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«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»

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and where η = vector of the generalized coordinate of the bridge-SDS system; ξ = generalized coordinate of the bridge; γ = coordinate of the vertical motion of SDSs; a superscript “T” represents 179 transpose of vectors and a superscript prime “’” represents a derivative with respect to time t; U = wind velocity; M, C, and S = mass, damping, and stiffness matrices, respectively; n1 = number of modes of the bridge; n2 = number of SDSs; I unit = unit matrix; G = external force vector from wind buffeting; and Q b = generalized buffeting force. All the components of the matrices can be found in Chapter 7 and are not repeated here.

–  –  –

A carefully designed SDS system can increase the flutter critical wind speed for the combined bridge–SDS system. The homogenous part of Eq. (8.1) is expressed in the state-space

format as:

–  –  –

where η and G = Fourier transformation of η and G, respectively; the impedance matrix F has the general form of Fij = −ω2 M ij + i ⋅ ωCij ( ω) + Sij ( ω), where subscripts i, j = 1 to (n1+n2) and i = −1.

The mean square of displacements in vertical, lateral and torsion directions is related to the

buffeting force spectra SQbi Qb j as follows:

–  –  –

where σr and σr are the root-mean-square (RMS) of displacement with and without SDS ˆ control, respectively.

8.4 Numerical Example: Humen Bridge-SDS system To better demonstrate the applicability of the developed procedure, the Humen Suspension Bridge is analyzed. This bridge with a main span of 888m is located in the south of China, where hurricane (typhoon) is a serious threat. The basic data of the bridge are shown in Table 7.1. Four modes (two symmetric and two asymmetric) are listed in the table (Lin and Xiang 1995). Existent analysis has shown that the midpoint of the main span has the largest vibration response and the 1st symmetric vertical bending mode and the 1st symmetric torsion mode are the two modes most prone to couple together. Therefore, these two symmetric modes are the most important modes for buffeting response (asymmetric modes contribute nearly nothing at the mid-span point) and flutter instability. For simplicity and for demonstration purpose, only these two symmetric modes are considered in the evaluation of the SDS control efficiency for bridge flutter instability and buffeting response. The mass of the SDS, expressed as the percentage of the bridge mass, ranges from 1.0 to 1.5% in the present study.

–  –  –

182 Using the complex eigenvalue modal analysis approach introduced earlier, the flutter instability of the bridge-SDS system was analyzed with varied frequencies of the SDS. It is found in Fig. 8.3 that when the frequency of the SDS is very low (close to zero), the SDS is actually acting as a static mass block on the bridge. Since the SDS is placed on one side of the cross-section at the mid-point of the span, it acts essentially as a static eccentric load when its frequency approaches to zero. In the case of 1% mass ratio, the flutter critical wind velocity can be improved from about 87 m/s (without SDS) to 93 m/s (R1= 7%). Such a result also agrees with the conclusion that eccentric mass can increase the flutter stability of the bridge (Phongkumsing et al. 2001). However, when the SDS frequency becomes quite high (larger than 0.5 Hz), the SDS has no effect on the flutter stability at all as reflected by the flat horizontal line that corresponds to the case without SDS (Ucr = 87 m/s).

The same tendency can also be observed in the cases of larger mass ratios (1.25% and 1.5%). These results suggest that the vibration of traditional vehicles may have insignificant effect on the flutter instability of bridges under wind action since the frequency of the vehicles (normally over 1.0 Hz) is relatively too high to affect the bridge stability. Fig.8.3 also indicates that the optimal SDS frequency is about 0.25 Hz. This frequency corresponds to the torsion modal oscillation frequency (modified from the natural frequency by aerodynamic forces).

Fig. 8.4 shows the acceleration peak response under different wind speeds in cases with and without SDS control for the case of 1.25% mass ratio with optimal SDS frequency (0.25 Hz, determined from flutter analysis). As expected, the service criteria (indicated by the horizontal dash lines) can be satisfied in a higher wind speed when SDS is used. More specifically, for the same control criteria, the service wind speed limit is raised from originally 50 m/s (without SDSs) to 61 m/s (with SDSs).

