. Myers and Spillers (1986) opposed that the subsequent solutions generated in response to the shape for the optimal fixed-fixed column according to Tadjbakhsh and Keller (1962) are smooth approximations of that singular shape. The designs of the sequence, which was obtained by Olhoff and Rasmussen (1977), are actually smooth approximations asymptotically approaching the Tadjbakhsh and Keller solution, which contain a singularity. Cox and Overton (1992) stated that the directional derivatives of repeated eigenvalues are not linear functional in direction arguments and rigorously justified by the method of explicitly calculating the generalized gradient of Clarke for some related functionals. The authors establish existence, derive necessary conditions, infer regularity, and construct and test an algorithm for the maximization of a column's Euler buckling load under a variety of boundary conditions over a general class of admissible designs. It is proven that symmetric clamped-clamped columns possess a positive first fundamental function and a symmetric rearrangement is introduced that does not decrease the column's buckling load. The works Rozvany 1975, 1977) and  expose the influence of the support conditions on the shape of the optimal column.
The numerical methods were applied for the effective visualization of the optimal solutions. Solid circular cantilevers subjected to tip axial force and own weight were treated by Hornbuckle and Boykin (1978). The optimization problem was handled via maximum principle due to Pontryagin. The relative mass reduction between the optimal and the reference designs was considred as the optimization gain. The numerically attained optimization gain was about 11.5%. Turner and Plaut (1980) considered clamped-clamped columns using an iterative procedure based on the optimality criterion accomplished by the finite element method. The column was divided into the uniform elements with equal lengths, and the resulting optimization gain was 27.6%. The optimization problem of composite reinforced beams subjected to bending and compression was studied numerically by Banichuk and Larichev (1981). Stability optimization of elastic columns against simultaneous compression and torsion was explored with numerical optimization methods by Banichuk and Barsuk (1982). Plaut et al. (1986) determined optimal designs for sandwich columns attached to elastic foundations. The application of the optimal control theory to buckling optimization was given by Goh et al. (1991). A simply supported column constructed from five piecewise constant segments was optimized with the design variables taken to be only the length and area of each segment. The optimization problem formulation contained many mathematical formulae to calculate derivatives of both the objective and constraint functions, and the obtained final suboptimal solutions were restricted to the same deficiency of assuming geometrically similar cross sections. Ishida and Sugiyama (1995) proposed an optimization algorithm, referred to as the constructive algorithm, applied to a finite element model. Numerical solutions were restricted to clamped-free and clamped-pinned columns having circular solid cross sections. The maximum buckling load of a 16-equally-spaced element model was calculated to be 31% higher than that of a uniform model. Manickarajah et al. (2000) also used the finite element method in conjunction with an iterative procedure for optimizing columns and plane frames against buckling. A local modification of each element was assessed by gradually shifting the material from the strongest part of the structure to the weakest one while keeping the structural weight constant. Maalawi (2002) presented the model, which considers columns that can be practically made of uniform segments with the true design variables defined to be the cross-sectional area, radius of gyration and length of each segment. The presented brief survey of the acknowledged results accounts several numerical approaches. The direct application of the established numerical methods allows to get the rapid image of the optimal shapes. However, none of the previous approaches explains the mathematical procedure from the viewpoint of the differential equations and leads to a closed-form solution.
The comprehensive analytical approaches were applied as well (Elishakoff 2000). Considerable progress in the case of multiple eigenvalues was arrived in Seiranyan (1984) and Olhoff and Seyranian (2008). The analytical solution of the problem of the optimal design of columns with the elastic clamping was given in Banichuk and Barsuk (1995). Optimum shapes were found for columns with various types of cross-sections. An asymptotic analysis was performed for the optimum solution in the cases of small and large values of the elastic clamping coefficient. The overview of the works on the optimization of the multiple eigenvalues was performed in Lewis and Overton (1996). In this article, the applications of multiple eigenvalue problems to structural analysis and combinatorial optimization are surveyed. In the series of works Egorov and Kondrat'ev (1996a, 1996b, 1996c performed the estimates for the eigenvalues and discussed the solution of the bimodal problem for the columns. The article (Seyranian and Privalova 2003) overviewed the state of the art for buckling columns optimization and contains the comprehensive list of references. Bimodal optimization was treated later by the use of maximum principle due to Pontryagin in Atanackovic and Seyranian (2008).
Nevertheless, the solution of the optimization problem for a rod, clamped at both ends, to the author's awareness, was not expressed in the form of the closed analytical expressions in terms of higher functions. Accordingly, the question of the existence of the optimal distribution of thickness among all admissible positive functions, which satisfy the isoperimetric condition, actually remains open for the bimodal solution. The most advanced solutions in the optimization problem with multiple eigenvalues contain the unsolved nonlinear equations for the Lagrangian multipliers. These equations profound definite integrals, which expected to be expressed in terms of higher transcendental functions. Namely, the Lagrangian multipliers should result as the solution of the nonlinear system of simultaneous equations which contain definite integrals of an elliptic type. This task remains apparently neither resolved in the closed form nor the existence of the solution was stated. Only numerial solutions of these nonlinear systems were studied.
The problem of establishing the existence of optimum control functions is an essential part of the theory of optimum processes. Moreover, this problem plays a central role in the solution of applied optimization problems. As has been pointed out by Young (1937), the theory of necessary conditions for optimality remains in a ingenuous form, if it is ambiguous whether there exists a solution of a variational problem within a specified class of admissible functions. If this problem remains open, numerical algorithms based on necessary conditions for optimality do not lead to the construction of an optimizing sequence of admissible elements, even if the necessary conditions select a single element which is a contender for optimality. Furthermore, efforts to presume the existence of solutions to optimization problems on the basis of physical arguments are not always reasonable. The mathematical optimization problem is a model of the concrete process, and we cannot be convinced that the modelwhich has been constructed even sufficiently adequatereplicates the physical state. Even if it is clear from physical thoughts that an optimal solution exists, this does not suggest that the existence theorem in the corresponding mathematical problem is valid. As noted in Athans and Falb (1966), the existence of a solution in the mathematical optimization problem is the first criterion for the model which has been created to give an adequate description of the physical process. Another mathematically valuable task is the declaration about the uniqueness of the solution.
In this manuscript, we present the closed-form solution of the optimization problem in Lagrange sense, which contains one auxiliary parameter. This parameter could be arbitrary assigned. For each given positive value of this parameter follows the isoperimetric inequality between the length, critical buckling load and the material volume of the rod. The proof of isoperimetric inequality is based on the standard optimality conditions for multiple eigenvalues, as deliberated in the above cited papers. Consequently, the existence of the optimal solution is stated for each positive value of the auxiliary parameter. The mass of the established optimal design is the monotonically decreasing function of the auxiliary parameter. However, there is no smallest positive real number. Accordingly, there is no attainable solution of the optimization problem for all admissible positive control functions, which represent the cross-sectional area of the buckling rod. The optimal cross-sections asymptotically transform with the vanishing auxiliary parameter to the certain limit shape.
Notably, that the application of an additional constraint on the smallest cross-section area over the axis of the column regularizes the optimization problem. This regularization leads to the attainable solution for the maximal buckling load in Nikolai sense.
Consequently, the tasks of this paper are to establish the method for the closed-form solution of the Emden-Fowler equation and the one-dimensional optimization equations; the dimensionless factors which avoids the Lagrange multipliers and leads to direct relations; the verification of the isoperimetric inequalities between the dimensionless factors and to find the answers of the existence and uniqueness of the optimal solution in Lagrange and Nikolai senses.

