How to Approach to Full Model Wind Tunnel Testing? | Wind Engineering

The below mentioned article will guide you about how to approach to full model wind tunnel testing.

Introduction to Full Model Wind Tunnel Testing:

Recent needs of going for the record to stretch the spans much farther are hardly competitive in the construction of suspension bridges as well as cable-stayed bridges. To provide longer, uninterrupted spans over terrain the structural systems of cable-supported crossings are more complicated to design and build.

As a matter of fact, the stretch of longer span length leads to more flexible structural features, which may cause a couple of never-experienced, complicated behaviors, governing its safety, in static as well as dynamic responses to wind action.

Among various plans of those very long span bridges, constructions of the Akashi Kaikyo Bridge, a suspension bridge of span of 960 + 1,990 + 960m, and the Tatara Bridge, a cable-stayed bridge of span of 270 + 890 + 320m, are well under way for their completion in 1998 and 1999 respectively, when they will be the longest of each type of bridge in the world.

Because of their span length never-experienced so far, numerous investigations on wind effects have been, as a matter of course, carried out from a couple of viewpoints accumulated from the past understandings. In case of the Akashi Kaikyo Bridge, a series of full aeroelastic model wind tunnel testings have just finished at a newly built large boundary layer wind runnel with 41m width and 30m length.

Those behaviors, such as flutter instability and gust responses, observed in the full model wind tunnel testings have been realized in three-dimensional fashion under statically deflected phase caused by wind loads, particularly in the higher wind speed range associated with the limit state.

Among them, the flutter instability occurred in the truss stiffening girder of the Akashi Kaikyo Bridge, showed a complicated, three-dimensional mode shape consisting of predominant torsional as well as vertical motions, slightly accompanied by horizontal motion, of which the description could be successfully done by means of a newly-developed three- dimensional flutter analysis method.

Thus, looking at the fact that a specified behavior was quite three-dimensionally correlated with the deflection and vibration, taking place at the same time, rather well performance in building the full aeroelastic wind tunnel model should be strictly verified again.

Here, in this contribution, some of governing factors interactions and influenced behaviors of wind effects on above-mentioned two bridges will be illustrated and discussed, focusing on how to approach to full model wind tunnel testing and to get better findings from those tests.

Flexibility of Long-Span Cable-Supported Bridges:

Looking at the reduction of natural frequency and critical wind speed of flutter with the stretch of span length, it is quite easy to understand how flexible are the long- span bridges in their structural and aerodynamic characteristics.

Fig. 1 shows the change of natural frequencies of the lowest torsional and vertical bending modes with increase of mid-span length in a couple of existing and proposed suspension bridges, and also that of critical wind speeds of so-called coupled flutter, calculated by using the preceding natural frequencies on assumption of flat plate unsteady aerodynamic forces, which may be often used for a good index to present wind resistibility of a specified long-span bridge.

It can be seen that it decreases from higher wind speed than 80m/s in conventional class of 1,000m mid-span length to 60m/s in 2,000m class, and furthermore down to 40m/s in 3,000m class, such lower level as expected frequently to occur.

Primary reason of this decrease can be referred to a significant reduction of natural frequencies, particularly to that of torsional ones because of the greater polar moment of inertia caused by increasing mass of main cables to stretch the longer spans over terrain. The torsional frequency tends to approach to the vertical heaving one, extremely to the limit of two parallel suspended cables from which a non-stiffened girder simply hangs.

Then, it is quite indispensable for a proposed very long-span bridge to overcome such an inevitable and flexible characteristics, in cooperation with developing aerodynamically more stable cross section of stiffening girder other than the flat-plate-like if not enough. In order to realize an appropriate high level of wind speed against possible flutter instability, say more than 70m/s, a couple of ideas for countermeasures have been actually developed and investigated.

A non-dimensional reduced wind speed U/fB, where U is wind speed, f torsional natural frequency and B width of stiffening girder, is a key factor to identify a specified aeroelastic behavior on objects in wind. Calculating that of a certain flutter instability at design wind speed to be verified for the safety, usually anticipated at same level in any case though, it increases nearly in proportion to mid-span length, because the torsional natural frequency reduces inversely but the width is nearly constant.

Higher reduced wind speed in case of a very long-span bridge, never-experienced so far, should require aerodynamic investigations from different viewpoints. This is a kind of evidence to show the change of targeted behavior due to that of reduced wind speed.

