Unsteady Aerodynamic Modeling and Flutter Analysis of Long-Span Suspension Bridges

2012 ◽  
Author(s):  
Andrea Arena ◽  
Walter Lacarbonara ◽  
Pier Marzocca
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Equations Of Motion ◽  
Suspension Bridge ◽  
Longitudinal Direction ◽  
Lagrangian Formulation ◽  
Suspension Bridges ◽  
Equilibrium Equations ◽  
Aeroelastic System ◽  
Flutter Analysis

A parametric one-dimensional model of suspension bridges is employed to investigate their static and dynamic aeroelastic behavior in response to a gust load and at the onset of flutter. The equilibrium equations are obtained via a direct total Lagrangian formulation where the kinematics for the deck, assumed to be linear, feature the vertical and the chord-wise displacements of the deck mean axis and the torsional rotations of the deck cross sections, while preserving their shape during rotation. The cables elasto-geometric stiffness contribution is obtained by condensing the equilibrium in the longitudinal direction assuming small horizontal displacements and neglecting the cable kinematics along the bridge chord-wise direction. The equations of motion are linearized about the prestressed static aeroelastic configuration and are obtained via an updated Lagrangian formulation. The equations of motion governing the structural dynamics of the bridge are coupled with the incompressible unsteady aero-dynamic model obtained by a set of reduced-order indicial functions developed for the cross section of a suspension bridge, here represented by a rectangular cross-section. The space dependence of the governing equations is treated using the Galerkin approach borrowing as set of trial functions, the eigenbasis of the modal space. The time integration is subsequently performed by using a numerical scheme that includes the modal reduced dynamic aeroelastic Ordinary Differential Equations (ODEs) and the added aerodynamic states also represented in ODE form, the latter being associated with the lag-state formulation pertinent to the unsteady wind-induced loads. The model is suitable to analyze the effect of a time and space non uniform gust load distributed on the bridge span. The obtained aeroelastic system is also suitable to study the onset of flutter and to investigate the sensitivity of the flutter condition on geometrical and aerodynamic parameters. The flutter instability is evaluated using appropriate frequency and time domain characteristics. The parametric continuum model is exploited to perform dynamic aeroelastic flutter analysis and gust response of the Runyang Suspension Bridge over the Yangtze river in China.

Effect of Slenderness Ratio on the Natural Frequencies of Pre-Twisted Cantilever Beams of Uniform Rectangular Cross-Section

1971 ◽  
Vol 13 (1) ◽  
pp. 51-59 ◽  
Author(s):  
B. Dawson ◽  
N. G. Ghosh ◽  
W. Carnegie
Keyword(s):  
Differential Equations ◽  
Shear Deformation ◽  
Cross Section ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Slenderness Ratio ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Rotary Inertia ◽  
Cantilever Beams

This paper is concerned with the vibrational characteristics of pre-twisted cantilever beams of uniform rectangular cross-section allowing for shear deformation and rotary inertia. A method of solution of the differential equations of motion allowing for shear deformation and rotary inertia is presented which is an extension of the method introduced by Dawson (1)§ for the solution of the differential equations of motion of pre-twisted beams neglecting shear and rotary inertia effects. The natural frequencies for the first five modes of vibration are obtained for beams of various breadth to depth ratios and lengths ranging from 3 to 20 in and pre-twist angle in the range 0–90°. The results are compared with those obtained by an alternative method (2), where available, and also to experimental results.

One-dimensional model of fourth order for rods with loading on lateral boundary: The case of rectangular cross section

2016 ◽  
Vol 22 (12) ◽  
pp. 2269-2287 ◽  
Author(s):  
Erick Pruchnicki
Keyword(s):  
Cross Section ◽  
Fourth Order ◽  
Displacement Field ◽  
Rectangular Cross Section ◽  
Transverse Dimension ◽  
Lateral Boundary ◽  
Dimensional Model ◽  
Equilibrium Equations ◽  
Lagrange Equations ◽  
One Dimensional

