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
MSC: 35G30, 74G05, 74G60, 74K10.
Keywords:
extensible elastic beam; suspension bridge; boundary value problems for nonlinear higherorder PDE; bifurcation and buckling1 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
In the case of a bridge with a single span (no intermediate pier), the problem can
be recast into a nondimensional setting, where its length is supposed to be unitary,
for simplicity. Let
Assuming that both ends of the bridge are pinned, the equation of the bending equilibrium looks like (see [1])
The term
Assume now that the bridge span is composed of N extensible beams whose internal ends are hinged to
We denote by
According to the assumptions, the deflection u of the whole span obeys
Furthermore, we assume that consecutive subspans are mutually clamped at the common
end, which in turn implies the continuity of
Figure 1. The joint connecting two consecutive subspans and the intermediate pier.
Let
Such solutions must fulfill the mutual groove condition
We finally observe that the total elongation of the span equals the sum of the elongations of each beam, namely
Throughout the paper, we assume a uniform distribution of the axial load, that is,
Our aim is to scrutinize the existence of suitable buckled solutions for u, which can be obtained by joining buckled solutions for
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 [47]).
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
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 thermoelastic theories. An example is the following quasilinear equation:
which is obtained by matching the modeling of the extensible elastic beam (see [9,10]) with the wellknown equation describing the motion of a damped suspension bridge (see [11,12])
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, [1618]).
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
2 Stationary states I
2.1 A single span without piers (
N
=
1
)
The set
It is worth noting that
When
Theorem 2.1 (see [7], Th. 4.1)
Whenκvanishes, the set
where
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 negative buckled solution,
• a positive buckled solution,
Proof For all values
which solves the WoinowskyKrieger problem (see [21])
If
Since
Figure 2. The buckled solutions
2.2 A bridge with a single pier (
N
=
2
)
Now we consider a bridge with a single pier at
When
Theorem 2.3When
Proof In order to establish the form of
where
Arguing as in Section 2.1, we exploit the explicit expression of solutions
where, for some
Of course, such solutions both exist provided that
and
Then, by joining w and v, we obtain the whole solution
The function
Figure 3. The buckled solutions
Computation of
Thus, the required value
which implies
It can be easily checked (see Figure 4) that
In order to compute the critical value of the axial force,
Figure 4. The value of
Summarizing, for any given
and
Then, when
In the limit
This means that solutions
2.3 A bridge with two piers (
N
=
3
)
When the bridge has two intermediate piers and three subspans, we shall construct
solutions
Theorem 2.4When
Proof The buckled solutions are constructed as follows. The former,
so that
In order to construct
Then, we apply the rescaling procedure with the scale factor
Taking into account that
so that
Accordingly, we obtain
where
Of course, such solutions exist at the same time, provided that
where
If this is the case, we havethe whole solution (see Figure 5a)
In order to construct
By means of a rescaling procedure with the scale factor
Accordingly, we find
where
So, we obtain
where
Of course, such solutions do exist together provided that
where
By means of (2.9) it is easy to check that
□
Figure 5. The buckled solutions
3 Stationary states II
In this section we generalize the problem to a bridge with N subspans and
Theorem 3.1For any
• In the odd case,
• In the even case,
ProofThe odd case. Solutions
where
CONSTRUCTION OF
so that
we stress that each restriction
is similar to
Therefore, on
while on
where
Such solutions exist provided that
By virtue of Lemma 3.2, which will be proved later, we get
CONSTRUCTION OF
so that
where
is similar to
so on
while on
with
The even case. In this case, we construct the solutions
CONSTRUCTION OF
so that
where
and
So, on
while on
where
Such solutions exist provided that
By virtue of Lemma 3.2, we get
CONSTRUCTION OF
so that
where
and
As a consequence, on
while on
where
Such solutions exist provided that
By virtue of Lemma 3.2, we get
□
Lemma 3.2 (Characterization of the bifurcation values)
For any given
where
Proof First, in view of (3.2), we have to prove that
which is equivalent to
By replacing the expression of κ given by (2.9), we obtain
which implies that (3.5) holds for all
which is equivalent to
By virtue of (2.9), we obtain
which is identically satisfied for all admissible values
and this is equivalent to
Applying (2.9) as in the previous cases, we obtain
□
Previous results can be summarized by means of a simple sketch which highlights the main bifurcation features (see Figure 6).
Figure 6. The bifurcation portrait: forNodd (on the left,
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])
Remark 3.3 In the limit
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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