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Steady-state solutions for a suspension bridge with intermediate supports

Claudio Giorgi* and Elena Vuk

Author Affiliations

DICATAM, Università degli Studi di Brescia, Via Valotti 9, Brescia, 25133, Italy

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Boundary Value Problems 2013, 2013:204  doi:10.1186/1687-2770-2013-204


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/204


Received:5 March 2013
Accepted:19 July 2013
Published:9 September 2013

© 2013 Giorgi and Vuk; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This work is focused on a system of boundary value problems whose solutions represent the equilibria of a bridge suspended by continuously distributed cables and supported by M intermediate piers. The road bed is modeled as the junction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M1">View MathML</a> extensible elastic beams which are clamped to each other and pinned at their ends to each pier. The suspending cables are modeled as one-sided springs with stiffness k. Stationary solutions of these doubly nonlinear problems are explicitly and analytically derived for arbitrary k and a general axial load p applied at the ends of the bridge. In particular, we scrutinize the occurrence of buckled solutions in connection with the length of each sub-span of the bridge.

MSC: 35G30, 74G05, 74G60, 74K10.

Keywords:
extensible elastic beam; suspension bridge; boundary value problems for nonlinear higher-order PDE; bifurcation and buckling

1 Introduction

In this paper, we investigate the solutions of a system of one-dimensional nonlinear problems describing the steady-states of an extensible elastic suspension bridge with M intermediate supports (piers). In particular, we assume that the road bed of the bridge (deck) is composed of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M1">View MathML</a> extensible elastic beams which are clamped to each other, pinned at their ends to each pier and suspended by a continuous distribution of flexible elastic cables. On account of the midplane stretching of the beams due to their elongations, a geometric nonlinearity appears into the bending equations of the deck.

In the case of a bridge with a single span (no intermediate pier), the problem can be recast into a non-dimensional setting, where its length is supposed to be unitary, for simplicity. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M3">View MathML</a> be a non-dimensional field which accounts for the downward deflection of the bridge in the vertical plane with respect to its reference configuration, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M4">View MathML</a> stand for its positive part, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M5">View MathML</a>

Assuming that both ends of the bridge are pinned, the equation of the bending equilibrium looks like (see [1])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M6">View MathML</a>

(1.1)

The term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M7">View MathML</a> accounts for the restoring force due to one-sided springs which models the supporting cables. Since we confine our attention to stationary conditions, we neglect the dynamical coupling between the deck and the main cable. The constant p represents a non-dimensional measure of the axial force acting at the ends of the span in the reference configuration. Accordingly, p is negative when the span is stretched, positive when compressed. The symbol ′ represents the derivative with respect to the argument.

Assume now that the bridge span is composed of N extensible beams whose internal ends are hinged to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M8">View MathML</a> intermediate piers. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M10">View MathML</a>, be the points at which these supports are located and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M11">View MathML</a>

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M13">View MathML</a>, the corresponding intervals between two subsequent piers and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M14">View MathML</a> the corresponding length, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M15">View MathML</a>

According to the assumptions, the deflection u of the whole span obeys

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M16">View MathML</a>

Furthermore, we assume that consecutive sub-spans are mutually clamped at the common end, which in turn implies the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M17">View MathML</a> across the supported points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M10">View MathML</a> (see Figure 1).

thumbnailFigure 1. The joint connecting two consecutive sub-spans and the intermediate pier.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M13">View MathML</a>, account for the downward deflection of the nth beam in the vertical plane, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M22">View MathML</a> be the axial load acting on it. Of course, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M23">View MathML</a> can be viewed as the restriction of u on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M24">View MathML</a>, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M25">View MathML</a>. Then, taking advantage of the nonlinear model (1.1) for a single supported span, each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M23">View MathML</a> solves the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M27">View MathML</a>

(1.2)

Such solutions must fulfill the mutual groove condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M28">View MathML</a>

(1.3)

We finally observe that the total elongation of the span equals the sum of the elongations of each beam, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M29">View MathML</a>

Throughout the paper, we assume a uniform distribution of the axial load, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M31">View MathML</a>. In addition, we let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M32">View MathML</a>

Our aim is to scrutinize the existence of suitable buckled solutions for u, which can be obtained by joining buckled solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M23">View MathML</a> on each sub-span, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M31">View MathML</a>. For later convenience, we denote such solutions by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M35">View MathML</a>. We prove that they exist provided that the lengths of the sub-spans are properly chosen.

