A targeted review on large deformations of planar elastic beams: extensibility,distributed loads,buckling and post-buckling

Abstract

In this paper, we give a targeted review of the state of the art in the study of planar elastic beams in large deformations, also in the presence of geometric nonlinearities. The main scope of this work is to present the different methods of analysis available for describing the possible equilibrium forms and the motions of elastic beams. For the sake of completeness, we start by giving an overview of the nonlinear theories introduced for approaching this argument and then we account for the variational principles and deformation energies introduced for modelling beams undergoing large deformations and displacements. We then consider different kinds of loads treated in the literature and the corresponding induced beam deformations. We conclude by accounting for the available analysis for stability and some considerations about problems where live loads are applied, as well as by describing some relevant numerical methods of use in the applications we have in mind. The selection criterion for the reviewed papers is dictated by the need to study large deformations and the dynamics of pantographic sheets. (Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc R Soc A 2016; 472(2185): 20150790), dell’Isola et al. (Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z Angew Math Phys 2015; 66(6): 3473–3498), Turco et al. (Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z Angew Math Phys 2016; 67(4): 1–28)].

Keywords

Nonlinear beams Euler–Bernoulli beam Timoshenko beam nonlinear elasticity large deformations of beams

1. Introduction

In this paper, we want to discuss the state of the art of large deformations of elastic beams in the framework of nonlinear elasticity. There are many works that analyse nonlinearity in large deformations of beams. We obviously cannot manage to make a complete view of so wide a panorama, but we have selected a part of the principal works that treat this topic and reordered them on the basis of their peculiarities and of the approach to the problem. As specified in the conclusions, we need to establish the state of the art in the specified context in order to study lattices of beams, which form the microstructures of some metamaterials. Concerning the method of approaching the problem, we have to say that it is possible to recognize two fundamental lines, based on the procedure used to obtain the governing equations: the variational method and the method of the balance of force and momentum. The variational method, first introduced by Euler [1] and Lagrange [2, 3], allows the equations of motion and the stability equilibrium conditions to be derived in a very simple and efficient way by minimizing the functional of energy (an example of the so-called principle of virtual work). For the simplicity of its application and for its powerful capabilities to study also the most complex structures, the variational method is obviously very often used when searching the equilibria of mechanical systems.

We divide the papers included in this review on the basis of their central target. In fact, every work could be categorized on the basis of more than one aspect: we tried to check the fundamental one. There are some works that treat large deformations of beams in a general way; in Section 2, we give an overview of these general papers [4 –16] and we also include in it some textbooks, written about nonlinear theories of beams [17 –20].

In Section 3, we introduce some articles [21 –31] that analyse the different loads that can be applied to a beam. So we will see how the beam is deformed if subjected to a point load or to a distributed one. In some cases, we also have more complex kinds of beam loading. This particular section is fundamental because it allows us to understand the relations between the externally applied forces and moments and the deformations of a beam.

We then present papers that analyse the stability of a beam, with different loads, in Section 4, as well as a textbook by Atanackovic [32] for a general overview. We discuss many papers on this fundamental argument [33 –42]. In fact, studying the stability of beams can be a very difficult and delicate topic and many aspects of the problems linked to the stability of a structure can be approached.

After having studied stability and instability, it is natural to approach the dynamic processes that can occur in beam theory [43 –52]. In fact, as stated by Bolotin [53], following the D’Alembertian spirit, ‘The dynamic stability of mechanical systems represents a specific stability of motion.’

Having in mind the aspects of stability introduced in the previous sections, we then present the works [54 –58] in which stretching, bending and shear deformations are particularly analysed. These kinds of work, like those on stability, incidentally, are mostly approached using a variational method to derive the governing equations. So we will see different ways to write down the energy of stretching, bending or shear and different points of view about this kind of problem. Particular attention will be paid to buckling and post-buckling problems, which are the central topic of many works [59 –66].

We proceed in the review with a section in which are presented some different kinds of beam [67 –73]. It could have been reasonable to put this section at the start of our paper, but we chose to put it near the end because in the study of large deformations of beams the fundamental topic is not the particular type of beam we are considering, but the nonlinearity we are treating and the origin of this nonlinearity. It is also clear that the large deformation problem is treated in these articles, but giving importance to the kinematic aspects of the problem, i.e. to the particular kinds of constraint applied to the beams (cantilever, simply supported, hinged, …).

Finally, having in mind targeted applications to metamaterials undergoing large deformations and displacements, which we intend to design by means of microlattices of constrained beams, we report some interesting papers in which some numerical methods are discussed.

In this review paper, we analyse only a minor part of the enormous material published on the topic of large deformations of beams. But we try to give a complete overview of the problems treated in the literature with a view to the particular application that we have in mind. For a more complete panorama of some aspects of the theory of large deformations of beam, one can refer to the textbook by Antman [17].

2. Large deflections of beams: kinematics, deformation and action

One who wants to approach problems relative to large deformations of beams should first analyse more general problems in nonlinear mechanics and elasticity. Different aspects of nonlinear theories can be found in interesting books, such as Antman [17], mainly for the analysis of nonlinear problems in elasticity.

Antman [17] assumes a complete variational postulation of mechanics and treats the particular case of planar beams in detail. The approach used in this textbook is to lay down a general theory for each kind of elastic body, by formulating specific problems and by introducing the needed mathematical methods. This kind of analysis is carried out for strings, rods, shells and 3D bodies. The followed scheme is generally applicable in every part of physics: formulation of the problem, analysis (i.e. mathematical methods adopted to describe and solve the problem) and interpretation. The arguments are exposed in a logical way. Antman starts by analysing the case of extensible strings. In this framework, the equivalence of the linear impulse–momentum law with the principle of virtual power is shown. The given proof is completely rigorous and technically simple. A large part is devoted to perturbation methods. Some effort is also devoted to the resolution of practical problems, such as strings loaded with vertical, normal and central loads or the catenary problem. As an application of perturbation methods the study of problems of travelling waves is presented. On this topic we want to cite the interesting papers of Gavrilov et al. [74] and Bersani et al. [75].

After this large study on elastic strings, Antman introduces planar problems and the theory of elastic rods. In this framework, he develops the bifurcation theory, by presenting some rigorous theorems, and studies some problems of buckling. Furthermore, particular attention is paid to dynamic aspects and to stability: an important aspect (one among several others) that is approached is the whirling of strings and rods. As an example of this last topic, we have a variational treatment of the following problem

N^{″} + λ \hat{ν} (N) = 0, N (0) = 0, N^{'} (1) = λR

(1)

which describes the radial steady states of a rotating uniform elastic rod of length 1, with its end s=0 free, and with its end s=1 attached to a rigid ring of radius R rotating about its centre with angular speed $\sqrt{λ ∕ ρA}$ . N is the tension in the rod and $\hat{ν} (N)$ is the stretch it produces. The variational principle is written by introducing the conjugate $W^{*}$ of the stored-energy function, which has the property that

\hat{ν} (N) = W_{N}^{*} (N)

(2)

If we have $λ > 0$ , then the variational principle consists of minimizing the functional

Φ [N] : = \int_{0}^{1} [\frac{1}{2} N^{'} {(s)}^{2} - λ W^{*} (N (s))] ds + λ R N (1)

(3)

over the closed convex set

K : = \{N \in W_{2}^{1} (0, 1) : N (0) = 0, N (s) \leq 0 \forall 0 \leq s \leq 1\} .

(4)

In the view of a variational approach to continuum mechanics, Antman introduces the classical multiplier rule and specifically the Lagrange multipliers. Always by referring to variational principles, Antman makes an overview of 2D and 3D elastic bodies: so, after the presentation of the theory of rods deforming in space, we can find a discussion on equilibria of shells and a general theory of 3D nonlinear elasticity. A conclusive section is devoted to the analysis of dynamic aspects in the general nonlinear theory of elasticity.

We also want to cite the textbooks by Fertis for a general study of nonlinear mechanics [18] and structural engineering [19], and by Luongo and Zulli [20] for the attention given to the mathematical aspects involved in the nonlinear theory of beams.