Differential equations of stability
Consider the problem of optimizing the stability of a rod. The total length of the single-length column is L ¼ 2l: We use the description "single-length column" to underline the difference to the "double-length column", which appears in the case of both clamped ends later. The function A ¼ AðxÞ designates the a-priori unknown cross-sectional area of the rod along the span Àl x l: We use the gothic letter A to differ the cross-sectional area from the exponent a: The principal moments of the cross-second order for the cross-section are I 1 , I 2 : If the principal moments of inertia I 1 , I 2 are different, the buckling occurs in the plane with the minimal moment of inertia of the cross-section. Accordingly, for the maximal buckling load both principal moments must of inertia be equal, I 1 ¼ I 2 ¼ J: Only this case will be studied in this manuscript. The second moment of area is given by the relation: where a is the positive integer. Thus, the quantity Ek a A a is the flexural stiffness EJ: The bending stiffness EJ x ð Þ is two times continuously differentiable function of the on the interval Àl x l: The exponent a takes the values of 1, 2, and 3 (Banichuk, 1990). The case a ¼ 2 corresponds to a similar variation of the form of the cross-section. For the simply connected cross-section with topological genus null, the optimal convex shape of the was determined (Ting 1963). Of all convex domains with the area A , the equilateral triangle yields the maximum of the product: ffiffiffiffiffiffiffi I 1 I 2 p ffiffi ffi 3 p

18
A 2 ¼ I~: Thus, the rod with the cross-section in form of an equilateral triangle delivers the maximum for minimal second moment of inertia for all convex domains of the same cross-sectional area A~, because: min I 1 , I 2 ð Þ ffiffiffiffiffiffiffi I 1 I 2 p max I 1 , I 2 ð Þ: For the cross-section in form of the equilateral triangle k 2 ¼ ffiffi ffi 3 p =18 % 0:9622: For the circular cross-section, the constant is k 2 ¼ 4p ð Þ À1 % 0:07957::: The rod with the circular, simply connected cross-section delivers correspondingly the minimum for critical eigenvalue: ffiffiffiffiffiffiffi Two other cases describe the situations in which the form of transverse cross-section undergoes the transformation one of the geometrical dimensions of the cross-section alternates proportionally to a design parameter. The technically important case is the thin-walled tubes with the variable thickness of wall tðxÞ and the mean diameter of tube DðxÞ: The second moments of is the thin-walled tubes is: The case a ¼ 1 corresponds the changeable wall thickness and constant mean diameter of the tube. The design variable is the thickness of the material.
The case a ¼ 3 corresponds the variable mean diameter of the tube and constant wall thickness. The design variable is the mean diameter of the tube.
The rod is placed horizontally (along x axis) and is compressed by forces F in x direction applied to its ends. If a rod is subjected to a gradually increasing load, when the load reaches a critical level, the rod may suddenly change shape and the component is said to have buckled. The sudden change in shape under load occures, which look like as the bowing of the initially straight axis of the rod. The origin of the coordinate system coincides with middle point of the rod, as shown on Figures 1 and 2. The x axis passes through the point of the rod's support. The deflection of the bent axis of the rod from the line of action of compressing forces is yðxÞ: The function yðxÞ is four times continuously differentiable on Àl x l: The bending moment reads: The bending moment m is two times continuously differentiable function on Àl x l: The stability of a compressed elastic rod with the certain end conditions is considered: This equilibrium equation is of the fourth order. It can be reduced to the second order, if instead of deflection y the bending moment m is used as the unknown variable. In terms of the bending moment the equilibrium equation for the bending of a compressed rod may be written For derivation of the boundary values in terms of bending moment the twice integration of the equilibrium equation is usually performed (Kamke 1939). Taking into account the assigned boundary conditions on displacements, the boundary conditions are formulated in terms of the bending moment only (Egorov and Kondrat'ev 1996c). The proof of isoperimetric inequalities exploits the variational method and the H€ older inequality. The isoperimetric inequalities for Euler's column were rigorously verified in (Kobelev 2016).

