No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Stability of dynamic response of suspension bridges
Abstract
The potential occurrence of internal parametric resonance phenomena has been recently indicated as a potential contributory cause of the appearance of critical dynamic states in long-span suspension bridges. At the same time, suspension bridges, in view of their flexibility, are prone to aeroelastic response, such as vortex shedding, torsional divergence and flutter. In this paper, a non-linear dynamic model of a suspension bridge is devised, with the purpose of providing a first attempt toward a unified framework for the study of aeroelastic and internal resonance instabilities. Inspired by the pioneering work of Herrmann and Hauger, the analyses have been based on a linearized formulation that is able to represent the main structural non-linear effects and the coupling given by aerodynamic forces. The results confirm that the interaction between aeroelastic effects and non-linear internal resonance leads to unstable conditions for wind speeds which can be lower than the critical threshold for standard aeroelastic predictions.
... However, it is known that cable-supported bridges can exhibit structural nonlinearities such as geometric nonlinearities [15,16], material nonlinearities [17,18] as well as localized nonlinearities like hydraulic buffers [17,19,20]. Additionally, it is worth noting that mathematicians have shown that dynamic vertical forcing can lead to large torsional oscillations due to nonlinear vertical-torsional mode coupling associated with geometric nonlinearities [21][22][23][24]. In [24], these large oscillations caused by structural coupling between the modes of vibration are referred as internal parametric resonance, a structural dynamic instability. ...
... Additionally, it is worth noting that mathematicians have shown that dynamic vertical forcing can lead to large torsional oscillations due to nonlinear vertical-torsional mode coupling associated with geometric nonlinearities [21][22][23][24]. In [24], these large oscillations caused by structural coupling between the modes of vibration are referred as internal parametric resonance, a structural dynamic instability. This instability is solely structural unlike flutter that is an aeroelastic phenomenon. ...
... Equations for the non-normalized flutter derivatives are easily obtained by comparing the first and second rows of Eqs. (23) and (24). For example, it is possible to find that 1 = 1∕2 2 ( * ∕ ). ...
Wind tunnel tests for cable-supported bridges have been utilized extensively to achieve stable and wind resistant bridge designs over the past decades. Section model tests are by far the most common tests carried out. This is mainly due to their simplicity and their cost effectiveness. However, in order to reach this level of simplicity, some generalizations are required. One of them is the assumption that the bridge structure behaves linearly. For most bridges, this seems reasonable, but for very long spans, the cable system plays a dominant role in providing the vertical and torsional stiffnesses. Hence, as the cable system behaves nonlinearly, very long spans show a stronger nonlinear behavior, especially for suspension bridges. Furthermore, the theoretical demonstrations that have been made with regard to structural dynamic instabilities associated with geometric nonlinearities in suspension bridges confirm the need for a better understanding of the possible interaction between nonlinear structural effects and aeroelastic effects. Therefore, this paper presents the theoretical developments for a new type of section model test for bridges that accounts for geometric nonlinearities of the bridge structure. This theory for nonlinear section model tests starts from two-mode nonlinear generalized stiffness parameters obtained using nonlinear pushover analysis, which need to be scaled using a specifically developed procedure. Using eleven numerical models of cable-supported bridges, the assumptions made in the theory for nonlinear tests are then validated. The proposed scaling procedure is also tested.
... Moreover, the excitation of one natural mode could give rise to another one. The importance of the energy transfer during nonlinear vibrations of suspension bridges has been discussed in [20][21][22], and this nonlinear phenomenon has been indicated as 'a potential contributory cause of the appearance of critical dynamic states in long-span suspension bridges' [22], especially under the action of such external forces as moving loads, earthquakes, and/or wind loading. ...
... Moreover, the excitation of one natural mode could give rise to another one. The importance of the energy transfer during nonlinear vibrations of suspension bridges has been discussed in [20][21][22], and this nonlinear phenomenon has been indicated as 'a potential contributory cause of the appearance of critical dynamic states in long-span suspension bridges' [22], especially under the action of such external forces as moving loads, earthquakes, and/or wind loading. ...
... In order to utilize the generalized method of multiple time scales for solving Eqs. (22) and (23) which describe the damped nonlinear vibrations, the viscosity coefficients β i should be represented as β i = ε k μ i , where μ i are finite values, with k = 1 and 2 for the 2:1 and 1:1 internal resonances, respectively. ...
The history of formulation of the efficient method for studying the nonlinear dynamic response of structures, damping features of which depend on natural frequencies of vibrations, is presented. This technique is the modified version of the method of multiple scales, the efficiency of which is illustrated by the examples of nonlinear vibrations of suspension bridges and plates subjected to the conditions of the internal resonances.
... Nevertheless, it is known that cable-supported bridges can show geometric nonlinearities [6] as well as material nonlinearities [8] and localized nonlinearities [3] in specific cases. Using advanced mathematical models of suspension bridges, mathematicians and engineers demonstrated that a dynamic vertical force can lead to large torsional oscillations due to nonlinear structural dynamic coupling [1,4]. This coupling is caused by the intrinsic nonlinear geometric behavior of the suspension bridge system. ...
... This coupling is caused by the intrinsic nonlinear geometric behavior of the suspension bridge system. In [4], it is mentioned that internal parametric resonance, a structural dynamic instability caused by the nonlinear structural coupling between modes of vibration, would be responsible. To the authors' knowledge, such large torsional oscillations due to nonlinear structural dynamic effects have never been witnessed on an actual bridge. ...
It has been demonstrated by mathematicians using nonlinear simplified models of suspension bridges that a vertical dynamic forcing can cause large torsional vibrations due to geometric nonlinearities of the bridge. Compared to the extensive research that has been conducted on aeroelastic instabilities in cable-supported bridges, the approach used by the mathematicians seems too simplistic, especially in terms of wind load modeling. However, they raise a point that is not considered in the wind design of cable-supported bridges, i.e., an interaction between nonlinear structural coupling and aeroelastic effects. In order to study the effect of structural geometric nonlinearities on the dynamic stability of very long cable-supported bridges, a special wind tunnel procedure was developed. Such procedure requires the development of a new experimental apparatus for section model tests of bridges that allows the representation of geometric nonlinearities and nonlinear structural coupling. Therefore, one of the first steps toward nonlinear section model tests was to design a new bridge rig for dynamic section model tests, which first needed to be validated for standard linear tests. Consequently, this paper will present a comparison of wind tunnel results for a single-box girder bridge model tested in two different wind tunnels, i.e., the main wind tunnel at the Université de Sherbrooke and the Boundary Layer Wind Tunnel Laboratory (BLWTL) at the University of Western Ontario. At the BLWTL, tests were conducted using the existing bridge rig and the new bridge rig designed for nonlinear tests. At first, the static aerodynamic coefficients and the flutter derivatives are compared. Then, the dynamic responses for smooth and turbulent flows obtained using the different experimental setups are analyzed. This comparison procedure allows the validation of the new bridge rig that will be used in the near future to assess the effect of structural nonlinearities on aeroelastic stability.KeywordsWind tunnelTest rig for bridges
... Let us mention studies through the Melan equation [85] (both analytically [39,62,66,69] and numerically [42,96,97,98]), through coupled oscillators [8,9,26,41,64], beam equations [12,14,22,24,25,26,53,54,56,58,63], plate equations with two degrees of freedom [10,23,28,29,38,46,49,65]; see also the monographs [55,60]. These different models and the overall mathematical description of certain phenomena found consensus among engineers [7,32,33,41,42,44,74,79,108,109]. ...
... The peak (maximum) represents a kind of "structural resonance" called lock-in in the engineering literature, see e.g. [33]. The height of the spikes is decreasing with respect to k and the maximum is moving to the right (larger ω). ...
We give a new full explanation of the Tacoma Narrows Bridge collapse, occurred on November 7, 1940. Our explanation involves both structural phenomena, such as parametric resonances, and sophisticated mathematical tools, such as the Floquet theory. Contrary to all previous attempts, our explanation perfectly fits, both qualitatively and quantitatively, with what was observed that day. With this explanation at hand, we set up and partially solve some optimal control and shape optimization problems (both analytically and numerically) aiming to improve the stability of bridges. The control parameter to be optimized is the strength of a partial damping term whose role is to decrease the energy within the deck. Shape optimization intends to give suggestions for the design of future bridges.
... To investigate the mechanism of large amplitude torsional vibration, researchers conducted the nonlinear dynamic analysis of suspension bridge using the models formulated from Hamilton or minimum potential energy principles [4][5][6]. Arena and Lacarbonara [7][8] proposed a mathematical suspension bridge model by adopting the theories of direct total Lagrangian formulation and the kinematic. ...
... In this study, the generalized configurations of suspension bridge are as shown in Fig. 1, the hangers along the span-wise direction are replaced by a continuum massless and inextensible (along the axial direction of hangers) membrane, which is similar to that used in Capsoni et al. [6]. Cables and deck are assumed hinge-supported at two ends. ...
Torsional vibration of long span suspension bridge can be commonly induced by various sources, and detailed investigation is required from both design and safety point of view. This study proposes a continuum model of the generalized suspension bridge using the Hamilton principle. This model can consider any inclination angle of main cables. The horizontal and vertical motions of main cables, and torsional motion of deck are considered individually in this model. The proposed model is verified by comparing its modal properties with those from a finite element model. The nonlinear primary resonances of the two lowest torsional modes are studied through multiple-scale method (MSM) under harmonic excitation. The effects of system parameters, such as inclination angle of hangers, sag-to-span ratio, tensile stiffness of main cable and torsional stiffness of deck, on the resonance responses are investigated in this study. Results show that, increasing inclination angle of main cable and torsional stiffness of deck can intensify the softening feature of the system. In contrast, increasing the sag-to-span ratio and tensile stiffness of main cable can strengthen the hardening feature.
... Some of them investigated the non-linear free and forced vertical-torsional coupling with the multiple scale method (MSM). [27][28][29][30][31][32] Capsoni et al. 33 investigated the torsional aero-elasticity and parametric resonance instabilities of suspension bridges via the MSM based on a non-linear continuum model. In these studies, although the full bridge models were proposed or adopted, most of them assumed the vibration of main cables only in the vertical plane. ...
... This assumption is widely adopted by the existing studies. 33 The equations of motion of the bridge system can then be obtained with the Hamilton's principle 40,41 as ...