In the literature are present not only books covering the general theory of large deformations of beams, but also many interesting papers that analyse several aspects of this topic [4 –16]

We have found that one of the most interesting works on nonlinear beam theory is that of Reissner [4], which is a sort of prototype of the study of deformations of nonlinear beams. Following Reissner [4], we see that, studying plane deformations of beams in large deformations, he introduces a new generalized form of constitutive relations in terms of axial force strain $ε$ , shear force strain $γ$ and bending strain $κ$

\begin{matrix} N & = f_{N} (ε, γ, κ) \end{matrix}

(5)

\begin{matrix} T & = f_{T} (ε, γ, κ) \end{matrix}

(6)

\begin{matrix} M & = f_{M} (ε, γ, κ) \end{matrix}

(7)

Once the constitutive relations are defined, we can write the virtual work of internal forces as

δ W^{i} = \int_{s_{1}}^{s_{2}} (M δκ + N δε + T δγ) ds

(8)

Together with the expression of external work, equation (8) will give the equilibrium (balance) equations, once integration by parts is performed.

On the basis of the work by Reissner, one can find in Irschik and Gerstmayr [5] and Humer and Irschik [6] a further investigation of beam theory. In Irschik and Gerstmayr [5], the general formulation is restricted to the Euler–Bernoulli model of the beam, so that one can neglect the shear force and write

\begin{matrix} N = f_{N} (ε, κ) \end{matrix}

(9)

\begin{matrix} M = f_{M} (ε, κ) \end{matrix}

(10)

The paper of Irschik and Gerstmayr [5] is the basis for that of Humer and Irschik [6], in which the problem of equilibrium and stability of extensible elastica is studied in the presence of a sliding constraint, so that the length of the deformed elastica is unknown. Humer and Irschik [6] prescribe a continuum-mechanicsbased derivation of the field equations for large displacements; the constitutive relations are determined by assigning the stressstrain relationship. Moreover, always referring to Reissner [4], the nonlinear local equilibrium equations

N^{'} + ϕ^{'} \frac{M^{'}}{Λ_{x 0}} + n = 0, ϕ^{'} N - {(\frac{M^{'}}{Λ_{x 0}})}^{'} - q = 0

(11)

are derived in the variational framework by using the principle of virtual work. Here, we have that $ε = Λ_{x 0} - 1$ is the normal force strain (and $Λ_{x 0}$ is the stretch of an element of the beam axis) and $κ = ϕ^{'}$ is the bending strain. These equations are written in terms of the external force per unit of length of the undeformed beam in the direction and orthogonal to the deformed beam, respectively, q and n. It is obtained that

N = Y A (Λ_{x 0} - 1), M = Y J_{y} ϕ^{'}

(12)

with Y the Young modulus, A the cross-sectional area and the cross-sectional moments of inertia

J_{y} = \int_{A} z^{2} dA, K_{y} = \int_{A} z^{4} dA

(13)

Therefore, a kind of micro–macro identification procedure is also presented in Humer and Irschik [6].

Abedinnasab et al. [7] introduce, more precisely, Hamilton’s principle. Using Hamilton’s principle

\int_{0}^{t} (δT - δ U^{g} - δ U^{e} + δ W^{nc}) dt = 0,

(14)

the governing equations of nonlinear Euler–Bernoulli beams with finite strains are derived. Here $δT$ is the variation of kinetic energy, while $δ U^{g}$ and $δ U^{e}$ are the variations of gravitational and elastic potential and $δ W^{nc}$ is the term relative to nonconservative forces and moments. Abedinnasab et al. make a strain formulation of the problem by using the exact complete Green–Lagrange strain tensor. Moreover, a comparison between the linear case and the nonlinear inextensible case is presented. In fact, Abedinnasab et al. add to the Hamilton’s principle a Lagrange multiplier $θ$ , corresponding to the inextensibility constraint: $δ (θ, e)$ , where e is the axial strain of the beam. An experimental verification of the nonlinear equations is also shown.

In Park and Gao [8], a new Euler–Bernoulli model is introduced using a modified couple stress theory. Park and Gao derive the equations of motion, minimizing the total potential energy; this model contains an internal material length scale and can capture the size effect, unlike the classical Euler–Bernoulli model. The strain energy is written as

U = \frac{1}{2} \int \int \int_{Ω} (σ : ϵ + m : χ) dv

(15)

where $σ$ is the stress tensor and $ϵ$ is the strain tensor, while $m$ is the deviatoric part of the couple stress tensor and $χ$ is the symmetric curvature tensor. $Ω$ is referred as the region occupied by the material. Using the displacement vector $u$ and the rotation vector $θ = \nabla \times u$ , the external forces work is expressed as

W = \int \int \int_{Ω} (f \cdot u + c \cdot θ) dv + \int \int \int_{∂Ω} (t \cdot u + s \cdot θ) da

(16)

where $f$ , $c$ , $t$ and $s$ are the body force, the body couple, the traction and the surface couple. With the procedure of Park and Gao, a stress energy is obtained

U = - \frac{1}{2} \int_{0}^{L} (M_{x} + Y_{xy}) w^{″} (x) dx

(17)

where the resultant moment $M_{x}$ and the couple moment $Y_{xy}$ are defined by

M_{x} = \int_{Ω} σ_{xx} z dΩ, Y_{xy} = \int_{Ω} m_{xy} dΩ

(18)

Another method developed to analyse elastica problems of beams using a matrix displacement approach is presented by Yang [9]. The displacement considered is not small if compared with the length of the beam and a midpoint tangent incremental method is used to predict the nonlinear path; the described method is applied to the case of a cantilever beam, to a diamond-shaped frame and to a square frame.

An important contribution to reformulate the theory of nonlinear elastic theory of beams is given by Steigmann. Steigmann and Faulkner [10, 11] present a fundamental work on the simplest theory of spatial rods in a variational setting. Steigmann and Faulkner derive the necessary conditions for minimizers of the potential energy. In particular, in Steigmann and Faulkner [10], attention is paid to the description of 3D deformations of elastic rods in the theory of Kirchhoff–Clebsch to model the flexural and the torsional response. The principal contribution of this work consists of the derivation of the Weierstrass and Legendre necessary conditions for rods. These conditions require that the vector describing the curvature and twist of the rods belongs to a domain of convexity of the strain energy function at every material point in a minimizing configuration; if the strain energy is nonconvex at some parts of its domain, then energy minimizers may have discontinuities in the curvature and twist. Faulkner and Steigmann [11] present, always in the scheme of the Kirchhoff–Clebsch theory, the determination of all the statically controllable deformations for the elastic inextensible rods. An interesting result is the demonstration that the most general deformation is the case in which a rod initially in the shape of a helix is deformed into another helix.

The previous results are applied in other papers by Steigmann and colleagues [12 –16] to the case of spatial lattices or networks of rods.

3. Distributed and concentrated external loads

The external loads contribute to determine the deformations of a beam. It is clear that if one wants to study the deformations of a beam, one must suitably account for the effects of the external forces. There are obviously different physical situations, which correspond to the different kinds of force applied to the beam. In the problem of large deformations, the energy due to the load plays a fundamental role in obtaining the equations of motion; this shows some further difficulties with respect to the linear elastic case. Indeed, the balance of moments and the balance of force can be estimated only a posteriori on the unknown equilibrium shape. A number of papers [21 –31] present different possibilities for loading a beam. One of the simplest distinctions we can make about different kinds of applied force is to divide them into concentric or point loads and distributed loads. We notice that we do not refer directly to forces, because we can also use the term ‘load’ to refer to a concentrated moment or to a moment distribution.

Xiao [21] analysed the case of a prismatic cantilever beam subjected to a uniformly distributed load, whose solution is derived in an approximate analytical way via a homotopic analysis method.

The homotopic analysis method, which is also a useful method for analysing systems in which soliton solutions occur, has been introduced by Liao [76]. This is a semi-analytical technique for solving nonlinear differential equations; it has the relevant property of being applicable to not only weakly but also strongly nonlinear problems. The aim of this method is to generate a convergent series: in this way the original nonlinear equation can be transformed into an infinite number of linear equations, without making any assumptions about small or large physical parameters. If one considers a general nonlinear differential equation

N [u (x)] = 0

(19)

where $N$ is a nonlinear operator, then it can be transformed to the so-called zeroth-order deformation equation

(1 - q) L [U (x; q) - u_{0} (x)] = c_{0} q N [U (x; q)]

(20)

where $L$ is an auxiliary linear operator; $u_{0} (x)$ is the initial guess for u(x); c₀ is a constant, called the convergence-control parameter; and $q \in [0, 1]$ is the embedding parameter derived from the homotopic theory. The function $U (x; q)$ has the property of being $U (x; 0) = u_{0} (x)$ and $U (x; 1) = u (x)$ so that, increasing q from 0 to 1, the solution $U (x; q)$ deforms from the initial guess $u_{0} (x)$ to the solution u(x). This function can be expanded in a Taylor series around q=0

U (x; q) = u_{0} (x) + \sum_{m = 1}^{\infty} u_{m} (x) q^{m}

(21)

and by choosing the convergence-control parameter c₀ the convergence of the series at q=1 can be assured. In this way we obtain the homotopic series solution

u (x) = u_{0} (x) + \sum_{m = 1}^{\infty} u_{m} (x)

(22)

which allows us to write down an equation for $u_{m} (x)$ and so obtain the searched infinite number of linear equations instead of the nonlinear one.

