Material and geometric nonlinear buckling of simply supported columns

Abstract

This study investigated the material and geometric nonlinear buckling of simply supported columns. The column body material followed Ludwick’s constitutive law. Rectangular and elliptical cross sections were considered, and corresponding generalized moments of inertia (GMIs) were explicitly formulated. By applying the GMIs, the governing differential equations and boundary conditions of post-buckling columns were derived based on the Bernoulli–Euler beam theory, and the buckling loads and elastica were computed using numerical methods. To derive the elastica of buckling columns, the differential equations were integrated using the Runge–Kutta method, and the eigenvalues of the buckling load were determined using the bisection method. In our numerical experiments, the GMI formulae were applied, and parametric studies were conducted to analyze post-buckling columns in terms of buckling load, equilibrium path, elastica, and nonlinear stress along the cross section.

1 Introduction

Recently, nonlinear elastic materials have been used as major structural members to support external compressive loads stably [1]. Ludwick-type materials are a family of representative nonlinear materials that follow a load-strain relationship of $\sigma = E\varepsilon^{1/n}$, where ($\varepsilon , \sigma$) denote the strain and stress, respectively, $n$ is the material constant, and $E$ is the modulus of elasticity [2]. The relationship $\sigma = E\varepsilon^{1/n}$, which is called the Ludwick constitutive law, can be used to calculate the bending moment $M = EI_{{\text{g}}} \kappa^{1/n}$, where $I_{{\text{g}}}$ is the generalized moment of inertia (GMI) and $\kappa$ is the curvature caused by $M$. It is important to formulate $I_{{\text{g}}}$ in advance for buckling analysis, which is the main focus of this study.

Over the past few decades, calculation methods for GMI $I_{{\text{g}}}$ have only been formulated by researchers for three cross-sectional planar areas. Lee [3] calculated $I_{{\text{g}}}$ as $I_{{\text{g}}} = [0.5^{(1 + 1/n)} n/\left( {2n + 1} \right)]ab^{{\left( {2 + 1/n} \right)}}$ for a rectangular cross section with width $a$ and height $b$. Brojan and Kosel [4] computed $I_{{\text{g}}}$ for a superellipsoidal cross section using a beta function that is explicitly expressed in an integral form. Lee and Lee [5] developed a calculation method for $I_{{\text{g}}}$ for regular polygonal cross sections, where $I_{{\text{g}}}$ was explicitly formulated using an integral formula. Five types of regular polygonal cross sections, namely triangles, squares, pentagons, hexagons, and circles, were considered as numerical examples. Kim et al. [6] formulated $I_{{\text{g}}}$ for an elliptical cross section with semi-axes $a$ and $b$ based on the results of Brojan and Kosel [7], which was expressed in terms of the area $A$, but not the semi-axes $a$ and $b$. Beyond the cross-sectional shapes mentioned above, additional formulas for $I_{{\text{g}}}$ have not yet been derived.

Regarding geometric nonlinear analysis, the first study focusing on elastica was conducted by Euler [8], where thin rods of linear elastic materials were investigated. Following the study by Euler, both linear elastic materials and nonlinear elastic materials were considered for elastica analyses. Additionally, several elastica problems related to geometric and material nonlinearities have been investigated. We now review representative studies that have analyzed the deformed elastica related to our study.

First, regarding geometrical nonlinear elastica problems for linear materials, elastica studies similar to Euler’s classical work are reviewed. Such studies include analyses of the large deflection of Euler beams using Bernoulli–Euler beam theory by Bisshopp and Drucker [9]; largely deformed elastica of Timoshenko beam-columns with internal joints, where the non-conservative loads were considered by Aristizabal-Ochoa [10]; shear deformable maximum column strength with constant volume and regular polygonal cross sections by Lee and Lee [11]; finite deflection of elastic rods following a hyperbolic tangent law by Oden and Childs [12]; large deflection of tapered cantilever beams considering distributed loads, concentrated loads, and tip coupling by Lee et al. [13]; Euler buckling loads with the corresponding buckled elastica of a non-prismatic column with a constant volume by Lee and Oh [14]; and bifurcation elastica of a slender beam subjected to axial external thrust by Berkey and Freedman [15]. Second, regarding nonlinear materials, several studies addressing elastica behavior have been conducted. For example, Giardina and Wei [16] and Anatolyevich and Yokovlevna [17] conducted two representative studies, where a material hardened by plastic deformation according to the Ramberg–Osgood constitutive law was considered.

In bending analysis, the material and geometric nonlinear behavior of structural members should be considered based on the GMI $I_{{\text{g}}}$. In the literature, $I_{{\text{g}}}$ has been applied to elastica analysis, which considers highly deformed geometric nonlinear beams/columns. We now review the most representative works on beams/columns made of the Ludwick-type materials considered in this study. Lewis and Monasa [2] studied a largely deformed cantilever beam and applied coupling to the free end, where the exact curvature of the deformed beam was considered. Lee [3] applied the shear force formula to derive differential equations for the elastica of a buckled cantilever beam. Lee and Lee [5] studied post-buckling elastica for regular polygonal nonlinear cantilever columns made of Ludwick-type materials. Kim et al. [6] computed the buckling load and buckled elastica of nonlinear cantilever columns with elliptical cross sections that which followed the Ludwick constitutive law. Brojan and Kosel [7] applied the relationship between strain and stress to a superellipsoidal cross section. Eren [18] studied cantilever beam elastica with a uniform load and vertical point load. Eren [19] calculated simple beam elastica with a uniform load as a function of curvature expressed by arc length functions of two different types. Jung and Kang [20] studied the deformed elastica of uniform rectangular columns composed of nonlinear fibrous materials subjected to horizontal and vertical point loads. Saetiew and Chucheepsakul [21] derived first-order simultaneous differential equations to compute the buckling load of a linearly tapered column. Brojan and Kosel [7] investigated the stability of Euler columns considering their buckled shapes. Borboni and Santis [22] investigated cantilever beams with asymmetrical cross sections subjected to nonlinear bending under vertical and horizontal point loads with moment coupling. Liu et al. [23] studied curved beam elastica considering the effects of axial expansion, which implicitly formulated the corresponding stress–strain relationships.

As discussed above, the subject of the GMI $I_{{\text{g}}}$ and nonlinear column buckling remains an important topic in classical mechanics. In the case of nonlinear column buckling, the conventional moment of inertia of a plane area $I$, which only applies to linear materials, is no longer valid for nonlinear members that follow the Ludwick constitutive law [3, 4]. Therefore, it is necessary to develop $I_{{\text{g}}}$ formulations for various cross-sectional shapes and apply $I_{{\text{g}}}$ to the buckling analysis of appropriate structural members for real-world engineering. From this perspective, this study has two main goals: reformulating $I_{{\text{g}}}$ in a manner that is applicable to nonlinear elastic column buckling and applying $I_{{\text{g}}}$ to the buckled column problem.