This study proposes an approach to set up a continuum full bridge model with spatially inclined cables based on the Hamilton principle. The dynamic governing functions, considering the geometric non-linearities of cables and deck, represent simultaneously the vertical motion of deck and vertical–horizontal motion of cable. With the comparison of the modal properties obtained from the model to those from the accurate model, results show that the proposed model is capable of accurately simulating the modal properties. The primary resonance responses and corresponding frequency-response curves are obtained through the multiple-scale-method. A finite element (FE) model is established, and the corresponding non-linear dynamic analysis in time domain is conducted. Comparing the results from two models, it can be checked that the proposed model is reliable. According to the results of the proposed model, it is found that the second-order shape functions (SOSFs) play a significant role in the system response. Once the non-linear vibration of the bridge becomes significant only considering the excited mode with using the classical Galerkin decomposition cannot correctly predict the structure response. The SOSFs can be classified into stationary and vibrating components. The vibrating component can deviate the time-series of response from the harmonic wave, and the stationary component directly determines the mean value of the time-series.
... For simulating this structural dynamic instability, simplified models or continuum models based on systems of nonlinear partial differential equations were used. Using a continuum model, it was demonstrated that flutter was likely responsible for Tacoma [10]. Nevertheless, the authors of [6][7][8][9] raise a point not considered in the wind design of long-span bridges, i.e., possible large oscillations caused by structural coupling between the modes of vibration because of structural nonlinearities. ...
... Nevertheless, the authors of [6][7][8][9] raise a point not considered in the wind design of long-span bridges, i.e., possible large oscillations caused by structural coupling between the modes of vibration because of structural nonlinearities. The nonlinear effect described previously seems to be attributed to internal parametric resonance, a dynamic instability [10]. The possible occurrence of internal resonance has also been demonstrated in cable-stayed bridges [11][12][13]. ...
Using simplified mathematical representations of suspension bridges, mathematicians have demonstrated that a vertical dynamic forcing can cause large torsional vibrations due to geometric nonlinearities of the bridge that appear to be a structural dynamic instability. Compared to the extensive research that has been conducted on the dynamic behavior of cable-supported bridges, the approach used by the mathematicians appears too simplistic. This is due to the fact that the dynamic force considered by the mathematicians is approximate compared to actual dynamic loadings on bridges, especially those originating from wind. However, they raise a point that is not considered in the wind design of cable-supported bridges, i.e., a possible nonlinear structural coupling between the modes of vibration that could be detrimental to the bridge performance. Therefore, this paper presents a preliminary investigation of nonlinear vertical-torsional coupling in long-span bridges using a simplified practical approach. The proposed method relies on the finite element method and nonlinear pushover analyses. Using this approach, the nonlinear structural coupling is assessed for the numerical models of five suspension bridges and two cable-stayed bridges. The method allows determining the nonlinear stiffness parameters of equivalent systems having between one and three degrees of freedom (lateral, vertical and torsional). Since the proposed technique relies on the modes of vibration and can account for the interaction between the vertical and torsional effects, it can be used to judge which ones of the bridges considered are likely to be the most susceptible to nonlinear mode coupling under wind loads. The analysis results for the seven bridges shows that the suspension bridge system has a greater nonlinear vertical-torsional coupling in comparison to the cable-stayed system. Additionally, it is demonstrated that the span length has an influence on the vertical-torsional coupling. The results also show that the nonlinear coupling is slightly affected by lateral effects.
... Recently, some authors (Capsoni, Ardito, and Guerrieri 2017;Lepidi and Gattulli 2016) underlined the possible activation of a complex dynamic response into suspended bridges connected with internal resonance conditions, even if the hangers' unilateral behavior is not considered. In Capsoni, Ardito, and Guerrieri (2017), the possibility of interaction between aeroelastic effects and nonlinear internal resonance into long span suspension bridges is highlighted showing that unstable conditions may occur for wind speeds lower than the critical threshold for standard aeroelastic predictions. ...
... Recently, some authors (Capsoni, Ardito, and Guerrieri 2017;Lepidi and Gattulli 2016) underlined the possible activation of a complex dynamic response into suspended bridges connected with internal resonance conditions, even if the hangers' unilateral behavior is not considered. In Capsoni, Ardito, and Guerrieri (2017), the possibility of interaction between aeroelastic effects and nonlinear internal resonance into long span suspension bridges is highlighted showing that unstable conditions may occur for wind speeds lower than the critical threshold for standard aeroelastic predictions. Lepidi and Gattulli (2016) proposed a section model which can consider the two "global" modes, due to the vertical and torsional vibration of the system, and four "local" modes, due to transversal oscillations of the main cables. ...
The aim of the present paper is to investigate the performance of suspended footbridges under pedestrian loads. Indeed, several Authors have underlined the possible activation of large amplitude oscillations into suspended footbridges due to the nonlinear behavior of the hangers. In fact, the last ones act as linear elastic springs in tension and do not react in compression. Consequently, if the whole suspended footbridge or parts of it undergo large amplitude oscillations, the initial hangers’ pretension stress may become zero and the slackening may start. In these cases, the stiffness of the footbridge deck decreases drastically, and a complex dynamic response may occur. Hence, the footbridge may show unexpected vertical and torsional oscillations that the “classical” models cannot predict; these models, in fact, assume a bilateral behavior for the suspended system and, consequently, no variations of the global stiffness during the motion. Here, the response of suspended footbridges is evaluated by using a continuous model obtained by adopting the nonlinear equivalent regularization technique proposed for long span suspended bridges. The dynamic analysis, performed by means of a perturbation method, shows the possibility of the coexistence of multiple solutions, some of which are characterized by high amplitudes and by the activation of the afore-described slackening phenomenon. The response is evaluated for several values of loading, mechanical and geometrical parameters, with the main aim of highlighting the characteristics and the stability of the investigated oscillations and obtaining information and/or design indications to prevent such phenomena.
Communicated by Dumitru Caruntu.
... The main motivation for starting the study of problems (I) and (II) is the analysis of parametric torsional instability for some recent suspension bridge models, where a finite dimensional projection of the phase space reduces the stability analysis at small energies of the model to the stability of a Hill equation such as 1.1. We refer the reader to Gazzola's book [19], to the papers [8,9,3,10,17], and to our previous works [30,31]. Other interesting applications arise in the study of the stability of nonlinear modes in some beam equations [18] or string equations [12,11]. ...
... An important issue in the mathematical modeling of suspension bridges is the phenomenon of energy transfer from flexural to torsional modes of vibration along the deck of the bridge. According to a recent field of research [3,8,19,17,10] internal nonlinear resonances giving rise to the onset of instability may occur even when the aeroelastic coupling is disregarded. In particular, in the fish-bone bridge model ( [19, ch. ...
We study the asymptotics for the lengths $L_N(q)$ of the instability tongues of Hill equations that arise as iso-energetic linearization of two coupled oscillators around a single-mode periodic orbit. We show that for small energies, i.e. $q\rightarrow 0$, the instability tongues have the same behavior that occurs in the case of the Mathieu equation: $L_N(q) = O(q^N)$. The result follows from a theorem which fully characterizes the class of Hill equations with the same asymptotic behavior. In addition, in some significant cases we characterize the shape of the instability tongues for small energies. Motivation of the paper stems from recent mathematical works on the theory of suspension bridges.
... For suspension bridges, it has also been demonstrated that dynamic vertical forcing can lead to large torsional oscillations as a result of structural nonlinearities, i.e., there is nonlinear vertical-torsional coupling due to the geometric nonlinearities (Arioli and Gazzola 2017). The phenomenon at cause would be internal parametric resonance, a structural dynamic instability (Capsoni et al. 2017). To the authors' knowledge, this structural instability has never been witnessed on an actual bridge. ...
Following the collapse of the Tacoma Narrows Bridge due to an aeroelastic instability, it has been common practice to test cable-supported bridges in a wind tunnel to check the soundness of bridge designs with respect to wind dynamic actions. Due to their simplicity, versatility and cost effectiveness, section model tests have become the standard approach for testing bridges. More advanced testing techniques , like full-aeroelastic model tests, are only utilized for validation purposes toward the end of the design process. Nevertheless, some generalizations with regard to the behavior of the bridge are necessary in section model tests in order to reach such simplicity. One of them is that they assume a linear structural behavior of the bridge structure. This might be inaccurate for very long cable-supported bridges as the structural behavior of such bridges is governed by their cable system, which is geometrically nonlinear. Considering that span lengths are getting longer, it is believed that it is needed to develop a better understanding of the influence of geometric nonlinearities on the wind response of bridges. Thus, this paper presents an experimental assessment of the effect of structural nonlinearities on the aeroelastic stability and wind response of cable-supported bridges. At first, the development of a new experimental apparatus for nonlinear section model tests of bridges is discussed. Then, the results of nonlinear section model tests conducted using the experimental apparatus are presented. Three different suspension bridge configurations are tested. The first one is for a single-box girder suspension bridge, and the second and third ones are for two twin-box girder suspension bridges having different span lengths. By comparing the results of linear tests to those of nonlinear tests, it is possible to assess the effect of structural nonlinearities. It is found that structural nonlinearities can have an effect on the critical velocity for flutter.
... In addition, vertical vibration from bridge decks can result in resonance oscillation on the cable, which is called parametric excitation [9,10]. Such hazardous vibrations should be properly assessed under service conditions [11,12]. To this end, structural health monitoring (SHM) systems have been implemented on bridges with various wired sensors, providing valuable information on the structural conditions and responses in near-real-time. ...
This study aims to monitor excessive vibrations in the stay-cables of long-span bridges, which can cause structural defects or discomfort to the public. Traditional methods like structural health monitoring (SHM) systems and computer vision techniques are impractical and costly for monitoring all stay-cables. Therefore, this study proposes a cost-effective solution using surveillance cameras (CCTV) commonly used for monitoring traffic conditions. Deep learning and computer vision techniques were used for semantic segmentation, dehazing, and tracking to address technical challenges in selecting feature points and image quality. Synthetic image generation was employed to obtain sufficient training images with pixel-wise annotations. The proposed framework was first validated using laboratory-scale experiments and applied to actual CCTV images collected under various environmental conditions. Parametric studies confirmed the efficiency of image synthesis and additional loss functions. The proposed method provides a viable alternative to monitor cable vibrations in long-span bridges using existing CCTV systems.