Xiao [21] therefore use the homotopic analysis method to arrive at the Euler–Bernoulli equation for the cantilever beam

\frac{dM}{ds} = EI \frac{d^{2} ϕ}{d s^{2}} = - qs \cos ϕ,

(23)

M being the bending moment, s the curvilinear abscissa, q the magnitude of the distributed load and $ϕ$ the angle of rotation. By introducing a dimensionless variable $z = s ∕ L$ , the following Dirichlet problem is obtained

\frac{d^{2} ϕ}{d z^{2}} + kz \cos ϕ = 0, ϕ^{'} (0) = 0, ϕ (1) = 0

(24)

with $k = q (L^{3} ∕ EI)$ . Clearly a curvilinear abscissa has been introduced; this implies that the extreme in 1 is constrained while the one in 0 is free. The solution is obtained via the homotopic analysis method. A similar problem is investigated by Wang et al. [22]. In fact, Wang et al. [22] use the same homotopic analysis resolution method to solve the nonlinear problem

\begin{matrix} \{\begin{matrix} θ^{″} + α \cos θ = 0 \\ θ (0) = 0 \\ θ^{'} (1) = 0 \end{matrix} \end{matrix}

(25)

introducing the decomposition

θ (ξ) = \sum_{k = 1}^{+ \infty} a_{k} ξ^{k}

(26)

where a nondimensional quantity $ξ = s ∕ L$ has been defined, with s the curvilinear abscissa and L the length of the beam in the reference configuration.

The rotation angle of the cross-section plane at the tip is denoted $ϑ_{B} = ϑ (1)$ . The vertical displacement of the free tip is given by

f_{B} = ℓ - 2 [E (μ) - E (ϕ, μ)] \sqrt{\frac{EI}{P}}

(27)

where $E (μ)$ is the complete elliptic integral of the second kind, $E (ϕ, μ)$ is the elliptic integral of the second kind, and

μ = \sqrt{\frac{1 + \sin ϑ_{B}}{2}}, ϕ = \arcsin (\frac{1}{\sqrt{2} μ})

(28)

If a large deformation is considered, one has to solve a nonlinear algebraic equation

\sqrt{β} = K (μ) - F (ϕ, μ)

(29)

where $K (μ)$ is the complete elliptic integral of the first kind and $F (ϕ, μ)$ is the elliptic integral of the first kind. To solve this equation, Wang et al. [22] again used a homotopic analysis method, as done by Xiao [21].

In general, the length of a beam depends on the deformation and therefore on the intensity of the external load. Humer [23] considered the large deformations of a slender beam under a concentrated force. On the basis of the Reissner variational principle [4], a local form of the balance equation is derived

\int_{l} [- (\frac{∂M}{∂s} + N Λ_{0} \cot ϕ) δϕ - (\frac{N}{\sin ϕ} + p) \frac{∂δw}{∂s}] ds + M δϕ |_{s = l} = 0

(30)

where M and N are the bending moment and the normal force, respectively, p is the tip force in the vertical direction and $Λ_{0}$ is defined by

Λ_{0} = \sqrt{{(1 + u^{'})}^{2} + {(w^{'})}^{2}}

(31)

The representation of the solution is given in terms of elliptic integrals. A similar case, a beam that can slide relative to its spatially fixed supports as soon as external forces are applied, is investigated by Humer and Irschik [24]. The extensible case is also approached. Referring to Reissner’s geometrically exact relations for the plane deformations of beams, we can write down some constitutive relations on the stress–strain level. The virtual work is expressed in terms of the deformation gradient $F$ and the first Piola–Kirchhoff stress tensor

δ W^{int} = \int_{ω} P : δ F dΩ

(32)

Numerical proofs are provided.

Large deflections of cantilever beams of Ludwick-type material are studied by Lee [25]. A Ludwick-type material can be characterized by a stress–strain curve like

σ = E ε^{1 ∕ n}

(33)

where $σ$ is the stress, $ε$ the strain and E and n given material constants. The beam under study is subjected to a combined load constituted of a uniformly distributed load w and a vertical concentrated force P at the free end.

The problem involves both geometrical and material nonlinearities, implying a complicated nonlinear differential equation. If k is the curvature, $dϕ ∕ ds$ , and $ϕ (s)$ is the rotation of any cross-section along the arc length with respect to the x axis, then

\frac{d^{2} ϕ}{d s^{2}} = \frac{- w (L - s) \cos ϕ - P \cos ϕ}{E I_{n} (1 ∕ n) {(k)}^{\frac{1 - n}{n}}} ϕ (0) = 0, ϕ^{'} (L) = k (L) = 0 .

(34)

This equation is solved using a fifth-order Runge–Kutta method.

A numerical analysis of the problem of a slender, uniform elastic beam, loaded under its own weight and built into a supporting wall, is presented by McMahon [26]. It is shown that there exists a critical length, which maximizes the lateral extension. The beam makes an angle $θ_{0}$ with the support; it is described by the problem

\begin{matrix} \{\begin{matrix} θ^{″} = - \frac{w}{EI} (l - s) \cos θ \\ θ (0) = θ_{0} \\ θ^{'} (l) = 0 \end{matrix} \end{matrix}

(35)

The critical length is estimated using numerical tools.

An analysis from a physical and analytical viewpoint of the equation

ω u^{‴} - (γ + \int_{0}^{1} {[u^{'} (x)]}^{2} dx) u^{″} = g

(36)

whose solutions are the equilibria of a thin extensible beam subjected to an external load, is provided in Zelati et al. [27], where the boundary problem with the conditions $u (0) = u (1) = 0$ and $u^{″} (0) = u^{″} (1) = 0$ is studied in a variational manner. Here $g \in L^{2} (0, 1)$ represents the lateral static load distribution, while $ω$ is a positive parameter of the order $h^{2} ∕ L^{2}$ , with $h < L$ the thickness of the beam, and $γ \in ℜ$ represents the longitudinal displacement of the ends and is positive when the beam is stretched and negative when it is compressed. The aim of the article is to understand how the steady-state solutions are affected by the presence of an external load. A weak solution of equation (36) is a function $u \in H^{2} (0, 1) \cup H_{0}^{1} (0, 1)$ , such that

ω \int_{0}^{1} u^{″} (x) w^{″} (x) dx + (γ + \int_{0}^{1} {[u^{'} (ξ)]}^{2} dξ) \int_{0}^{1} u^{'} (x) w^{'} (x) dx = \int_{0}^{1} g (x) w (x) dx

(37)

for every test function $w \in H^{2} (0, 1) \cup H_{0}^{1} (0, 1)$ . Equation (36) represents the static counterpart of many different evolution equations arising from both elastic and viscoelastic theories, for example the quasilinear equation describing small transverse deflections of a Euler–Bernoulli beam proposed by Woinowsky-Krieger [77]:

\partial_{tt} u (x, t) + ω \partial_{xxxx} u (x, t) - (γ + \int_{0}^{1} {[\partial_{ξ} u (ξ, t)]}^{2} dξ) \partial_{xx} u (x, y) = g (x)

(38)

The most important results in Zelati et al. [27] involve (i) providing a detailed variational derivation of equation (36) to obtain a closed-form solution for the post-buckling configurations and (ii) analysing the homogeneous case with an explicit formula for the solutions, for all values of g and $ω$ .

Morgan and Cannell [28] considered a complex loading in the particular case of tree trunks and branches. This article is an interesting example of an application of cantilever beam theory, which is applied in the framework of large deflections and with any patterns of distributed loading, point loading or externally applied couples. The large-deflection problem is solved by using the elementary theory for untapered beams, i.e. by considering a cantilever as a series of untapered segments, each of them with a small deflection, and building up to a large deflection at the free end. The mathematical apparatus used in solving the presented problem is known as the transfer matrix method.