In this study, the GMIs for regular elliptical as well as rectangular cross section were formulated analytically and explicitly, and when applied to cantilever beam elastica, a uniformly distributed load was considered. The results for the stress and strain imposed on the cross section along the depth of the section, which are some of the most important design criteria for material and geometric nonlinear columns in actual engineering, are presented based on numerical experiments.

We begin by reformulating $I_{{\text{g}}}$ for rectangular and elliptical cross sections based on theories from the literature [3,4,5]. This includes the derivation of governing differential equations with corresponding boundary conditions for buckled column elastica with a simply supported end and the development of numerical solution methods. Finally, based on numerical experiments, parametric studies of the GMI $I_{{\text{g}}}$ and elastica of buckled columns are conducted and discussed extensively.

Our elastic analysis is based on the following assumptions: no initial imperfections exist in the column axis, the column material is inextensible/incompressible, and the plane of the cross section remains flat, even after buckling.

2 GMI for Ludwick-type material cross sections

2.1 Configuration of cross sections

Prior to buckling analysis, for columns following the Ludwick constitutive law, the calculation method for $I_{{\text{g}}}$ is summarized based on theory from the open literature [3,4,5] and reformulated for the rectangular and elliptical cross sections considered in this study. Formulating $I_{{\text{g}}}$ for a nonlinear elastic column is an essential preprocessing step.

Figure 1a presents an arbitrary planar cross section symmetrical about the $w$ and $h$ axes, where an elliptical area with semi-axes of $a/2$ and $b/2$ is presented as an example. The bending moment $M$ is applied to the cross section through which the strain $\varepsilon$ is linearly distributed and the stress $\sigma$ for a nonlinear material is nonlinearly distributed along the cross-sectional depth $h$, as shown in Fig. 2b, where $n$ is the material constant and $E$ is the Young’s modulus.

Fig. 1

a Symmetrical cross section acted to $M$ and b stress $\varepsilon$ and strain $\sigma$ distributed along depth of cross section

Fig. 2

Configuration of a rectangular and b elliptical cross section

For Ludwick-type materials, the relationship between $\sigma {-}\varepsilon$ and the curvature $\kappa$ is given as follows [24]:

$$ \varepsilon = \kappa h, $$

(1)

$$ \sigma = E\varepsilon^{1/n} = E\left( {\kappa h} \right)^{1/n} $$

(2)

The infinitesimal force ${\text{d}}F$ on an area ${\text{d}}A$ at the coordinate $h$ can be obtained as

$$ {\text{d}}F = \sigma {\text{d}}A = E\left( {\kappa h} \right)^{1/n} wdh, $$

(3)

where $w$ is a function of $h$ (i.e., $w = f\left( h \right)$) for any closed curve along the cross-sectional boundary.

The bending moment $M$ imposed on area $A$ can be obtained using Eq. 3 as

$$ M = \smallint {\text{d}}M = \smallint \left( {{\text{d}}F} \right)h = E\kappa^{1/n} \mathop \smallint \limits_{ - b/2}^{b/2} wh^{1 + 1/n} {\text{d}}h = EI_{{\text{g}}} \kappa^{1/n} , $$

(4)

where the GMI $I_{{\text{g}}}$ is defined as

$$ I_{{\text{g}}} = \mathop \smallint \limits_{ - b/2}^{b/2} wh^{1 + 1/n} {\text{d}}h $$

(5)

Note that if $n = 1$ (i.e., linear elastic material), then $I_{{\text{g}}}$ is reduced to $I_{{\text{g}}} = I$, which is the conventional moment of inertia of the cross section.

The typical properties of $n$ and $E$ for the Ludwick-type materials considered in this study are.

• $n = 1$ and $E = 207$ GPa for steel (linear elastic material),

• $n = 2.16$ and $E = 458.5$ MPa for annealed copper,

• $n = 4.785$ and $E = 455.8$ MPa for the NP8 aluminum alloy,

where a linear material such as steel ($n = 1$) can be considered in terms of the GMI $I_{{\text{g}}}$.

To apply $I_{{\text{g}}}$ to column buckling, rectangular and elliptical cross sections were considered in this study. The configurations of each cross-sectional feature with corresponding width and depth dimensions are presented in Fig. 2a and b, respectively.

2.2 Formulation of GMI $I_{{\text{g}}}$

2.2.1 Rectangular cross section

GMI $I_{{\text{g}}}$ was reformulated to consider column buckling. Applying Eq. 5 to the cross section in Fig. 2a yields $I_{{\text{g}}}$[3] as

$$ I_{{\text{g}}} = \frac{{0.5^{(1 + 1/n)} n}}{2n + 1}ab^{2 + 1/n} . $$

(6)

The GMI $I_{{\text{g}}}$ is initially expressed in terms of $a$ and $b$, as shown in Eq. 6. For column buckling analysis, it is desirable to express $I_{{\text{g}}}$ as an area $A$, rather than in terms of $a$ and $b$. This is because the magnitude of $A$ differs for different cross-sectional shapes with the same $a$ and $b$, and it is effective to compare buckling behaviors for the same $A$. Therefore, the ratio of $b$ to $a$, which is defined as the aspect ratio $\alpha$ in Fig. 2a, is defined as

$$ \alpha = \frac{b}{a}. $$

(7)

Then, the area $A$ is expressed as $A = ab$, and $a$ and $b$ are solved for a uniform area $A$ as

$$ a = \sqrt {\frac{A}{\alpha }} ,\, b = \sqrt {\alpha A} . $$

(8)

Subsequently, considering Eq. 8, $I_{{\text{g}}}$ in Eq. 6 is rearranged in terms of $n$, $\alpha$, and $A$ as follows:

$$ I_{{\text{g}}} = \frac{{0.5^{(1 + 1/n)} }}{2 + 1/n}\alpha^{n1} A^{n2} = \beta A^{n2} ,\,\beta = \frac{{0.5^{1 + 1/n} }}{2 + 1/n}\alpha^{n1} , $$

(9)

where $n_{1} = \left( {1 + 1/n} \right)/2$ and $n_{2} = \left( {3 + 1/n} \right)/2$. In Eq. 9, $\beta$, which is called the GMI coefficient, is dimensionless, where $I_{{\text{g}}}$ is computed using the given $A$.

2.2.2 Elliptical cross section

Applying Eq. 5 to the elliptical cross section in Fig. 2b yields $I_{{\text{g}}}$ as [4]

$$ I_{{\text{g}}} = \frac{{2ab^{n + 2} }}{n + 3} \times B\left( {\frac{1}{2},{ }\frac{n + 2}{2}} \right), $$

(10)

where $B$ is a beta function given explicitly in a definite integral form.

