Steadystate solutions for a suspension bridge with intermediate supports
 Claudio Giorgi^{1}Email author and
 Elena Vuk^{1}
https://doi.org/10.1186/168727702013204
© Giorgi and Vuk; licensee Springer 2013
Received: 5 March 2013
Accepted: 19 July 2013
Published: 9 September 2013
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 $N=M+1$ extensible elastic beams which are clamped to each other and pinned at their ends to each pier. The suspending cables are modeled as onesided 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 subspan of the bridge.
MSC:35G30, 74G05, 74G60, 74K10.
Keywords
1 Introduction
In this paper, we investigate the solutions of a system of onedimensional nonlinear problems describing the steadystates 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 $N=M+1$ 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.
The term $k{u}^{+}$ accounts for the restoring force due to onesided 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 nondimensional 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.
Our aim is to scrutinize the existence of suitable buckled solutions for u, which can be obtained by joining buckled solutions for ${u}_{n}$ on each subspan, $n=1,2,\dots ,N$. For later convenience, we denote such solutions by ${u}_{(N)}=({u}_{1},\dots ,{u}_{N})$. We prove that they exist provided that the lengths of the subspans are properly chosen.
1.1 Early contributions
In recent years, an increasing attention has been payed to the analysis of buckling, vibrations and postbuckling 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 inplane 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 $k=0$, 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 ${u}^{+}$ with u in (1.1), we obtain a simpler model which was scrutinized in [1, 5].
Free and forced vibrations of (1.5) were recently scrutinized in [13, 14], whereas the longtime 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 $N=1$. As is well known, the analysis of the corresponding set ${\mathcal{S}}_{\kappa}$ 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 $N=1,2,3$ subspans and are stated in Section 2. In Section 2.1 we scrutinize the case of a single span without piers ($N=1$) and we prove that increasing the value of the lateral load p, first a negative ${u}_{(1)}^{}$, then a positive ${u}_{(1)}^{+}$ buckled solution appear at equilibrium. In Section 2.2 a bridge with a single pier ($N=2$) is considered. When the position of the pier is allowed to be asymmetric (${I}_{1}\ne {I}_{2}$), we establish a condition on $\epsilon ={I}_{1}$ in order that buckled static solutions exist. In particular, we prove that $\epsilon \to 1/2$ as $k\to 0$. Taking advantage of these results, the analysis of a bridge with two symmetricallyplaced piers ($N=3$) is performed in Section 2.3, where buckled static solutions are proved to exist provided that ${I}_{1}={I}_{3}\ne {I}_{2}$ fulfills a suitable condition. In Section 3 we deal with the general problem of a bridge with $M=N1$ 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 ${C}^{2}(0,1)$. Each of them is constructed by rescaling and suitably collecting positive and negative solutions, ${u}_{(1)}^{+}$ and ${u}_{(1)}^{}$. 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 ($N=1$)
It is worth noting that ${\mathcal{S}}_{\kappa}$ is bounded in ${H}^{2}(0,1)\cap {H}_{0}^{1}(0,1)$ for all $b\in \mathbb{R}$, $\kappa \in {\mathbb{R}}^{+}$ (see [20]).
When $\kappa =0$, a general result was established in [7] for a class of nonvanishing sources. In [4], the same strategy with minor modifications was applied to a problem close to (2.1), where the onesided springs are replaced by unyielding ties. We summarize here the results concerning stationary solutions in the case $\kappa =0$.
Theorem 2.1 (see [7], Th. 4.1)
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 $\kappa >0$, besides the null solution ${u}_{0}$, which exists for all $b\in \mathbb{R}$, problem (2.1) admits

a negative buckled solution, ${u}_{(1)}^{}$, if$b>1$;