Boundary conditions
The following boundary conditions are usually formulated (Kamke 1939;Timoshenko and Gere 1961). The boundary conditions are demonstrated on Figures 1 and 2. The boundary conditions on Figure 1 are valid for the single-length rod. The rod, which is clamped on both ends, requires more attention. This rod is shown in its original double length on Figure 2. I. The length of the rod is L 2l: The boundary conditions for the rod which is clamped at x ¼ Àl and free at x ¼ l are the following In terms of moments the boundary conditions (I) are II. The length of the rod is L 2l: If the rod is clamped x ¼ Àl and hinged at x ¼ l, the boundary conditions assume the form yj x¼Àl ¼ 0, y 0 j x¼Àl ¼ 0, EJy 00 j x¼l ¼ 0, yj x¼l ¼ 0: In terms of moments the boundary conditions (II) are III. The length of the rod is L 2l: The rod is hinged at x ¼ l and x ¼ Àl, thus the boundary conditions are: In terms of moments the boundary conditions (III) read as: IV. The length of the rod is L 2l: The end x ¼ l is clamped and its transversal movement is restricted. The other end x ¼ Àl can freely move in the direction, orthogonal to the axis of the rod. However, the direction of this end remains parallel to the axis of undeformed rod in course of deformation: In terms of moments the boundary conditions (IV) are V. The length of the rod is 2L 4l: If the ends x ¼ ÀL and x ¼ L are clamped and their movements transversal to the axis are restricted, the boundary conditions are VI. The length of the rod is L=2 l: This auxiliary rod plays role of symmetry element for the establishing of the optimal solution for the cases IV and V. Instead of boundary conditions, the symmetry conditions for the admissible buckling forms will be applied. One buckling form is mirror-symmetric with respect to the plane, normal to the axis of the rod in the top point x ¼ 0: The second buckling form is point-symmetic respectively to the top point x ¼ 0: The different number of waves of buckling functions for diverse boundary conditions causes the challenging symbolization. The rod, which is clamped on both ends (case V), is shown in its original double length on Figure 2. Figure 2 displays also the clamped-transversally free rod of the single-length (case IV). Due to symmetry, both rods possess the same buckling force. The case VI is similar to the case V, but the rod IV is halved in the middle. In the bisection point, the section is clamped, but can freely displace to the transversal direction. As stated above, the critical buckling forces in both cases IV and V are equal due to the mirror symmetry of the optimal solution for the case IV with respect to the point x ¼ 0: In its turn, the eigenfunctions of the solution VI are point-symmetric with respect to its middle section x ¼ L=2: The distribution of the thickness for the optimal rod will be mirror symmetric to this section, as we demonstrate below. The half of the rod IV is reffered to as the rod VI. This case plays the auxiliary role as the symmetry element for the establishing of the optimal solution. Tadjbakhsh and Keller (1962) assumed that the boundary conditions correspond to the case I, such that the shorter rod is clamped at x ¼ l and free at x ¼ 0: If this assumption should be valid, the thickness of the rod degenerates at the top point. Consequently, the optimal rods VI and V should have correspondingly one and two singularities, as shown on Figure 2. This assumption contradicts the restriction of the positiveness of the thickness. The family of the optimal solutions with the positive thickness will be presented below.
Figures 1 and 2 exhibit the common boundary conditions for the above-mentioned cases. Figure 2 makes evident the application of the symmetry considerations for the reduction of the solution procedure. In this manuscript, we study the optimal solution for the symmetry element the rod VI, namely for its half 0 x l (case VI).
In this manuscript, the variational method for establishing the isoperimetric inequality will be applied. We study the cases I to IV and an auxiliary case VI. The mathematical treatment of case V follows from the solution of the cases IV and V and will not be pr ecised below. We use for brevity for integrals of functions u x ð Þ, À l x l the notation: The variational principle for the stability of a rod with a given A can be written as (Reddy, 2002): In the dimensionless form of the variational principle (4), the Rayleigh's quotient is: The admissible buckling moment for the Rayleigh's quotient is m ¼ mðxÞ: The admissible functions mðxÞ are all functions, having piecewise continuous first derivatives, satisfying definite boundary conditions.
The actual buckling moment is m ¼ mðxÞ: The actual buckling moment m delivers the minimal value for the Rayleigh's quotient. Euler's equation for variational problem (4) is (1).