Similarly, the aspect ratio $\alpha$ for an elliptical cross section is defined as

$$ \alpha = \frac{b}{a}. $$

(11)

Then, the area $A$ is expressed as $A = \pi ab/4$, and $a$ and $b$ are solved for a uniform area $A$ as

$$ a = 2\sqrt {\frac{A}{\alpha \pi }} ,\quad b = 2\sqrt {\frac{\alpha A}{\pi }} . $$

(12)

Subsequently, considering Eq. 12, the GMI $I_{{\text{g}}}$ in Eq. 10 is rearranged explicitly in a definite integral form in terms of $n$, $\alpha$, and $A$ as

$$ I_{{\text{g}}} = \frac{{4\alpha^{n1} A^{n2} }}{{\pi^{n2} }} \mathop \smallint \limits_{0}^{1} \zeta^{2n1} \sqrt {1 - \zeta^{2} } d\zeta = \beta A^{n2} ,\,\beta = \frac{{4\alpha^{n1} }}{{\pi^{n2} }} \mathop \smallint \limits_{0}^{1} \zeta^{{2n_{1} }} \sqrt {1 - \zeta^{2} } d\zeta , $$

(13)

where the GMI coefficient $\beta$ can be calculated numerically using the Runge–Kutta method, which is the direct integration method considered in this study.

As indicated in Eqs. 9 and 13, $\beta$ depends on the cross-sectional shape (rectangular or elliptical), $n$, and $\alpha$. Note that $I_{{\text{g}}}$ has physical dimensions of $I_{{\text{g}}} = \left[ A \right]^{{n_{2} }} = \left[ L \right]^{3 + 1/n}$, as shown in Eqs. (9) and (13). If $A$ is given in cm², then the units of $I_{{\text{g}}}$ are cm⁴ with $n = 1$ for steel (the linear material), cm^3.463 with $n = 2.16$ for annealed copper, and cm^3.209 with $n = 4.785$ for the NP8 aluminum alloy.

2.3 Numerical example of GMI $I_{{\text{g}}}$

The GMI coefficient $\beta$ is computed using Eqs. 9 and 13 for the given values of $n$ and $\alpha$, and then, the value of $I_{{\text{g}}}$ is calculated from $\beta$ using Eqs. 9 and 13 for the given area $A$ of a rectangular or elliptical shape.

Table 1 compares the GMI $I_{{\text{g}}}$ results from this study to those presented in references [3,4,5,6] to verify the reformulation of $I_{{\text{g}}}$ developed in this study. The two results for $I_{{\text{g}}}$ obtained by changing the values of $\alpha$ for two typical nonlinear materials ($n = 2.16$ and $4.785$ with $A = 15$ cm²) are in good agreement, as shown in Table 1, indicating that the reformulation of $I_{{\text{g}}}$ developed in this study yields correct results.

Table 1 Comparison GMI $I_{{\text{g}}}$ ($A =$ 15 cm²) between this study and references

It is noteworthy to compare the GMI $I_{{\text{g}}}$ results for rectangular and elliptical cross sections, which are often used for structural members in practical engineering. Table 2 presents a comparison of the three material constants $n$ for the same $\alpha$ and $A (15$ cm²) values. The $I_{{\text{g}}}$ value of a rectangle is always approximately 5% greater than the $I_{{\text{g}}}$ value of an ellipse, regardless of the magnitudes of $n$ and ${ }\alpha$. Note that the ratios of 1.0472, 1.0495, and 1.0475 for given $n = 1, 2.16$, and $4.785$ are similar, but not identical. Additionally, although all results are not shown here, the ratio for rectangular to elliptical shapes is always the same, regardless of $\alpha$.

Table 2 Comparison of GMI $I_{{\text{g}}}$ ($A =$ 15 cm²) between cross-sectional shapes

The GMI coefficient $\beta$ expressed in Eqs. 9 and 13 depends on the cross-sectional shape, material constant $n$, and aspect ratio $\alpha$. Once a surface map of $\beta$ is graphically derived by varying $n$ and $\alpha$, then $I_{{\text{g}}} = \beta A^{n2}$ in Eqs. 9 and 13 can be easily calculated for a given area $A$. Surface maps of $\beta$ are reported for $0 < n \le 10$ and $0 < \alpha \le 1.5$ for the rectangle in Fig. 3a and ellipse in Fig. 3b. These surface maps indicate that $\beta$ increases with increases in $n$ and $\alpha$. According to these results, the greatest values of $\beta$ occur at the greatest values of $n = 10$ and $\alpha = 1.5$ in $0 < n \le 10$ and $0 < \alpha \le 1.5$. As expected, for cross sections with the same $n$ and $\alpha$, the $\beta$ value of the rectangular cross section is always approximately 5% greater than the $\beta$ value of the elliptical cross section, as indicated in Table 2.

Fig. 3

Surface map $\beta$ by varying $n$ and $\alpha$ of a rectangular and b elliptical cross section

A practical example of calculating the GMI $I_{{\text{g}}}$ in dimensional units is illustrated in Fig. 4, which presents $I_{{\text{g}}}$ versus $\alpha$ curves according to the cross-sectional shape with $A = 20$ cm². The $I_{{\text{g}}}$ versus $\alpha$ curves are nonlinear because $I_{{\text{g}}}$ increases as $\alpha$ increases, indicating that the greater the depth of the cross section, the higher the bending stiffness. When $\alpha = 1$, meaning $a = b$, $I_{{\text{g}}}$ is the same relative to the $w$ axis and $h$ axis. In Fig. 4a, $I_{{\text{g}}}$ with $\alpha > 1$ corresponding to the weak axis, whereas $I_{{\text{g}}}$ with $\alpha < 1$ corresponds to the strong axis. It is important not to overestimate the buckling load $P_{cr}$, so it is better not to apply $I_{{\text{g}}}$ to the strong axis ($\alpha < 1$) when calculating $P_{{{\text{cr}}}}$. The $I_{{\text{g}}}$ value of a rectangular cross section is approximately 5% greater than that of an elliptical cross section. Figure 4b presents $I_{{\text{g}}}$ versus $\alpha$ curves according to the material constant $n$ with $A = 20$ cm² for an elliptical cross section, where $I_{g}$ increases with an increasing $\alpha$. The $I_{{\text{g}}}$ versus $\alpha$ curves are nonlinear for $n = 2.16$ and 4.785 (i.e., nonlinear materials), whereas the curve is linear for $n = 1$(i.e., linear material).

