Nonlinear aerostatic stability analysis of suspension bridges

Su, Luo & Yun (2010) performed an aerostatic reliability of bridges with long spans. According to their study, the response surface Monte Carlo method (RSMCM) is suggested to be suitable for the analysis of the reliability of both the aerostatic stability and the aerostatic response for various types of long-span bridge structures. In these bridges, the nonlinear effects that come about as a result of the geometric nonlinearity as well as loads that are dependent of deformation are taken into due consideration. In their study, the geometric parameters, aerostatic coefficients of the girders used on the bridge as well as the parameters of the materials used are treated as random variables. The response surface Monte Carlo method (RSMCM) is pointed out to possess a higher accuracy as compared to the other more traditional approaches of response surface method. It also has the merit of reduced costs of computation as compared to the usual Monte Carlo Method. The method proposed by Su, Luo & Yun (2010) is applicable to the analysis of the reliability of aerostatic response as well as the stability of aerostatics of the Ting Kau Bridge, Hong Kong and they obtain and present reasonable results that truly illustrate the method's effectiveness. The work of Su, Luo & Yun (2010) also explored the deterministic nature of the process of aerostatic analysis of bridges having a long-span. Their analysis revealed that it is crucial for long-span bridges to be effectively designed in order to withstand the static forces that are static and caused by the design mean velocity of wind. On top of this, bridges designed that way are affected readily by various forms of aeroelastic effects that include the aerostatic instability. This aerostatic instability include forces that cause torsional divergence, fluttering, galloping, lateral buckling, vortex-induced oscillation as well as buffeting that takes place of forces that are self-excited as earlier noted by Simiu and Scanlan (1996). In the analysis by Su, Luo & Yun (2010) on the other hand, the only forces that were considered are the ones that are due to aerostatic responses as well as the aerostatic instability for the case of long-bridges subjected to static wind loads. To be specific, they considered the variations of both the aerostatic responses as well as the critical wind velocities due to aerostatic instability. These were considered as a result of the random influence of the parameters that are geometric in nature, the parameters of the materials as well as the structure's aerostatic coefficients. In order to achieve this, the deterministic analyses of both the aerostatic stability and the aerostatic response for the bridges having long-spans were considered. They were however necessary for the purpose of acting as the numerical tests that are taken in the process of probabilistic analysis. An earlier study by Hirai et al. (1967) indicated that the lateral-torsional buckling of various types of suspension bridge can take place under the influence of static wind loads in tests carried out in wind tunnels involving a full model of the bridge. Several other studies have been dedicated to the nonlinear buckling of long-span bridges under the influence of static wind loads when considered theoretically. The works of Boonyapinyo et al. (1994), Ma et al. (2009) as well as Xie et al. (1997) effectively came up with a correlation between the mathematical models as well as the algorithms that are employed in the aerostatic stability analysis as well as the aerostatic response analysis of long-span bridges. Their study employed a setup consisting of static wind loads as a result of mean wind velocity on special bridge decks that have three major components; lift, drag as well as torque moments of force. The expressions used in their determination are the ones derived by Simiu and Scanlan (1996).