The interesting case of a combined tip point loading is analysed by Tari [29], who applies a method based on an automatic Taylor expansion technique to determine the parametric large-deflection components of an Euler–Bernoulli cantilever beam. The starting point is to write down the characteristic equation of the beam’s deflection:

\frac{d^{2} θ (s)}{d s^{2}} = α^{2} \sin (θ (s) - ϕ), θ (0) = 0, \frac{d θ (1)}{ds} = β

(39)

where $α^{2}$ and $β$ are two constants written in terms of a general force f ₀ with an angle $ϕ$ and a general momentum M₀

α^{2} = \frac{f_{0} L^{2}}{EI}, β = \frac{M_{0} L}{EI}

(40)

The displacement components are

\frac{dx (s)}{ds} = \cos (θ (s)), x (0) = 0

(41)

and

\frac{dy (s)}{ds} = \sin (θ (s)), y (0) = 0

(42)

To solve the system, the angular deflection $θ (s)$ should be identified first; then one would proceed to solve for the displacement components. After having introduced a new angular deflection $ψ (s) : = θ (s) - ϕ$ , one can proceed with the automatic Taylor expansion technique, writing

ψ (s) = c_{0} + c_{1} (s - s_{0}) + \lim_{M_{\infty} \to \infty} \sum_{i = 2}^{M_{1}} \frac{1}{i!} \frac{d^{i} ψ (s)}{d s^{i}} {(s - s_{0})}^{i}

(43)

and for the displacement components

\begin{matrix} x (s) & = x_{M_{2}} (s_{0}) + \lim_{M_{\in} \to \infty} \sum_{i = 2}^{M_{2}} \frac{1}{i!} \frac{d^{i} x (s)}{d s^{i}} {(s - s_{0})}^{i} \end{matrix}

(44)

\begin{matrix} y (s) & = y_{M_{3}} (s_{0}) + \lim_{M_{∋} \to \infty} \sum_{i = 2}^{M_{3}} \frac{1}{i!} \frac{d^{i} y (s)}{d s^{i}} {(s - s_{0})}^{i} \end{matrix}

(45)

where

\begin{matrix} x_{M_{2}} (s_{0}) = \lim_{M_{\in} \to \infty} \sum_{i = 2}^{M_{2}} \frac{{(- 1)}^{i + 1}}{i!} \frac{d^{i} x (s_{0})}{d s^{i}} s_{0}^{i} \end{matrix}

(46)

\begin{matrix} y_{M_{3}} (s_{0}) = \lim_{M_{∋} \to \infty} \sum_{i = 2}^{M_{3}} \frac{{(- 1)}^{i + 1}}{i!} \frac{d^{i} y (s_{0})}{d s^{i}} s_{0}^{i} \end{matrix}

(47)

The solutions obtained are shown to be independently and efficiently adoptable for very large loading conditions.

Also, Wang [30, 31] studies the case of deflection of a cantilever and simply supported beam. In Wang [30], there is a theoretical discussion of the nonlinear equation

D \frac{d^{2} θ}{d s^{2}} = - ws \cos θ

(48)

with D the flexural rigidity of the beam and w the load intensity. Equation (48) is solved to obtain the maximum deflection $y_{\max}$ and the maximum elongation $L_{\max}$ (h is the horizontal projection of the beam length). In fact, the Euler equation gives, for an initially straight beam,

\frac{d θ}{ds} = - \frac{M}{D}

(49)

which can be simply rewritten as

\cos θ d θ = - \frac{M}{D} dξ

(50)

where $ξ$ is the horizontal distance measured from the free end of the beam, $θ$ is the slope and M is the bending moment. If one of the angles $θ$ is given, then the slope at all sections can be determined and we can simply calculate

y_{\max} = \int_{0}^{h} \tan θ ds, L_{max} = \int_{0}^{h} \sec θ ds

(51)

In Wang [31], a simple numerical method to analyse the nonlinear bending of beams subjected to concentric load is presented.

4. Analysis of stability

The analysis of stability (and instability) of nonlinear beams is a fundamental aspect in the study of beam theory. We have found the textbook by Atanackovic [32], in which this topic is well presented and analysed, very interesting and complete. It has to be noted that this work follows the spirit of the Soviet school.¹ The textbook is structured in the following way: a review of all the basic equations is first proposed, after which the book is divided into three main parts, which approach the stability analysis in three different ways.

In the part concerning the basic equations, the main attention is first paid to shear stress planar deformations; afterwards Atanackovic introduces spatial deformations, distinguishing the cases of inextensibility and extensibility. The invoked methods for stability analysis are the adjacent equilibrium method (also known as the Euler method), the energy method and the dynamic method.

In the adjacent equilibrium method, we have two basic definition of stability:

A1. An equilibrium configuration of an elastic rod is stable if, under a given load and given boundary conditions, there are no other neighbouring (infinitesimally close) equilibrium configurations.

A2. An equilibrium configuration of an elastic rod is unstable if, under a given load and given boundary conditions, at least one more, infinitesimally close, equilibrium configuration exists.

An important aspect in analysing the equations that describe equilibrium is that we have to face the problem of finding the solution of a system of nonlinear equations

F (y (ξ), λ) = 0, ξ \in (0, 1)

(52)

where $F = (F_{i}), i = 1, 2, \dots, n$ , is a nonlinear operator, $y (ξ) = (y_{i})$ is a vector of dependent variables and $λ$ is an m -dimensional vector denoting parameters. Assume that $F (0, λ) = 0$ . We furthermore add boundary conditions

B (y (ξ), λ) = 0, ξ \in [0, 1]

(53)

where $B = (B_{i})$ is a given operator. The central problem is now to determine the existence and the number of solutions for a given $λ$ . The values of $λ$ at which the number of solutions changes are called bifurcation values. At $λ$ , some possibilities can occur (the bifurcations).

The energy method is mainly based on variational principles. The definitions given for the Euler method can be rewritten in this case as:

B1. An equilibrium configuration of an elastic rod under conservative loads is stable if and only if the (total) potential energy assumes a weak proper minimum value at the equilibrium configuration in the class of finite virtual displacements satisfying the kinematic constraints.

B2. An equilibrium configuration of an elastic rod under conservative loads is infinitesimally stable if and only if the (total) potential energy assumes a weak proper minimum value at the equilibrium configuration in the class of infinitesimal virtual displacements satisfying the kinematic constraints.

Since the potential energy is a functional, we have to use the methods to find the minimum of a functional (mainly variational principles).

The dynamic method for the study of the stability of an equilibrium configuration of elastic rods can be considered a special case of a study of the stability of motion of elastic bodies. The basic principle under this approach is Liapunov’s definition of stability, which can be written as:

C1. An equilibrium configuration of an elastic rod is stable if all motions of the body, caused by the change of initial conditions, remain in the vicinity of the equilibrium configuration for all of the time.

The mathematical apparatus needed for the application of this kind of principle is based on a formulation of the problem in which the motion of the elastic rod is studied via a system of partial differential equations of the type

\frac{\partial ψ_{i}}{∂t} = F_{i} (ψ_{k}, s, t), i = 1, 2, \dots, n

(54)

where we have n functions $ψ_{i} (s, t)$ defined for each $s \in [0, L]$ and for all times $t \in [t_{0}, \infty)$ . For fixed $t = t_{1}$ , the functions $ψ_{i} (s, t_{1})$ define the state of the rod and belong to a normed linear space Y (Banach or Sobolev).

The background we have now presented allows us to give a simpler review of the many articles that are targeted on the stability of nonlinear beams [33 –42].