Fig. 4

${I}_{\mathrm{g}}$ Iversus $\alpha$ curves: a by cross-sectional shape and b by material constant $n$

3 Application of GMI $I_{{\text{g}}}$ to buckled column elastica

3.1 Mathematical modeling of column buckling

The GMI $I_{{\text{g}}}$ was applied to post-buckling column elastica. We present and discuss the mathematical modeling of elastica, numerical solution methods, and numerical experiments. In this study, the buckling loads of a simple column made of Ludwick-type materials were evaluated.

Figure 5a presents a simple support column of length $l$ made of a Ludwick-type material that is loaded with an external load $P$ at the top end, which moves downward after buckling. The column contains no initial imperfections, meaning it is perfectly straight until buckling begins. When $P$ is smaller than the buckling load $P_{{{\text{cr}}}}$ (i.e., $P < P_{{{\text{cr}}}}$), the axis of the column represented by the dotted curve remains a straight line. As $P$ gradually increases and eventually reaches $P_{{{\text{cr}}}}$ (i.e.,$P = P_{{{\text{cr}}}}$), the column buckles and elastically deforms. The shape of the highly deformed column after buckling, which is called an elastica, is illustrated by the solid curve in rectangular coordinates ($x,y$) originating from the base end $o$. The arc length at ($x, y$) measured from the base end is displayed in s. Based on the inextensibility/incompressibility of the column, the elastica arc length matches the original length $l$. At any ($x, y$), the rotational angle of the cross section is denoted as $\theta$ and the vertical deflection at the top end ($s = l$) is represented by $\Delta_{{\text{v}}}$. The $\theta$ value at the base end ($s = 0$) is denoted as $\theta_{i}$. The internal force consists of $\left( {N, Q, M} \right)$ at any ($x,y$) in Fig. 5, which denotes the axial force, shear force, and bending moment, respectively. Figure 5b presents the trigonometric relationships of a post-buckling column element at coordinates ($x, y$) and the sign convention of the positive internal forces of $\left( {N, Q, M} \right)$ imposed on the element.

Fig. 5

a Geometry of simply supported column and b trigonometric relationships of buckled column element and its sign conventions of $\left(N, Q, M\right)$

The internal forces of $\left( {N, Q, M} \right)$ at any $\left( {x, y} \right)$ can be expressed using the post-buckling equilibrium state presented in Fig. 5a as follows:

$$ N = P\cos \theta , $$

(14)

$$ Q = P\sin \theta , $$

(15)

$$ M = Py. $$

(16)

The trigonometric relationships of the buckled column element presented in Fig. 5b are based on the Bernoulli–Euler beam theory as follows:

$$ \frac{{{\text{d}}x}}{{{\text{d}}s}} = \cos \theta , $$

(17)

$$ \frac{{{\text{d}}y}}{{{\text{d}}s}} = \sin \theta . $$

(18)

By using the exact curvature of $\kappa = {\text{d}}\theta /{\text{d}}s$ and GMI $I_{{\text{g}}} \left( { = \beta A^{n2} } \right)$, the bending moment $M$ can be expressed as follows:

$$ M = EI_{{\text{g}}} \kappa^{1/n} = \beta EA^{n2} \left( {\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)^{1/n} . $$

(19)

Combining Eqs. 16 and 19, and rearranging in terms of ${\text{d}}\theta /{\text{d}}s$ yields

$$ \frac{{{\text{d}}\theta }}{{{\text{d}}s}} = \left( {\frac{Py}{{\beta EA^{n2} }}} \right)^{n} . $$

(20)

To discuss numerical results in a dimensionless form, dimensionless system parameters are defined as follows:

$$ \lambda = \frac{s}{l}, $$

(21)

$$ \xi = \frac{x}{l}, $$

(22)

$$ \eta = \frac{y}{l}, $$

(23)

$$ \delta_{{\text{v}}} = \frac{{\Delta_{{\text{v}}} }}{l}\left( { = 1 - \xi_{\lambda = 1} } \right), $$

(24)

$$ p = \frac{{Pl^{2n1} }}{{EA^{n2} }}, $$

(25)

where $\left( {\lambda , \xi , \eta } \right)$ are dimensionless coordinates, $\delta_{{\text{v}}}$ is the dimensionless vertical deflection at the top moveable end ($s = l$), and $p$ is the normalized load parameter.

By combining the system parameters in Eqs. 17, 18, and 20, we can derive a dimensionless form of the governing differential equations for a post-buckling column that follows the Ludwick constitutive law as

$$ \frac{{{\text{d}}\xi }}{{{\text{d}}\lambda }} = \cos \theta , $$

(26)

$$ \frac{{{\text{d}}\eta }}{{{\text{d}}\lambda }} = \sin \theta , $$

(27)

$$ \frac{{{\text{d}}\theta }}{{{\text{d}}\lambda }} = \left( {\frac{p\eta }{\beta }} \right)^{n} . $$

(28)

Considering the boundary conditions for the initial problems in Eqs. 26–28, the coordinates $\left( {x, y} \right)$ of the base end ($s = 0)$ are zero and have the following dimensionless form:

$$ \xi = 0, \eta = 0\,{\text{at}}\,\lambda = 0 $$

(29)

One of the initial boundary conditions, namely the rotation $\theta_{i}$ at the bottom end ($\lambda = 0$), is not considered as an initial value in the differential Eqs. 26–28 and is derived later through numerical analysis.

At the top end ($s = l)$, the coordinate $y$ is zero, and its dimensionless form is

$$ \eta = 0\;{\text{at}}\;\lambda = 1 $$

(30)

The internal forces $\left( {N, Q, M} \right)$ of a post-buckling column are normalized as follows:

$$ \overline{n} = \frac{{Nl^{2n1} }}{{EA^{n2} }} = p\cos \theta , $$

(31)

$$ \overline{q} = \frac{{Ql^{2n1} }}{{EA^{n2} }} = p{\text{sin}}\theta , $$

(32)

$$ \overline{m} = \frac{{Ml^{1/n} }}{{EA^{n2} }} = p\eta . $$

(33)

3.2 Numerical solution methods and verification

There are no closed-form solutions for Eqs. 26 and 28. Therefore, to solve these differential equations numerically, approximate numerical methods were developed to compute the buckling loads and post-buckling elastica. The input column parameters were the cross-sectional shape (rectangular or elliptical), span length$l$, cross-sectional dimensions of $a$ and $b$, material properties of $\left( {n,E} \right)$, and load $P$. The corresponding dimensionless parameters $\alpha$, $\beta$, and $p$ are calculated using these dimensional parameters. To compute the elastica of $\left( {\xi , \eta } \right)$, the Runge–Kutta method [25], which is a direct numerical integral method, was used to integrate Eqs. 26–28. To determine the unknown $\theta_{i}$ at the base end ($\lambda = 0$), a numerical method for solving nonlinear equations, such as the bisection method [25], can be applied. This type of solution method for deriving an unknown $\theta_{i}$ has been frequently used to solve the initial and boundary value problems in the literature [14]. For interested readers, the numerical method for calculating the elastica of $\left( {\xi , \eta , \theta } \right)$ is described below.