Fig. 8.5 shows the acceleration reduction ratio versus the different SDS frequencies under a wind speed of 60 m/s. It is found that the maximum reduction ratio can be about 28%, 32% and 35% for the SDS mass ratio of 1.0%, 1.25%, and 1.5%, respectively. However, the SDS with high or low frequency has small reduction effect on the buffeting (acceleration) response. This means that, similarly to the flutter instability, a well-designed SDS with its frequency tuned to the optimal one will have an efficient control effect on the vertical accelerations. In this particular example, the SDS should have a frequency around 0.2 Hz for an optimal buffeting control for wind velocity of 60 m/s.

In that case, the SDS with a pre-tuned frequency becomes a kind of TMD.

The numerical results discussed above have shown that a well-designed SDS system can raise flutter velocity and reduce buffeting vibrations. Movable SDS control devices are thus specifically designed for extreme situations when a hurricane forecast is made and the control need is identified.

The preliminary design concept of a vehicle-type of control approach is introduced as follows.

To avoid the problem of vehicle instability or lateral overturning, the proposed vehicle-type of SDS (Fig. 8.6) has almost the same height as a typical truck. Lever-type design (Gu et al. 2002), which can greatly reduce the required vertical clearance of a typical vertical control device, is adopted here. The mass block uses iron to minimize the required space. Each vehicle-type of SDS has a gross mass of 10,000 kg. The dimensions of a vehicle-type of SDS can be seen in Fig. 8.6 (a).

The SDSs can be spread along the bridge as shown in Fig. 8.6(b), which is similar to the case of traditional multiple tuned mass dampers (MTMDs).

–  –  –

As in the case of designing MTMDs, the optimal frequency bandwidth ratio (Bf), the ratio of the central frequency of MTMD series (fav) to the modal frequency of the concerned mode, and the 186 damping ratio are the three main design variables (Gu et al. 2001). Optimal variables can usually be obtained through numerical searching. There are also some approximate design formulas to help attain the optimal frequency bandwidth ratio and central frequency of MTMDs (Kareem and Kline, 1994). The procedure of searching optimal variables is not repeated here and the reader can be referred to (Gu et al. 2001; Kareem and Kline 1994).

The number of SDSs is directly related to the control effect and the investment on control facilities. Using the Humen Bridge as example, Table 8.1 compares the control efficiencies corresponding to one-lane and two-lane SDS placements and various numbers of SDSs. All of the optimal design variables with respect to different numbers of SDSs are listed in the table as well. In the table, Bf = fhighest − f lowest is the bandwidth ratio of the even frequency distribution of SDSs (fhighest fav and flowest are the highest and lowest frequencies among all SDSs, respectively), fav is the average frequency of multiple SDSs and ζt is the damping ratio for all of the SDSs. The control performances are quantified using two variables: the new Usev and the new Ucr. The former is the new service wind velocity limit under which service criteria shown in Fig. 8.4 can be satisfied and the latter is the new critical flutter wind velocity with SDSs. It has been shown in Table 8.1 that the control performance basically changes positively with the increase of the numbers of SDSs and for most cases, significant control effects can be observed.

It is noted that this exploratory research is to investigate an alternative to improve the bridge performance in extreme events for long-span bridges during hurricanes. Placement of movable SDS system on bridge during evacuation will certainly block traffic and is not a perfect solution.

However, it is better than otherwise to completely close bridge in evacuation or see the bridge being damaged or collapse. In extreme case, to protect the bridge otherwise from being damaged or failure, the movable SDSs can also be placed on the bridge when the traffic is completely closed. Certainly, some issues as how to fix the movable SDSs on the bridge under strong wind need to be addressed before actual implementation.

8.5 Concluding Remarks To maximize the service capability and maintain the safety of the bridge itself, a movable/temporary passive control approach has been proposed based on a general formulation of the bridge-SDS system. The effect of vehicles on the dynamic performance of long-span bridges subjected to wind action is then investigated with the Humen Suspension Bridge. The following

conclusions can be drawn:

- A well-designed movable vehicle-type of control facility can effectively and conveniently increase the maximum wind velocity limit for bridge service in hurricane evacuations. For the case of 1.0% mass ratio, it reduces the peak acceleration by around 28% and simultaneously increases the flutter critical wind speed by about 14%.