A wind tunnel test by the use of sectional model indicated that torsional flutter, usually observed in the truss stiffening girder of conventional 1,000m mid-span class suspension bridges, changed to so-called motion-coupled flutter, recognized in a shallow flat-plate-like girder, at such high wind speed range as verified for the Akashi Kaikyo Bridge of 2,000m class.

Along with the reduction of natural frequencies with the stretch of span length, a significant structural coupling can be found out in natural vibration mode shapes between vertical, horizontal and torsional components of motion, particularly when deflected and twisted under the action of drag, lift and pitching moment forces by a strong wind. For instance, there happens a contribution of horizontal and vertical motions in the torsion-predominant modes, and that of torsional motion in the horizontal sway-predominant modes.

Then, paying an attention to above mentioned existence of motion-coupled flutter on a very long-span suspension bridge, it is quite natural that those effects of essential coupled vibration mode shapes must be carefully taken into account in its verification by means of full model wind tunnel testings.

Stretching the spans of cable-stayed bridges longer, those natural frequencies are also reduced. Fig. 2 illustrates that a couple of lower modes are compared in the mid-span range of 300 to 1,000m for normal three-span bridges. Cable-stayed bridges can have a wider choice of play with the form structurally as well as aesthetically.

According to the type of their towers and cables, those bridges are classified to several ways of combination; single, delta or twin tower type, two- vertical-planes, two-inclined-planes or single-plane position of cables in space, number of cables and their arrangement. And, structurally, a couple of other selections are provided with support conditions of the decks, including existing or no auxiliary piers in side-spans, and those between the decks and the towers.

Among them, it is quite clear that the reduction of torsional frequencies in case of a single-plane stay is the most remarkable with the stretch of span length. Significant structural coupling in the natural vibration mode shapes is also seen, particularly between horizontal and torsional motions.

Now, to verify and measure the aerodynamic vibrations and forces on a bridge, there are a couple of combinations of wind tunnel model of a bridge and a smooth/simulated turbulent flow (+ a topographic model). In process of practical experiments the purpose of the wind tunnel testings is to find out desirable configurations of bridge deck proposals first and then the screened proposal becomes to be brushed up.

Although the combination of three-dimensional full bridge model and simulated turbulent flow is the most preferable in the modeling, the cost of this choice is very high. Besides it may be very difficult to find out the best configuration of a bridge deck.

Therefore, usual two-dimensional sectional model test is a practical selection for that purpose, although interpretation of the wind tunnel test results might be complicated. Usually, this complicated interpretation must be very essential in the wind resistant design. On stretching the span length of a bridge beyond the past experiences, however, the interpretation is probably not so simple, because of possible correlation caused by some characteristics mentioned above, that full aeroelastic bridge model wind tunnel testing comes to be carried out, also by modeling the topography if necessary.

Choice of Truss Stiffening Girder on Akashi Kaikyo Bridge:

The Akashi Kaikyo Bridge, which crosses over an open sea terrain of the Akashi Kaikyo Straits with a shore-to-shore distance of approximately 4 km in an inland sea of Japan, is a very long-span truss-stiffened suspension bridge for six road lanes with a midspan of 1,990 m and two sidespans of 960 m. Construction has started in the spring of 1988 to bring to complete in 1998. The main cables erection is now under way to terminate in the spring of 1995.

As mentioned in the preceding, with the suspended deck crossings stretched to longer, more flexible structural features are so indispensable that static as well as dynamic wind effects are of primary importance for the structural design of this bridge.

First of all, flutter instability has been a serious problem for possible selection of deck configurations, because a very long-span, never-experienced, made the entire structure extremely flexible, as seen in predicted lower torsional natural frequency, and prescribed design code required a rather high critical wind speed which naturally stemmed from wind environment at frequent typhoon attack region.

Therefore, from the very beginning of the project, numerous investigations have been repeated on the aerodynamic problem by means of conventional two- dimensional sectional model wind tunnel testings in a smooth flow.

After passing beyond preliminary stage, along with conventional truss girders, a couple of box girders and their variations were also proposed, aiming at the improvement of aerodynamic stability as well as sophisticated design. Single closed box girders failed to meet the code requirement for flutter, provided that conventional design was made for prescribed loadings and design conditions.