Dimensionless factors for efficiency evaluation of optimization
Consider hereafter columns having a certain volume of material: The volume V remains equal for all competitive designs of the column. For the boundary value conditions I to IV, there are different values of masses of the optimal rods. The comparison of the results is not straightforward. The influence of the exponent A influences the estimations for the optimization effects. Another argument for the introduction of the invariant optimization factors lay on the line for the clarification of the analytical method. In the variational calculus is common to get one factor as the optimization objective and the others as the a-priori given constraints. To convert it into an unconstrained problem, the method of Lagrange multipliers is commonly used. For the column the optimality expresses with the isoperimetric inequalities. The resulting unconstrained problem with Lagrange multiplies increases number of variables. The new number of unknown variables is the original number of variables plus the original number of constraints. The constraints are usually solved for some of the variables in terms of the others, and the former can be substituted out of the objective function, leaving an unconstrained problem in a smaller number of variables. This method of solution of leads to the nonlinear algebraic equations for Lagrange multiplies. Moreover, the nonlinear algebraic equations comprise unsolved definite integrals. Furthermore, the integrands of the definite integrals contain the Lagrange multipliers as parameters. The integrands have poles, such that for an arbitrary setting of Lagrange multipliers the integrals are improper in Riemann sense. Thus, the nonlinear equations in the most cases do not possess the closed analytical solutions and are solvable solitary numerically.
Instead of dealing with the Lagrange multipliers, we introduce the certain invariant factors. Consider the columns with the same form of cross-sections, fixing the exponent a: For each fixed value of a, we introduce two factors: These two dimensionless factors (F A , F B ) are the commensurable physical quantities are of the same kind and can be directly compared to each other, even if they are originally expressed in differing units of measure. We use the indices to distinguish the optimization of the total volume of the column V and the total stiffness of the column e with the same given single-length L: For some arbitrary powers p 1 , p 2 , p 3 , p 4 , the factors F A , F B alter for any affine transformation of the column. The affine transformation of the column is the product of two elementary transformations, namely homothety and scaling. The homothety of ratio f multiplies lengths by the factor. Thus, f is the ratio of magnification or dilation factor or scale factor or similitude ratio. The cross-section function A scales by another factor ., such that for the affine transformed column the cross-section function will be .A: Apparently, the eigenvalue K alters in course of the affine transformation of the column. If the column will be to twice long but the cross-section function and volume of the column remain constant, the eigenvalue will be 16 times smaller. If the length does not alter, but the volume of the column doubles, the eigenvalue will be four times high.
We use the factors F A , F B for the comparisons of different designs. The critical buckling load F ¼ k a EK inherits the factor k a and is proportional to this value. Evidently, that the ratios of the buckling loads for different designs with the same form of the cross-sections do not depend on the constants k a .
With the methods of dimensional analysis, we can immediately determine the distinctive choice of powers and To avoid the faults, we point out, that L denotes the single-length, the double length of the column is 2L, and l ¼ L=2 is the half-length, or the length of the symmetry element. Figure 2 demonstrates this situation. Because its length of rod V differs from the length of rods in other cases I to IV, the direct comparison is delicate and will be omitted. We try to compare different boundary conditions I, II, II, IV and VI. We analyze the possible types of cross-sections, particularly a ¼ 1, 2, 3: The values L, V , e are correspondingly the length, volume, and total stiffness of the singlelength column, as shown on Figure 1, case I to IV. Similarly, the values l, v, stay for length, volume, and total stiffness of the symmetry element (half-length column), which is displayed on Figure 2, case VI. We have L ¼ 2l, V ¼ 2v, e ¼ 2 for the rods, which are composed from the same symmetry element IV.
The invariants F A and F B are calculated for the single-length column (cases I to IV). The factors F A , F B do not alter for any affine transformation of the column. Thus, the factors F A , F B are invariant to the affine transformation of the column and provide a natural basis for the comparison of different designs. Moreover, the cases I to III were correctly studied by Tadjbakhsh and Keller (1962). Thus, the cases I and II will not be discussed for breifness below. We show only the new closed-form solution with an arbitrary parameter a only for the case III.
With the above factors, the estimation of the effect of mass optimization is straightforward. For this purpose, we consider the reference design with the constant cross-section along the span. The invariant factors for the reference design areF A ,F B : The eigenvalue of the rod is set to K ¼ 1: For all exponents a and for the boundary conditions III with both ends hinged both factors are:F For all exponents a and for the boundary conditions IV with both ends clamped both factors are equal:F The sense of both factors clarifies without difficulty. The greater the factors are, the higher the buckling force for the given length and volume of the column. For example, the buckling force of the reference clamped column is four times the buckling force of the reference column with the hinged ends.
The dual formulations are typical the optimization of buckling column as well. For the dual formulations, the masses of the columns for the fixed lengths and fixed buckling forces are compared. The volumes and masses of the optimal and reference columns relate to each other as the inverse roots of the order a of the factors F A : Specifically, the column with the higher value of the factor F A possesses the lower mass for the same length and buckling load.
Correspondingly, the ratio of the total stiffness e is the inverse ratio of factors We use systematically the method of dimensionless factors for the optimization analysis. The applied method for integration of the optimization criteria delivers different length and volumes of the optimal columns. Instead of the seeking for the columns of the fixed length and volume, we directly compare the columns with the different lengths and cross-sections using the invariant factors.

Optimality conditions for the Sturm type boundary conditions
The eigenvalue problems for the ordinary differential equation with the above given boundary conditions are self-adjoint. The conditions (1) and (2) are of the Sturm type; see Hazewinkel (2001) and Zettl (2005). There exists an infinite set of eigenvalues, all eigenvalues are real and positive and can be arranged as a monotonic sequence, and each eigenvalue is simple: K 1 < K 2 < :: < K 2kÀ1 < K 2k < :::: The column with the thickness distribution A, which satisfies the necessary optimality condition and the static equation with boundary conditions I, II, or III could be proved to be optimal for boundary conditions of the Sturm type. Here, c is the Lagrange multiplier of the variational calculus problem (8) with the isoperimetric condition (5).
The optimal solution for boundary conditions of the Sturm type (S) in Lagrange sense produces the optimum function: Accordingly, the optimal problem for boundary conditions of the Sturm type (S) in Nikolai sense applies an additional constraint: We proceed with the solution of the optimization problem in Lagrange sense (S). The applied dimensional analysis is based on the rescaling of the optimal shape and does not require the determination of the Lagrange multiplier. We set the value as c L ¼ 1: 5.2. Closed-form solution of the optimization problem for the Sturm type boundary conditions