1.1 Early contributions

In recent years, an increasing attention has been payed to the analysis of buckling, vibrations and post-buckling dynamics of nonlinear beam models (see, for instance, [2,3]). As far as we know, most of the papers in the literature deal with approximations and numerical simulations, and only few works are able to derive exact solutions (see [4-7]).

The investigation of solutions to BVP (1.1), in dependence on p, represents a classical nonlinear buckling problem in the literature on structural mechanics. The notion of buckling, introduced by Euler more than two centuries ago, describes a static instability of structures due to in-plane loading. In this respect, the main concern is to find the critical buckling loads, at which a bifurcation of solutions occurs, and their associated mode shapes, called postbuckling configurations. In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M36">View MathML</a>, a careful analysis of the corresponding buckled stationary states was performed in [7] for all values of p in the presence of a source with a general shape (see also [8]). By replacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M4">View MathML</a> with u in (1.1), we obtain a simpler model which was scrutinized in [1,5].

In addition, it is worth noting that (1.1) represents the static counterpart of quite many different evolution problems arising both from elastic, viscoelastic and thermo-elastic theories. An example is the following quasilinear equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M38">View MathML</a>

(1.4)

which is obtained by matching the modeling of the extensible elastic beam (see [9,10]) with the well-known equation describing the motion of a damped suspension bridge (see [11,12])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M39">View MathML</a>

(1.5)

Free and forced vibrations of (1.5) were recently scrutinized in [13,14], whereas the long-time behavior of (1.4) was described in [15] for all values of p.

Obviously, solutions to BVP (1.1) represent the steady states of a lot of models more general than (1.4), for instance, when either the rotational inertia (as in the Kirchhoff theory) or some kind of damping are taken into account. In particular, (1.1) works either when external viscous forces are added or when some structural dissipation phenomena occur in the deck, as in thermoelastic and viscoelastic beams (see, for instance, [16-18]).

When the geometric nonlinear term into (1.1) is disregarded, the existence of nontrivial (positive) solutions to the corresponding system were established in [19] by the variational method. Therein, some nonlinearly perturbed versions were also scrutinized, but the set of assumptions made there no longer holds when the full model is considered.

1.2 Outline of the paper

To the best of our knowledge, this is the first paper in the literature dealing with exact solutions to the doubly nonlinear BVPs (1.2), even for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M40">View MathML</a>. As is well known, the analysis of the corresponding set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M41">View MathML</a> of their stationary solutions takes a great importance in the longterm dynamics of the corresponding evolution system, especially when its structure is nontrivial [20]. The main results of this paper concern the steady states analysis of a bridge with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M42">View MathML</a> sub-spans and are stated in Section 2. In Section 2.1 we scrutinize the case of a single span without piers (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M40">View MathML</a>) and we prove that increasing the value of the lateral load p, first a negative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44">View MathML</a>, then a positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45">View MathML</a> buckled solution appear at equilibrium. In Section 2.2 a bridge with a single pier (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M46">View MathML</a>) is considered. When the position of the pier is allowed to be asymmetric (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M47">View MathML</a>), we establish a condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M48">View MathML</a> in order that buckled static solutions exist. In particular, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M49">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M50">View MathML</a>. Taking advantage of these results, the analysis of a bridge with two symmetrically-placed piers (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M51">View MathML</a>) is performed in Section 2.3, where buckled static solutions are proved to exist provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M52">View MathML</a> fulfills a suitable condition. In Section 3 we deal with the general problem of a bridge with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M8">View MathML</a> piers, and we discuss separately the cases when the number M of the piers is either odd or even. All buckled solutions are determined in a closed form and belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M54">View MathML</a>. Each of them is constructed by rescaling and suitably collecting positive and negative solutions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44">View MathML</a>. For any given N, a general explicit formula is established to compute the bifurcation values as a function of k.

2 Stationary states I

2.1 A single span without piers (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M40">View MathML</a>)

The set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M41">View MathML</a> of the bridge equilibria in this case consists of all (weak) solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M59">View MathML</a> to the following boundary value problem on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M60">View MathML</a> (see Eq. (1.1))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M61">View MathML</a>

(2.1)

It is worth noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M41">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M63">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M65">View MathML</a> (see [20]).

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M66">View MathML</a>, a general result was established in [7] for a class of non-vanishing sources. In [4], the same strategy with minor modifications was applied to a problem close to (2.1), where the one-sided springs are replaced by unyielding ties. We summarize here the results concerning stationary solutions in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M66">View MathML</a>.