We propose deducing from three-dimensional elasticity a one dimensional model of a beam when the lateral boundary is not free of traction. Thus the simplification induced by the order of magnitude of transverse shearing and transverse normal stress must be removed. For the sake of simplicity we consider a beam with rectangular cross section. The displacement field in rods can be approximated by using a Taylor–Young expansion in transverse dimension of the rod and we truncate the potential energy at the fourth order. By considering exact equilibrium equations, the highest-order displacement field can be removed and the Euler–Lagrange equations are simplified.

scholarly journals Simplified Analytical Method for Optimized Initial Shape Analysis of Self-Anchored Suspension Bridges and Its Verification

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Myung-Rag Jung ◽  
Dong-Ju Min ◽  
Moon-Young Kim
Keyword(s):  
Analytical Method ◽  
Bending Moment ◽  
Suspension Bridge ◽  
Suspension Bridges ◽  
Equilibrium Equations ◽  
Initial State ◽  
Initial Shape ◽  
Main Cable ◽  
Initial Shape Analysis ◽  
Bridge Models

A simplified analytical method providing accurate unstrained lengths of all structural elements is proposed to find the optimized initial state of self-anchored suspension bridges under dead loads. For this, equilibrium equations of the main girder and the main cable system are derived and solved by evaluating the self-weights of cable members using unstrained cable lengths and iteratively updating both the horizontal tension component and the vertical profile of the main cable. Furthermore, to demonstrate the validity of the simplified analytical method, the unstrained element length method (ULM) is applied to suspension bridge models based on the unstressed lengths of both cable and frame members calculated from the analytical method. Through numerical examples, it is demonstrated that the proposed analytical method can indeed provide an optimized initial solution by showing that both the simplified method and the nonlinear FE procedure lead to practically identical initial configurations with only localized small bending moment distributions.

scholarly journals Cross-Sectional Analysis of the Resistance of Rc Members Subjected to Bending With/without Axial Force

2021 ◽  
Author(s):  
Marek Lechman
Keyword(s):  
Cross Section ◽  
Axial Force ◽  
Rectangular Cross Section ◽  
Bending Moment ◽  
Equilibrium Equations ◽  
Cross Sectional ◽  
Cross Sectional Analysis ◽  
Nonlinear Stress ◽  
The Cross ◽  
Sectional Analysis

The paper presents section models for analysis of the resistance of RC members subjected to bending moment with or without axial force. To determine the section resistance the nonlinear stress-strain relationship for concrete in compression is assumed, taking into account the concrete softening. It adequately describes the behavior of RC members up to failure. For the reinforcing steel linear elastic-ideal plastic model is applied. For the ring cross-section subjected to bending with axial force the normalized resistances are derived in the analytical form by integrating the cross-sectional equilibrium equations. They are presented in the form of interaction diagrams and compared with the results obtained by testing conducted on RC columns under eccentric compression. Furthermore, the ultimate normalized bending moment has been derived for the rectangular cross-section subjected to bending without axial force. It was applied in the cross-sectional analysis of steel and concrete composite beams, named BH beams, consisting of the RC rectangular core placed inside a reversed TT welded profile. The comparisons made indicated good agreements between the proposed section models and experimental results.

Numerical Computation of Flow in Rotating Ducts

1977 ◽  
Vol 99 (1) ◽  
pp. 148-153 ◽  
Author(s):  
A. K. Majumdar ◽  
V. S. Pratap ◽  
D. B. Spalding
Keyword(s):  
Experimental Data ◽  
Kinetic Energy ◽  
Finite Difference ◽  
Numerical Computation ◽  
Cross Section ◽  
Dissipation Rate ◽  
Rectangular Cross Section ◽  
Longitudinal Direction ◽  
The Other ◽  
Constant Area

A finite-difference procedure is employed to predict the turbulent flow in ducts of rectangular cross-section, rotating about an axis normal to the longitudinal direction. The flows were treated as “parabolic” and the turbulence model used involved the solution of two differential equations, one for the kinetic energy of the turbulence and the other for its dissipation rate. Agreement with experimental data is good for a constant-area duct at low rotation, but less satisfactory for a divergent duct at larger rotation. It is argued that a “partially-parabolic” procedure will be needed to predict the latter flow correctly.