Auxiliary solution of generalized Emden-Fowler equation
For the basic solution of the optimization problem, we use the auxiliary ordinary differential equation of the second order: Equation (15) is the generalization of the Emden-Fowler equation (Berkovic, 1997). The Lane-Emden equation was used to model the thermal behavior of a spherical cloud of gas within the framework of the classical thermodynamics. The Lane-Emden-Fowler type equations are used for description of a number of physical phenomena, originally phase transitions in critical thermodynamic systems of spherical geometry and Ginzburg-Landau theory of phase transitions.
We solve the Emden-Fowler equation (15) using the method of integration factors. The integration factor for Eq. (15) is: Multiplication of Eq. (15) by -1 leads to its first integral: dx dn S À K S P n P þ C 1 ¼ 0: The first integral delivers the solution of Eq. (15): For the values (17) evaluates in terms of the hypergeometric function 2 F 2 (Abramowitz and Stegun 1983): The values of the function and its first derivatives on the ends of the interval for 0 n 1 are: x dx dn n¼1 ¼ 0: The integral of the function x n ð Þ evaluates in closed form as well (Gradshteyn and Ryzhik 2014) With the notation the formulas (18) and (23) express as:

Solution of the optimization problem
In this section, the solution of the optimization problem for the admissible values of the parameter a is studied. The closed-form solution outcomes for an arbitrary permissible value of parameter a. For the application of the above results to the optimization of elastic elements, the dependent and independent variables in the equations (13), (14) are to be exchanged: In the new variables, Eq. (25) turns into the Emden-Fowler equation (15) with the following parameter: According to Eq. (11), the general solution of Eq. (25) for a > 1 is: The values and C are the integration constants of Eq. (25). To avoid the solution of the nonlinear transcendental equations for the integration constants we favor to exploit the symmetry thoughts. The sense of the constant is the moment in the middle of the column: Owing to the symmetry the equations with respect to the point x ¼ 0, the function mðxÞ must be an even function of the variable x: This condition requires for the integration constant: With this value, the integral (26) evaluates in the closed form with the hypergeometric function for a > 1 as (Gradshteyn and Ryzhik 2014): The solution (27) could be also expressed in terms of the Jacobi polynomials P b, c ð Þ a x ½ (Gradshteyn and Ryzhik 2014), Section 8.96: Thus, the axial coordinate is the explicit function of the new independent parameter m: According the boundary conditions III, the moment vanishes on the hinged ends. From these conditions, the length of the half-length rod VI determines from Eq. (28) as (Gradshteyn and Ryzhik 2014): The inversion of Eq. (28) delivers the moment m as the function of coordinate x: The easiest way is to calculate x over m and exchange the abscissae and ordinates in the resulting graphs. The functions of moments m over the coordinate x are shown for a ¼ 2 on Figure 1. The moment in the middle of the rod is given for each curve ( ¼ 2, 3, 4Þ: The critical value K is equal for all curves. As the result, the single-lengths L s ¼ 2l s of the rods are appear to be different. The curves for a ¼ 3 area shown on Figure 3. The higher transcendental functions (22) reduce in the cases a ¼ 1, 2, 3 to the elementary functions (Table 1).
In its turn, the area of cross-section of the optimal column results from Eq. (7) is the function of the new independent variable: The optimal cross-section will be designated with the capital letter A, leaving the small letter a for any admissible cross-section function. The corresponding distributions of cross-section areas along the span are plotted for a ¼ 2 on Figure 3 and for a ¼ 3 on Figure 4.
The moment of inertia for the optimal cross-section reads: The volume of the single-length column is V s ¼ 2v s . To find the volume of the half-length element VI, we evaluate the proper integral of the cross-section area of the column for all values of a, higher that one (Gradshteyn and Ryzhik 2014): We define one another constant that was referred above as a total stiffness e of the column. The total stiffness e of the single-column expresses as an integral of the moment of inertia of the cross-sections along the single-length of the column. The total stiffness expresses as e s ¼ 2 s , where s is the total stiffness of the symmetry element VI. To find the total stiffness of the half-length element VI, we evaluate another proper integral of the cross-section area A a s ð Þ for all values of a, higher that one (Gradshteyn and Ryzhik 2014): We demonstrated above the closed-form solution of the both hinged ends (case III). The other two solutions with the Sturm type boundary conditions (cases I and II) could be derived from the basic solution using traditional scaling methods, as already disclosed in Tadjbakhsh and Keller (1962).

Isoperimetric inequalities
For the optimal column with the hinged ends III, the invariant factors are: These ratios are shown on Figure 5. Together with the ratios of factors F A:S =F AðIIIÞ and F B:S =F BðIIIÞ , the limit value ffiffiffiffiffi 12 p p À1 is displayed (Table 2). Notable, that the setting a ¼ 0 leads to the rod of the constant cross-section. For this value, the factors F A:S =F AðIIIÞ and F B:S =F BðIIIÞ will be equal to 1.
With Eq. (26), the expressions for the volume and stiffness ratios reads for a > 1 : and The mass and stiffness ratios Eqs. (33), (34) are displayed on Figure 6. The optimality expresses in form of the isoperimetric inequalities between the invariant factors: The equality holds if and only if the column is an affine transformation of the column with the optimal shape.
The isoperimetric inequality was stated in the cited article of Tadjbakhsh and Keller: If the eigenvalues are simple, the optimal column has the largest buckling load among all the columns with the same volume of material: The equality in (35) sign holds only for the column with the cross-area distribution A ¼ A n ð Þ .