a positive buckled solution, ${u}_{(1)}^{+}$, if$b>1+\kappa $.
2.2 A bridge with a single pier ($N=2$)
When ${I}_{1}={I}_{2}$, it is easy to check that this condition cannot be satisfied by any buckled solution. Then we choose ${I}_{1}\ne {I}_{2}$. Let ${x}_{1}=\epsilon $ be the point at which the pier is located, so that ${I}_{1}=\epsilon $. For the sake of definiteness, let ${u}_{1}>0$ and ${u}_{2}<0$.
Theorem 2.3 When $\kappa >0$, problems (1.2) for $N=2$ admit two buckled solutions, called ${u}_{(2)}^{+}$ and ${u}_{(2)}^{}$, provided that $\epsilon ={\epsilon}_{\kappa}$ and $b>{b}_{\kappa}$, where the values of ${\epsilon}_{\kappa}$ and ${b}_{\kappa}$ are defined in (2.9) and (2.10), respectively.
where $w(y)={u}_{1}(\epsilon y)$ and $v(z)={u}_{2}([1\epsilon ]z+\epsilon )$. Obviously, the null solution $v=w=0$ exists for all $b\in \mathbb{R}$. On the contrary, nontrivial solutions occur under special conditions.
Then, when $b\le 4$, the set ${\mathcal{S}}_{\kappa}$ contains only the trivial solution ${u}_{0}$. □
This means that solutions ${u}_{(2)}^{+}$ and ${u}_{(2)}^{}$ tend to coincide with the second bifurcation branch of problem (2.2), as expected.
2.3 A bridge with two piers ($N=3$)
When the bridge has two intermediate piers and three subspans, we shall construct solutions ${u}_{(3)}=({u}_{1},{u}_{2},{u}_{3})$, where ${u}_{1}$, ${u}_{2}$ and ${u}_{3}$ solve (1.2). It is easy to check that no buckled solution exists when ${I}_{1}={I}_{2}={I}_{3}$. Then we choose ${I}_{1}={I}_{3}\ne {I}_{2}$ and construct a buckled solution by joining three (suitably rescaled) functions which have the form of either ${u}_{(1)}^{+}$ or ${u}_{(1)}^{}$.
Theorem 2.4 When $\kappa >0$, problems (1.2) for $N=3$ admit two buckled solutions, ${u}_{(3)}^{+}$ and ${u}_{(3)}^{}$, provided that $b>{c}_{\kappa}^{+}$ and $b>{c}_{\kappa}^{}$, respectively, where the values of ${c}_{\kappa}^{+}$ and ${c}_{\kappa}^{}$ are defined in (2.12) and (2.15).
so that ${w}_{k}>0$ and ${v}_{k}<0$ for $k=0,1,2$.
3 Stationary states II
In this section we generalize the problem to a bridge with N subspans and $M=N1$ piers. The existence of buckled solutions is investigated in connection with the length of the subspans. 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 ${u}_{(1)}^{+}$ or ${u}_{(1)}^{}$. 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 3.1 For any $\kappa >0$, $N\in \mathbb{N}$, (1.2) admits two buckled solutions.

In the odd case, $M=2m1$, there exist${u}_{(2m)}^{+}$and${u}_{(2m)}^{}$provided that$b>{b}_{\kappa}(m)$, where the value of${b}_{\kappa}(m)$is characterized in (3.5).

In the even case, $M=2m$, there exist${u}_{(2m+1)}^{+}$and${u}_{(2m+1)}^{}$provided that$b>{c}_{\kappa}^{+}(m)$and$b>{c}_{\kappa}^{}(m)$, respectively, where the values of${c}_{\kappa}^{+}(m)$and${c}_{\kappa}^{}(m)$are characterized in (3.6).
where ${\epsilon}_{\kappa}$ is given by (2.9).
with ${A}_{2m}^{+}$, ${A}_{2m}^{}$ defined as in (3.1).
The even case. In this case, we construct the solutions ${u}_{(2m+1)}^{+}$ and ${u}_{(2m+1)}^{}$, where $m=M/2=(N1)/2$.
□
Lemma 3.2 (Characterization of the bifurcation values)
where ${\epsilon}_{\kappa}$ and ${b}_{\kappa}$ are computed by (2.9) and (2.10), respectively.
□
Finally, it is worth noting that by removing the coupling between the roadbed and the cable, we recover wellknown results (see, for instance, [5, 8])
Declarations
Authors’ Affiliations
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