Theorem 2.1 (see [7], Th. 4.1)

Whenκvanishes, the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M68">View MathML</a>is finite for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M64">View MathML</a>. Letting<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M70">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M68">View MathML</a>has exactly<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M73">View MathML</a>elements:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M74">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M75">View MathML</a>

The general case is much more complicated. Since the scheme devised in [5,7] does not work in the present situation, we obtain here a limited result.

Theorem 2.2 (Existence of buckled solutions)

When<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76">View MathML</a>, besides the null solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M77">View MathML</a>, which exists for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M64">View MathML</a>, problem (2.1) admits

a negative buckled solution, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M79">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M80">View MathML</a>;

a positive buckled solution, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M81">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M82">View MathML</a>.

Proof For all values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M64">View MathML</a>, problem (2.1) has at least the trivial solution, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M84">View MathML</a>. For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M80">View MathML</a>, besides the null solution, we obtain also the negative buckled solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M86">View MathML</a>

which solves the Woinowsky-Krieger problem (see [21])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M87">View MathML</a>

(2.2)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M88">View MathML</a>, there exists only the null solution. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M89">View MathML</a> will be referred to as the smallest bifurcation value. By paralleling [5], we infer that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M90">View MathML</a>, system (2.1) admits the positive buckled solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M91">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76">View MathML</a>, we can easily check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M93">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M94">View MathML</a> (see Figure 2).  □

thumbnailFigure 2. The buckled solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M95">View MathML</a>(a) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M96">View MathML</a>(b) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M97">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M98">View MathML</a>.

2.2 A bridge with a single pier (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M46">View MathML</a>)

Now we consider a bridge with a single pier at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M100">View MathML</a> and two sub-spans. The whole solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M101">View MathML</a> is obtained by joining their deflections, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M103">View MathML</a>, which solve problems (1.2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M104">View MathML</a> and fulfill the mutual groove condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M105">View MathML</a>

(2.3)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M106">View MathML</a>, it is easy to check that this condition cannot be satisfied by any buckled solution. Then we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M47">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M108">View MathML</a> be the point at which the pier is located, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M109">View MathML</a>. For the sake of definiteness, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M111">View MathML</a>.

Theorem 2.3When<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76">View MathML</a>, problems (1.2) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M46">View MathML</a>admit two buckled solutions, called<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M114">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M115">View MathML</a>, provided that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M116">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M117">View MathML</a>, where the values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M119">View MathML</a>are defined in (2.9) and (2.10), respectively.

Proof In order to establish the form of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M103">View MathML</a>, we rescale the domains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M123">View MathML</a> by virtue of the transformations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M125">View MathML</a>, respectively. Hence, we get the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M126">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M127">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M128">View MathML</a>. Obviously, the null solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M129">View MathML</a> exists for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M64">View MathML</a>. On the contrary, nontrivial solutions occur under special conditions.

Arguing as in Section 2.1, we exploit the explicit expression of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44">View MathML</a>, respectively, and we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M133">View MathML</a>

where, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M134">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M135">View MathML</a>

Of course, such solutions both exist provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M136">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a> satisfies the continuity condition (2.3), namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M138">View MathML</a>

(2.5)

Then, by joining w and v, we obtain the whole solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M139">View MathML</a>

(2.6)

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M140">View MathML</a>, which is obtained by symmetrizing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M141">View MathML</a> with respect to the vertical line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M142">View MathML</a>, solves (1.2) under the same conditions (see Figure 3). Its expression is given by replacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M145">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M146">View MathML</a> into (2.6),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M147">View MathML</a>

(2.7)

thumbnailFigure 3. The buckled solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M148">View MathML</a>(a) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M149">View MathML</a>(b) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M150">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M151">View MathML</a>.

Computation of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M119">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76">View MathML</a>, in order to compute the unknown value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a>, which ensures the existence of buckled solutions, we exploit the continuity condition (2.5), namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M156">View MathML</a>

Thus, the required value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a> has to satisfy the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M158">View MathML</a>

(2.8)

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M159">View MathML</a>

(2.9)

It can be easily checked (see Figure 4) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M160">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M161">View MathML</a>

In order to compute the critical value of the axial force, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M119">View MathML</a>, we observe that by virtue of (2.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M163">View MathML</a>

(2.10)

thumbnailFigure 4. The value of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M164">View MathML</a>as a function ofκ.