Optimality conditions for the mixed type boundary conditions
The eigenvalue problems with the boundary conditions of the mixed type were discussed in Hazewinkel (2001), Zettl (2005), and Agarwal and O'Regan (2008). For the columns with boundary conditions of the mixed type, the above method will be applied for solution of the optimization problem (M). The eigenvalue problem for the ordinary differential equation (13) with boundary conditions IV and V is self-adjoint. However, the boundary conditions IV are not of the Sturm type and the eigenvalues are not necessarily simple.
Specifically, the conditions IV are of mixed type. There exists an infinite set of eigenvalues, and all eigenvalues are real. The eigenvalue problem has a double zero eigenvalue with associated fundamental functions (see, Seiranyan, 1984 andKaraa, 2003). Only the positive eigenvalues of the buckling problem have physical meaning. The positive eigenvalues can be arranged as two sequences: K 1 < K 3 < :: < K 2kÀ1 < K 2kþ1 < ::: K 2 < K 4 < :: < K 2k < K 2kþ2 < ::: (36) such that for k ¼ 1, 2, 3::::: Thus, the positive eigenvalues of the boundary value problem are simple or double. Each admissible function A > 0 must satisfy the isoperimetric condition: The derivation of necessary optimality conditions in the case of multiple eigenvalues was performed in the known papers of Bratus' and Seiranian (1983), Seiranyan (1984), and Bratus' (1991).
In the cited papers was presumed, that there exist the functions U, W, and A such that: The constant c L in the necessary optimality condition (39) plays the role of Lagrange multiplier for the isoperimetric condition. We can arbitrarily set this constant, using the applied method of dimensional analysis. Thus, the constant c L will be set to 1.
The functions U, W satisfy the boundary conditions of type IV and the orthogonality condition In Kobelev (2016), was proved that the column, that obeys the necessary optimality conditions (39), has the largest buckling load among all the columns with the same weight, assuming the bimodal condition is fulfilled for the optimal distribution of thickness A:

Equations of optimization problem with mixed type boundary conditions
The corresponding optimality conditions in the case of multiple eigenvalues were given in Seiranyan (1984). The standard method for determination of the optimal shapes in this case leads to the nonlinear boundary value problem and reveals no closed-form solution for the shape of the column in terms of higher transcendental functions.
For the closed-form solution of the optimization problem in the case IV, we study two simultaneous equations: The function } is the complex function of the real variable x : The functions M ¼ } j j and h portrays the amplitude and the phase of the complex moment }: These functions are the scalar real functions of the real variable x: For brevity of formulas, we use below the auxiliary dimensionless parameters P ¼ 2 aþ1 and P ¼ 1 þ R: The cross-sectional area A of the optimal rod is the function of M, according to the necessary optimality condition (13) or (39): The substitution of (43) in Eqs. (41), (42) leads to one complex differential equation: There are the real @ R ¼ Re @ and the imaginary parts @ I ¼ Im @ of Eq. (44). With the phase function h, we get two simultaneous real differential equations: At first, we solve the Eq. (45). The boundary conditions for Eq. (45) are: The boundary conditions (47) ensure, that the function @ R is even and the function @ I is odd. The solution of (45) reads: Furthermore, we simplify Eq. (47). The substitution of the function from Eq. (48) in Eq. (46) leads to one nonlinear ordinary differential equation of the second order for M x ð Þ :

The shape of the optimal column
For closed-form solution, the dependent and independent variables in the equation (49) must to be exchanged. As the result, we come to the generalized Emden-Fowler equation for x ¼ x M ð Þ : There is one notable difference between Eqs. (50) and (25). A distinguishing feature of Eq. (50) is that its right side is not a homogeneous function of the independent variable M : This circumstance leads to the bulky formulas for the closed-form solution.
We proceed now with the analytical solution. For the beginning, one boundary condition of Eq. (50) is assumed to be xð1Þ ¼ 0: This means, that we temporary fixing the origin of coordinate at the cross-section with the area 1. With this boundary condition, two solutions of Eq. (50) read: Apparently, that M x ð Þ is the even function of x due to the symmetry of the problem with respect to the origin of coordinate system. The significant task is to determine the integration constant C 1 from the a-priori symmetry conditions. The length of the half-size column is equal to l ¼ L=2: For the beginning we use only one, namely the positive solution of two in Eq. (51). Let the positive function (51) assumes the maximum value in the point M ¼ l: In other words, x l ð Þ ¼ l, because the axial coordinate runs from Àl to l. The integrand of Eq. (51) must have an integrable singularity in this point. Consequently, the denominator of the integrand must vanish in the point t ¼ l: From this condition, the integration constant results as: Notable, is that with the certain value (52) for integration constant, the equation (51) delivers the solution of the optimization problem in integral form. Substantial, that the setting (52) satisfies the boundary conditions (3). The integrals (51) permit the representation in terms of the higher transcendental functions for a equal to 1,2, and 3.
From now we study the dominant for applications case a ¼ a 0 2, P ¼ 2=3: The first important step is the factorization of the integrand in Eq. (51). Substitution of (52) in (51) leads to the positive solution: We go further and attempt to express the solution (53) in terms of the higher transcendental functions.
For this purpose, the integration variable will be replaced, s 2 ¼ t 3 : The integral (53) turns into: The polynomial F s ð Þ is of the fourth order. The roots of the polynomial F are indicated as E i, i ¼ 1, 2, 3, 4: The remarkable attribute of the polynomial F is substantial for the further solution. The polynomial F allows the factorization to the product of the polynomials of the first and the third order: is displayed with the solid lines.

