• Korean Journal of Mathematics
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• v.15 no.1
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• pp.1-11
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• 2007

It is well known that a suspension bridge may display certain oscillations under external aerodynamic forces. Under the action of a strong wind, in particular, a narrow and very flexible suspension bridge can undergo dangerous oscillations. So it would be very contributive to determine under what conditions a similar situation cannot occur, and find out safe parameters of the bridge construction. In this paper, we investigate relations between the multiplicity of solutions and nonlinear terms in this suspension bridge equation using critical point theorem and linking theorem.

• Korean Journal of Mathematics
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• v.19 no.2
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• pp.233-242
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• 2011

We get a theorem that there exist at least two solutions for the piecewise linear suspension bridge equation with variable coefficient jumping nonlinearity and Dirichlet boundary condition when the variable coefficient of the nonlinear term crosses first two successive negative eigenvalues. We obtain this multiplicity result by applying Leray-Schauder degree theory.

• Korean Journal of Mathematics
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• v.16 no.1
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• pp.1-24
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• 2008

Let $Lu=u_{tt}+u_{xxxx}$ and E be the complete normed space spanned by the eigenfunctions of L. We reveal the existence of six nontrivial solutions of a nonlinear suspension bridge equation $Lu+bu^+=1+{\epsilon}h(x,t)$ in E when the nonlinearity crosses three eigenvalues. It is shown by the critical point theory induced from the limit relative category of the torus with three holes and finite dimensional reduction method.

• Journal for History of Mathematics
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• v.18 no.2
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• pp.107-116
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• 2005

A suspension bridge is an example of a nonlinear dynamical system, especially systems with the so called jumping nonlinearity. The fact that we deal with a serious and topical problem is demonstrated for example by the collapse of the Tacoma Narrow suspension bridge. So it would be very contributive to determine under what conditions a similar situation cannot occur and find out safe parameters of the bridge construction. In this paper, we show various possibilities how to model the behaviour of suspension bridge. Then we introduce our own results concerning existence and uniqueness of time-periodic solutions.

• Wind and Structures
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• v.7 no.6
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• pp.421-447
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• 2004

This paper presents a time domain approach for predicting buffeting response of long suspension bridges under skew winds. The buffeting forces on an oblique strip of the bridge deck in the mean wind direction are derived in terms of aerodynamic coefficients measured under skew winds and equivalent fluctuating wind velocities with aerodynamic impulse functions included. The time histories of equivalent fluctuating wind velocities and then buffeting forces along the bridge deck are simulated using the spectral representation method based on the Gaussian distribution assumption. The self-excited forces on an oblique strip of the bridge deck are represented by the convolution integrals involving aerodynamic impulse functions and structural motions. The aerodynamic impulse functions of self-excited forces are derived from experimentally measured flutter derivatives under skew winds using rational function approximations. The governing equation of motion of a long suspension bridge under skew winds is established using the finite element method and solved using the Newmark numerical method. The proposed time domain approach is finally applied to the Tsing Ma suspension bridge in Hong Kong. The computed buffeting responses of the bridge under skew winds during Typhoon Sam are compared with those obtained from the frequency domain approach and the field measurement. The comparisons are found satisfactory for the bridge response in the main span.

• Interaction and multiscale mechanics
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• v.3 no.4
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• pp.309-320
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• 2010

With taking the geometric nonlinearity of bridge structure into account, a framework is presented for predicting the dynamic responses of a long-span suspension bridge subjected to running train and turbulent wind. The nonlinear dynamic equations of the coupled train-bridge-wind system are established, and solved with the Newmark numerical integration and direct interactive method. The corresponding linear and nonlinear processes for solving the system equation are described, and the corresponding computer codes are written. The proposed framework is then applied to a schemed long-span suspension bridge with the main span of 1120 m. The whole histories of the train passing through the bridge under turbulent wind are simulated, and the dynamic responses of the bridge are obtained. The results demonstrate that the geometric nonlinearity does not influence the variation tendency of the bridge displacement histories, but the maximum responses will be changed obviously; the lateral displacement of bridge are more sensitive to the wind than the vertical ones; compared with wind velocity, train speed affects the vertical maximum responses a little more clearly.

• Structural Engineering and Mechanics
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• v.56 no.6
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• pp.939-957
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• 2015

In this paper, a new approach based on the continuum model is proposed to estimate the main cable tension force of suspension bridges from measured natural frequencies. This approach considered the vertical vibration of a main cable hinged at both towers and supported by an elastic girder and hangers along its entire length. The equation reflected the relationship between vibration frequency and horizontal tension force of a main cable was derived. To avoid to generate the additional cable tension force by sag-extensibility, the analytical solution of characteristic equation for anti-symmetrical vibration mode of the main cable was calculated. Then, the estimation of main cable tension force was carried out by anti-symmetric characteristic frequency vector. The errors of estimation due to characteristic frequency deviations were investigated through numerical analysis of the main cable of Taizhou Bridge. A field experiment was conducted to verify the proposed approach. Through measuring and analyzing the responses of a main cable of Taizhou Bridge under ambient excitation, the horizontal tension force of the main cable was identified from the first three odd frequencies. It is shown that the estimated results agree well with the designed values. The proposed approach can be used to conduct the long-term health monitoring of suspension bridges.

• Journal of the Korea Computer Industry Society
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• v.3 no.5
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• pp.569-574
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• 2002

We model the torsional oscillation of a suspension bridge, which is the forced sine-Cordon equation on a bounded domain. We use finite difference method to solve nonlinear partial differential equation numerically. The partial differential equation has multiple periodic solutions. Whether the span oscillates with small or large amplitude depends oかy on its initial displacement and velocity. Moreover, we observe that the qualitative properties are consistent with the behavior observed at the Tacoma Narrows Bridge on the day of its collapse.

• Bulletin of the Korean Mathematical Society
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• v.33 no.4
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• pp.503-512
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• 1996

In this paper we investigate a relation between the multiplicity of solutions and source terms in a nonlinear suspension bridge equation in the interval $(-\frac{2}{\pi}, \frac{2}{\pi})$, under Dirichlet boundary condition $$ (0.1) u_{tt} + u_{xxxx} + bu^+ = f(x) in (-\frac{2}{\pi}, \frac{2}{\pi}) \times R, $$ $$ (0.2) u(\pm\frac{2}{\pi}, t) = u_{xx}(\pm\frac{2}{\pi}, t) = 0, $$ $$ (0.3) u is \pi - periodic in t and even in x and t, $$ where the nonlinearity - $(bu^+)$ crosses an eigenvalue $\lambda_{10}$. This equation represents a bending beam supported by cables under a load f. The constant b represents the restoring force if the cables stretch. The nonlinearity $u^+$ models the fact that cables expansion but do not resist compression.

• Structural Engineering and Mechanics
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• v.62 no.1
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• pp.85-95
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• 2017

This paper presents an explicit analytical iteration method for form-finding analysis of suspension bridges. By extending the conventional analytical form-finding method predicated on the elastic catenary theory, two nonlinear governing equations are derived for calculating the accurate unstrained lengths of the entire cable systems both the main cable and the hangers. And for the gradient-based iteration method, the derivation of explicit calculation for the Jacobian matrix while solving the nonlinear governing equation enhances the computational efficiency. The results from sensitivity analysis show well performance of the explicit Jacobian matrix compared with the traditional finite difference method. According to two numerical examples of long span suspension bridges studied, the proposed method is also compared with those reported approaches or the fundamental criterions in suspension bridge structural analysis, which eventually confirms the accuracy and efficiency of the proposed approach.