Some sufficient conditions on the stability of Euler elasticae for a beam fixed at one end and guided at the other are proposed by Jin and Bao [33], introducing a conjugate point theory in the calculus of variations. In the conjugate point theory, it is possible to introduce a functional $J$ defined by

\begin{matrix} \{\begin{matrix} J (θ) = \int_{0}^{1} F (x, θ, θ^{'}) dx \\ θ (0) = θ (1) = 0 \end{matrix} \end{matrix}

(55)

where $θ$ is the independent variable function and $θ^{'} = d θ ∕ dx$ . Suppose now that $θ$ is an extremum of functional $J$ , so that $δ J (θ) = 0$ . It is possible to associate with the previous system a further equation, the Jacobi equation

\begin{matrix} \{\begin{matrix} \frac{d}{dx} (\frac{\partial^{2} F}{\partial {θ^{'}}^{2}} u^{'}) - (\frac{\partial^{2} F}{\partial θ^{2}} - \frac{d}{dx} \frac{\partial^{2} F}{∂θ∂ θ^{'}}) u = 0 \\ u (0) = 0, u^{'} (0) = 1 \end{matrix} \end{matrix}

(56)

Jin and Bao [33] then use three theorems from Smirnov to prove the stability of the functional $J (θ)$ . On the basis of these theoretical results, the potential energy of an Euler column is written in a way that makes it possible to apply the conjugate point theory

\begin{matrix} \{\begin{matrix} Π (θ) = \int_{0}^{L} \frac{EI}{2} {(\frac{d θ}{ds})}^{2} ds - N \int_{0}^{L} (1 - \cos θ) ds \\ θ (0) = θ (L) = 0 \end{matrix} \end{matrix}

(57)

where L, N, EI and $θ$ are, respectively, the length of the column, the axial load, the flexural rigidity and the tangent angle of the lateral deflection curve. $Π$ is therefore a functional of $θ$ . According to the Lagrange theorem, one equilibrium is stable if and only if the potential energy of the system takes the minimum value in it. By introducing $k = L \sqrt{(N ∕ EI)}$ and $x = s ∕ L$ , we rewrite the system as follows:

\begin{matrix} \{\begin{matrix} Π (θ) = \int_{0}^{1} [\frac{1}{2} {(θ^{'})}^{2} + k^{2} \cos θ] dx \\ θ (0) = θ (1) = 0 \end{matrix} \end{matrix}

(58)

so the Jacobi equation associated with the system is

\begin{matrix} \{\begin{matrix} u^{″} + u k^{2} \cos θ \\ u (0) = 0, u^{'} (0) = 1 \end{matrix} \end{matrix}

(59)

In the last part of the article, some properties of the equilibrium of this Jacobi equation are given. Henderson and Neukirch [34] study the equilibrium configurations of an inextensible, unshearable, isotropic, uniform prismatic rod when it is subjected to end loads and clamped boundary conditions. To approach this problem, first the equilibrium and the boundary value problem of Kirchhoff equations are studied. In a second stage, a discretized boundary values problem is considered and its solution is investigated by a continuation method. Finally, we find the analysis of the properties of the solution manifold.

Ito and Kunimatsu [35] study the linear differential equation describing the motion of a Timoshenko beam with some viscous internal damping and certain boundary inputs,

\frac{dz (t)}{dt} = - Az,

(60)

in a Hilbert space. We have some relevant results: (i) the linear operator A generates an analytic semigroup; (ii) in some cases, the type of boundary inputs degrades the degree of stability in the system. Consider the Timoshenko equation

\begin{matrix} ρ \frac{\partial^{2} u}{\partial t^{2}} = κ (\frac{\partial^{2} u}{\partial s^{2}} - \frac{∂ϕ}{∂s}) \end{matrix}

(61)

\begin{matrix} I_{ρ} \frac{\partial^{2} ϕ}{\partial t^{2}} = EI \frac{\partial^{2} ϕ}{\partial s^{2}} + κ (\frac{∂u}{∂s} - ϕ) \end{matrix}

(62)

where $t \in [0, \infty)$ and $s \in [0, L]$ , with $u (s, t)$ the deflection of the beam from the equilibrium (which is given by $u (s, t) = 0$ ), $ϕ (s, t)$ the slope of the deflection curve when the effect of shear force in each section is neglected, $ρ$ the mass per unit length, $I_{ρ}$ the mass moment, E the Young modulus, I the moment of inertia of the cross-section and $κ = kSG$ , with S the cross-sectional area, G the shear modulus and k a numerical factor depending on the slope of the cross-section. Suppose that equation (61) can be transformed to a linear evolution equation

\frac{dz (t)}{dt} = - A z (t)

(63)

in a suitable space X. We can substitute the following proportionalities between stress and strain

σ = Eϵ, τ = Gβ

(64)

where $σ$ , $ϵ$ , $τ$ and $β$ are, respectively, the normal stress, the normal strain, the shear stress and the shear strain, with other relations in which the viscosity of the beam is considered by adopting the Voigt model

σ = Eϵ + C_{M} \frac{dϵ}{dt}, τ = Gβ + C_{V} \frac{d β}{dt}

(65)

with $C_{M}$ and $C_{V}$ , the moduli of viscosity for bending and shear. Then we can rewrite equation (61) as

\begin{matrix} ρ \frac{\partial^{2} u}{\partial t^{2}} = k (\frac{\partial^{2} u}{\partial s^{2}} - \frac{∂ϕ}{∂s}) + C_{V} kA (\frac{\partial^{3} u}{∂t∂ s^{2}} - \frac{\partial^{2} ϕ}{∂t∂s}) \end{matrix}

(66)

\begin{matrix} I_{ρ} \frac{\partial^{2} ϕ}{\partial t^{2}} = EI \frac{\partial^{2} ϕ}{\partial s^{2}} + C_{M} I \frac{\partial^{3} ϕ}{∂t∂ s^{2}} + k (\frac{∂u}{∂s} - ϕ) + C_{V} kA (\frac{\partial^{2} u}{∂t∂s} - \frac{∂ϕ}{∂t}) \end{matrix}

(67)

After that it is possible to introduce the following linear operator:

\begin{matrix} (\begin{matrix} 0 & - 1 & 0 & 0 \\ - p \partial^{2} & - γ \partial^{2} & p∂ & γ∂ \\ 0 & 0 & 0 & - 1 \\ - dp∂ & - dγ∂ & - q \partial^{2} + dp 1 & - δ \partial^{2} + dγ 1 \end{matrix}) \end{matrix}

(68)

with $γ$ , $δ$ , d, p and q positive constants. The domain of A is

D (A) = \{z = (v_{1}, v_{2}, ϕ_{1}, ϕ_{2}) \in {(H^{2})}^{4} : z satisfies the boundary condition\}

(69)

Ito and Kunimatsu [25] have two interesting results: (i) the operator $- A$ generates an analytic semigroup on X; (ii) it is proven that the solution of equation (63) decays exponentially in norm as the time t increases.

Lazarus et al. [36] present a theoretical and numerical framework to compute bifurcations of equilibria and the stability of slender elastic rods. The 3D kinematics of the rod are treated in a geometrically exact way, parametrizing the position of the centreline and making use of quaternions to represent the orientation of the material frame. Quaternions allow us, in fact, to write equilibrium equations in a quadratic form and to solve them efficiently using an asymptotic numerical continuation method. Consider a material frame, which rotates in the fixed reference with a rotation velocity $Ω (s)$ when following the centreline at unit speed. The velocity $Ω (s)$ is given in terms of the Darboux vector:

Ω (s) = κ_{1} (s) d_{1} (s) + κ_{2} (s) d_{2} (s) + κ_{3} (s) d_{3} (s)

(70)

where $κ_{1}$ and $κ_{2}$ are material curvatures that represent the extent of rotation of the material frame with respect to the directions $d_{1}$ and $d_{2}$ and $κ_{3}$ represents the material twist that quantifies the rotation of the material frame with respect to the tangent $d_{3}$ . To write the material curvature and twist in an explicit form, the Darboux vector must be rotated into the local frame

κ (s) = R^{T} (s) Ω (s)

(71)

with

κ (s) = (\begin{matrix} κ_{1} (s) \\ κ_{2} (s) \\ κ_{3} (s) \end{matrix})

(72)

where $R (s) \in ℜ^{3 \times 3}$ is the rotation matrix of the Euclidean 3D space. A quaternion is a number $q_{1} i + q_{2} j + q_{3} k + q_{4}$ , where the imaginary part $q_{1} i + q_{2} j + q_{3} k$ is an element of the vector space $ℜ^{3}$ and the real part q₄ is a scalar. We have Hamilton’s well-known relation:

i^{2} = j^{2} = k^{2} = ijk = - 1

(73)

Unit quaternions are a particularly convenient mathematical notation for representing orientations of objects in three dimensions; they are defined by the relation

q_{1}^{2} + q_{2}^{2} + q_{3}^{2} + q_{4}^{2} = 1

(74)

The material curvatures $κ_{1} (s)$ and $κ_{2} (s)$ and the material twist $κ_{3} (s)$ can then be derived solely in terms of the Euler parameters

ϰ_{κ} (s) = 2 B_{κ} q (s) q^{'} (s)

(75)

where $B_{k}$ are the skew-symmetric matrices

B_{1} = (\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & - 1 & 0 & 0 \\ - 1 & 0 & 0 & 0 \end{matrix}), B_{2} = (\begin{matrix} 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \end{matrix}), B_{3} = (\begin{matrix} 0 & 1 & 0 & 0 \\ - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & - 1 & 0 \end{matrix})