... Zhou et al. [34] analytically investigated the solution to the deformation of suspension bridges under the temperature loads. Numerical procedures for nonlinear dynamic analysis were also developed in the previous work [35][36][37]. Based on the deterministic analysis, Imaia and Frangopolb [19,20] highlighted the need to consider the probabilistic safety assessment of the main cable, instead of the traditional design with FOS. ...
... Existing mathematical model of suspension bridge considering vertical-torsional vibrations can be classi¯ed into full-span model 18,19 and section model. [20][21][22] Very recently, a six-degree of freedom section model has been proposed, 23 that can consider the horizontal and vertical motion of main cables on both sides in addition to the vertical and torsional motion of deck. ...
Asymmetric stiffness in transverse direction of suspension bridge can be easily induced by many causes during its long-term service. Such phenomenon may cause the coupling effect between vertical and torsional vibrations. A cross-section model of suspension bridge with seven-degree of freedom is proposed, to investigate the asymmetry effect on the dynamic behavior of the system. Corresponding modal analysis is firstly carried out. Results show that the asymmetric stiffness will induce veering phenomenon when natural frequency loci of vertical and torsional modes approach each other. In the veering region, mode hybridization phenomenon can be observed between these two modes. In addition, asymmetry-induced nonlinear vibration of hybrid vertical and torsional modes is studied using the extended incremental harmonic balance method. The effect of asymmetry extent is also investigated in this study. Results show that both hybrid modes can be excited by either the vertical or torsional excitation. Moreover, the energy can be transferred between these two modes, because of the nonlinear stiffness introduced by the significant swaying motion of hanger and cable.
... The vibrations in the horizontal plane can arise from the wind influence, if its gusts alternate with a certain interval. Torsional vibrations of the superstructure arise from the asymmetric application of the vertical and horizontal loads [10,11]. ...
The research paper is devoted to the study of the practical dynamic impact of the tram rolling stock on city bridges. The authors analyzed the current state and prospects for the tram rail lines` development in conjunction with bridge crossings in Novosibirsk and Rostov-on-Don. It is considered the requirements for the dynamic impact accounting of the tram traffic specified in domestic and foreign regulatory documents. Using a mobile measuring complex, which includes accelerometer sensors, an experiment was carried out, and according to the obtained results the natural frequencies of vertical, horizontal transverse and horizontal longitudinal vibrations of the 71-619K and 71-605 tram cars were determined. The technique used in the experiment makes it possible to obtain the initial data for the dynamic calculation of structures for the tram traffic, to clarify the real dynamics of the rolling stock, depending on the state of the rail track on the bridge. The data obtained can be used in vibration diagnostics when assessing the state of an engineering structure.
... Finally, unifying theories have been proposed since the work of Herrmann and Hauger 12 . More recently, Capsoni et al. 13 presented a nonlinear dynamic model to provide a unifying approach to aeroelastic and internal resonance instabilities. The richness and depth of the mentioned approaches increase our understanding of the dynamic behaviour of suspension bridges. ...
A simplified nonlinear model for the dynamic behaviour of suspension bridges is proposed, based on sinusoidal shape functions. The model is applied to the understanding of torsional instability. Numerical experiments are performed using the structural parameters of the Tacoma Narrows Bridge, with initial conditions similar to those observed before the collapse. Each simulation spans about 30 minutes. In line with previous work, numerical results show how torsional instability depends on total energy, with a specific energy level required for each set of initial conditions to reach instability. The time before the onset of instability varies for each initial condition. Interestingly, the model indicates a lack of energy transfer to non-initially-excited even modes. The present numerical investigation confirms recent results, while the proposed model provides a simple and intuitive tool to simulate the relationship between design parameters and potential instability phenomena.
... Suspension bridges have been increasingly favored in the engineering sector due to technical maturity, reasonability of stress modes, structural aesthetics, and adaptability to varying terrains (Sun et al. 2015;Cao et al. 2017). All these features make the suspension bridge lucrative among all types of large-span bridges (Sun et al. 2004;Kim et al. 2006;Capsoni et al. 2017). As an important construction connecting the two banks, a super-long span suspension bridge has to meet the growing traffic volume. ...
Suspension bridges with three cable planes provide an excellent solution to the suspension bridges’ downwarp problem with ultra-wide decks, which implies their enormous demand and popularization prospect in the engineering scenarios. In this paper, a method for uniform allocation of dead load in the transverse direction to three cables of the suspension bridge with three cable planes (SB-3CP), which transforms the spatial stress mode into a plane model to simplify the calculation, is proposed. By altering the cross-sectional area of each hanger, the axial forces of the three hangers in the same cross section of the SB-3CP are equal. Therefore, the cross-sectional area and shape of the three cables are equal, improving the suspension bridge outlook in both the cross-sectional and facade views of the suspension bridge. Meanwhile, under the uniform allocation of the dead load, the side main cable bears a higher share of the dead load, which is conducive to improving the entire bridge’s torsional rigidity. In this study, conditions for compatibility of deformation and energy conservation are utilized to derive the relationship between the axial rigidity of the three hangers in the same cross sections. The effects of axial rigidity of hangers, flexural rigidity of the deck in the transverse direction, and length between the hanging points on the difference in the axial rigidities of the three hangers are analyzed and discussed in detail. Finally, an SB-3CP with a main span of 2,320 m and a width of 75 m was taken as an example. The cross-sectional areas of each hanger of the bridge were calculated using the proposed method. Then, its accuracy was validated through the finite-element analysis. This method’s design effect was verified by comparing the differences in the main cable diameter and torsional rigidity between the SB-3CP with uniform and nonuniform dead load distributions.
... In addition to the requirements mentioned above, however, new demands come into being for the cable tray, such as the large span, the lightweight, and the ease of construction [27] along with the progress in industrial technologies, especially with the idea and recognition of the resource-conserving and the environment-friendly societies. It should be pointed out that various meshless methods advance very fast in the regime of numerical simulations [28][29][30] in recent years, which can yet be regarded expectantly as an effective means in the simulation of cable trays. ...
In order to realize the optimal design of the cable supporting system for the purpose of material saving and energy saving and green manufacturing, the strength-stiffness ratio is proposed in the paper in nondimensional form, which defines quantitatively the relation between the static load strength and stiffness of the cable tray. On the premise of ensuring service safety, the correlation between the strength and stiffness of the cable tray under static load is discussed extensively through the theoretical analysis of the mechanical model. The weakest link in the carrying capacity of the cable tray as well as the issue that needs to pay attention is proposed in the process of design and the test of the cable tray. A reasonable strength-stiffness ratio will help to make full use of the potential of material strengths. The value of the strength-stiffness ratio is obtainable by means of the finite element method or by the loading test of the cable tray. It is shown through the analysis that the value of the strength-stiffness ratio being setting in the range close to but less than 1 will make comparatively reasonable material utilization and will help the deflection test going smoothly to obtain a relatively safer allowable working load for the cable tray.
... Gwon and Choi (2018a, b) calculated the dynamic responses of the suspension bridge under the moving load, considering the influence of hanger elongation and spatial shape of the main cable. Capsoni et al. (2017) analyzed the parametric resonance and wind-induced vibration of the suspension bridge. ...
This study derives the differential equations of free vertical bending and torsional vibrations for two and three-pylon suspension bridges using d'Alembert's principle. The respective algorithms for natural vibration frequency and vibration mode are established through the separation of variables. In the case of the three-pylon suspension bridge, the effect of the along-bridge bending vibration of the middle pylon on the vertical bending vibration of the entire bridge is considered. The impact of torsional vibration of the middle pylon about the vertical axis on the torsional vibration of the entire bridge is also analyzed in detail. The feasibility of the proposed method is verified by two engineering examples. A comparative analysis of the results obtained via the proposed and more intricate finite element methods confirmed the former feasibility. Finally, the middle pylon stiffness effect on the vibration frequency of the three-pylon suspension bridge is discussed. It is found that the vibration frequencies of the firstand third-order vertical bending and torsional modes both increase with the middle pylon stiffness. However, the increase amplitudes of third-order bending and torsional modes are relatively small with the middle pylon stiffness increase. Moreover, the second-order bending and torsional frequencies do not change with the middle pylon stiffness.
... The complexity of this behavior has drawn much research attention on its nonlinear characteristics. Most existing studies [32][33][34] are based on the modal discretization modeling, where only one or two modes can be included in the analysis. This poses a problem of accuracy with the limited modes studied. ...
Nonlinear vibration analysis of complex structures is still a challenging topic due to limitations of existing algorithms. This paper proposes a highly efficient method to address this problem where the structure has only a portion of components with geometric or material nonlinearity under service load. Incremental harmonic balance (IHB) method serves as the main tool. The linear and nonlinear parts of the structure are separated with the introduction of interface forces at the boundaries. The analysis of the linear and nonlinear parts is interacting via the interface forces with the compatibility condition. Two numerical examples are analyzed for illustration of the method, i.e. a frame with base isolation system and a suspension bridge model. Various resonances are noted in the vibration of the structures suggesting the need of analyzing the nonlinear vibration of any structure with nonlinear components. The accuracy of the proposed method is validated with comparison the obtained results with that by Runge-Kutta method. Results show that the proposed method provides a neat and efficient tool for this purpose, when the nonlinear components are far less than the linear components in the system.
... A new mathematical explanation for the origin of torsional oscillations was given in [3] through the introduction of suitable Poincaré maps: these oscillations appear whenever there is a large amount of energy within the bridge and this happens due to the nonlinear behavior of structures. The model in [3] was fairly simplified, but the very same conclusion was subsequently reached in more sophisticated models [4,5,7,13,22,23]. A further purpose of the paper is to study the torsional instability of the deck through the model with convexified cables. ...
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, 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.
... Surprisingly, the MOF composed of the longer organic linkers (abdc) was more resilient to direct compression of the framework, and pressure compliance was attributed to framework-dynamic behaviour, much like a suspension bridge, whereby some flexibility in the linking struts gives rise to a more stable structure. [12] Similarly, recent work by Suslick et al., has shown that UiO-66(Zr), which has shorter terephthalate linkers, undergoes bondbreakage under pressure, and is even less stable. [13] Ligandflexibility in UiO-MOFs can also be induced by guest inclusion, leading to changes in fluorescence emission spectra, with potential for use of these materials as sensors. ...