Cheng, Jiang, Xiao and Xiang (2001) conducted a nonlinear aerostatic analysis of a suspension bridge (Jiang Yin) that is located over the Yangtse River in China. They proposed a nonlinear technique to be used in the analysis of the aerostatic stability of the suspension bridge on the basis of the three components of geometric linearity and the load of the winds. A special computer program called NASAB that is based on this method was developed. Both the accuracy as well as the efficiency of this computer program were examined via numerical techniques. The effects of certain crucial parameters on Jiang Yin bridge's aerostatic stability were evaluated. The outcome indicated that when the effects of the wind angle of attack on the slope that exists on the curve of pitch moment coefficient is great to a point that is considered significant for the bridge's stability analysis. A strong linearity was found to exist for the displacement response when the bridge is subjected to displacement-dependent wind loads. Their analysis also indicated that the aerostatic instability of Jiang Yin bridge can effectively exhibit what is referred to as the asymmetric flexural -- torsional instability in a given space. The angle of incidence of the wind angle as well as the load cables of the wind does have a great effect on the bridge's aerostatic stability. According to their study, the recent decades have seen the development of suspension bridges to be one of the most popular all over the world. They also pointed out that bridges with longer spans are being designed and constructed with examples like Messina bridge located in Italy having a length of 3300 m and Gibraltar bridges that exists between Morocco and Spain having a length of 3,500m becoming realities. They however pointed out that there are several new problems that are being faced in the design of these longer bridges possessing more flexible girders. Aerostatic stability was mentioned as a concern in these bridges. This is because aerostatic instability is dominant whenever a deformed shape of a given structure effectively produces a large increase in the magnitude of the three major components of the wind loads that are displacement-dependent and effectively distributed along the structure. Earlier on it was believed that oncoming wind velocity that is responsible for flutter is much lower as compared to the velocity of wind load (static) for the suspended kind of bridges. Cheng, Jiang, Xiao and Xiang (2001) further pointed out that several engineers who design and construct the suspension bridges paid little attention to the potential instability of aerostatic origins. Very few studies have been conducted to determine the aerostatic stability of various bridges. Past literature however indicate that aerostatic instability is indeed a problem for all suspension bridges of certain length and rigidity. Hirai et al. (1967) pointed out that the torsional divergence of all forms of suspension bridges is likely to occur when they are subjected to static wind loads when the tests are performed in wind tunnels of models of full bridges. Additionally, a study of a wind tunnel located in Tongji University (Booyapinyo et al.,2004) indicated that the phenomenon affects the suspension bridges. They therefore pointed out the necessity of investigating the aerostatic stability of various types of suspension bridges. The other earlier aerostatic stability analyses of the long-span suspension bridges relied heavily on a linear approach. The linear approach advanced on the basis of the assumption of the more linearized derivative of the moment of pitching as well as the linear structural stiffness matrix as pointed out by Simiu & Scalan (1986) as well as Xiang et al. (1996). The linearized approach however never accounts the nonlinear effects that come up as a result of the structure of the bridge as well as the three major components of wind load. The critical wind velocity that causes the aerostatic instability can never be easily determined. The other factors that are not considered are the mode of the instability, the coupling effect as well as the wind velocity. The bridge's the wind velocity -- deformation path as a result of the velocity of the wind to the divergence can never be easily traced. Currently, the nonlinear analyses techniques are heavily employed. This is attributed to the increasing computing power that makes it possible to merge other technologies like the nonlinear finite element methods (NFEM) for the analysis of the nonlinear effects that results due to the structure of the bridge as well as the major components of load of wind for the analysis of the aerostatic responses using NFEM. Cheng, Jiang, Xiao and Xiang (2001) pointed out that the nonlinear method is based on the concept of bifurcation point instability and that a nonlinear method that combines both the eigenvalue analysis as well as the updated bound algorithms is also in use (Booyapinyo et. al,2004). The operation involves the analysis of the nonlinear lateral-torsional buckling of the cable-stayed bridges that are induced by wind. Cable-stayed bridges as well as the suspension bridge are pointed out as never being perfect structural designs/systems due to the fact that their elements like girders as well as towers are normally exposed to bending moments as well as axial forces. They also usually sustain a lot of dead loads as well as initial deformations which causes the initial deformations as well as stresses to be present in all of their members. This therefore renders the bifurcation point instability null and void for the cable strayed as well as suspension bridges (Ren,1999). Cheng, Jiang, Xiao and Xiang (2001) pointed out that in theory, the analysis of the aerostatic stability of such kinds of bridges should be regarded as a limit point instability challenge. In their paper, which is based on the limit point instability concept; Cheng, Jiang, Xiao and Xiang (2001) presented a nonlinear finite element method (NFEM) in order to evaluate in a more direct manner, the critical velocity of the wind for all cases of suspension bridges suffering from aerostatic instability. In these bridges, the three main components of the wind loads together with the geometric nonlinearity are taken into consideration. This required the employment of a specialized computer program called NASAB which had its basis on the nonlinear technique was tested for numerical examples. Jiang Yin suspension bridge's aerostatic stability was investigated via the NFEM. The outcome was found to give information on the critical wind velocity of the NFEM which was noted to be greater that the magnitude obtained via the linear technique. The cause of the difference was explained. In their study, they also explored certain other aerostatic stability parameters necessary for various bridges.