(76)

With the previous positions, we can write the elastic energy as in the following:

E (q (s)) = \sum_{k = 1}^{3} \frac{E_{k} I_{k}}{2} \int_{0}^{L} 2 {(B_{κ} q (s) q^{'} (s) - {\hat{ϰ}}_{κ} (s))}^{2} ds

(77)

with $E_{1} = E_{2} = E$ the Young modulus, $E_{3} = G$ the shear modulus, I₁ and I₂ the moments of inertia along the principal directions of curvature of the cross-section and I₃ the moment of twist. By varying the energy, it is possible to obtain the strong form of equilibrium equations as second-order differential equations, simply solvable using numerical tools

\begin{matrix} 0 = p (s) + μ^{'} (s) \end{matrix}

(78)

\begin{matrix} 0 = τ (s) - m (s) + 2 α (s) q (s) - 2 D (q (s)) μ (s) \end{matrix}

(79)

where $p (s)$ is the density of forces, $μ$ are the Lagrange multipliers ensuring the condition of inextensibility of slender elastic rods, $m$ is the density of moments, $α (s)$ is the Lagrange multiplier that imposes the norm of quaternions to one, $D (q (s))$ is an antisymmetric operator and

τ (s) = \sum_{k = 1}^{3} (G_{k} (s) 4 B_{k} q^{'} (s) + G_{k}^{'} (s) 4 B_{k} q (s))

(80)

with

G_{k} (s) = E_{k} I_{k} (B_{κ} q (s) q^{'} (s) - {\hat{ϰ}}_{κ} (s))

(81)

A semi-analytical method for investigating the stability of the planar equilibrium of an inextensible elastic rod under end-loading conditions is adopted in Levyakov and Kuznetsov [37]. The functional of strain energy of the rod in the presence of isoperimetric constraints is given by

W = \frac{1}{2} \int_{0}^{1} {(θ^{'} - ℓ κ_{0})}^{2} dξ + p (\int_{0}^{ℓ} \cos θ dξ - a) + q (\int_{0}^{ℓ} \sin θ dξ - b)

(82)

Again, a numerical application is presented. Other aspects of the same problem are investigated in Levyakov [38], where the attention is focused on the sign of the functional of the energy. The aim of the aforementioned work is to analyse the stability of equilibrium configurations of a flexible rod. The associated Jacobi’s equation can be integrated analytically, obtaining a solution in closed form (through elliptical functions).

The stability of post-critical equilibrium in a simply supported column loaded with an axial force is investigated by Kuznetsov and Levyakov [39]. The total potential energy presented is given by

W = \frac{1}{2} \int_{0}^{1} {θ^{'}}^{2} dξ - q (1 - \int_{0}^{1} \cos θ dξ)

(83)

where a nondimensional quantity $ξ ∕ s ∕ ℓ$ has been introduced. From the energy in equation (83), the following nonlinear equation can be derived

θ^{″} + q \sin θ = 0

(84)

To solve the last equation, Kuznetsov and Levyakov [39] use elliptical integrals of the first and second type. The paper provides some numerical simulations.

Sachkov and Levyakov [40] consider the study of stability conditions for inflectional Euler’s elasticae centred at vertices. The problem of equilibrium configurations for the elastic inextensible rod is written as

\begin{matrix} ẋ = \cos θ, ẏ = \sin θ, \dot{θ} = u \end{matrix}

(85)

\begin{matrix} q = (x, y, θ) \in M = ℜ_{x, y}^{2} \times S_{θ}^{1}, u \in ℜ \end{matrix}

(86)

\begin{matrix} J = \frac{1}{2} \int_{0}^{L} u^{2} (s) ds \to \min \end{matrix}

(87)

where, as usual, x and y are the coordinates of a point of an elastica, $θ$ is the angle between the tangent to the elastica and the x axis and the integral J is proportional to the bending energy of the elastica. The elasticae can be divided into two classes:

inflectional elastica;

noninflectional elastica.

To approach the problem of inflectional elastica, Sachkov and Levyakov [40] introduce a new parametrization involving Jacobian elliptic functions, epsilon functions and elliptic integrals of the first and second kinds. As result, the stability conditions are obtained from a mathematical point of view and are then compared with experiments.

Maddocks [41] presents an analysis of the stability of planar configurations of a nonlinearly elastic rod that is buckled under the action of a dead load. The study is enforced by a variational approach, which enables some interesting properties to be discovered:

secondary bifurcation from the first buckled mode;

differences between stability for 2D and 3D variations;

stability influences of resistance to twist.

In the ambit of the dynamics of beams the case considered by Ozturk et al. [1] should be cited, in which the in-plane stability of nonuniform cross-sectioned thin curved beams with uniformly distributed dynamic loads is analysed. The problem is approached using both analytical and numerical tools. In particular, an equation of motion of the Mathieu–Hill type is obtained. In fact, the dynamic response of the beam for a conservative system is written as Lagrange’s equation, where external forces are given in terms of time-dependent potentials

M \overset{\cdot\cdot}{q} + [K_{e} - P (t) K_{g}] q = 0

(88)

where $M$ is the global mass matrix, $K_{e}$ and $K_{g}$ are the global elastic and geometric stiffness matrices and P(t) is the periodic load. Moreover, various zones of instability are studied for a uniform fixed–fixed curved Timoshenko beam, using finite-element models.

To end this section well, we add an overview of some interesting articles on the dynamics of beams. In fact, the dynamic stability of mechanical systems, according to Bolotin’s definition [53], represents a specific stability of motion. Having in mind this interesting concept, we now introduce some work in which the dynamical properties of the systems analysed are predominant with respect to the others.

We will present a quick overview of the most interesting works on dynamics aspects in nonlinear beam theory [43 –52]. We will see different effects that arise in beam theory due to the dynamic analysis. For example, flexible beams become stiffer if subjected to high-speed rotations. This phenomenon, known as geometric stiffening, is studied in Trindade and Sampaio [43], where the nonlinear problem is approached using a nonlinear strainâŁ“displacement relation.

A geometrically nonlinear analysis of flexible sliding beams is taken into account in Behdinan et al. [44], where equations are derived through an extension of HamiltonâŁ™s principle. After a formal analysis, the problem is studied using numerical tools with GarlekinâŁ™s method.

The works of Goriely [45 –47] present a well-founded analysis of nonlinear dynamics of filaments. In the scheme of the time-dependent version of the Kirchhoff model in the Euler angle frame, a direct proof of the existence of dynamical instabilities is provided and consequently a selection mechanism for the shape of unstable filaments. A particular attention is paid to the effect of boundary conditions on the instability threshold.

Other interesting results in dynamics of strings and beams can be found [48 –52].

5. Deformations in beams: bending, stretching, shear, buckling and post-buckling

In this section, we present some works on the possible deformations that can arise in beams. First, we concentrate on the simplest kinds of deformation, bending, stretching and shear [54 –58]; afterwards, we present some articles on buckling and post-buckling [59 –66].

The case of an elastic beam subject to three-point bending is studied in Batista [54] using Jacobi elliptical functions, while the shear-deformable case is presented in Mohyeddin and Fereidoon [55]. In fact, in this latter article, exact solutions for the system of nonlinear differential equations are obtained for horizontal, vertical and angular displacements.

In O’Grady J and Foster [56], a peridynamic state-based model is shown to represent the bending of an Euler–Bernoulli beam. In fact, O’Grady J and Foster [56] want to study the failure of solid materials. It is well known that the peridynamic model is based on integral equations instead of ODP. In this work, the model is derived by inserting a rotational spring between bonds; it can be shown that it is equivalent to Eringen’s nonlocal elasticity, reducing to the classical Euler–Bernoulli beam equations as a peridynamic horizon approach to 0. If $ϑ$ is the angle between the bonds and every bond-vector is connected by a rotational spring to its opposite originating from the same point, then the potential energy of the spring is

w (ξ) = ω (ξ) α (1 + \cos ϑ)

(89)

for the bond pair $ξ$ and $- ξ$ . The target is to determine the value of $α$ , imposing the constraint that the peridynamic model has strain energy density equivalent to a classical Euler–Bernoulli beam in a local limit. For any material point in the peridynamic beam, the strain energy density is

W = \frac{α}{2} κ^{2} \int_{- δ}^{δ} ω (ξ) ξ^{2} dξ + O (δ^{5})

(90)

The $α$ value obtained is $α = EI ∕ m$ , with

m = \int_{- δ}^{δ} ω (ξ) ξ^{2} dξ .

