1. Introduction
The Bernoulli [
1] and Euler [
2] theory of beams is a standard introductory subject in textbooks on elasticity [
3,
4,
5,
6,
7,
8,
9,
10,
11] and leads to the phenomenon of buckling, which has been considered in several conditions: (i) geometric and material non-linearities [
12]; (ii) in combination with shear [
13,
14] that is more significant for thin-walled beams [
15,
16,
17,
18,
19]; (iii) constraints [
20,
21,
22], such as hyper or non-local elasticity [
23,
24]; (iv) vibrations [
25,
26], that can be excited by unsteady applied forces [
27,
28,
29,
30,
31,
32,
33], leading to control problems [
34]; (v) steady mechanical [
35] or thermal [
36,
37,
38] effects; and (vi) vibrations of tapered beams [
39,
40,
41,
42,
43,
44,
45,
46,
47,
48,
49,
50,
51,
52,
53,
54,
55], with multiple applications like airplane wings and flexible aircraft and helicopters [
56,
57,
58,
59,
60,
61,
62,
63,
64]. Among this wide range of topics related to the buckling of elastic beams, the present paper focuses on geometric non-linearities associated with a large slope of the elastica.
The equation of the elastica of a beam is usually written in one of the two forms: (i) in Cartesian coordinates,
with the
x-axis along the undeformed beam; or (ii) in curvilinear coordinates,
with the arc length
s as a function of the angle of inclination. The linear theory assumes for (
1a) a small slope,
where the prime in this paper denotes the derivative with
x, and implies that the maximum deflection is small, compared with the length (
Figure 1),
However, the latter condition of small maximum deflection relative to length (
2b) does not imply [
65] linearity (
2a) in the case of “ripples” with a large slope (
Figure 2). The condition of linearity can be expressed in terms of a small angle of inclination,
that is equivalent to
The Euler–Bernoulli theory of beams states that the bending moment
M is proportional to the curvature
k,
that is the product of the Young modulus
E of the material by the moment of inertia
I of the cross-section. For a beam of constant cross-sections made of a homogeneous material, the bending stiffness
is constant. In the case of a uniform beam [
66], that is, with constant bending stiffness, geometric non-linearities can arise from the curvature,
,that is, the rate of change of the angle of inclination with the arc length
The curvature is given by
and thus, it is only in the case of a small slope (
2a) that the equation of the elastica is linear:
If the slope of the elastica is not small, its shape is specified by non-linear ordinary differential equations [
67,
68,
69,
70,
71,
72,
73,
74,
75,
76,
77,
78,
79].
The linear theory shows that if a critical buckling load
is reached, a beam in the tension deforms and gains a buckled shape, and higher critical loads,
with
, lead to a succession of harmonics
. In the present paper, it is shown that geometric non-linearities associated with a large slope,
do not affect the critical buckling load, but change the shape of the elastica that becomes a superposition of harmonics of the linear case,
The coefficients
are determined in this paper for the three cases of (i) cantilever, (ii) clamped, and (iii) pinned beams, and the shape of the elastica is illustrated taking into account non-linear geometric effects associated with a large slope. Before proceeding to discuss non-linear geometric effects in the Euler–Bernoulli theory [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11], the preceding classification of the references [
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45,
46,
47,
48,
49,
50,
51,
52,
53,
54,
55,
56,
57,
58,
59,
60,
61,
62,
63,
64] is complemented by a brief discussion of some additional references. The method of the elastica for non-linear beams, schematized in
Figure 3, involves the solution of ordinary differential equations [
66,
67,
68,
69,
70,
71,
72,
73,
74,
75,
76,
77,
78,
79]. The exact analytical solutions can be obtained using elliptic functions [
80,
81,
82,
83,
84,
85] for simpler loading cases.
The articles [
86,
87] show that large deflections are specified by a combination of incomplete and complete elliptic integrals. The results of these two articles are given with limited accuracy, because at that time, the calculations were not performed with digital computers. More accurate results of elliptic integrals are presented, for instance, in [
88,
89]. Furthermore, to obtain highly accurate numerical results, these problems may also be solved using the shooting-optimization technique. The aforementioned two methods for solving large deflections of beams are presented in [
90], where an inextensible elastic beam is hinged at one end, while the other end is assumed to be a frictionless support where the beam can slide freely. Additionally, the beam is under a moment gradient and the moment at each end of the beam can be varied from zero to a full moment by a scaling parameter. In [
90], the elastica theory serves to formulate the elliptic-integral method, and the results are obtained by an iterative process, while the governing set of differential equations are needed for the shooting-optimization technique and are numerically integrated using the fourth-order Runge–Kutta method. The results obtained from both methods are in close agreement to each other. The paper [
91] continues in this line of research, but considers the double curvature bending of the elastica under two applied moments in the same direction applied at the supports, and complements earlier studies that confined bending to one of the single curvature-type bendings. The two aforementioned methods are used. The elliptic integral technique provides analytical solutions to the governing non-linear differential equation for elasticas, while the shooting optimization method numerically integrates the equation using the fifth-order Cash–Karp Runge–Kutta method. Both methods provide almost the same stable and unstable equilibrium solutions and, for some cases of the unstable equilibrium configuration, the elastica can form a single loop or snap-back bending.