Druckinduzierte bathochrome Verschiebungen in den Fluoreszenzemissions‐ und UV/Vis‐Absorptionsspektren eines zweifach verzahnten Hf‐MOF werden durch Rotation des zentralen Phenylrings des Linkers ausgelöst. Einkristall‐Röntgenbeugung wurde bei Drücken bis 2.1 GPa vorgenommen; die Ergebnisse ermöglichen die Korrelation der Linkerrotation mit der Emissionsmodulation.
Abstract
Conformational changes of linker units in metal‐organic frameworks (MOFs) are often responsible for gate‐opening phenomena in selective gas adsorption and stimuli‐responsive optical and electrical sensing behaviour. Herein, we show that pressure‐induced bathochromic shifts in both fluorescence emission and UV/Vis absorption spectra of a two‐fold interpenetrated Hf MOF, linked by 1,4‐phenylene‐bis(4‐ethynylbenzoate) ligands (Hf‐peb), are induced by rotation of the central phenyl ring of the linker, from a coplanar arrangement to a twisted, previously unseen conformer. Single‐crystal X‐ray diffraction, alongside in situ fluorescence and UV/Vis absorption spectroscopies, measured up to 2.1 GPa in a diamond anvil cell on single crystals, are in excellent agreement, correlating linker rotation with modulation of emission. Topologically isolating the 1,4‐phenylene‐bis(4‐ethynylbenzoate) units within a MOF facilitates concurrent structural and spectroscopic studies in the absence of intermolecular perturbation, allowing characterisation of the luminescence properties of a high‐energy, twisted conformation of the previously well‐studied chromophore. We expect the unique environment provided by network solids, and the capability of combining crystallographic and spectroscopic analysis, will greatly enhance understanding of luminescent molecules and lead to the development of novel sensors and adsorbents.
... Surprisingly, the MOF composed of the longer organic linkers (abdc) was more resilient to direct compression of the framework, and pressure compliance was attributed to framework-dynamic behaviour, much like a suspension bridge, whereby some flexibility in the linking struts gives rise to a more stable structure. [12] Similarly, recent work by Suslick et al., has shown that UiO-66(Zr), which has shorter terephthalate linkers, undergoes bondbreakage under pressure, and is even less stable. [13] Ligandflexibility in UiO-MOFs can also be induced by guest inclusion, leading to changes in fluorescence emission spectra, with potential for use of these materials as sensors. ...
Conformational changes of linker units in metal‐organic frameworks (MOFs) are often responsible for gate‐opening phenomena in selective gas adsorption and stimuli‐responsive optical and electrical sensing behaviour. Herein, we show that pressure‐induced bathochromic shifts in both fluorescence emission and UV/Vis absorption spectra of a two‐fold interpenetrated Hf MOF, linked by 1,4‐phenylene‐bis(4‐ethynylbenzoate) ligands (Hf‐peb), are induced by rotation of the central phenyl ring of the linker, from a coplanar arrangement to a twisted, previously unseen conformer. Single‐crystal X‐ray diffraction, alongside in situ fluorescence and UV/Vis absorption spectroscopies, measured up to 2.1 GPa in a diamond anvil cell on single crystals, are in excellent agreement, correlating linker rotation with modulation of emission. Topologically isolating the 1,4‐phenylene‐bis(4‐ethynylbenzoate) units within a MOF facilitates concurrent structural and spectroscopic studies in the absence of intermolecular perturbation, allowing characterisation of the luminescence properties of a high‐energy, twisted conformation of the previously well‐studied chromophore. We expect the unique environment provided by network solids, and the capability of combining crystallographic and spectroscopic analysis, will greatly enhance understanding of luminescent molecules and lead to the development of novel sensors and adsorbents.
... Many articles have been written on analyses of rail bridge vibrations. Such research has mainly focused on bridge conditions [4], such as a resonance mechanism explained by Xia et al. [5], vibration mode-coupling and intermittent contact loss and vibration instability in a large motion bistable compliant mechanism by Nikman et al. [6,7], the stability of dynamic response by Capsoni et al. [8], the dependence of bridge vibration parameters on cross winds by Xu et al. [9] and the use of bridge-track-vehicle element by Cheng et al. [10]. ...
This article analyses the dispersion of vibration accelerations of a railway bridge during the passage of a train, and presents an analysis of their parameters after the application of the theory of covariance functions. The measurements of vibration accelerations at the fixed points of the beams of the overlay of the bridge were recorded in the time scale as digital arrays (matrices). The values of inter-covariance functions of the arrays of data of measurements of digital vibration accelerations and the values of auto-covariance functions of the individual arrays, changing the quantization interval in the time scale, were calculated. The compiled software Matlab 7 in the operator package environment was used in calculations. This article aims at determining the interdependencies of results of vibrations of bridge points rather than at the impact which a train makes on a bridge without emphasizing the modal parameters of the bridge. The aforementioned interdependencies make it possible to predict the results of hard-to-reach points.
... Structure totalement immergée (Figure 1.2(a)) : rayonnement acoustique d'un sous-marin [Wei et al., 2012] [Yao et al., 2017], stabilité des ailes d'avion [Kamakoti et Shyy, 2004], comportement d'un parachute [Tezduyar et Osawa, 2001], tenue des ouvrages d'art vis-à vis des effets du vent [Capsoni et al., 2017], déplacement des poissons [Hovnanian, 2012], instruments de musique [Derveaux et al., 2002], dimensionnement de structures soumises aux courants marins ou d'éoliennes [Lee et al., 2017]... Structure partiellement contenue (Figure 1.2(b)) : rayonnement acoustique de la coque d'un navire [Leblond et al., 2009], éolien offshore [Yan et al., 2016] [Calderer et al., 2018]... ...
La maîtrise du bruit et des vibrations est un objectif fréquemment rencontré dans le domaine industriel. Qu’il s’agisse de questions de confort ou de sécurité, les domaines d’applications sont nombreux et variés : transport, BTP, ingénierie civile et militaire…Dans cette thèse, un problème de vibroacoustique interne avec couplage fluide-structure est étudié. Il s’agit d’une cavité remplie de fluide dont les parois sont constituées d’une structure sandwich viscoélastique. Les difficultés numériques associées à ce modèle portent sur la non linéarité du matériau et sur les propriétés des opérateurs matriciels manipulés (conditionnement, non symétrie).Le calcul des vibrations du système dissipatif couplé nécessite une valeur initiale, choisie comme la solution du problème conservatif. Cette solution n’étant pas aisée à déterminer, deux solveurs aux valeurs propres basés sur la Méthode Asymptotique Numérique (MAN) sont proposés pour résoudre le problème des vibrations libres du système conservatif. Associant des techniques de perturbation d'ordre élevé et de continuation, la MAN permet de transformer le problème non linéaire de départ en une suite de problèmes linéaires, plus simples à résoudre. Les solutions obtenues sont ensuite utilisées comme point initial pour déterminer la réponse libre du système dissipatif. Un solveur de Newton d’ordre élevé, basé sur les techniques d’homotopie et de perturbation est développé pour résoudre ce problème. Enfin, le régime forcé est étudié.Pour toutes les configurations envisagées, les résultats obtenus mettent en évidence des performances numériques améliorées par rapport aux méthodes classiquement utilisées (Arpack, Newton…).
... A large number of analytical methods for calculating classical suspension bridges with rigid cables are based on nonlinear calculations according to the distorted scheme (Arco & Aparicio, 2001;Clemente, Nicolosi, & Raithel, 2000;Gimsing & Georgakis, 2012;Idnurm, 2006;Jennings, 1987;Kim & Thai, 2010;Kulbach, 2007;Wollmann, 2001). Also, plenty of studies on the dynamic characteristics of suspension bridges have been carried out (Capsoni, Ardito, & Guerrieri, 2017;Goremkins, Rocens, Serdjuks, & Sliseris, 2013;El Ouni & Kahla, 2012;Sousa, R., Souza, R. M., Figueiredo, & Menezes, 2011;Treyssede, 2010). ...
The article presents the results of the numerical analysis of the asymmetrical one-pylon suspension bridge built-in rigid cables. The models for the suspension bridge with the cables of different rigidity are analyzed by comparing vertical displacements, bending moments and strains in the structural members of the bridge. The numerical analysis was performed by examining the bridge under symmetrical and asymmetrical loading and different erection methods. The stress-strain state of a single asymmetrical pylon with the cables of different rigidity and the rational relationship between cable rigidity and girder stiffness has been established.
... A new mathematical explanation for the origin of torsional oscillations was given in [2] through the introduction of suitable Poincaré maps: these oscillations appear whenever there is a large amount of energy within the bridge and this happens due to the nonlinear behavior of structures. The model in [2] was fairly simplified, but the very same conclusion was subsequently reached in more sophisticated models [3,4,6,11,16,17]. A further purpose of the paper is to study the torsional instability of the deck through the model with convexified cables. ...
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.
... A new mathematical explanation for the origin of torsional oscillations was given in [2] through the introduction of suitable Poincaré maps: these oscillations appear whenever there is a large amount of energy within the bridge and this happens due to the nonlinear behavior of structures. The model in [2] was fairly simplified, but the very same conclusion was subsequently reached in more sophisticated models [3,4,6,11,16,17]. A further purpose of the paper is to study the torsional instability of the deck through the model with convexified cables. ...
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.
Modal discretization is commonly applied for dynamic analysis of non-linear continuum system. Considering the possible coupling effect between modes is necessary to obtain accurate results. In this case, the system may become increasingly complex, as the number of adopted modes can be a lot. Such complexity will lead to the difficulty of solution finding. This paper proposes a generic technique to simplify the governing functions by making non-linear stiffness matrix symmetric. The symmetric non-linear stiffness matrix is constructed by utilizing the mode shape vectors. The proposed procedure can theoretically guarantee non-linear stiffness matrix symmetric. The incremental harmonic balance (IHB) method is served as the main tool for finding solutions of systems. Dynamic analysis of axially moving beam and generalized suspension bridge are presented in this study for illustration. Results proved that the neighboring modes are critical during the resonance of target mode, which suggests the necessity of including sufficient modes for non-linear dynamic analysis. By applying the proposed technique, it is found that calculating time of IHB method can greatly shortened, especially for the case included modes becomes large. Results show that the time consumption with using the proposed method can be half of that not using it, when more than 5 modes are considered in the calculation.