(91)

We just want to point out that in this article, O’Grady J and Foster [56] wrongly cite Silling as the first to develop peridynamics, while it has been widely shown that this theory was first introduced in continuum mechanics by Piola (refer to dell’Isola et al. [78, 79] for more information).

The dynamic of a beam, free to move relative to one of its supports, is derived by Humer [57]. There is presented a variational approach to the problem of large deformations of the sliding beam. The analysis is based on Reissner’s work [4]. The principle of virtual work can also be derived using the Biot’s tensor of nominal stress [58]. It is related to the Biot’s stress tensor by a linear relation, which implies the following differential equation

EI θ^{″} + P \sin θ - \frac{P^{2}}{EA} \cos θ \sin θ = 0

(92)

In this theory, the extensibility of the beam is considered.

The notion of buckling, introduced by Euler more than two centuries ago, describes a static instability of structures due to in-place loading [59 –65].

Barbero et al. [59] present an experimental verification of buckling-mode interaction in composite columns. This kind of interaction is a very actual and interesting problem; in fact, it can induce a tertiary post-buckling path. In this article, a comparison with experiments for intermediate-length pultruded wide-flange columns subject to uniaxial compression is also provided. Nonlinear geometric models are applied by Filipich and Rosales [60] to study the post-buckling of extensible elastic rods.

A very interesting work, which studies the buckling and post-buckling of elastic rods, is that of Humer [61]. In this case, the buckling is influenced by both the axial compressibility and shear deformations. The study in this article is founded on Reissner’s study [4]. The main point of this work consists of rewriting the equation for the bending moment M(x)

\frac{∂M}{∂x} + (1 + ϵ) (P \sin ϕ + V \cos ϕ) + γ (P \sin ϕ - V \cos ϕ) = 0

(93)

where P is the compression force applied at the material point $x = L$ and V is the reaction force in y -direction, into the following equation

B \frac{\partial^{2} ϕ}{\partial x^{2}} + (P \sin ϕ + V \cos ϕ) [1 - (\frac{1}{D} - \frac{1}{S}) (P \sin ϕ - V \cos ϕ)] = 0

(94)

where D is the extensional stiffness, S the shear stiffness and b the bending stiffness, introduced in the equation, as in Reissner [4]

N = Dϵ, T = Sγ, M = Bχ

(95)

Equation (94) is solved and analysed by the use of elliptic integrals.

An exact nonlinear formulation of the equilibrium of elastic prismatic rods subjected to compression and planar bending is presented by Mazzilli [62]. In this work, the method of multiple spatial scales is used to survey the post-buckling regime for five classical Euler buckling cases. The main equation is equation (94), which can be written in a nondimensional way and approximated up to the order $ϵ^{3}$ ( $0 < ϵ ≪ 1$ ), as

\frac{d^{2} ϕ}{d ξ^{2}} + α_{1} ϕ + ϵ α_{2} ϕ^{2} + α_{3} ϕ^{3} = ϵ α_{0}

(96)

where the $α_{i}$ are related to D, S, B, P and V. The method now involves giving an asymptotic expansion

ϕ (ξ) = ϵ ϕ_{1} (ξ_{0}, ξ_{1}, ξ_{2}, \dots) + ϵ^{2} ϕ_{2} (ξ_{0}, ξ_{1}, ξ_{2}, \dots) + ϵ^{3} ϕ_{3} (ξ_{0}, ξ_{1}, ξ_{2}, \dots)

(97)

Unlike time scales, the spatial scales can be introduced accordingly

ξ_{p} = ϵ^{p} ξ, p = 0, 1, 2, \dots

(98)

The functions $ϕ_{q}$ are determined when equations of increasing orders of $ϵ$ are solved and secular terms are eliminated. So by introducing the following derivative operators

\begin{matrix} \frac{d}{dξ} = D_{0} + ϵ D_{1} + ϵ^{2} D_{2} + ϵ^{3} D_{3} + \dots \end{matrix}

(99)

\begin{matrix} \frac{d^{2}}{d ξ^{2}} = D_{0}^{2} + ϵ 2 D_{0} D_{1} + ϵ^{2} (D_{1}^{2} + 2 D_{0} D_{2}) + ϵ^{3} (2 D_{0} D_{3} + 2 D_{1} D_{2}) + \dots \end{matrix}

(100)

with

D_{p}^{k} = \frac{\partial^{k}}{\partial ξ_{p}^{k}}, k = 1, 2, p = 0, 1, 2, \dots

(101)

we can obtain the equation for $ϕ_{1}$

D_{0}^{2} ϕ_{1} + α_{1} ϕ_{1} = α_{0}

(102)

while for the terms of order $ϵ^{2}$ we have

D_{0}^{2} ϕ_{1} + α_{1} ϕ_{2} = - 2 D_{0} D_{1} ϕ_{1}

(103)

and for terms of $ϵ^{3}$ we have

D_{0}^{2} ϕ_{3} + α_{1} ϕ_{3} = - 2 D_{0} D_{2} ϕ_{1} - α_{2} ϕ_{1}^{2} - α_{3} ϕ_{1}^{3}

(104)

This technique is then applied to specific examples.

Okay et al. [63] apply the variational iteration method for finding buckling loads and mode shapes of a heavy column. The variational iteration method is a method for solving a wide range of problems whose mathematical models yield linear and nonlinear differential equations.

The nonlinear large-deflection analysis and post-buckling behaviours of beams and columns are investigated by Vega-Posada [64]; in particular, laterally braced and unbraced slender beams and columns.

Yuan and Wang [65] presented an interesting analysis of buckling and post-buckling of extensible beam-columns using a numerical tool. Usually numerical integration does not produce convergent post-buckling solutions, so in this paper a differential quadrature by numerical integration is proposed.

Finally, Wang et al. [66] presented a Hencky bar-chain model for buckling and vibration analyses of Euler–Bernoulli beams with elastic end restraints.

6. Further relevant results

In this section, we give some further relevant results. In fact, there are some papers that are not so easy to classify on the base of specific features, but are instead very interesting [67 –73].

A Kirchhoff-type beam is analysed by Ma [67]. The fundamental equation is, in general, written as follows:

ü + u^{ı v} + M (\int_{0}^{L} | u^{'} (x, t) |^{2} d x) u^{″} = 0

(105)

This equation models the vibrations of a beam clamped at x = 0 and supported at $x = L$ under nonlinear boundary conditions. Ma [67] demonstrates the existence of a global solution and the exponential decay of the energy. Other hints on this approach are given in Wang et al. [80] and Ma [81].

Banerjee et al. [68] presented the problem of large deflections of cantilever beams with geometric nonlinearity, which they approach in two different ways: a nonlinear shooting method and an Adomian decomposition method. The obtained results are validated using elliptic integral solutions.

The nonlinear shooting method converts a boundary value problem into an initial value problem with an assumed curvature at the fixed end. The considered initial value problem is

\begin{matrix} \{\begin{matrix} \frac{d^{2} ϕ}{d s^{2}} = - \frac{P}{EI} (\cos ϕ + n \sin ϕ) \\ ϕ |_{s = 0} = 0, \frac{dϕ}{ds} |_{s = 0} = m_{k} \end{matrix} \end{matrix}

(106)

where P and nP are the horizontal and vertical components of the force and $m_{k}$ is assumed to be the first derivative at the fixed end at the k th iteration of the Runge–Kutta method utilized for the integration.

The Adomian method, instead, allows us to decompose the solution u of a nonlinear differential equation

Λu + Πu + Nu = g

(107)

where $Λ$ is an invertible linear operator, $Π$ the remaining linear part and N a nonlinear operator, into a sum

u = \sum_{n = 0}^{\infty} u_{n}

(108)

where u₀ is the complete solution for $Λu = g$ .