Mikkelsen & Jakobsen (2010) performed a flutter analysis on the Hardanger bridge. In their analysis, they investigated the aeroelastic stability of the bridge which is a suspension bridge having a main span of about 1310 meters and was being constructed in Norway. The wind and structure system was described via a state-space format in a multimodal flutter analysis format. The outcome of the multimodal flutter analysis on the basis of the ambient vibration data had earlier in been reported by Jakobsen and Hjorth-Hansen (2007). Their work on the otherhand had its basis on the motion-dependent loads that were obtained from the forced vibration wind-tunnel tests using a 3 DOF-section model. The resulting aeroelastic loads were then approximated using the rational function approach that is closely associated with the frequency independent system matrix. This eliminated the need for iterating the eigen-values.

Xin-jun (2005) on the other hand performed an advanced aerostatic analysis for suspension bridges that are lengthy. He employed an advanced method aerostatic analysis that relies on the consideration of the geometric nonlinearity, spatial non-uniformity of the wind speed as well as the nonlinear wind-structures. The example of suspension bridge that was considered for this work was Runyang Bridge that runs over the Yangtze River. The parameter that were investigated included the effects of the nonlinear interaction of wind, the spatial uniformity of the wind speed as well as the wind load of the cable on the overall behavior of the suspension bridge.

Quite a number of studies regarding nonlinear aerostatics stability analysis of suspension bridges have been conducted. For instance, Liu et al. (2004, p.56) conducted an analysis on aerostatic responses of the cable-stayed bridges that are having long life spans. He took into consideration the geometric parameter's uncertainties and again, he took into consideration the aerostatic coefficients of the major girder. On the other hand, nonlinear impacts because of the interactions from wind-structures and geometric nonlinearity were not taken into consideration since the cost of computation was so high because the technique needs a fresh round of the analysis of FE for every sampling check. Cheng et al., (2004, p.780) also carried out a stochastic study of aerostatic stability for suspension bridges through the use of MCM that had its basis on cycle solutions. This needs very minimal work of computation. On the contrary, the cycle solutions are majorly appropriate for the study of torsional divergence, and therefore it is hard to widen the technique to analyze other kinds of bridges.

Similarly, the aerostatic behaviors of suspension bridges having long-span was expansively studied by Boonyapinyo et al. (1994, p.500), Xiao and Cheng et al. (2002, p.45).

The aerodynamic stability and static stability of bridges that are cable-stayed have been researched by numerous authors. Aerostatic instability may be grouped into two kinds depending on static instability modes: lateral-torsional buckling and torsional divergence. The two aerostatic instability phenomena were studied by Boonyapinyo (1994, p.504).

Simiu and Scanlan (1978) developed a linear technique to critically analyze the long span bridge's torsional divergence. Similarly, Xiang et al. (1996, p.) wrote the method that was used by them. These two techniques had their basis on the suppositions of linear structural inflexibility matrix and also of linearized pitch moment. Hence, critical velocity of wind that causes aerostatic instability can't be calculated accurately, the instability mode and also the coupling impacts can't be taken into consideration. The quick development of computers and also the finite-element methods that are nonlinear has resulted into the consideration of the nonlinear impacts that arises from the structures of the bridge and also the three wind load elements, and to assess the aerostatic reaction by NFEM. Boonyapinyo et al. (1994, p.501) also used a nonlinear technique that merges eigenvalue scrutiny and also bound algorithms so as to examine the wind-induced non- linear lateral-torsional buckling of cable-stayed bridges. On the other hand, it is having three disadvantages. First, the notion of divergence point instability with its basis on eigenvalue study may not be valid for the bridges that are cable-stayed. This is due to the fact that: its components like the towers and girder are subject to bending moments and also axial forces; before the application of wind loads, the bridges may have sustained very heavy built-in loads of construction in order that the first stresses and deformations are present in all the members.