Non-linear vibration is a crucial issue for suspension bridge due to its slenderness and low damping. This study proposes a fully mathematical model of a suspension bridge considering arbitrary inclined angle of main cables by using principle of Hamilton. This is the first mathematical model that is capable of analyzing the nonlinear coupled vertical-torsional vibration of generalized bridge configurations with independently considering the motion of main cables and girder. Galerkin method is adopted for the discretization of the model, and the multi-scale method is employed to acquire modulation equations. The accuracy of the non-linear vibration obtained from proposed mathematical model are validated by a FE model. The internal resonance analysis for such a suspension bridge with spatial layout of main cables between vertical and torsional modes with frequency ratio of 2:1, and between two torsional modes respectively dominated by deck and main cables with frequency ratio of 1:1 are successively investigated in the first time within a narrow detuning frequency domain. The phenomenon of nonlinear resonance was discussed with different inclination angle of main cables and detuning parameters. Some novel and practical conclusions were obtained. Results show that 2:1 internal resonance between vertical and torsional modes may be induced only when two forces with frequencies corresponding to involved modes are applied simultaneously. In addition, significant 1:1 internal resonance is observed, with inducing Hopf bifurcation. Understanding such possible situations is favorable to the design of bridge structure.
p>It has been shown that the nonlinear differential equations representing the structural system of a suspension bridge exhibit nonlinear modal coupling that can lead to large torsional vibrations of the bridge deck. Such nonlinear coupling could play a role in the stability of cable-supported bridges under wind effects. Therefore, this paper presents an investigation of nonlinear modal coupling in cable-supported bridges with an emphasis on coupling between pairs of non- analogous modes, i.e., modes having a weak correlation along the bridge deck between the verti- cal displacement and torsional rotation. A procedure for assessing nonlinear coupling that relies on nonlinear generalized stiffness parameters is utilized for this purpose. Results of nonlinear gen- eralized stiffness analysis for suspension bridges indicate that non-analogous modes have a weak- er nonlinear coupling compared to analogous modal pairs.</p
In suspension bridges, flutter is the primary source of instability, commonly controlled by a tuned mass damper (TMD). In torsional motions, such as flutter, the shape of the mass block, indicating its distribution around the torsion axis, is critical; indeed, an optimal distribution results in a more effective and lighter device. The flutter analysis of the Vincent Thomas suspension bridge was performed using the multi-mode method in this article. Then, a new optimal configuration was used to avoid it; in addition to common parameters such as mass ratio, damping ratio, and frequency ratio, a set of parameters called shape variables was also considered. These parameters take into account the manner in which mass is distributed around the torsion axis. The performance of the aforementioned configuration was compared to that of the other recommended TMD configurations. Additionally, a general formulation of TMD was represented, which included the other configurations. Finally, the effectiveness of TMD was evaluated in comparison to other control systems. The results indicated that the optimal configuration was reliable and that it reduced the mass ratio by up to 0.2 percent by optimizing its distribution around the torsion axis compared to other existing configurations.
The article is dedicated to the actual dynamic impact study of rolling stock on urban bridges. The requirements for recording the dynamic impact specified in domestic and foreign regulatory documents are considered. Using a mobile measuring complex, the results of which determine the natural frequencies of vertical, horizontal transverse, and horizontal longitudinal vibrations. The technique used in the experiment allows us to obtain initial data for the dynamic calculation of structures, to clarify the real rolling stock dynamics depending on the state of the rail track on the bridge. Urban rolling stock is a relevant and necessary type of public transport that meets modern city requirements. The bridge crossings construction will ensure the connection of the urban areas separated by obstacles into a single network. When designing such structures, it is important to take into account the effects of actual dynamic coercitive forces. The dynamics study allows, on the one hand, to reasonably set the dynamic loads impact on bridges, on the other hand, to diagnose the structures state by dynamic parameters. In this work, using a specialized mobile measuring complex, the authors measured the oscillation parameters on moving vehicles. As a conducted experimental studies result, a methodology for determining the dynamic characteristics of a moving load and directly urban bridges has been demonstrated. As a result of the conducted experimental studies, it can be concluded that the proposed method allows obtaining initial data for the dynamic analysis of structures for the transport load. In addition, fixing the transport load dynamic impact during operation allows you to clarify the real rolling stock dynamics, depending on the state of the rail track on the bridge.
A new type of suspension bridge is proposed based on the gravity stiffness principle. Compared with a conventional suspension bridge, the proposed bridge adds rigid webs and cross braces. The rigid webs connect the main cable and main girder to form a truss that can improve the bending stiffness of the bridge. The cross braces connect the main cables to form a closed space truss structure that can improve the torsional stiffness of the bridge. The rigid webs and cross braces are installed after the construction of a conventional suspension bridge is completed to resist different loads with different structural forms. A new type of railway suspension bridge with a span of 340 m and a highway suspension bridge with a span of 1020 m were designed and analysed using the finite element method. The stress, deflection of the girders, unbalanced forces of the main towers, and natural frequencies were compared with those of conventional suspension bridges. A stiffness test was carried out on the new type of suspension bridge with a small span, and the results were compared with those for a conventional bridge. The results showed that the new suspension bridge had a better performance than the conventional suspension bridge.
In this paper, a new-style cable net structure consisting of a wind-resistance main cable, wind-resistance secondary cable and pulleys is proposed to improve the wind-resistance stability of narrow suspension bridges. In the improved structure, the wind-resistance secondary cable which slides on pulleys to achieve uniform distribution of cable force. However, the calculation for this structure is very difficult due to the sliding cable, which means the unstressed length of the cable between the pulleys is not fixed. To this end, a multi-node wind cable element considering the cable sliding on nodes is proposed for the wind-resistance secondary cable. Additionally, an analytical model of the wind cable element is established according to the structural features. The unstressed state method, virtual work principle and influence matrix method are all used to deduce and verify the tangent stiffness matrix of the element. As a result, the exact nodal forces can be calculated according to the static equilibrium conditions and the proposed element implemented for the nonlinear finite element analysis of the suspension bridge. Numerical and bridge examples are used to demonstrate that the new element can successfully simulate the sliding of the cable on the pulleys, and be applied to the nonlinear finite element calculation like any normal element. By means of this method, a wind-resistance secondary cable with any cable section connected by pulleys to the wind-resistance main cable and stiffening girder, can be accurately simulated with only one wind cable element. Hence, the calculation problem of the wind-resistance cable is resolved and overall calculation accuracy and efficiency are improved.
Nonlinear vibration of suspension bridge is a topic requiring comprehensive investigation. This study is focused on the internal resonance of the suspension bridge consisting of spatially inclined main cables and hangers by proposing a mathematic full bridge model. The vertical vibration of the bridge deck is studied. Specifically, the dynamic governing functions of this model, which can consider the geometric non-linearity of cables and deck. Firstly, the modal property of the continuum model is investigated, and the scenarios of modal frequency ratio between different modes equals to 3:1 are obtained. Then, the 3:1 internal resonances among two lowest symmetric or asymmetric vertical modes of deck with the external harmonic excitation are investigated by the Multiple Scales Method. The corresponding modulation equations explaining the internal resonance of two modes are obtained and solved by pseudo-arc-length method for exploring the nonlinear dynamic behaviors. The internal resonance phenomenon is discussed with various detuning parameters. Some novel and interesting conclusions have been made. Knowledge of such possible occurrences can be beneficial in the design of the bridge structure.
Suspension type cable system is an important type of cable bearing system, its dynamic problems have always been the key to the structural design, health monitoring, and vibration control. In view of this, an exact dynamic analysis method for suspension type cables is proposed by the author. The dynamic characteristic of the suspension cable system is studied by a full-scale cable test. By comparing with experimental results and finite element solutions, the accuracy and universality of the proposed method are verified. The results show that the calculation result agrees well with the measured results, and the maximum relative error of each condition is basically no more than 2%. Through parameter identification, the bending stiffness, cable force, and the installation position of the weights are identified and corrected, and the calculation errors after parameter identification are controlled within 1%, which significantly improves the calculation accuracy and further verifies the accuracy of the proposed method.
This paper aims to propose a novel model for predicting the nonlinear heave-torsion coupled flutter instability of flat bridge decks. The nonlinear oscillatory system during coupled post-flutter instabilities was modeled as a weak perturbation of the classical linear flutter theory. A novel nonlinear self-excited force model was then proposed by introducing extra nonlinear terms in classical linear model to consider the effects of nonlinear aerodynamic damping, amplitude-dependent vibration mode and the coupling of aerostatic deformation. An efficient algorithm for parameters identification and solving of the nonlinear coupled oscillator was also developed. The effectiveness of the proposed analytical framework was validated through an elastically-supported sectional model test in estimating the self-sustained vibrations during post-critical states of a typical closed-box bridge deck.
Nonlinear force driven coupled vertical and torsional vibrations of suspension bridges, when the frequency of an external force is approaching one of the natural frequencies of the suspension system, which, in its turn, undergoes the conditions of the one-to-one internal resonance, are investigated. The generalized method of multiple time-scales is used as the method of solution. The damping characteristics are described by the rheological model involving a fractional derivative, which is interpreted as a fractional power of the differentiation operator. The main subject of this research is the numerical analysis of the influence of the fractional damping on the dynamic behavior of a suspension combined system.
Although the transverse vibration of the cantilever structure is of great interest, the internal resonance of the vertical cantilever structure is always ignored. In this paper, nonlinear free vibration and 1/3 super-harmonic resonance of a hanging cantilever beam are firstly presented with 3:1 internal resonance caused by gravity. By employing the temporal multi-scale method, mode responses in these two vibrations are obtained. The harmonic balanced method is used for solving the excitation component and the whole deflection in super-harmonic resonance. For nonlinear free vibration, beat phenomenon is found and the effect of mode interaction is determined by gravity, damping and initial perturbation. For super-harmonic resonance, the amplitude of the first two modes can be quite large and exceed the excitation component if damping is not strong. Besides, the softening-type frequency curves are predominated by the inertia nonlinearity. Compared with the excitation component and the whole deflection, the mode responses are more easily affected by gravity. If the value of gravity parameter is a little bit lower than the one in the condition of strict internal resonance, the energy transmission, multiple solutions and bifurcations and three complex types of frequency curves can be found in mode responses. Hysteresis and saturation phenomena in mode responses can be discovered as well. Results from analytical methods are almost identical to those from numerical ways. In summary, the internal resonance of slender hanging cantilever structures should be aroused more attention, for hanging cantilevers are common in practical engineering and considerable mode responses can be induced by large initial perturbation in weakly damped free vibration or by low excitation frequency and large excitation amplitude in super-harmonic resonance. In addition, the complex dynamics can be worthwhile to judge the occurrence of internal resonance and its impact on structures. Since gravity has tensile effects on hanging cantilevers, these phenomena may also occur in beams with axial tension.