Experimental Study of the Nonlinear Torsional Flutter of a Long-Span Suspension Bridge with a Double-Deck Truss Girder

2021 ◽  
pp. 2150102
Author(s):  
Ming Li ◽  
Yanguo Sun ◽  
Yongfu Lei ◽  
Haili Liao ◽  
Mingshui Li
Keyword(s):  
Wind Tunnel ◽  
Model Test ◽  
Wind Tunnel Test ◽  
Suspension Bridge ◽  
Suspension Bridges ◽  
Nonlinear Flutter ◽  
Long Span ◽  
Aeroelastic Model ◽  
Flutter Analysis ◽  
Section Model

The purpose of this study is to investigate the nonlinear torsional flutter of a long-span suspension bridge with a double-deck truss girder. First, the characteristics of nonlinear flutter are studied using the section model in the wind tunnel test. Different aerodynamic measures, e.g. upper and lower stabilizers and horizontal flaps, are applied to improve the flutter performance of the double-deck truss girder. Then, the full bridge aeroelastic model is tested in the wind tunnel to further examine the flutter performance of the bridge with the optimal truss girder. Finally, three-dimensional (3D) flutter analysis is performed to study the static wind-induced effects on the nonlinear flutter of the long-span suspension bridge. The results show that single-degree-of-freedom torsional limit cycle oscillations occur at large amplitudes for the double-deck truss section at the attack angles of [Formula: see text] and [Formula: see text]. The upper and lower stabilizers installed on the upper and lower decks, respectively, and the flaps installed near the bottoms of the sidewalks can all effectively alleviate the torsional flutter responses. Meanwhile, it is found that the torsional flutter responses of the truss girder in the aeroelastic model test are much smaller than those in the section model test. The 3D flutter analysis demonstrates that the large discrepancies between the flutter responses of the two model experiments can be attributed to the additional attack angle caused by the static wind-induced displacements. This finding highlights the importance and necessity of considering the static wind-induced effects in the flutter design of long-span suspension bridges.

scholarly journals Nonlinear Aeroelastic in-Plane Behavior of Suspension Bridges under Steady Wind Flow

2020 ◽  
Vol 10 (5) ◽  
pp. 1689 ◽  
Author(s):  
Simona Di Nino ◽  
Angelo Luongo
Keyword(s):  
Elastic Beam ◽  
Suspension Bridge ◽  
Multiple Scale ◽  
Continuous Model ◽  
Suspension Bridges ◽  
Wind Flow ◽  
Aeroelastic System ◽  
Natural Modes ◽  
Structural Nonlinearities ◽  
Limit Cycle Amplitude

The nonlinear aeroelastic behavior of suspension bridges, undergoing dynamical in-plain instability (galloping), is analyzed. A nonlinear continuous model of bridge is formulated, made of a visco-elastic beam and a parabolic cable, connected each other by axially rigid suspenders, continuously distributed. The structure is loaded by a uniform wind flow which acts normally to the bridge plane. Both external and internal damping are accounted for the structure, according to the Kelvin-Voigt rheological model. The nonlinear aeroelastic effects are evaluated via the quasi-static theory, while structural nonlinearities are not taken into account. First, the free dynamics of the undamped bridge are addressed, and the natural modes determined. Then, the nonlinear equations ruling the dynamics of the aeroelastic system, close to the bifurcation point, are tackled by the Multiple Scale Method. This is directly applied to the partial differential equations, and provides the finite-dimensional bifurcation equations. From these latter, the limit-cycle amplitude and its stability are evaluated as function of the mean wind velocity. A case study of suspension bridge is analyzed.

An Approximate Theory Governing Symmetric Motions of Elastic Rods of Rectangular or Square Cross Section

1968 ◽  
Vol 35 (2) ◽  
pp. 333-341 ◽  
Author(s):  
Paul Hertelendy
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Equations Of Motion ◽  
Longitudinal Mode ◽  
Circular Cylinders ◽  
Wave Number ◽  
Approximate Theory ◽  
Fourier Synthesis ◽  
Experimental Frequency ◽  
Shear Modes

Variational equations of motion are developed for symmetric motions of linear elastic bars of rectangular cross section. In the finite term approximation, sufficient terms are retained to allow a longitudinal mode, two thickness-stretch modes, and two thickness-shear modes of vibration in an infinite bar of square cross section. Modes for complex wave numbers are also investigated. Adjustment factors in the strain energy and kinetic energy potentials are used to match exact and experimental solutions. Experimental frequency versus wave number results for four modes are reduced by Fourier synthesis and compared both to the approximate theory and to the exact solution for circular cylinders. Theory is intended to predict behavior of thick rectangular bars for which the plane stress solution is not accurate.