• (1) Set the column parameters of the cross-sectional shape $n, \alpha$, and $p$ (and calculate $\beta$ for the given $n$ and $\alpha$).

• (2) Assume a trial unknown $\theta_{i}$ at $\lambda = 0$. The first trial $\theta_{i}$ is $\theta_{i} = 0$.

• (3) Subject the boundary conditions in Eq. 29 to Eqs. 26 to 28 and integrate Eqs. 26 to 28 at $0 \le \lambda \le 1$ using the Runge–Kutta method. Subsequently, the trial solution $\left( {\xi ,\eta ,\theta } \right)$ along the arc length $\lambda$ is obtained.

• (4) In this calculation, evaluate $E_{{\text{r}}} = \eta_{\lambda = 1}$ for the boundary condition in Eq. 30 considering the following convergence criterion:

  $$ \left| {E_{{\text{r}}} } \right| \le 1 \times 10^{ - 8} . $$

  (34)

If the criterion in Eq. 34 is satisfied, then calculation is stopped after an additional calculation of $\left( {\overline{n},\overline{q},\,\overline{m}} \right)$.

5) If the above criterion is not met, then increase the trial $\theta_{i}$ by ${ }\Delta$ and return to step 2 with the advanced trial $\theta_{i} ( \leftarrow \theta_{i} + \Delta )$.

6) Monitor whether the sign of $E_{{{\text{r}}1}} \times E_{{{\text{r}}2}}$ has changed during steps 2–5, where $E_{{{\text{r}}1}}$ and $E_{{{\text{r}}2}}$ are $E_{{\text{r}}}$ values corresponding to the previous and current trials, respectively. If the sign of $E_{r1} \times E_{r2}$ does not change until $\theta = \pi$, then the given $p$ is smaller than $p_{{{\text{cr}}}}$ and the column is still straight. Because $\theta_{i}$ cannot exceed $\pi$, we stop calculating. Here, the buckling load parameter $p_{{{\text{cr}}}}$ is defined as

$$ p_{{{\text{cr}}}} = \frac{{P_{{{\text{cr}}}} l^{2n1} }}{{EA^{n2} }}. $$

(35)

7) When changing the sign of${ E}_{\mathrm{r}1}\hspace{0.17em}\times \hspace{0.17em}{E}_{\mathrm{r}2}$, the target solution for ${\theta }_{i}$ is between ${\theta }_{{i}_{1}}$,${E}_{\mathrm{r}1}$, and $\theta_{{i_{2} }}$ with $E_{r1}$. A more advanced new trial ${\theta }_{{i}_{3}}$ for the target solution ${\theta }_{i}$ compared to $\theta_{{i_{1} }}$ and $\theta_{{i_{2} }}$ is evaluated using the bisection method [25] as

$$ \theta_{i3} = \frac{{\theta_{i1} + \theta_{i2} }}{2} . $$

(36)

8) Once the step 7 is completed, repeat the steps above until the following criterion is satisfied:

$$ \frac{{\theta_{i1} + \theta_{i2} }}{2} \le 1 \times 10^{ - 5} . $$

(37)

9) If the criterion in Eq. 37 is satisfied, compute $\left( {\overline{n}, \overline{q}, \overline{m}} \right)$ and stop the calculation. If the sign of $E_{r1} \times E_{r2}$ for a given $p$ in the steps above does not change until $\theta_{i} = \pi$, then the given $p$ is less than $p_{{{\text{cr}}}}$, meaning the column does not buckle, as mentioned in step 6.

A numerical solution method for calculating the buckling load parameter $p_{{{\text{cr}}}}$ was developed based on the fact that the buckling loads $p_{{{\text{cr}}}}$ for nonlinear columns following the Ludwick constitutive law are jump buckling loads [5]. A column under a load parameter $p < p_{{{\text{cr}}}}$ does not buckle, meaning $\theta_{i} = 0$, whereas a column with $p > p_{{{\text{cr}}}}$ buckles, meaning $\theta_{i} \ne 0$. Therefore, it is clear that $p$ will be $p_{{{\text{cr}}}}$, which is the smallest $p$ that $\theta_{i}$ has encountered. A numerical method for calculating $p_{{{\text{cr}}}}$ can be developed in the same manner by adopting the numerical method for a buckled column described above. Starting with $p = 0$, we increase $p$ by the increment $\Delta p( <$ $\pi )$ and steps 2–5 are performed until ${ E}_{\mathrm{r}1}\hspace{0.17em}\times \hspace{0.17em}{E}_{\mathrm{r}2}$ changes its sign. $\Delta p$ can be arbitrarily selected from $0\hspace{0.17em}<\hspace{0.17em}\Delta p\hspace{0.17em}<\hspace{0.17em}\pi$. Then, ${p}_{\mathrm{cr}}$ is located between two adjacent ${p}_{a}$ and ${p}_{b}$, between which the previous sign of ${ E}_{\mathrm{r}1}\hspace{0.17em}\times \hspace{0.17em}{E}_{\mathrm{r}2}$ does not change, and the current sign of ${ E}_{\mathrm{r}1}\hspace{0.17em}\times \hspace{0.17em}{E}_{\mathrm{r}2}$ changes. In this calculation, $p_{{\text{a}}}$ is $p_{2}$ from the previous iteration, whereas $p_{{\text{b}}}$ is $p_{1}$ from the current iteration. Therefore, $p_{{\text{b}}} > p_{{\text{a}}}$. Steps 2–5 are iterated from $p_{{\text{a}}}$–$p_{{\text{b}}}$ in small increments $\Delta p\left( { < p_{{\text{b}}} - p_{{\text{a}}} } \right)$ until the following convergence criterion is satisfied:

$$ p_{{{\text{cr}}}} \approx p_{{\text{b}}} ,\;{\text{if}}\, D = \frac{{p_{{\text{b}}} - p_{{\text{a}}} }}{{p_{{\text{b}}} }} \le 1 \times 10^{ - 5}. $$

(38)

Figure 6 presents a typical example of determining $p_{{{\text{cr}}}}$ for a rectangular cross section with $n = 2.16$ and $\alpha = 0.5$. In the first iteration with $i = 1$, $p_{{\text{b}}}$ for $\Delta p = 0.1$ is $p_{{\text{b}}} =$ 0.6 with $D = \left( {0.6 - 0.5} \right)/0.1 = 1 > 1 \times 10^{ - 5}$, where the sign of $E_{{{\text{r}}1}} \times E_{{{\text{r}}2}}$ changes, implying that the solution $p_{{{\text{cr}}}}$ lies within $0.5\left( { = p_{{\text{a}}} } \right) < p_{{{\text{cr}}}} < 0.6\left( { = p_{{\text{b}}} } \right)$. Finally, for $i = 6$, $p_{{{\text{cr}}}}$ is determined as $p_{{{\text{cr}}}} \left( { = p_{{\text{b}}} } \right) = 0.53644$ with $E_{{\text{r}}} = 1.95 \times 10^{ - 6} < 1 \times 10^{ - 5}$.