We study the asymptotics for the lengths LN(q) of the instability tongues of Hill equations that arise as iso-energetic linearization of two coupled oscillators around a single-mode periodic orbit. We show that for small energies, i.e. q→0, the instability tongues have the same behavior that occurs in the case of the Mathieu equation: LN(q)=O(qN). The result follows from a theorem which fully characterizes the class of Hill equations with the same asymptotic behavior. In addition, in some significant cases we characterize the shape of the instability tongues for small energies. Motivation of the paper stems from recent mathematical works on the theory of suspension bridges.
To study the distribution and regularity of the longitudinal force of continuously welded rail on a railway bridge, a railway suspension bridge with a span of (84 + 84 + 1092 + 84 + 84) m is taken as the engineering background. Considering the main cable, suspender, steel truss, orthotropic plate and adjacent bridge structure, a model of the interaction between the suspension bridge and four-line tracks is established. Distribution patterns of the longitudinal forces on the suspension railway bridge with length exceeding 1000 m are explored. On this basis, the influences of design parameters on the longitudinal force are discussed. The results show that the track expansion force at the bridge end can be greatly reduced after the installation of rail expansion devices at the end of the suspension bridge. As the vertical stiffness of the suspension bridge is relatively low, the deflection force instead of the expansion force becomes the control load. When the four-line loading is simultaneously loaded, the vertical rail displacement is 2.4 m, and the track deflection stress is up to 103.0 MPa. The relative displacement between the suspension bridge and track is about 10.8 mm owing to the vertical load, which could cause the ballast track to buckle.
The flutter control effectiveness of passive aerodynamic countermeasures for long-span bridges is much dependent on the configuration of these countermeasures. The present study investigated the nonlinear flutter behaviors of a twin-box girder bridge with application of four common types of countermeasures, i.e. upward vertical central stabilizers (UVCS), downward vertical central stabilizers (DVCS), external guide vanes (EGV) and internal guide vanes (IGV). Particularly, the displacement bifurcation, power spectrums, oscillation configurations and failure modes of the bridge with these four countermeasures were systematically studied using a nonlinear aerodynamic force model (NAFM). The model predications were validated using wind tunnel testing data. It shows that the oscillation configuration of the bridge without countermeasures is an anti-symmetrical bending-torsional coupled motion with a hard-flutter phenomenon. Furthermore, the installation of 0.4 h/H UVCS or DVCS could lead to a relatively higher critical flutter wind velocity through prolonging the limit cycle oscillation of soft flutter, especially for 0.4 h/H UVCS. In addition, the flutter performance of the bridge with EGV dominated by the torsional DOF in soft flutter was superior to that with IGV dominated by the vertical DOF in soft flutter.
This contribution investigates the influence of parametric excitation on the dynamic stability of a microelectromechanical system. In systems with just a single degree of freedom, parametric excitation causes the oscillator to exhibit unstable behavior within certain intervals of the parametric excitation frequency. In multi-degree of freedom systems on the other hand, unstable behavior is caused within a wider range of intervals of the parametric excitation frequency. Moreover, such systems show frequency intervals of enhanced stability, an effect known as anti-resonance phenomenon. Both types of phenomena, the parametric resonance and anti-resonance, are
The limit cycle oscillations (LCOs) exhibited by long-span suspension bridges in post-flutter condition are investigated. A parametric dynamic model of prestressed long-span suspension bridges is coupled with a nonlinear quasi-steady aerodynamic formulation to obtain the governing aeroelastic partial differential equations adopted herewith. By employing the Faedo-Galerkin method, the aeroelastic nonlinear equations are reduced to their state-space ordinary differential form. Convergence analysis for the reduction process is first carried out and time-domain simulations are performed to investigate LCOs while continuation tools are employed to path follow the post-critical LCOs. A supercritical Hopf bifurcation behavior, confirmed by a stable LCO, is found past the critical flutter condition. The analysis shows that the LCO amplitude increases with the wind speed up to a secondary critical speed where it terminates with a fold bifurcation. The stability of the LCOs within the range bracketed by the Hopf and fold bifurcations is evaluated by performing parametric analyses regarding the main design parameters that can be affected by uncertainties, primarily the structural damping and the initial wind angle of attack.
The spectacular collapse of the Tacoma Narrows Bridge has attracted the attention of engineers, physicists, and mathematicians in the last 74 years. There have been many attempts to explain this amazing event, but none is universally accepted. It is however well established that the main culprit was the unexpected appearance of torsional oscillations. We suggest a mathematical model for the study of the dynamical behavior of suspension bridges which provides a new explanation for the appearance of torsional oscillations during the Tacoma collapse. We show that internal resonances, which depend on the bridge structure only, are the source of torsional oscillations.
A frequency domain approach for non-linear bridge aerodynamics and aeroelasticity, based on the Volterra series expansion, is introduced in this paper. The Volterra frequency-response functions (VFRFs) and the associated linear equations are formulated utilizing a topological assemblage scheme and are identified utilizing an existing full-time-domain non-linear bridge aerodynamics and aeroelasticity analysis framework. A two-dimensional sectional model of a long-span bridge is used to illustrate this approach. The results show a good comparison between the time-domain simulation and the proposed frequency-domain model. The availability of VFRFs enables to gain a qualitative insight to the non-linear bridge aerodynamics and aeroelasticity.
Suspension bridges are widely used as engineering structures to across long spans and give rise to the usage of domains under the bridge. In the finite element analyses of suspension bridges, it is assumed that the structure is built and loaded in a second. However, this type of analysis does not always give the reliable and healthy solutions. Because, construction period of this type of the structures continue along time and loads may be changed during this period. Therefore, construction stages and time dependent material properties should be considered in the analysis to obtain the reliable and healthy results.
The first part of the study deals with an iteration scheme for the nonlinear static analysis of suspension bridges by means of tangent stiffness matrices. The concept of tangent stiffness matrix is then introduced in the frequency equation govverning the free vibration of the system. At any equilibrium stage, the vibrations are assumed to take place tangent to the curve representing the force-deflection characteristics of the structure. The bridge is idealized as a three dimensional lumped mass system and subjected too three orthogonal components of earthquake ground motion producing horizontal, vertical and torsional oscillations. For purposes of illustration the results of analytical and experimental analyses for the Istanbul Bogazici Bridge have been presented.
Dimensionless charts, useful for the the understanding of the static behaviour of single-span suspension bridges, are generated in this paper by applying numerical methods to the equations of the deflection theory. Charts include displacements and bending moments under concentrated loading and maximum displacements and bending moments under distributed loading. Based on these charts, explicit formulae useful for design are given. Extension to the analysis of three span suspension bridges is performed. The accuracy of this extension is checked by analysing 19 existing and hypothetical suspension bridges.
Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equations. Free and forced vibrations with damping are investigated in detail. Amplitude and phase modulation equations are obtained. The dependence of nonlinear frequency on amplitude is described. Steady-state solutions are analyzed. Frequency–response equation is derived and the jump phenomenon in the frequency–response curves resulting from non-linearity is considered. Effects of initial amplitude
and phase values, amplitude of excitation, and damping coefficient on modal amplitudes, are determined.
A method is developed based on a linearized theory and the finite-element method. The method involves two distinct steps: (1)Specification of the potential and kinetic energies of the bridge, (2)use of finite element technique to: (a)discretize the structure into equivalent systems of finite elements; (b)select the displacement model most closely approximating the real case; (c)derive the element and assemblage stiffness and inertia properties; and finally (d)form the matrix equations of motion and the resulting eigenproblems. A numerical example is presented to illustrate the applicability of the analysis and to investigate the dynamic characteristics of laterally vibrating suspension bridges. This method eliminates the need to solve transcendental frequency equations, simplifies the determination of the energy stored in different members of the bridge, and represents a simple, fast, and accurate tool for calculating the natural frequencies and modes of lateral vibration by means of a digital computer.
A generalized theory of free torsional vibration for a wide class of suspension bridges with double lateral systems is developed, taking into account warping of the bridge deck cross section and the effect of torsional rigidity of the towers. The analysis is based on a linearized theory and on the use of the digital computer. A finite element approach is used to determine vibrational properties in torsion. Simplifying assumptions are made, and Hamilton's principle is used to derive the matrix equations of motion. The method is illustrated by numerical examples. The objective of the study is to clarify the torsional behavior of suspension bridges and to develop a method of determining a sufficient number of natural frequencies and mode shapes to enable an accurate practical analysis.
Electrochemical biosensor based on interdigitated electrodes was prepared for the sensitive determination of thyroid stimulating hormone (TSH). The interdigitated electrodes used as the sensing structure of biosensor were fabricated by microelectromechanical systems (MEMS) technology. The TSH protein was immobilized to the capture and detection antibodies by enzyme linked immunosorbent assay (ELISA). The enzymatic silver deposition reaction occurred on the glass surface between the interdigitated electrodes. Then the deposition of silver was dispersed on the substrate to make the interdigitated electrodes connected electrically. The TSH concentration could be calculated according to the electrical conductance of interdigitated electrodes. TSH concentrations ranging from 0.02 mIU/L to 100 mIU/L can be readily determined by the electrochemical biosensor based on interdigitated electrodes. A detection limit of 0.012 mIU/L was achieved, which is below the guidelines recommended by the National Academy of Clinical Biochemistry. The proposed electrochemical biosensor can also be used in the detection of other hormones, which are critical for disease early diagnosis.
The methodology of the free vibration analysis of suspension bridges with horizontal decks is refined and extended, utilizing a continuum approach to include both the coupled vertical-torsional vibration and the effect of cross-sectional distortion. Variational principles are used to obtain the coupled equations of motion in their most general, and nonlinear form. The general equations are linearized, eliminating the coupling effect, and solved for a specific bridge; the resulting natural frequencies and modes of vertical and torsional vibrations are compared with those calculated using the finite element approach and with those estimated from low-amplitude full-scale ambient vibration tests which were conducted on the bridge.