Summarizing, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M166">View MathML</a> no buckled solution exists, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M167">View MathML</a> there exists a unique value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a>, which produces two buckled solutions. As a consequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M119">View MathML</a> represents the bifurcation value. According to (2.10), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M170">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M171">View MathML</a>. In addition, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76">View MathML</a>, we can easily obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M173">View MathML</a>

Then, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M174">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M41">View MathML</a> contains only the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M77">View MathML</a>. □

In the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M177">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M178">View MathML</a>

This means that solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M180">View MathML</a> tend to coincide with the second bifurcation branch of problem (2.2), as expected.

2.3 A bridge with two piers (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M51">View MathML</a>)

When the bridge has two intermediate piers and three sub-spans, we shall construct solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M182">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M185">View MathML</a> solve (1.2). It is easy to check that no buckled solution exists when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M186">View MathML</a>. Then we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M52">View MathML</a> and construct a buckled solution by joining three (suitably rescaled) functions which have the form of either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44">View MathML</a>.

Theorem 2.4When<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76">View MathML</a>, problems (1.2) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M51">View MathML</a>admit two buckled solutions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M192">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M193">View MathML</a>, provided that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M194">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M195">View MathML</a>, respectively, where the values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M196">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M197">View MathML</a>are defined in (2.12) and (2.15).

Proof The buckled solutions are constructed as follows. The former, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M198">View MathML</a>, has two positive components with the same shape on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M199">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M200">View MathML</a>, and a negative one on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M201">View MathML</a>. The latter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M202">View MathML</a>, is negative in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M199">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M200">View MathML</a>, but positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M201">View MathML</a>. Accordingly, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M206">View MathML</a>

so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M207">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M208">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M209">View MathML</a>.

In order to construct <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M198">View MathML</a>, we exploit the same argument as in Section 2.2. First, we argue that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M211">View MathML</a>

Then, we apply the rescaling procedure with the scale factor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M212">View MathML</a> and, by virtue of the transformations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M214">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M215">View MathML</a>

(2.11)

Taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M116">View MathML</a> is uniquely and implicitly given by (2.9), the corresponding value of ξ becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M217">View MathML</a>

so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M218">View MathML</a>

Accordingly, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M219">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M220">View MathML</a>

Of course, such solutions exist at the same time, provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M221">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M222">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M223">View MathML</a> represents the bifurcation value. By virtue of (2.9), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M224">View MathML</a>

(2.12)

If this is the case, we havethe whole solution (see Figure 5a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M225">View MathML</a>

(2.13)

In order to construct <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M202">View MathML</a>, we proceed as before by letting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M227">View MathML</a>

By means of a rescaling procedure with the scale factor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M228">View MathML</a>, the transformations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M229">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M230">View MathML</a> lead to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M231">View MathML</a>

(2.14)

Accordingly, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M232">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a> is given by (2.9), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M234">View MathML</a>

So, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M235">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M236">View MathML</a>

Of course, such solutions do exist together provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M237">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M238">View MathML</a> represents the bifurcation value. By virtue of (2.9), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M239">View MathML</a>

(2.15)

By means of (2.9) it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M240">View MathML</a>. The complete solution is given by (see Figure 5b)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M241">View MathML</a>

(2.16)

 □

thumbnailFigure 5. The buckled solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M242">View MathML</a>(a) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M243">View MathML</a>(b) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M97">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M245">View MathML</a>.

3 Stationary states II

In this section we generalize the problem to a bridge with N sub-spans and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M8">View MathML</a> piers. The existence of buckled solutions is investigated in connection with the length of the sub-spans. Indeed, it is easy to check that no buckled solution exists when all of them are of the same length. Then a buckled solution may be obtained by collecting and joining N (suitably rescaled) functions of the same form as either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44">View MathML</a>. To this end, we are forced to consider separately the cases when the number M of the piers is either odd or even. In the former case, indeed, we adopt a strategy which is close to that applied in Section 2.2. In the latter, we iterate the procedure devised in Section 2.3.

Theorem 3.1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M250">View MathML</a>, (1.2) admits two buckled solutions.

In the odd case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M251">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M252">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M253">View MathML</a>provided that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M254">View MathML</a>, where the value of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M255">View MathML</a>is characterized in (3.5).

In the even case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M256">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M257">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M258">View MathML</a>provided that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M259">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M260">View MathML</a>, respectively, where the values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M261">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M262">View MathML</a>are characterized in (3.6).