To raise critical wind speed, it should be necessary either to increase the use of steel for greater torsional rigidity in spite of economical disadvantage, or to adopt aerodynamic devices such as open slots and a median barrier. Some double box girders with a wide open slot and a median barrier met the requirement, but they were not advantageous because estimated dead load of the girder was heavier than that of a truss structure.

In addition, they also suffered from severe vortex excitation. Compound stiffness girder was an attempt to ensure aerodynamic stability by using conventional girder configuration, which consisted of two single box sections of same width and different depth. They were arranged in such a way as shallow sections at middle portions of each span to attain sufficient stability and deep sections at end portions to provide appropriate rigidity.

Through these investigations, finally, the decision was made to adopt a properly designed truss structure, taking account of various factors such as structural details, erection methods over frequent navigation route, maintenance cost, etc.

Further efforts have been continued to improve the performance of the truss stiffening girder from viewpoints of economy, aerodynamic stability and design practice, for instance:

(a) Effects of truss width, carrying the same floor deck,

(b) Location and numbers of built-in members such as inspection ways, electric power line and water supply piping,

(c) Configuration of detailed members as perforated curb,

(d) Interaction effects of main cables with the girder, resulting to make the flutter more unstable with the relative distance reached to the same as truss height, and

(e) Effects of a kind of stabilizer, increasing means of flutter stability, with solid short plates installed at the center on and/or under road deck.

In general, the flutter instability, recognized in sectional wind tunnel tests at this stage, was of a heaving-and-torsion coupled type in the most stable case, or in the higher range of wind speed close to that for design verification, although it turned to a torsion predominant type, or so-called torsional flutter, resulting in lower critical wind speed of 40 to 50 m/s, in the most unstable case. The torsional flutter used to be observed in the truss girder of conventional span length of 1,000m class. The change to coupled type in the most stable case was really significant.

The cross section of a truss stiffening girder, finally judged the most stable and reasonable for adoption, was such a simplified, contemporary truss type with a kind of stabilizer installed at the center as shown in Fig. 3.

Meantime, the code requirement for wind effects has also been investigated for this specified bridge in the form of improvement and revision of the former code (1976) and manual (1980), applied to conventional class-span bridges constructed so far, along with considering extension of the scope which never-experienced scale of the bridge brought out.

A new code was established in 1990, where the requirement against flutter instability was prescribed in the form of critical wind speed U_c satisfied with the relationship in the 10 min. mean:

in which U_z = (z/10)^1/8U_ref = 60 m/s; wind speed at mean height of the girder z = 80 m and U_ref = 46 m/s; a reference wind speed at 10 m height over open sea level for 150 years return period; and a, angle of attack of air flow, specified by taking into account of the effects of wind speed fluctuation over open sea, probable torsional displacement caused by wind loads and experimental error on mounting sectional wind tunnel models as well as the fact that flutter instability is quite susceptible to it.

The coefficient 1.2 is a principal safety factor considering reliability of experimental testings, design and construction of the structure, social importance of the structure and a destructive feature of flutter instability, while the coefficient µF = 1.08 is a conditional safety factor considering spatial gust effect of wind speed fluctuation under assumption of the built-up time (30 sec. for this case) of flutter compared with 10 min. mean wind speed.

Here, as a result of requirement of two-dimensional sectional model test in a smooth flow, most of effects of wind speed fluctuation come to be dealt with as quasi-static effect. In case of the wind tunnel test in a simulated turbulent flow, the conditional coefficient must be µf = 1.0 because of its original definition, and then the critical wind speed shall be larger enough than 72 m/s, only for a horizontal angle of attack if the torsional displacement of the girder is not so significant.

In the old code (1976) the wind load includes both time averaged wind effect and influence of gust response in stress of structural members. However, as far as calculated results for the truss girder proposed are concerned, after loading of those wind loads, allowable torsional amplitude to be added is not large enough to permit significant vibration, particularly in the lateral bracings of truss girder.

In comparison among a couple of gust effects, usually, horizontal gust response is predominant. In case of this very long-span bridge, however, the gust effects due to vertical and torsional responses are not negligibly small. These two results made us add a separate detailed consideration on vertical and torsional gust effects on the various members to the new code (1990). It took into account these effects in two steps of the design procedure.

At the first step as static wind resistant design, a simple wind load was selected in the form of a typical lateral load, very similar to the old. There may be other procedures for detailed wind load especially in the loading pattern. Predominant wind load effects, however, are still due to lateral static load and horizontal gust response.