Continuing in this line of investigation, the paper [
92] considers the large deflection problem of variable deformed arc-length beams, also with a uniform flexural rigidity, but under a point load. In [
92], the ends are partially elastically supported against rotation (it covers both the cases of hinged or clamped ends). Both previously mentioned methods are also used, and the results obtained are in close agreement. This kind of problem highlights the possibility of two equilibrium states for a given load, implying the possibility of a snap-through phenomenon, the existence of a critical load, and a maximum arc-length for equilibrium. The analytic elastica solution of slightly curved cantilever beams, fixed at one end, while being deflected under couples and forces of various directions, is evaluated in [
93] using elliptic integrals. It has been shown that for some cases, the solution is very sensitive to small errors in the calculation of elliptic integrals. An analytic elliptic solution for the post-buckling response of a linear-elastic and hygrothermal beam, subjected to an increase in temperature and/or moisture content, is presented in [
94]. In [
94], the beam is pinned at both ends, and therefore the extensibility of the beam cannot be ignored. Additionally, it shows that the critical load is a maximum and, in the post-buckling regime, the magnitude of the load decreases. The beam theory can be extended to more complex structures [
95].
Other methods using the elastica approximation are useful for more complex loadings. The paper [
96] determines a parametric solution to the elastic pole-vaulting problem, where the pole is taken to be a thin uniform elastic column with the upper end being subjected to lateral and transverse forces and a bending moment at the same time as the bottom end is free to pivot during the vaulting. The parametric solution is given in terms of tabulated elliptic integrals. The investigation [
97] gives a closed-form solution for the problem of a non-linear elastica and buckling analysis of a straight bar, due to concentrated and uniformly distributed loads, while the flexural rigidity varies along the bar. It achieves an integral closed-form solution of the equation governing the equilibrium of the bar, by applying successive functional transformations. The paper [
98] presents the buckling analysis of an elastic continuous bar on several rigid supports subjected to end-compressive forces, and assumes that the compressive forces and flexural rigidities vary from one span to the next. The closed-form solution expressed by elliptic integrals is derived for each span. The same authors presented an analytic solution for the problem of non-linear elastic buckling of a straight bar subjected to bending compression due to forces and couples at the ends superimposed to a uniformly applied transverse load along its length in [
99], by applying functional transformations. Additionally, the same authors analysed the problem of non-linear buckling for a straight uniform bar, fixed at its base and free at its upper end, due to the bar’s own weight in [
100]. It yields reliable results in agreement with the physical problem. The same procedure was used in [
101] to study the problem of non-linear buckling for a straight bar of uniform cross-sections and flexural rigidity, lying on a continuous elastic medium, and subjected to terminal point-loads and bending moments. In all the works described above, the effects of transverse deformation due to axial, lateral, and transverse forces are negligible.
The paper [
102] constructs an exact parametric analytic solution for the full non-linear differential equations of the cantilever elastica due to end loads, end couples, and also including the effects of transverse deformation, completing, for instance, the work [
96]. Translational or rotary springs may be used [
103] to brace a beam increasing its critical buckling load, or to have the opposite effect of decreasing the critical buckling load to facilitate demolition. The buckling can also be facilitated or opposed by supporting the beam on a continuous bed of springs [
66]. A beam of variable cross-sections can taper in two directions [
104], for example, in the case of a pyramidal beam representing an airplane wing with chords much larger than the thickness affecting the natural frequencies of bending modes.