Fig. 6

Convergence analysis according to iteration number $i$ to obtain buckling load parameter ${p}_{\mathrm{cr}}$

To calculate the buckling load parameter $p_{{{\text{cr}}}}$ and buckling elastica $\left( {\xi , \eta , \theta } \right)$ using the analytical and numerical methods described above, two FORTRAN programs were designed. These programs contain a subroutine that calculates the GMI $I_{{\text{g}}}$. For validation purposes, a comparison of the buckling load $P_{{{\text{cr}}}}$ values calculated in this study and the reference values for steel columns ($n = 1$ and $E = 270$ GPa) for $l = 1$ m and $A = 20{ }$ cm² are presented in Table 3. The references presented here are the only results published in the literature. Although the $P_{{{\text{cr}}}}$ value in this study was approximated using numerical methods, it matches well with the closed solution from reference [24] (within five digits of accuracy).

Table 3 Comparisons* of buckling load $P_{{{\text{cr}}}}$ between this study and reference

3.3 Numerical experiments

Numerical experiments were conducted for the buckling load parameter $p_{{{\text{cr}}}}$ and elastica $\left( {\xi , \eta } \right)$ using the FORTRAN programs developed in this study. As discussed in Sect. 2.1, three different material constants were considered in our numerical experiments: $n = 4.785$ for the NP8 aluminum alloy, $n = 2.16$ for annealed copper, and $n = 1$ for steel. In our numerical experiments, only the fundamental critical buckling load parameters $p_{{{\text{cr}}}}$ of the columns were calculated because a higher buckling load is less important than the critical $p_{{{\text{cr}}}}$ [26]. Therefore, only the post-buckling equilibrium paths versus $p_{{{\text{cr}}}}$ are presented.

It is relevant to compare the buckling loads $p_{{{\text{cr}}}}$ between rectangular and elliptical cross sections, which are often used for structural members in practical engineering. Table 4 presents comparisons of $p_{{{\text{cr}}}}$ values for three different values of n with the same $\alpha$, $l( = 1$ m), and $A( = 20$ cm²). The $p_{{{\text{cr}}}}$ value of the rectangular section is always approximately 5% greater than that of the elliptical section, regardless of the magnitudes of $n$ and $\alpha$. It is noteworthy that the ratios in this table are the same as those in Table 2, where the GMI $I_{{\text{g}}}$ values are compared. Additionally, even for values not shown here, the ratios of values for rectangular to elliptical cross sections are always the same, regardless of $\alpha$.

Table 4 Comparisons* of buckling load $P_{{{\text{cr}}}}$ between cross-sectional shapes

We now discuss the various effects of the dimensionless system parameters on the buckling load parameter $p_{{{\text{cr}}}}$. Figure 7a presents the $p_{{{\text{cr}}}}$ versus aspect ratio $\alpha$ curves obtained by varying the material constant $n$. As expected, $p_{{{\text{cr}}}}$ increases with an increase in $\alpha$ because a larger $\alpha$ results in a larger $I_{{\text{g}}}$ for the cross section. The buckling load parameter $p_{{{\text{cr}}}}$ curves for nonlinear materials with $n = 4.785$ and $n = 2.16$ are nonlinear, whereas the $p_{{{\text{cr}}}}$ curve for the linear material with $n = 1$ is linear. One can see that the order of the magnitudes of $p_{{{\text{cr}}}}$ values for columns with three different $n$ values depends on $\alpha$. For example, the magnitude order for $\alpha = 0.5$ is from $n = 2.16$ to $n =$ 4.785 to $n =$ 1, whereas the orders for $\alpha = 0.1$ and 0.9 differ from the order for $\alpha = 0.5$. The buckling load $p_{{{\text{cr}}}}$ in dimensional units for an elliptical column with $l = 1$ m and $A = 20$ cm² with a varying $\alpha$ is presented in Fig. 7b. Here, the conversion from the dimensionless $p_{{{\text{cr}}}}$ to the dimensional buckling load $P_{cr}$ was calculated using Eq. 35 as $P_{{{\text{cr}}}} = \left( {EA^{n2} /l^{2n1} } \right)p_{{{\text{cr}}}}$. $p_{{{\text{cr}}}}$ increases with an increase in $\alpha$, similar to Fig. 7a. The increasing slope of $p_{{{\text{cr}}}}$ is steeper at smaller $\alpha$ values. The dimensionless $p_{{{\text{cr}}}}$ does not exhibit large deviations between the three different columns, whereas the dimensional $p_{{{\text{cr}}}}$ exhibits large deviations. This is because $p_{{{\text{cr}}}}$ is related to the Young’s modulus $E$, as shown in Eq. (35), which exhibits large deviations for the three different materials. As a result, the corresponding $p_{{{\text{cr}}}}$ is highly dependent on $E$, even if the non-dimensional $p_{{{\text{cr}}}}$ does not vary significantly among the three different materials.

Fig. 7

Buckling curves of a dimensionless ${p}_{\mathrm{cr}}$ and b dimensional ${p}_{\mathrm{cr}}$ by changing $\alpha$

For nonlinear elastic members, the slenderness ratio $s_{{\text{r}}}$ can be expressed as

$$ s_{{\text{r}}} = \frac{l}{{\sqrt {I_{{\text{g}}} /A^{z} } }}, $$

(39.1)

where $z$ is an exponential constant, so $s_{{\text{r}}}$ is dimensionless. Then, $z$ is derived such that $z = n_{1}$ and $s_{{\text{r}}}$ is finally expressed as

$$ s_{r} = \frac{l}{{\sqrt {I_{g} /A^{n1} } }} = \frac{l}{{\sqrt {\beta A} }}. $$

(39.2)

The area $A$ and length $l$ are expressed in terms of $s_{{\text{r}}}$ for a given $\beta$ as

$$ A = \frac{{l^{2} }}{{\beta s_{r}^{2} }},\,l = s_{r} \sqrt {\beta A} . $$

(40)

To visualize the effects $s_{{\text{r}}}$ on the buckling load $P_{{{\text{cr}}}}$, $P_{{{\text{cr}}}}$ versus $s_{{\text{r}}}$ curves are presented for two cases. Figure 8a presents the curves with a varying $A$ and Fig. 8b presents the curves with a varying $l$ according to Eq. 40. The remaining column parameters are also presented in this figure. As expected, as $s_{{\text{r}}}$ increases, $P_{{{\text{cr}}}}$ decreases, exhibiting a very large decreasing rate. For example, when comparing $P_{{{\text{cr}}}} = 21.33$ kN with $s_{{\text{r}}} = 50$ and $P_{{{\text{cr}}}} = 0.1754$ kN to $s_{{\text{r}}} = 200$ for $n = 2.16$ in Fig. 8a, the difference is 121.6 (= 21.33/0.1754) times, which is relatively large. This indicates that $s_{{\text{r}}}$ is one of the most important factors for column design.