Three-Dimensional Vibrations of a Suspension Bridge Under Stochastic Traffic Flows and Road Roughness

2016 ◽  
Vol 16 (07) ◽  
pp. 1550038 ◽  
Author(s):  
Xinfeng Yin ◽  
Yang Liu ◽  
Shihui Guo ◽  
W. Zhang ◽  
C. S. Cai
Keyword(s):  
Dynamic Performance ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Coupled System ◽  
Suspension Bridge ◽  
Suspension Bridges ◽  
Traffic Flows ◽  
Longitudinal Vibrations ◽  
Coupled Equations ◽  
Road Roughness

When studying the vibration of a bridge–vehicle coupled system, most researchers mainly focus on the vertical vibration of bridges under moving vehicular loads, while the lateral and longitudinal vibrations of the bridges and the stochastic characteristics of the traffic flows are neglected. However, for long-span suspension bridges, neglecting the bridge’s three-dimensional (3D) vibrations under stochastic traffic flows can cause considerable inaccuracy in predicting the dynamic performance. This study is mainly focused on establishing a new methodology fully considering a suspension bridge’s vertical, lateral, and longitudinal vibrations induced by stochastic traffic flows under varied road roughness conditions. A new full-scale vehicle model with 18 degrees of freedom (DOFs) was developed to predict the longitudinal and lateral vibrations of the vehicle. An improved Cellular Automaton (CA) model considering the influence of the next-nearest vehicle was introduced. The bridge and vehicles in traffic flow coupled equations are established by combining the equations of motion of both the bridge and vehicles using the displacement relationship and interaction force relationship at the patch contacts. The numerical simulations show that the proposed method can rationally simulate the 3D vibrations of the suspension bridge under stochastic traffic flows.

Seismic Control of a Self-Anchored Suspension Bridge Using Fluid Viscous Dampers

2020 ◽  
pp. 2150025
Author(s):  
Dongming Feng ◽  
Jingquan Wang
Keyword(s):  
Energy Dissipation ◽  
Shear Force ◽  
Bending Moment ◽  
Suspension Bridge ◽  
Longitudinal Direction ◽  
Damping Coefficient ◽  
Suspension Bridges ◽  
Seismic Control ◽  
Fluid Viscous Dampers ◽  
Connection System

A self-anchored suspension bridge balances forces internally without external anchorage requirements, making it suitable for sites where anchorages would be difficult to construct. It often adopts either a full-floating or a semi-floating tower-girder connection system, which may result in large displacement responses along bridge longitudinal direction during earthquakes. This study investigated the efficacy of using the fluid viscous damper (FVD) for seismic control of a single-tower self-anchored suspension bridge. First, the energy dissipation behaviors of the FVD under sinusoidal excitations were studied. It revealed that besides the damper parameters (i.e. damping coefficient and velocity exponent) of an FVD itself, the energy dissipation capacity also relies on the characteristics of external excitations. Therefore, optimum damper parameters added to a structure should be determined on a case-by-case basis. Parametric study was then carried out on the prototype bridge, which indicated a tendency of decreasing the longitudinal deck/tower displacements and tower forces with increasing damping coefficient [Formula: see text] and decreasing velocity exponent [Formula: see text]. Compared with the linear FVD, the nonlinear FVD with a smaller velocity exponent can develop more rectangular force-displacement loops and thus achieve better energy dissipation performance. With selected optimum damper parameters (i.e. [Formula: see text][Formula: see text]kN[Formula: see text]m[Formula: see text][Formula: see text]s[Formula: see text] and [Formula: see text]) for the two FVDs added between the deck and the tower, the longitudinal deck and tower displacements could be reduced by 54%, while the peak bending moment and shear force at the tower base could be reduced by 30% and 19%, respectively. It is concluded that the nonlinear FVD can provide a simple and efficient solution to reduce displacement responses of self-anchored suspension bridges while simultaneously reducing the bending moment and shear force in the tower.

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