Variational formulation of the Melan equation

Gazzola, Filippo; Wang, Yongda; Pavani, Raffaella

doi:10.1002/mma.3962

Cited by 13 publications

(10 citation statements)

References 9 publications

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“…To highlight the level of applicability of the results shown, an application to crossed suspended bridges is presented, following some adjustments on the models suggested in [12].…”

Section: Discussionmentioning

confidence: 99%

“…To comply with the conceptual meaning of the model, the authors will adopt x as independent variable for this section, instead of t, as previously, as in these models, as x represents displacement. Such suspension crossed bridges can be approached via a coupled system of two fourth order differential equations, following the same principle as the original model suggested in [12], assuming the adapted form…”

Section: Bending Of Crossed Suspension Bridgesmentioning

confidence: 99%

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On Coupled Systems of Lidstone-Type Boundary Value Problems

Sousa

Minhós

Fialho

2021

Mathematical Modelling and Analysis

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This research concerns the existence and location of solutions for coupled system of differential equations with Lidstone-type boundary conditions. Methodology used utilizes three fundamental aspects: upper and lower solutions method, degree theory and nonlinearities with monotone conditions. In the last section an application to a coupled system composed by two fourth order equations, which models the bending of coupled suspension bridges or simply supported coupled beams, is presented.

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“…To highlight the level of applicability of the results shown, an application to crossed suspended bridges is presented, following some adjustments on the models suggested in [12].…”

Section: Discussionmentioning

confidence: 99%

Section: Bending Of Crossed Suspension Bridgesmentioning

confidence: 99%

On Coupled Systems of Lidstone-Type Boundary Value Problems

Sousa

Minhós

Fialho

2021

Mathematical Modelling and Analysis

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“…A model inspired to the Melan equation [24] and to its variational formulation [21] was studied in [17], in which the main span of the suspension bridge was considered as a combined system of two perfectly flexible strings (the cables) linked to the deck through inextensible hangers. On the contrary a model with fixed cables and extensible hangers was considered in [16], focusing on the difficult choice of the slackening nonlinearities.…”

Section: Discussionmentioning

confidence: 99%

A new model for suspension bridges involving the convexification of the cables

Crasta,

Falocchi,

Gazzola

2018

Preprint

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The final purpose of this paper is to show that, by inserting a convexity constraint on the cables of a suspension bridge, the torsional instability of the deck appears at lower energy thresholds. Since this constraint is suggested by the behavior of real cables, this model appears more reliable than the classical ones. Moreover, it has the advantage to reduce to two the number of degrees of freedom (DOF), avoiding to introduce the slackening mechanism of the hangers. The drawback is that the resulting energy functional is extremely complicated, involving the convexification of unknown functions. This paper is divided in two main parts. The first part is devoted to the study of these functionals, through classical methods of calculus of variations. The second part applies this study to the suspension bridge model with convexified cables.

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“…The presence of the nonlocal term makes challenging the study of the equation from both the theoretical as from the numerical point of view, see e.g. [10,11,21]; although (1.1) cannot be derived from the variation of the corresponding energy [11], von Kármán-Biot [23] call the Melan equation (1.1) the fundamental equation of the theory of the suspension bridge. This equation is our starting point, we propose a more reliable model for suspension bridge in which we have two strings (the cables) linked to the same deck, through inextensible hangers, see Section 2.1.…”

Section: Introductionmentioning

confidence: 99%

Torsional instability and sensitivity analysis in a suspension bridge model related to the Melan equation

Falocchi

2019

Communications in Nonlinear Science and Numerical Simulation

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Inspired by the Melan equation we propose a model for suspension bridges with two cables linked to a deck, through inextensible hangers. We write the energy of the system and we derive from variational principles two nonlinear and nonlocal hyperbolic partial differential equations, involving the vertical displacement and the torsional rotation of the deck. We prove existence and uniqueness of a weak solution and we perform some numerical experiments on the isolated system; moreover we propose a sensitivity analysis of the system by mechanical parameters in terms of torsional instability. Our results display that there are specific thresholds of torsional instability with respect to the initial amplitude of the longitudinal mode excited.

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Variational formulation of the Melan equation

Cited by 13 publications

References 9 publications

On Coupled Systems of Lidstone-Type Boundary Value Problems

On Coupled Systems of Lidstone-Type Boundary Value Problems

A new model for suspension bridges involving the convexification of the cables

Torsional instability and sensitivity analysis in a suspension bridge model related to the Melan equation

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