Following this introduction (
Section 1) to the Euler–Bernoulli theory of beams, the core of the paper focuses on geometric non-linearities associated with a large slope of the elastica. The equation of the elastica of a uniform beam (
Section 2) is obtained without restriction on the slope of the elastica (
Section 2.1). The well-known solutions for the linear case of a small slope are briefly recalled (
Section 2.2) because they supply the harmonics for non-linear corrections (
Section 2.3). The linear and non-linear cases are also compared, as concerns the boundary condition with small and large slopes, respectively, at the free end of a cantilever beam (
Section 2.4). The cantilever beam is considered first (
Section 3) to obtain the non-linear shape of the elastica (
Section 3.1) and to compare the linear approximation with non-linear corrections of all orders (
Section 3.2). The non-linear effects on the shape of the elastica are illustrated using the representation as a superposition of linear harmonics, by truncating the series in an analytic approximation (
Section 3.3), and adding a larger number of terms in a numerical computation (
Section 3.4). The non-linear buckling is also considered for clamped and pinned beams (
Section 4), starting with the non-linear effects of a large slope (
Section 4.1), that do not affect the critical buckling load (
Section 4.2), but do change the shape of the elastica by the generation of harmonics (
Section 4.3), illustrated by numerical calculations (
Section 4.4). The conclusion (
Section 5) highlights the use of linear buckling harmonics to specify the shape of the elastica for non-linear buckling with a large slope.
5. Conclusions
For a cantilever or pinned or clamped beam, the linear buckling (using the linear approximation) corresponds to a succession of increasing axial loads, given, respectively, by (
74b), (
75b), and (
76b), and corresponding harmonics, given, respectively, by (
74a), (
75a), and (
76a) for the buckled shape of the elastica. Buckling first occurs for the smallest axial load corresponding to the fundamental buckled shape. The non-linear effect is to add harmonics to the fundamental mode; therefore, the first consequence is: (i) not changing the critical buckling load, that remains the lowest; (ii) changing the buckled shape of the elastica by superimposing on the linear fundamental mode its harmonics with specified amplitudes. The non-linear shape of the buckled elastica has been illustrated: (a) for cantilever, pinned and clamped beams, respectively, in the
Figure 8,
Figure 9,
Figure 10,
Figure 11,
Figure 12,
Figure 13,
Figure 14,
Figure 15 and
Figure 16; (b) each figure consists of four panels, one each for the fundamental mode
, and for the next three modes
; (c) the first of each set of the three figures, namely, the
Figure 8,
Figure 11 and
Figure 14, shows the effect of changing amplitudes among four values; (d) the second of each set of the three figures, namely, the
Figure 9,
Figure 12 and
Figure 15, shows the effect of changing the length of the beam among four values; (e) the last of each set of three figures, namely, the
Figure 10,
Figure 13 and
Figure 16, indicates the magnitude of non-linear effects relative to the linear approximations. In all cases, the non-linear effects are larger for higher-order modes of shorter beams, leading to “ripples” with a large slope (
Figure 2) compared with the smoother or less undulated fundamental mode (
Figure 1).
The
Table 3 compares the values of the successive loads (the first five orders) that can buckle the beam for the three cases studied, between the linear and lowest-order non-linear approximations. Because the expressions to deduce the buckling loads are exactly the same in the two approximations, the
Table 3 shows that the critical values obtained in this paper are exactly the same as that in the literature [
3,
4,
7,
66] which considers linearisation of the equations.
The critical buckling load can be changed by using translational or rotational springs that favour or oppose buckling [
103], and the shape of the buckled elastica is further modified by transverse concentrated or distributed forces [
65]. The two aspects of (i) the critical buckling load and (ii) the shape of the buckled elastica are implicit in the vast literature on non-linear buckling of beams, and have been made explicit using the theory of Euler–Bernoulli beams in its simplest form. The
Table 4,
Table 5 and
Table 6 show the maximum numerical absolute errors between the linear approximation, used in the vast literature, and the lowest-order non-linear approximation, used in this paper, for several lengths
L of the beam, for the first four orders of buckling
n and for each type of beam. For all the three types of beams, the difference is more significant for shorter beams and for higher orders of buckling.
The solution of (
8) shows that the exact non-linear shape of the elastica is a superposition of harmonics of the linear problem (
9) where: (i) the fundamental buckling mode is determined from the linear approximation
; and (ii) the generation of harmonics is a non-linear effect. The present approach to the non-linear theory of bending with a large scale of Euler–Bernoulli beams thus uses an approach that is different from the classical and more recent research, in that it represents non-linear effects as a generation of harmonics.
The representation of the non-linear buckled elastica by a series of linear harmonics is an alternative to the classical solutions in terms of elliptic functions. This is an example of the fact that the solution of the same problem can have quite different representations. Two equivalent representations can be quite different in terms of the information they highlight, and this is the case here. There are three main differences: (i) the use of a series of elementary functions is simpler than the use of special functions; (ii) the elliptic functions are difficult to visualize, whereas the linear harmonics are more intuitive; (iii) the decomposition into linear harmonics shows, through their amplitudes, which are excited most, and give a greater contribution to the final shape of the elastica. The latter information (iii) is totally missing from the solution in terms of elliptic functions. Among the different solutions of the same problem, it is often the simplest one that is most illuminating.