Length, volume, and total stiffness of optimal column
For the optimal column with boundary conditions of mixed type (M), the half-length l m , half -volume v m and the half-total stiffness m allow the representations in terms of integrals: The formulas (58) to (60) are based on the implicit parametrization of the axial coordinate, volume and stiffness. The parametrization implies the area of the cross-section as an independent variable. To prove the above formulas, we can spot, that the dummy variable s indicates the area of the cross-section under the integral sign. Thus, the integrand in (58) represents the element of axial length dl: Therefore, if we multiply the element of length dl by area s and integrate it in the interval E 2 , E 1 ½ , we get in Eq. (59) the single-volume of the rod: V m ¼ 2v m : Intergrals (58) to (60) express with the elliptical functions. The derivation of the algebraic formulas for these intergrals is similar to the derivation Eq. (57). The evaluation of integral v m v m M, P ð Þ for the right side of the column reads (Gradshteyn and Ryzhik 2014): The auxiliary functions Eq. (61) are the following: We study below the most interesting case for a 0 2: The local bending stiffness is the squared area s 2 : The integral of the squared area s 2 over dl delivers the total stiffness e m ¼ 2 m in Eq. (60). The similar expressions result for the total stiffness as well:  The formulas for auxiliary functions of M, P ð Þ are analogous to the recipes, given in Eqs. (57) and (61). The terms rather bulky and are not displayed. Other two cases, a 0 1 and for a 0 3 deliver the comparable closed-form solutions as well. We omit the formulas of these solutions for briefness of the manusript.

Fundamental functions for buckling moments
Once the function x M ð Þ is determined, the phase # M ð Þ in Eq. (44) follows from (48) after the swapping of independent variables. The improper integral converges and expresses in terms of elliptic functions (Gradshteyn and Ryzhik 2014). Finally, the phase reads for . M l as follows: In Eq. (61), the following notations are used for the auxiliary function W M, P ð Þ The fundamental functions for moments are: The constant# is used for the normalization of the moments:

Fundamental functions for buckling displacements
The next task is to determine the fundamental functions for the lateral displacements of the rod in the moment of the buckling. As an exception, we show the buckling functions of the doublelength rod. The buckling displacements y 1 x ð Þ, y 2 x ð Þ, are the even and odd functions of the axial coordinate À L < x < L: These functions are the unknowns in the ordinary differential equations: Beside the unknown functions, Eqs. (65) contain several functions of x : J ¼ A a , m 1 , m 2 : These functions were already determined in Section 5.6. It is necessary to build the rod of the doublelength from the symmetry elements of the half-length, stacking them along the axis of the rod.
The fundamental functions for the displacements y 1 , y 2 result from the solution of the eigenvalue problem (65). The functions for second moment of inertia and for the moments seem to be too bulky for the closed-form solution. The solution of the boundary value problem was performed with the numerical methods of program MAPLE 2020 (Maple 2020). The results are displayed on Figure 14. The dashed lines demonstrate the even fundamental functions y 2 for definite values of parameter l ¼ 2, 3, and 4: The dotted curves show the odd fundamental functions y 1 for the same values of the parameter l:

Asymptotic solutions
For the estimation of the optimization effects, we determine for a ¼ 2 the limit form of the column with mixed boundary conditions. We consider one asymptotic case, for which the parameter l indefinitely increases (A). The shape of the optimal column appears as an hourglass figure with the infinitely slender waist . ! 0: For the formal description, we have to determine at first the asymptotic limits of the roots: parameters, as the exact analytical solutions. The colors correspond the colors of the lines for the exact solutions, but the approximate limit solutions are drawn with the dash style. The singlelength column is an hourglass figure symmetric to its middle section. With the vanishing narrow cross-section, the effectiveness of the optimal solution continually increases and asymptotically approaches its upper limit. Finally, we get the estimations of the invariants in the limit case of hourglass figure with the infinitely narrow tackle for the mixed boundary conditions (A): The values l a , v a , and a stay for the asymptotic expressions of length, volume, and total stiffness of the half-length column, which is displayed on Figure 2, case VI. The values L a , V a , and e a are the corresponding asymptotic formulas for length, volume, and total stiffness of the single-length column, as shown on Figure 2, case IV. With these values, the invariants F A and F B for the single-length column (case IV) are calculated to: The suitable setting for the eigenvalue is once again K ¼ 1: The length, volume, and total stiffness of the double-length column (case V) are twice the length, volume, and total stiffness of the single-length column Eq. (68). Accordingly, the invariants F A , F B of the double-length column are four times greater than the invariants of the single-length column, Eq. (69).

Isoperimetric inequalities
We evaluate the invariant factors F A:c , F B:c for the columns with the constant cross-section using the standard formulas of technical mechanics. Hereafter, we consider the single-length columns. With expressions (59)-(61) and (69), we can determine the invariant factors for the optimal columns with the mixed boundary conditions: The comparison of the factors (70) leads to the estimation of the masses for the column with the same lengths and the same critical buckling load: The mass estimation (70) is in good accordance with the numerically evaluated value. The mass of the optimal column V IV in relation to the mass of the reference columnṼ IV from the exact solution is shown on Figure 15 with the red color. This value reduces continuously with the increasing parameter. The asymptotical limit of the ratio V IṼ V IV for the infinitely high values of l is: and is equal to the corresponding value from the approximate solution (70). The analogous tendency has the ratio of total stiffness e of the optimal rod to the total stiffness of the constantsection rod: Notable, that the mass of the optimal rod reduces continuously with the increasing value of l: The optimum could not be attained with an any finite value of the parameter l: The waist of the hourglass . decreases with the growing values of l, but does not vanish for any finite value of the parameter l: Another argument for this observation follows from the comparison of the invariant factors. Figure 16 displays the invariant mass parameter F A:IV for the optimal column as the function of the parameter l: This value increases asymptotically to its limit value: The plots on Figure 17 show the similar behavior of the invariant stiffness parameter F B:IV : With the increasing parameter l and vanishing parameter ., the parameter F B:IV monotonically increases toward its upper limit: For the optimization problem in Lagrange sense, the isoperimetric inequality could be formulated as the strict inequality (Chavel, 2001). This isoperimetric inequality is sharp. The optimal solution could not be attained for any positive value of parameter .: The analogous consideration, but using completely different arguments, was deliberated by Cox and McCarthy (1998). In other words, the original Lagrange problem possesses no attainable solution from the both points of view of the optimal control theory and variational calculus.
Nevertheless, the reasonable solution could be established for the practical applications. Namely, the certain cross-section possesses the smallest area. The additional restriction to the minimal area of the cross-section presumes: As mentioned above, the formulation of the optimization problem with the additional restrictions to the cross-sections of the column (73) is referred to as the problem in Nikolai sense. The solution of optimizationa problem in Nikolai sense is appropriate for the engineering applications, because in this instance the stress in the narrowest cross-section remains limited.
The formula (57) delivers the attainable solution of the optimization problem in Nikolai sense. The setting in Eq. (57) presents the unique solution of the optimization problem for the mixed-type of boundary conditions. Thus, for the optimization problem in Nikolai sense, the isoperimetric inequality formulates as the not-strict, sharp inequality.