- It is noted that this exploratory research is to investigate an alternative to improve the bridge performance in extreme events for long-span bridges during hurricanes. Placement of movable SDS system on bridge during evacuation will certainly block traffic and is not a perfect solution. However, it is better than otherwise to completely close bridge in evacuation.

–  –  –

10 0.8 0.94 0.15 0.06 55 92 16 1.25 0.93 0.16 0.05 61 99 20 1.6 0.93 0.16 0.05 68 110 26 2.1 0.93 0.18 0.04 74 115

–  –  –

10 0.8 0.95 0.15 0.06 60 95 16 1.25 0.95 0.15 0.06 67 103 20 1.6 0.95 0.16 0.05 73 112 26 2.1 0.94 0.16 0.05 78 117

–  –  –

9.1 Summary and Conclusions The dissertation can be roughly classified as three interrelated parts: (a) background knowledge of the problem for a deeper insight of long-span bridge aerodynamics; (b) investigation on the interaction of vehicle-bridge-wind systems; and (c) mitigation of excessive responses of bridges.

(a) Deeper insight of long-span bridge aerodynamics (Chapters 2 and 3) The major objective of the dissertation is to investigate the dynamic performance of the bridge and transportation under the wind loading. To achieve this goal, some research related to bridge aerodynamics has been conducted first. Since modal coupling is very common in modern bridges under wind loading, knowing the coupling characteristics among modes is extremely important, in order to select only those appropriate modes for coupled aerodynamic analysis and to better understand aerodynamic behavior. With incorporating only those modes which are actually needed in the aerodynamic analysis, the whole process can be greatly simplified. Such simplification makes a big difference in calculation efforts when many vehicles are modeled together with the bridge in the second part of the dissertation. On the other hand, to have knowledge about modal coupling of the bridge under wind action is also desirable. However, there is no quantitative method to assess this coupling effect so far.

In this study, a general modal coupling quantification method is introduced through analytical derivations of coupled multimode buffeting analysis. As a result, a Modal Coupling Factor is proposed to quantitatively assess the modal coupling effect and to select key modes. An approximate method for predicting the coupled multimode buffeting response is also proposed.

The main conclusions of this study are:

● With the proposed Modal Coupling Factor, modal coupling effect between any two modes can be quantitatively assessed, which will help to better understand the modal coupling behavior of long-span bridges under wind action.

● Such an assessment procedure will also help to provide a quantitative guideline in selecting key modes that need to be included in coupled buffeting and flutter analyses.

● The proposed approximate method for predicting the coupled multimode buffeting response, derived through a closed-form formula, is with acceptable accuracy compared with the “accurate” approach.

After modal coupling assessment and key mode selection techniques have been developed, they are applied to the hybrid aerodynamic analyses of buffeting and flutter. Most existent works deal with these two analyses separately, with different approaches in the frequency domain. To consider the vehicle moving and interaction with the bridge, analyses in the time domain are necessary. Most existent approaches in the time domain are based on direct finite element modeling of the structures, and are used for buffeting analysis only. A hybrid analysis approach 189 is introduced in the present study to facilitate the following analysis on vehicle-bridge-wind system. In the meantime, such an approach is also helpful to understand the progressive natures of the two phenomena, buffeting and flutter, in a more consistent way and the main conclusions

are as follows:

• A hybrid approach based on complex eigenvalue modal analysis and time domain analysis provides a more convenient tool for unified buffeting and flutter analyses in a time domain.

• The proposed approach provides a convenient way to numerically replicate the transition of vibrations from multi-frequency buffeting or pre-flutter free vibration to a single-frequency flutter. This transition has been observed in wind tunnel tests and is commonly accepted as a fact.

• The mechanism of the flutter occurrence and the actual transition process from multifrequency dominated buffeting to the single-frequency dominated flutter are well illustrated through the numerical results of the Humen Bridge.

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