Fig. 8

${s}_{\mathrm{r}}$ versus ${P}_{\mathrm{cr}}$ curves: a with changing $A$ for rectangular cross section and b with changing $l$ for elliptical cross section

It is important to gain a complete understanding of the load parameter $p\left( { > p_{{{\text{cr}}}} } \right)$ versus $\left( {\delta_{{\text{v}}} , \theta_{i} } \right)$ curve, which is called the equilibrium path. Figure 9 presents the equilibrium paths of rectangular cross sections with $\alpha = 0.5$ for three different materials. The equilibrium paths with respect to $\delta_{{\text{v}}}$ are presented in Fig. 9a. For $n = 2.16$, if $p < 0.5364\left( { = p_{{{\text{cr}}}} } \right)$, then $\delta_{{\text{v}}}$ does not deform and the column does not buckle. However, as $p$ gradually increases and eventually reaches $p_{{{\text{cr}}}} = 0.5364$, $\delta_{v}$ abruptly increases from $\delta_{{\text{v}}} = 0$ to $\delta_{{\text{v}}} = 0.4813$, causing fatal buckling of the column with unexpected and dangerous consequences. When the column buckles, there are possible two equilibrium paths: the unstable path indicated by the dashed line and stable path indicated by the solid line. The column only follows the unstable path, which is a pass-through path, from its initial straight state until it reaches the final steady equilibrium state of the post-buckling column (see Fig. 10b). As described previously, in the solution method for calculating $p_{{{\text{cr}}}}$, the unstable and stable paths meet at $p_{{{\text{cr}}}}$, resulting in a sudden jump from $\delta_{{\text{v}}} = 0$ in the initially straight state to $\delta_{{{\text{v}},{\text{pcr}}}} = 0.4813$ in the buckled state. The unstable path is strongly nonlinear and $\delta_{{\text{v}}}$ decreases as $p_{{{\text{cr}}}}$ increases. Additionally, the stable path is also strongly nonlinear and as $p_{{{\text{cr}}}}$ increases, $\delta_{{\text{v}}}$ increases, approaching the maximum physical value of a simple column at $\delta_{{\text{v}}} = 2$. For a linear elastic column with $n = 1$ (e.g., steel), the column remains straight until buckling at $p = 0.4112\left( { = p_{{{\text{cr}}}} } \right)$, meaning there is no unstable path and $\delta_{{\text{v}}}$ does not abruptly jump, unlike the nonlinear materials with $n = 2.16$ and $n =$ 4.785. Therefore, for linear columns, buckling can be detected immediately after it occurs, facilitating the avoidance of catastrophic collapse caused by buckling. After buckling with a stable path, the equilibrium path is strongly nonlinear, where $\delta_{v}$ increases with an increasing $p$. The equilibrium paths of the initial rotation $\theta_{i}$ are presented in Fig. 9b. These paths are also nonlinear, similar to the equilibrium path for $\delta_{{\text{v}}}$, where $\theta_{i}$ increases as $p$ increases and eventually approaches $\theta_{i} = \pi$.

Fig. 9

Equilibrium paths for rectangular cross section: a vertical deflection ${\delta }_{\mathrm{v}}$ and b initial rotation ${\theta }_{i}$

Fig. 10

Elastica for elliptical cross section: a buckling elastica by $n$ and b post-buckled column elastica for $n=2.16$ with a varying $p$

In practical engineering fields, it is necessary to understand the elastica behavior of buckling and post-buckling processes for column design. In Fig. 10a, for the elliptical cross section, the buckling elastica $\left( {\xi ,\eta } \right)$ of the steel ($n = 1$), annealed copper ($n = 2.16$), and NP8 aluminum alloy ($n = 4.785$) columns with $\alpha = 0.5$ are presented. The larger the value of $n$, the more severe the elastica. For $n = 1$, the column remains straight at the buckling load $p_{cr} = 0.393$, whereas for $n = 2.16$ and $4.785$, the buckling elastica deforms significantly, which corresponds to the equilibrium paths shown in Fig. 9. For $n = 2.16$, a very large vertical deflection of $\delta_{{\text{v}}} = 0.187\left( { = 1 - 0.813} \right)$ occurs at $p_{{{\text{cr}}}} = 0.51$, which is denoted by, leading to abrupt deformation immediately after column buckling. Because these sharp deformations of buckling elastica are unpredictable and cause catastrophic collapse, it is critical to exercise caution when determining $p_{{{\text{cr}}}}$ in nonlinear column design. The post-buckling elastica of the elliptical cross section for $\alpha = 0.5$ and $n = 2.16$ is presented in Fig. 10b, where four elastica with $p = 0.709, 0.603, 0.534$, and $0.511\left( { = p_{{{\text{cr}}}} } \right)$ are included. The dashed elastica with $p\left( { > p_{{{\text{cr}}}} } \right) = 0.709, 0.603 ,$ and $0.534$ is unstable before the bucking elastica for $p_{{{\text{cr}}}} = 0.511$ occurs. An unstable dashed elastica represents a path that cannot exist naturally. In the unstable region before the buckling elastica, an elastica with $p( > p_{{{\text{cr}}}} )$ is present in the upper region close to the unbuckled straight column and all elastica with $p > p_{{{\text{cr}}}} \left( { = 0.511} \right)$ exist in this upper region. As discussed above, the top end of the column catastrophically moves to the deformation coordinates (0.406, 0.267), which are indicated by ▶ in Fig. 10a, immediately after column buckling. After passing through the unstable elastica, the post-buckling elastica, which are indicated by the solid line, becomes stable. In this lower stable region, four buckled elastica with $p = 0.511\left( { = p_{{{\text{cr}}}} } \right), 0.534, 0.603$, and $0.709$ can be observed. Here, the top end of the elastica with $p > 0.603$ can theoretically be located below the base end because the vertical deflection $\delta_{{\text{v}}}$ becomes $\delta_{{\text{v}}} = 1$ when $p = 0.603$. For example, see the elastica with $p = 0.709( > 0.603)$.