The classical two-degree-of-freedom (2-d-o-f) "sectional model" is of common use to study the dynamics of suspension bridges. It takes into account the first pair of vertical and torsional modes of the bridge and describes well global oscillations caused by wind actions on the deck, yielding very useful information on the overall behaviour and the aerodynamic and aeroelastic response; however, it does not consider relative oscillations between main cables and deck. On the contrary, the 4-d-o-f model described in the two Parts of this paper includes longitudinal deformability of the hangers (assumed linear elastic in tension and unable to react in compression) and thus allows to take into account not only global oscillations, but also relative oscillations between main cables and deck. In particular, when the hangers go slack, large nonlinear oscillations are possible; if the hangers remain taut, the oscillations remain small and essentially linear: the latter behaviour has been the specific object of Part I (Sepe and Augusti 2001), while the present Part II investigates the nonlinear behaviour (coexisting large and/or small amplitude oscillations) under harmonic actions on the cables and/or on the deck, such as might be generated by vortex shedding. Because of the discontinuities and strong nonlinearity of the governing equations, the response has been investigated numerically. The results obtained for sample values of mechanical and forcing parameters seems to confirm that relative oscillations cannot a priori be excluded for very long span bridges under wind-induced loads, and they can stimulate a discussion on the actual possibility of such phenomena.
When people discuss large torsional oscillations of suspension bridges like the one at the Tacoma Narrows, they invariably linearize the trigonometry out of the problem, unwittingly making a small-angle assumption. In this paper, I re-derive the equations for a torsionally oscillating plate, choosing the physical constants in accordance with the historical record. I show how nonlinearity from the trigonometry embedded in the problem leads naturally to large amplitude motions sustained by small forces. These motions are an excellent match for the historical record. I also show how large vertical (nontorsional) motions can lead, via a sudden instability, into the purely torsional motions of the type recorded on famous historical film footage.
The classical two-degree-of-freedom (2-d-o-f) "sectional model" is currently used to study the dynamics of suspension bridges. Taking into account the first pair of vertical and torsional modes of the bridge, it describes well global oscillations caused by wind actions on the deck and yields very useful information on the overall behaviour and the aerodynamic and aeroelastic response, but does not consider relative oscillation between main cables and deck. The possibility of taking into account these relative oscillations, that can become significant for very long span bridges, is the main purpose of the 4-d-o-f model, proposed by the Authors in previous papers and fully developed here. Longitudinal deformability of the hangers (assumed linear elastic in tension and unable to react in compression) and external loading on the cables are taken into account: thus not only global oscillations, but also relative oscillations between cables and deck can be described. When the hangers go slack, large nonlinear oscillations are possible; if the hangers remain taut, the oscillations are small and essentially linear. This paper describes the model proposed for small and large oscillations, and investigates in detail the limit condition for linear response under harmonic actions on the cables (e.g., like those that could be generated by vortex shedding). These results are sufficient to state that, with geometric and mechanical parameters in a range corresponding to realistic cases of large span suspension bridges, large relative oscillations between main cables and deck cannot be excluded, and therefore should not be neglected in the design. Forthcoming papers will investigate more general cases of loading and dynamic response of the model.
This paper presents the development of indicial functions (IFs) for two-dimensional bridge deck sections. A new set of IFs predicted for the cross section of the Great Belt Bridge (GBB) are discussed. Two approaches, based on the time- and frequency-domain descriptions, are applied. In time domain, IFs are determined by imposing an instantaneous change to a system state variable, e.g., the angle of attack (AOA) of air flow to the bridge section. In frequency domain, indicial representation is derived from the aeroelastic (or flutter) derivatives typically used to define the self-excited harmonic forces in an eigenvalue problem and generated by imposing a sinusoidal motion to the bridge section and performing a sweep in the frequency range of oscillations of the section. IFs are thus evaluated by exploiting the reciprocal relations that exist between them and the aerodynamic derivatives. To determine the aerodynamic response of a bridge cross section due to a step change in the AOA and to calculate the flutter derivatives from sinusoidal oscillations, the meshless discrete vortex method implemented in DVMFLOW® is adopted. The results from the proposed work can be applied in the development of reduced-order models (ROM) of aerodynamic loads suitable to investigate fluid–structure interaction (FSI) problems associated with practical analysis of wind effects on long-span bridges, including phenomena such as flutter, vortex-induced vibration, buffeting and galloping. The IFs reported in this paper illustrate the importance of flow separation and vortex shedding and their dependence on the magnitude of the change in a state variable such as, in particular, the AOA for the cases reported herein.
This paper presents a continuum model for the nonlinear coupled vertical and torsional vibrations of suspension bridges with arbitrary damage in one main cable and, after pursuing a suitable linearization of the equations of motion, an investigation of damage effects on modal parameters. Damage is modeled as a diffused loss of cross-section representing the typical effect of fretting fatigue and it is introduced in the formulation by enforcing relevant literature results providing analytical solution for the static response of damaged suspended cables. The coupled nonlinear equations of motion of the damaged bridge, including the effects of shear deformation, rotary inertia and warping of the cross-section of the girder, are derived by application of Hamilton׳s principle. In this way, the equations of motion available in the literature for undamaged suspension bridges are generalized to the presence of arbitrary damage in one main cable and the resulting eigenfrequencies and eigenfunctions are derived in an analytical fashion. An extensive parametric investigation is finally presented to discuss damage effects on eigenfunctions and eigenfrequencies under variation of practically meaningful parameters.
A fully nonlinear model of suspension bridges parameterized by one single space coordinate is proposed to describe overall three-dimensional motions. The nonlinear equations of motion are obtained via a direct total Lagrangian formulation and the kinematics, for the deck-girder and the suspension cables, feature the finite displacements of the associated base lines and the flexural and torsional rotations of the deck cross-sections assumed rigid in their own planes. The strain-displacement relationships for the generalized strain parameters, the elongations in the cables, the deck elongation, and the three curvatures, retain the full geometric nonlinearities. The proposed nonlinear model with its full extensional-flexural-torsional coupling is employed to study the torsional divergence caused by the static part of the wind-induced forces. Two suspension bridges are considered as case studies: the Runyang bridge (main span 1,490 m) and the Hu Men bridge (main span 888 m) in China. The evaluation of the onset of the static instability and the post-critical behavior takes into account the prestressed condition of the bridge subject to dead loads. The dynamic bifurcation that occurs at the onset of flutter is also studied accounting for the prestressed equilibrium state about which the equations of motion are obtained via an updated Lagrangian formulation. Such a bifurcation is investigated in the context of the parametric nonlinear model considering the model parameters of the Runyang Suspension Bridge together with its aeroelastic derivatives. The calculated critical wind speeds for the onset of the static and dynamic bifurcations are compared with the results obtained via linear analysis and the main differences are highlighted. Parametric sensitivity studies are carried out to assess the influence of the design parameters on the instabilities associated with the bridge aeroelastic response.
The basic characteristics of the nonlinear free flexural-torsional vibrationsof two suspension bridges are examined. The Golden Gate Bridge andthe Vincent-Thomas Bridge were chosen to represent both a relatively long-anda relatively short-span suspension bridge's vibrations. The amplitude-frequencyrelationships of the first six modes (symmetric and antisymmetric) of both verticaland torsional vibrations for each bridge are presented. The case when oneof the linear natural frequencies of vertical vibration is equal to, or approximatelyequal to, another linear natural frequency of torsional vibration, is considered.This case revealed that the two modes are strongly coupled. Finally, a comparison between the analytical results obtained via the perturbation analysisand those obtained by the numerical integration of the governing coupled nonlinear equations of motion is presented. The agreement is reasonably good.
A general theory and analysis of t h e nonlinear free coupled verticaltorsionalvibrations of suspension bridges with horizontal decks are presented.Approximate solutions are developed by using the method of multiple scalesvia a perturbation technique. The amplitude-frequency relationships for any Isingle set of coupled vertical-torsional modes are presented for three cases: (1) [When the large-amplitude vertical vibration is dominating the motion; (2) when «large-amplitude torsional vibration is dominating; and (3) when one of the lin-,'ear natural frequencies of vertical vibration is equal to, or approximately equal • ito, another linear natural frequency of torsional vibration, and the two modes f iare strongly coupled; this contrasts with the linear solution, which predicts that the two modes are uncoupled.
This paper presents the results from earthquake performance assessment and retrofit investigations for Fatih Sultan Mehmet and Bosporus suspension bridges, with main span lengths of 1090 and 1074m in Istanbul. In the first part of the study, sophisticated three-dimensional finite element model of two suspension bridges were developed and the results of the free vibration analysis were presented. The models contain detailed structural components of the bridges and geometric non-linearity with cable sagging and stress stiffening, cumber of the deck and set-back of the towers. These components affect the natural frequencies and the corresponding mode shapes of the bridges. In the second part of the study, the seismic performance evaluation of two suspension bridges was undertaken. For performance assessments, non-linear 3-D finite-element time history analysis of with multi-support scenario earthquake excitation was used. Displacements and stresses at critical points of the bridges were investigated. Their earthquake performance under the action of scenario earthquake (site-specific ground motion that would result from the Mw=7.5 scenario earthquake on the Main Marmara Fault) were estimated and comparison with actual design data were also presented. Although both suspension bridges were originally designed for much lower earthquake loads they exhibited satisfactory performance. Finally, suggestions for retrofit need were made and retrofit design with hysteretic dampers for the Bosporus suspension bridge was calculated.
The dramatic Tacoma Narrows bridge disaster of 1940 is still very much in the public eye today. Notably, in many undergraduate physics texts the disaster is presented as an example of elementary forced resonance of a mechanical oscillator, with the wind providing an external periodic frequency that matched the natural structural frequency. This oversimplified explanation has existed in numerous texts for a long time and continues to this day, with even more detailed presentation in some new and updated texts. Engineers, on the other hand, have studied the phenomenon over the past half‐century, and their current understanding differs fundamentally from the viewpoint expressed in most physics texts. In the present article the engineers’ viewpoint is presented to the physics community to make it clear where substantial disagreement exists. First it is pointed out that one misleading identification of forced resonance arises from the notion that the periodic natural vortex shedding of the wind over the struct...