The very-large-deflection behaviour of prismatic and nonprismatic cantilever beams subjected to various types of loading problem is presented by Dado and Al-Sadder [69]. The fundamental innovation introduced is the representation of the angle of rotation of the beam, which is substituted by a polynomial on the position variable along the deflected beam axis. A variable flexural stiffness EI(s) subjected to variable distributed loads in global $x, y$ -axes is introduced. If $q_{x}$ and $q_{y}$ are the distributed loads and $F_{x}$ and $F_{y}$ the concentrated tip loads, then the main equation of the problem is

EI (s) \frac{d^{2} ϕ}{d s^{2}} + E \frac{dI (s)}{ds} \frac{dϕ}{ds} + [\int_{s^{'} = s}^{s^{'} = L} q_{y} (s^{'}) d s^{'} + F_{y}] \cos ϕ - [\int_{s^{'} = s}^{s^{'} = L} q_{x} (s^{'}) d s^{'} + F_{x}] \sin ϕ

(109)

which is the differential equation governing the large-deflection behaviour of a nonprismatic cantilever beam. Its solution is obtained by approximating the angle of rotation $ϕ$ along the deflected beam axis by a polynomial of the nondimensional parameter $ξ = s ∕ L$

ϕ (ξ) = \sum_{i = 0}^{N} a_{i} ξ^{i}

(110)

where $a_{i}$ are unknown coefficients and N is the order of the polynomial approximating the angle of rotation.

The experimental measures in the case of a cantilever beam can be found in Dowell et al. [70]. An integral way to solve the large-deflection cantilever beam problem can be applied to cases of complex load and varying beam properties, as in Chen [71]. When the curvature is constant in a cantilever beam under a follower load, the problem can be approached as in Nallathambi et al. [72], by using a sort of shooting method.

Ahmed et al. [73] present an analysis of the elastic field of a stiffened simply supported fibre-reinforced composite beam using an analytical scheme based on a displacement-potential field. The equilibrium equation is a fourth-order partial differential equation, whose solution is determined in terms of Fourier series. A potential $ψ (x, y)$ is introduced, which can be used to define the placement components

u_{x} = \frac{\partial^{2} ψ}{∂x∂y}, u_{y} = - \frac{1}{X_{11}} [E_{1}^{2} \frac{\partial^{2} ψ}{\partial x^{2}} + G_{12} (E_{1} - ν_{12} E_{2}) \frac{\partial^{2} ψ}{\partial y^{2}}]

(111)

where $Z_{11} = ν_{12} E_{1} E_{2} + G_{12} (E_{1} - ν_{12}^{2} E_{2})$ , E₁ and E₂ are the elastic moduli in the x and y directions, $ν_{12}$ is the Poisson ratio and $G_{12}$ the in-plane shear modulus. The condition to be satisfied by the orthotropic plane problem of elasticity can be written in terms of $ψ$

E_{1} G_{12} \frac{\partial^{4} ψ}{\partial x^{4}} + E_{2} (E_{1} - 2 ν_{12} G_{12}) \frac{\partial^{4} ψ}{\partial x^{2} \partial y^{2}} + E_{2} G_{12} \frac{\partial^{4} ψ}{\partial y^{4}} = 0

(112)

whose solution is, in terms of Fourier series

ψ = \sum_{m = 1}^{\infty} Y_{m} \cos ax + κ y^{3}

(113)

with $Y_{m}$ a function of only y, $a = mπ ∕ L$ and $κ$ an arbitrary constant; for lateral stiffness, the solution can be found as

ψ = \sum_{m = 1}^{\infty} Y_{m} \sin ax

(114)

7. Some papers on applicable numerical methods

Because of the dimension-reduction process, beam theory is most probably the first continuum theory where deformation energy depends on the second gradient of displacement. This reduction process is already implicit in the original paper by Euler [1] and the subsequent developments by Lagrange [2, 3] and is based on the representation of a displacement field defined in a 3D domain in terms of a displacement field defined in a 1D domain plus the assumption of orthogonality of sections in every configuration. For this reason, the numerical integration schemes suitable for the theory of beams require the choice of a more sophisticated set of shape functions to be used to discretize the equilibrium minimization problem. Therefore, once one has formulated higher-gradient continuum plate or shell or 3D body theories, the knowledge obtained in the well-established and old theory of beams may prove to be extremely useful in the study of such generalized continuum models.

The target of this review paper is to acquire and reorder the available results preparatory to the design and study of pantographic sheets, moving either in their plane or deforming in 3D. The continuum models [82] proposed for this mechanical system are forcedly of second gradient and therefore the same difficulties found for the numerical study of beam deformations are to be confronted when numerically determining the equilibrium shapes of the aforementioned metamaterials.

For this proper aim it can be useful to refer to a number of articles that represent a very valuable introduction to isogeometric methods [84 –90].

8. Conclusions

The motivation for this targeted review comes from a recent research project dealing with so-called pantographic lattices or structures [82–83, 91 –103]. Actually, the advancement of 3D printing technologies made it possible to realize relatively ‘small structural architectures’ constituted of beams and elastic constraints. Once eventually homogenized [104], these structures may produce nonstandard generalized continua, if suitable microgeometrical or micromechanical hypotheses are verified. See the methods of Pideri and Seppecher [105] as a paradigmatic example.

It is an interesting topic in the history of science to understand how the feedback from technological advancement induces novel theoretical progress and opens new scientific perspectives by changing conceptual paradigms [78, 106 –109].

Moreover, as described by Russo [106] and Eugster and dell’Isola [107], it happens more frequently than we could expect that some results are lost and then rediscovered in subsequent works. Therefore, in the present paper we try to (i) find an exhaustive list of results relevant for our target and (ii) to find suitably complete and ‘ancient’ sources for every relevant result.

The main assumption in the most frequent homogenization procedure concerns the exclusive use of internal clamping devices for interconnecting used beams in the microstructure. When, instead, for instance internal pivots are introduced [110, 111], homogenization cannot produce standard Cauchy continua [112 –117]. In the effort of designing metamaterials, aforementioned limiting assumptions are fruitfully removed to obtain ‘more exotic’ mechanical behaviours. The microscopic analysis that must be performed in this context cannot be limited to small displacements or to small deformations of considered beams. The necessary inspections of the available literature induced and shaped the present review, motivating the choice of considered papers and the organization of treated matter.

Indeed, (i) in most advanced architectured microstructures, the considered beams are, in general extensible, shearable and flexible; (ii) in many pantographic structures, considered beams are loaded with linearly distributed dead and live loads; (iii) both macrostructures and microstructures can experience buckling phenomena and undergo post-buckling deformations.

It has to be remarked that some direct theories of shells and plates having a kinematics and deformation energy richer than those of standard Mindlin–Kirchhoff–Love theories have been presented [118 –128]. These models belong to the same class used for modelling pantographic structures; we found the theoretical results presented in the aforementioned papers really useful for framing our study.

The presented matter organizes some of the most relevant contributions in the previously listed arguments. Special attention has been addressed in a concluding section to the deformations induced on a beam by a system of deformable springs distributed along the beam. Indeed, once isolated, one beam in a pantographic structure is loaded by a distributed force field resulting from the interaction of a transverse array of beams.

Footnotes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

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How contact interactions may depend on the shape of Cauchy cuts in N th gradient continua: approach ‘à la D’Alembert’. Z Angew Math Phys 2012; 63(6): 1119–1141.

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‘Hypertractions and hyperstresses convey the same mechanical information Continuum Mech Thermodyn (2010) 22: 163176’ by Prof. Podio Guidugli and Prof. Vianello and some related papers on higher gradient theories. Continuum Mech Thermodyn 2011; 23(5): 473–478.

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On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch Appl Mech 2010; 80(1): 73–92.

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On the bending of viscoelastic plates made of polymer foams. Acta Mech 2009; 204(3–4): 137.

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Vibration analysis of non-linear 6-parameter prestressed shells. Meccanica 2014; 49(8): 1751–1761.

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Shell-like structures: non-classical theories and applications, volume 15. Berlin: Springer, 2011.

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Actual developments in the nonlinear shell theory – state of the art and new applications of the six-parameter shell theory. Shell Struct: Theory Appl 2014; 3: 3–12.

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On constitutive inequalities in nonlinear theory of elastic shells. Z Angew Math Mech 2007; 87(2): 94–101.

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Mechanics of advanced materials for lightweight structures. Proc IMechE, Part C: J Mech Eng Sci 2011; 225(11): 2481–2496.

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Altenbach

Rayleigh variational principle and vibrations of prestressed shells. Shell Struct Theory Appl 2014; 3: 285–288.

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The Rayleigh and Courant variational principles in the six-parameter shell theory. Math Mech Solids 2015; 20(7): 806–822.

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Large deformations of inelastic shells. Key Eng Mater 2013; 535: 76–79.

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Greco

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Consistent tangent operator for an exact Kirchhoff rod model. Continuum Mech Thermodyn 2015; 27(4–5): 861.