ProofThe odd case. Solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M263">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M265">View MathML</a>, are assumed to change the sign alternately on the sub-spans. The superscript + (−) means that the solution is positive (negative) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M199">View MathML</a>. In order to construct them, we join 2m rescaled functions of the same form as either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M45">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M44">View MathML</a>. Arguing as in the step <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M46">View MathML</a>, we let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M270">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a> is given by (2.9).

CONSTRUCTION OF<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M263">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M273">View MathML</a>

so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M274">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M275">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M276">View MathML</a>

we stress that each restriction

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M277">View MathML</a>

is similar to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M114">View MathML</a>, rescaled by a factor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M279">View MathML</a>. In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M280">View MathML</a>

Therefore, on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M281">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M282">View MathML</a>

while on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M283">View MathML</a> we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M284">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M285">View MathML</a>

(3.1)

Such solutions exist provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M286">View MathML</a>

(3.2)

By virtue of Lemma 3.2, which will be proved later, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M287">View MathML</a>

CONSTRUCTION OF<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M264">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M289">View MathML</a>

so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M290">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M291">View MathML</a>. Arguing as before, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M292">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a> is given by (2.9) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M294">View MathML</a>

is similar to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M115">View MathML</a>, rescaled by a factor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M279">View MathML</a>. In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M297">View MathML</a>

so on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M281">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M299">View MathML</a>

while on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M283">View MathML</a> we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M301">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M302">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M303">View MathML</a> defined as in (3.1).

The even case. In this case, we construct the solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M304">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M305">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M306">View MathML</a>.

CONSTRUCTION OF<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M304">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M308">View MathML</a>

so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M274">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M275">View MathML</a>. Arguing as in the step <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M51">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M312">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M313">View MathML</a> is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M314">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M315">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M316">View MathML</a>

So, on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M281">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M318">View MathML</a>

while on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M283">View MathML</a> we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M320">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M321">View MathML</a>

Such solutions exist provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M322">View MathML</a>

(3.3)

By virtue of Lemma 3.2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M323">View MathML</a>

CONSTRUCTION OF<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M305">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M325">View MathML</a>

so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M290">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M291">View MathML</a>. Arguing as in the step <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M51">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M329">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M330">View MathML</a> is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M331">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M332">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M333">View MathML</a>

As a consequence, on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M281">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M335">View MathML</a>

while on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M283">View MathML</a> we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M337">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M338">View MathML</a>

Such solutions exist provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M339">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M340">View MathML</a>

(3.4)

By virtue of Lemma 3.2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M341">View MathML</a>

 □

Lemma 3.2 (Characterization of the bifurcation values)

For any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M76">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M343">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M344">View MathML</a>

(3.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M345">View MathML</a>

(3.6)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M119">View MathML</a>are computed by (2.9) and (2.10), respectively.

Proof First, in view of (3.2), we have to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M348">View MathML</a>

which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M349">View MathML</a>

By replacing the expression of κ given by (2.9), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M350">View MathML</a>

which implies that (3.5) holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M343">View MathML</a>. Then, in view of (3.4), we have to prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M352">View MathML</a>

which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M353">View MathML</a>

By virtue of (2.9), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M354">View MathML</a>

which is identically satisfied for all admissible values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M343">View MathML</a>. Hence, (3.6)1 holds. Finally, in view of (3.3), we need to prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M357">View MathML</a>

and this is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M358">View MathML</a>

Applying (2.9) as in the previous cases, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M359">View MathML</a>, which is identically satisfied for all admissible values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M118">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M343">View MathML</a> so that (3.6)2 follows. Moreover, starting from (3.6) and taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M362">View MathML</a>, it is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M363">View MathML</a>

 □

Previous results can be summarized by means of a simple sketch which highlights the main bifurcation features (see Figure 6).

thumbnailFigure 6. The bifurcation portrait: forNodd (on the left,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M364">View MathML</a>) andNeven (on the right,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M365">View MathML</a>).

Finally, it is worth noting that by removing the coupling between the road-bed and the cable, we recover well-known results (see, for instance, [5,8])

Remark 3.3 In the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M177">View MathML</a>, from (2.9) it follows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M367">View MathML</a>, so that buckled solutions exist even if all sub-spans are equal. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/204/mathml/M368">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

CG conceived the study, participated in its design, verified all calculations and drew all figures, EV participated in the design of the study, performed the proofs of theorems and lemmas, carried out all calculations and drafted the manuscript.

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