Cable-supported bridges typically exhibit minimal torsional motion under traffic and wind loads. If symmetry of the bridge about the deck's centerline is suddenly lost, such as by the failure of one or more cables or hangers (suspenders), torsional motion of the deck may grow and angles of twist may become large. The initiation of the disastrous torsional oscillations of the original Tacoma Narrows Bridge involved a sudden lateral asymmetry due to loosening of a cable band at midspan. The effects of these types of events on two-degree-of-freedom and four-degree-of-freedom section models of suspension bridges are analyzed. Vertical and rotational motions of the deck, along with vertical motions of the cables, are considered. A harmonic vertical force and an aerodynamic moment proportional to angular velocity are applied to the deck. Resistance is provided by translational and rotational springs and dashpots. Flutter instability and large oscillations occur under the aerodynamic moment, which provides “negative damping.” In order to model the occurrence of limit cycles, nonlinear damping of the van der Pol type is included in one case, and nonlinear stiffness of the hangers in others. The frequencies of the limit cycles are compared to the natural frequencies of the system.
We consider a forced nonlinear wave equation on a bounded domain which, under certain physical assumptions, models the torsional oscillation of the main span of a suspension bridge. We use Leray-Schauder degree theory to prove that, under small periodic external forcing, the undamped equation has multiple periodic solutions. To establish this multiplicity theorem, we prove an abstract degree theoretic result that can be used to prove multiplicity of solutions for more general operators and nonlinearities. Using physical constants from the engineers' reports of the collapse of the Tacoma Narrows Bridge, we solve the damped equation numerically and observe that multiple periodic solutions exist and that whether the span oscillates with small or large amplitude depends only on its initial displacement and velocity. Moreover, we observe that the qualitative properties of our computed solutions are consistent with the behavior observed at Tacoma Narrows on the day of its collapse.
Nonlinear free vibrations of suspension bridges with a bisymmetrical stiffening girder are investigated by using the method of multiple scales in two cases: when a two-to-one or one-to-one internal resonance takes place. The influence of the initial conditions on the vibrational processes’ character is shown. Three types of the energy-exchange mechanism between vertical and torsional modes are determined: two-sided energy exchange (a periodic motion); one-sided energy interchange (an aperiodic motion); and, energy exchange does not occur (stationary vibrations). The solutions describing the one-sided energy exchange are demonstrated to correspond with soliton-like solutions in the form of a single kink. On the hydrodynamic analogy, the qualitative method of analysis of suspension bridge non-linear free vibrations is proposed. This procedure allows one to determine the types of vibrational process, to investigate the stability of each vibrational regime, to define the character of amplitude and phase dependences from the initial conditions etc. Using five examples, the non-linear free vibrations of the Golden Gate Bridge in San Francisco are considered for the cases of two-to-one and one-to-one internal resonances. All quoted examples verify the qualitative hydrodynamic analysis.
This article takes a new approach to understanding the aerodynamics failure mechanism responsible for the collapse of the Tacoma Narrows Bridge 60 years ago. It is demonstrated that the instability mechanism is associated with the formation and drift of large vortices from the upwind edge of the bridge girder cross section. An empirical model of the vortex formation and drift process is formulated that allows critical wind speed for onset of the instability to be estimated. The model is supported by numerical simulations and experimental results. It is shown that a structural modification of the Tacoma Narrows Bridge cross section would have suppressed the vortex formation or counteracted the aerodynamic actions and thus rendered the bridge more aerodynamically stable.
The mechanical behaviour of suspension bridges is characterised by nonlinearities due to the main cables geometric effects and to the inability of the hangers to sustain compressive loads. The nonlinear effects due to hanger slackening ale expected to increase in suspension footbridges due to lightweight decks, that is, low dead to live load ratio, and to shallow plate-girder decks with very low flexural and torsional stiffness. In this paper a new section model is proposed to study the limit of hanger linearity in lightweight suspension footbridges. The model is inspired to a four degrees-of-freedom model already proposed in the literature, but is expressed with a new formalism that allows some interesting properties to be outlined. Specifically, the expression of a particular frequency, herein called relative antiresonance frequency, as a function of the model generalised properties is derived: if the system is loaded with a harmonic force having that frequency, the linear behaviour of the hangers is assured for every value of the force amplitude. The proposed section model is applied to a footbridge benchmark subject to the pedestrian harmonic load and results are compared with :hose obtained through a nonlinear dynamic analysis on a 3D Finite Element model of the bridge.
In 1940, the original Tacoma Narrows Bridge was completed on June 10 and opened to traffic on July 1. On November 7, the deck collapsed. Before that day, significant vertical oscillations had occurred, but no torsion. The bridge as built was stable with respect to torsional motion under the winds of November 7 and previous winds with higher speeds. However, snap loads in the diagonal ties attached to the north midspan cable band helped to loosen the band, and the frictional resistance between the band and the north suspension cable passing through it was overcome. The cable began to slip through the band. For this new structural system, with longitudinal motion of the north cable, the wind speed was higher than the critical speed for torsional flutter, and torsional motion was initiated. Approximately 700 cycles of torsional oscillations occurred during the hour prior to the collapse. In the present study, the snap loads on the cable band are discussed first. Then a continuum model of the central span (deck, cables, and hangers) is formulated. The longitudinal motions of the cables are included, so that the slippage can be incorporated. Known information from the observed steady-state torsional motion is utilized with assumed forms of the vertical cable displacements, and the governing equations provide the horizontal cable displacements, the dynamic tensions in the cables, the vertical and torsional motions of the deck, and the resultant lift force and pitching moment (including damping) acting on the deck during its final hour.
An ambient vibration survey of the Humber Bridge was carried out in July 2008 by a combined team from the UK, Portugal and Hong Kong. The exercise had several purposes that included the evaluation of the current technology for instrumentation and system identification and the generation of an experimental dataset of modal properties to be used for validation and updating of finite element models for scenario simulation and structural health monitoring. The exercise was conducted as part of a project aimed at developing online diagnosis capabilities for three landmark European suspension bridges.
Two-dimensional viscous incompressible flow past five generic bridge deck cross sections are investigated by means of the discrete vortex method. The analyses yields root mean square lift coefficients and Strouhal numbers for fixed cross sections and aerodynamic derivatives for the cross sections undergoing forced oscillatory cross wind and twisting motion. Fair agreement is established between the present simulations and wind tunnel test results reported in the literature.
This paper presents a method for modeling cable supported bridges for nonlinear finite element analysis. A two-node catenary cable element, derived using exact analytical expressions for the elastic catenary, is proposed for the modeling of cables. The cable element tangent stiffness matrix and internal force vector are evaluated accurately and efficiently using an iterative procedure. The reliability and efficiency of the formulations used are demonstrated by studying the behavior of the Great Belt suspension bridge during girder erection and the behavior of a cable-stayed bridge. Eigenfrequency analyses are also conducted and the results show good agreement when compared with previously published data.
The classical continuum model for the linear vertical vibrations of a suspension bridge (Bleich et al., 1950 [1]) is re-examined. The primary objective of the study is to extend the definitive analytical and numerical results of Irvine and Caughey (1974) [2], Irvine and Griffin (1976) [3] and Irvine (1980, 1981)
[4] and [5] for the natural frequencies, mode shapes, and modal participation factors for an extensible suspension cable, which depend on one dimensionless parameter related to the elasticity of the cable, to the case of a stiffened suspension bridge in which the response depends also on a second dimensionless parameter related to the stiffness of the girder. The continuum suspension bridge model is also used to understand the pattern of variation of mode shapes as a function of cable elasticity and girder stiffness, which has been shown by West et al. (1984) [6] to be considerably more complex than that for a suspension cable. Finally, the threshold amplitudes of free vibrations that would result in the incipient slackening of the hangers are determined.
A numerical framework for full-bridge aeroelasticity is presented, based on unsteady cross-sectional load models and on the finite-element modeling of the structure. A frequency-domain approach based on aeroelastic derivatives and nonlinear complex eigenvalue analysis is compared to its equivalent time-domain counterpart based on indicial functions and direct integration of the equations of motion. A version of the time-domain load model consistent with the quasi-steady limit behavior is developed and a procedure for the numerical identification of the indicial functions from measured aeroelastic derivatives is presented. The aeroelastic stability analysis is chosen as benchmark. A numerical example is offered where the equivalency of the two approaches is proved for a full-bridge model. Advantages and disadvantages of the two techniques are discussed.
Design techniques based upon sensitivity analysis are not usual in the current design of suspension bridges. However, sensitivity analysis has been proved to be a useful tool in the car and aircraft industries. Evaluation of sensitivity analysis is a mandatory step in the way towards an efficient automated optimum design process which would represent a huge jump in the conception of long span bridges. Some of the authors of this paper were pioneers in establishing a methodology for obtaining the sensitivity analysis of flutter speed in suspension bridges a few years ago. That approach was completely analytical and required the evaluation of many matrices related to the phenomenon. In those works the total mass of the deck was considered as constant and such a circumstance supposed a limitation of the method. In the present paper the complete analytical formulation of the sensitivity analysis problem in bridges considering variable deck mass is presented, as well as its application to the design problem of the Great Belt Bridge. Analytical evaluation of sensitivities is a time demanding task, and in order to avoid excessive computation times, distributed computing strategies have been implemented which can be considered as an additional benefit of this approach. For the application example, it has been found that deck cross-section area and torsional inertia are the structural properties with the greatest influence on the flutter performance.
The dependence between static instability and kinetic instability (flutter) on autoparameteric resonance is studied by taking compressibility into account in a model of a cantilever beam under the action of a follower force. It is shown that both instabilities are formally special cases of instabilities known as subharmonic and combination resonances.
A Practical Treatise on Suspension Bridges
- D B Steinman
D.B. Steinman, A Practical Treatise on Suspension Bridges, Wiley, New York, 1953.
The Mathematical Theory of Vibration in Suspension Bridges
- F Bleich
- C Mccullogh
- R Rosecrans
- G Vincent
F. Bleich, C. Mccullogh, R. Rosecrans, G. Vincent, The Mathematical Theory of Vibration in Suspension Bridges, U.S. Government Printing Office, 1950.
Suspension bridge vibration: continuum formulation
A.M. Abdel-Ghaffar, Suspension bridge vibration: continuum formulation, ASCE J. Eng. Mech. Div. 108 (1982) 1215–1232.