Next, we discuss the internal forces ($N, Q, M$), and distributed strain $\varepsilon$ and stress $\sigma$ of buckled columns. The dimensionless forms of $\left( {\overline{n}, \overline{q}, \overline{m}} \right)$ can be converted into the dimensional form of ($N, Q, M$) using Eqs. 31–33. For example, a diagram of ($N, Q, M$) with $n = 4.785$ and $E = 455.8$ MPa for the NP8 aluminum alloy is presented in Fig. 11, where the input parameter areas are the elliptical cross section, $l = 1$ m, $a = 5.642$ cm, and $b = 4.514$ cm ($\alpha = 0.8, A = 20$ cm², and $I_{{\text{g}}} = 19.98$ cm^3.209). Based on these column parameters, the buckling load, $P_{{{\text{cr}}}}$, is calculated as $P_{{{\text{cr}}}} = 13.223$ kN. As expected, at the mid-arc of the elastica (i.e., $s = 0.5$ m), $N_{{\text{m}}}$ is $N_{{\text{m}}} \left( { = P_{{{\text{cr}}}} } \right) = 13.223$ kN and $Q_{{\text{m}}} = 0$ because $\theta_{{\text{m}}} = 0$. At the base end ($s = 0$), $N_{{\text{o}}}$ is $N_{{\text{o}}} = 0.667$ kN and $Q_{{\text{o}}} = 13.206$ kN with $\theta_{i} = 1.621$ rad, where $P_{{{\text{cr}}}} \left( { = 13.223 {\text{kN}}} \right) = \sqrt {N_{{\text{o}}}^{2} + Q_{{\text{o}}}^{2} }$. Additionally, as expected, at the mid-arc, $M_{{\text{m}}}$ with $y_{{{\text{s}} = 0.5 {\text{m}}}} = 0.445$ m is the maximum value because $M_{{\text{m}}} = P_{{{\text{cr}}}} \times y_{{{\text{s}} = 0.5 {\text{m}}}} = 5.882$ kNm.

Fig. 11

Buckling internal forces $(N, Q, M)$ for elliptical cross section with $n=4.785$ and $\alpha =0.8$

Finally, the distribution of ($\varepsilon ,\sigma$) for the buckling elastica at the mid-arc ($s = 0.5 {\text{m}}$), where the maximum bending moment $M_{{\text{m}}}$ occurs, is presented in Fig. 12. The input column parameters are the same as those in Fig. 11. Given $N$ and $M$, the distribution of ($\varepsilon ,\sigma$) along the cross-sectional depth $h$ can be obtained as follows:

$$ \varepsilon = \varepsilon_{{\text{N}}} + \varepsilon_{{\text{M}}} = \left( {\frac{\left| N \right|}{{EA}}} \right)^{n} \pm \left( {\frac{\left| M \right|}{{EI_{g} }}} \right)^{n} h, - \frac{b}{2} \le h \le \frac{b}{2}, $$

(41)

$$ \sigma = E\varepsilon^{1/n} . $$

(42)

Fig. 12

Buckling strain $\varepsilon$ and stress $\sigma$ at mid-arc for elliptical cross section

For the values of $N_{{\text{m}}} = 13.223$ kN and $M_{{\text{m}}} = 5.882$ kNm at the mid-arc in Fig. 11, the corresponding ($\varepsilon_{{\text{N}}} , \varepsilon_{{\text{M}}}$) can be obtained as $\varepsilon_{{\text{N}}} = + 1.596 \times 10^{ - 9}$ and $\varepsilon_{{{\text{M}},{\text{h}} = \pm {\text{b}}/2}} = \pm 0.278$, from which $\varepsilon = + 0.278$ and $\varepsilon = - 0.276$ at the two edges with $h = \pm b/2$ ($b = 4.514$) were obtained using Eq. 41. Note that $\varepsilon_{{\text{N}}} = + 1.596 \times 10^{ - 9}$ is very small and almost negligible compared to $\varepsilon_{{{\text{M}},{\text{h}} = \pm {\text{b}}/2}}$, and $\varepsilon$ is nearly equal to $\varepsilon_{{\text{M}}}$, implying that the change in the neutral axis position caused by $\varepsilon_{{\text{N}}}$ is negligible. Considering the values of $\varepsilon = + 0.278$ and $\varepsilon = - 0.276$ at the two edges, the linear relationship of $\varepsilon = \kappa h$ can be plotted, as shown in Fig. 12, where the curvature of $\kappa = 0.123/{\text{cm}}$ was calculated using Eq. 19. The nonlinear stress equation $\sigma = E\varepsilon^{1/n}$ was obtained and the result is presented in Fig. 12. The $\sigma_{{{\text{max}}}}$ values at the two edges were calculated as $\sigma_{{{\text{max}}}} = + 348.7$ MPa at the top edge and $\sigma_{{{\text{max}}}} = - 348.3$ MPa at the bottom edge. Because the shape of the nonlinearly distributed stress is almost rectangular, it is more advantageous to use the total cross-sectional area for the stress $\sigma$ compared to the case of linear materials. As shown in Fig. 12, this is beneficial for ensuring column safety when designing nonlinear elastic columns.

4 Concluding remarks

This study investigated the geometric and material nonlinear buckling of simple columns made of Ludwick-type materials. Before modeling nonlinear column buckling, a GMI ${I}_{\mathrm{g}}$ calculation method was derived based on theories from the open literature and reformulated for rectangular and elliptical cross sections. The buckling of a simple column was considered to apply the reformulated ${I}_{\mathrm{g}}$ to bending analysis. Nonlinear differential equations that govern the buckled column elastica were derived and numerically solved to obtain the buckling load and elastica of the column. An iterative numerical method was developed to calculate the elastica, where the Runge–Kutta method was used to integrate the governing equations and the bisection method was used to determine the rotational angle at the bottom hinged end. Numerical parametric experiments on GMI ${I}_{\mathrm{g}}$ for rectangular and elliptical cross sections were discussed in detail. As a major focus of this study, parametric studies on buckling behavior, including buckling load, equilibrium path, buckling and buckled elastica, and the strain and stress distribution of a buckled column, were conducted and discussed in detail. The buckling loads of materials and geometric nonlinear simple columns were reported for the first time in this paper. Additionally, a slenderness ratio applicable to nonlinear elastic members was derived and applied to the buckling problem. The results of this study should be useful for buckling analysis and nonlinear elastic column design. Further studies on the GMI ${I}_{g}$ for asymmetric cross-sectional shapes are required, as are further studies on support conditions other than the simply supported columns considered in this study.