Conclusions
The optimization of a column, compressed by axial forces was solved in closed form. The alternative designs were characterized by positive cross-sectional area functions. The critical values of buckling were equal for all alternative designs of the columns. The optimization problem in Lagrange sense searches the minimal mass of the column. The method of dimensionless factors was used for the optimization analysis. The applied method for integration of the optimization criteria delivers different length and volumes of the optimal columns. Instead of the seeking for the columns of the fixed length and volume, the columns with the different lengths and cross-sections are compared using the invariant factors. The isoperimetric inequalities were rigorously justified by means of the H€ older inequality about the mean values. We demonstrated that Euler's column with boundary conditions of mixed type, which is governed by necessary optimality conditions in the bimodal case, possesses the largest buckling load among all columns with the same weight. The optimal single-size column with one fully clamped end and other moving clamped end has the shape of hourglass, as shown on Figure 2, case V. The optimal full-length column with both clamped ends comprise two single-size columns (Figure 2, case VI). The closed-form solution contains one auxiliary parameter l: For each positive value of the parameter l, the necessary optimality conditions are resolved in closed form. Using the method of H€ older inequalities, for each given positive value of parameter l the isoperimetric inequality for the buckling eigenvalue is uniquely specified (Kobelev 2016). The statement of the isoperimetric inequality is based on the customary optimality conditions for multiple eigenvalues. Accordingly, the existence of the optimal solution is guaranteed for each positive value of the auxiliary parameter. The mass of the established optimal design is the monotonically decreasing function of the auxiliary parameter l: Because there is no smallest positive real number, there is also no attainable solution of the optimization problem in Lagrange sense. In this problem, we search the solution among the positive functions. With the disappearing parameter l, the optimal cross-sections asymptotically adjust to its limit shape.
The solution of the alternative, regularized Nikolai problem with the additional restrictions to the cross-sections is appropriate for the engineering applications, because the stress in the narrowest cross-section remains limited. The solution of optimization problem in Nikolai sense leads to the not-strict, sharp isoperimetric inequality.
List of symbols m actual buckling moment of an arbitrary single-length column m 1 , m 2 actual moments of bimodal problem m admissible buckling moment of an arbitrary single-length column M ¼ } j j amplitude of the complex moment A arbitrary cross-sectional area U i P ð Þ, i ¼ 0, ::, 4 auxiliary functions, Eq. (57) F axial force in the column } ¼ m 1 þ im 2 ¼ M Á exp i h ð Þ complex bimodal moment @ complex differential equation (44) I 1 , I 2 cross-section's principal moments of the cross-second order F A , F B dimensionless factors X ¼ m= dimensionless parameter K eigenvalue of the dimensionless eigenvalue problem k a factor that characterizes the form of cross-section u h i ¼ Ð l Àl u n ð Þdn: integrals of functions u x ð Þ , C integration constants of Eq. (25) F A:c , F B:c invariant factors for the columns with the constant cross-section 2L length of an arbitrary double-length column L length of an arbitrary single-length column ðL ¼ 2lÞ l length of t an arbitrary symmetry element (half-length column) L a length of the optimal single-length column in the asymptotic case (A) l s length of the optimal symmetry element (half-length column, S) l m length of the optimal symmetry element (half-length column, M) l a length of the optimal symmetry element (half-length column, A), the asymptotic case A min minimal area of the cross-section in Nikolai problem, A min 2 3 ¼ P min A optimal cross-sectional area R, P, S parameters of Emden-Fowler equation R¼ 1Àa 1þa , P ¼ 2 aþ1 , S ¼ À2 parameters of Emden-Fowler equation for the unimodal column h phase of the complex moment @ R ¼Re @ , @ I ¼ Im @ real and imaginary parts of the equation @ E i , i ¼ 1, ::, 4 roots of the equation (54) in the asymptotic case E i , i ¼1, ::, 4 E 1 ¼l 2=3 , E 2 ¼. total stiffness of an arbitrary single-length column total stiffness of an arbitrary symmetry element (half-length column) s total stiffness of the optimal symmetry element (half-length column, S) E m total stiffness of the optimal symmetry element (half-length column, M) a total stiffness of the optimal symmetry element (half-length column, A), the asymptotic case e a total stiffness of the optimal symmetry element (single-length column, A), asymptotic case V volume of an arbitrary single-length column v volume of an arbitrary symmetry element (half-length column) V a volume of the optimal single-length column in the asymptotic case (A) v s volume of the optimal symmetry element (half-length column, S) v m volume of the optimal symmetry element (half-length column, M) v a volume of the optimal symmetry element (half-length column, A) in the asymptotic case

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On behalf of all authors, the corresponding author states that there is no conflict of interest.