Therefore, it will be very convenient to propose the deck configuration under this load and to make various structural and aerodynamic comparisons before the detailed gust analysis. After this designing procedure, detailed gust response estimation of the full bridge system shall be carried out as one of aerodynamic verifications. This process shall be repeated until all the related design requirements in the code come to be satisfactory.

Full Aeroelastic Model Wind Tunnel Test Program:

As mentioned, a much simpler stiffening truss girder of contemporary type was finally proposed, instead of any other massive one of conventional type, after satisfying a couple of requirements for wind effects, such as wind-induced displacements, flutter instability and gust responses to be verified, although those were carried out only by sectional model wind tunnel test in a smooth flow, but also through a conventional analytical calculation for individual motion, upon an appropriately assumed analytical model of the bridge structure. In order to raise the critical wind speed of flutter instability over the prescribed wind speed, however, a kind of stabilizer and/or center barrier had to be installed at the center part of road deck.

As for such a proposal of stiffening truss girder to be proceeded towards its final design in detail, a series of full aeroelastic model wind tunnel tests were to be planned in order to carry out final verification of the wind resistant safety, because of following reasons:

(1) Models and methods for the requirements to be verified have been never applied, even if the best at that time, to such a very long-span suspension bridge of very long period never-experienced so far. Those have been derived only from the understanding and experiences of conventional long-span bridges though.

(2) Behaviors of the bridge may be of quite three-dimensional fashion in its deformation and vibration mode shape, at higher wind speed range in particular associated with its limit state, showing such a way as, first a large lateral deflection along with windward-down twist due to wind loads, on which superposed random gust responses of primary three directions, and finally a coupled flutter instability of complicated mode shape taking place on them.

Some of those three-dimensional effects are caused by followings:

(a) Deformation of horizontal deflection and twist accompanied with vertical deflection due to wind loads are quite varying along the bridge axis, resulting in the largest at the center of mid-span.

(b) Aerodynamic interference between truss girder and main cables depends on their interval, which may also be varying by their relative location partly caused by wind loads.

(c) There exists a complicated coupling in natural vibration mode shapes, between torsional and horizontal motions in particular, also including the contribution of vertical motion in case of laterally-deformed structure due to wind loads, according to the three-dimensional frame model analysis for the bridge.

(d) Effect of stabilizing devices to suppress the flutter instability must be confirmed, with all the spans installed or only the mid-span.

(e) Onset of coupled flutter instability in three-dimensional fashion may come to be quite different from that observed in two-dimensional sectional model.

Since the aim of the program was verification of the safety directly from the model responses, special care was taken to realize a good geometric and elastic simulation of the long-span bridge model as well as a good natural wind simulation. To realize a good simulation in a couple of key points such as geometric configuration and elastic characteristics of the bridge model as well as natural turbulent wind, a lot of examinations and comparisons were made along with the consideration for cost constraint of wind tunnel construction.

Then, the geometric scale of the model was chosen as 1/100. The size of this scaled model was comparable to that of the sectional models, so that the structural details of the truss girder and auxiliary members, including connection devices of each element, could be modeled with accuracy. For this program, a large boundary layer wind tunnel was especially constructed by Honshu-Shikoku Bridge Authority at the Public Works Research Institute, Ministry of Construction, to install a very long full aeroelastic bridge model of total length about 40 m.

Wind Tunnel and Turbulence Generation:

In full aeroelastic model wind tunnel testings, the determination of test similarity is very important. For structures where the resistance to deformation is influenced by the action of gravity, such as suspension bridges on which the tensile force of main cables caused by dead load plays a primary role, F_r number shall be satisfied with because the acceleration of gravity must be equal in model and in prototype.

Then, the scale of wind speed is the square root of the length scale. Actually, the maximum wind speed available in the facility was set to be 12 m/s by taking a margin for pressure loss in simulated boundary layer turbulent flow, which could allow sufficiently for the range of wind speed to verify flutter instability as required in the code (78 m/s in a smooth flow in this case).

The primary dimensions of test section were chosen 41 m wide, 4 m high and 30 m long in order to install an about 40 m long full bridge model as well as to generate well simulated boundary layer turbulent flows which could allow for appropriate thickness over the bridge model in the rear.

The wind tunnel is basically of the section Eiffel type, but the air is circulated within the building. Other properties and arrangements of the facility were specified by investigating the test results of a half model of 1/16 scaled pilot wind tunnel.

To evaluate the aerodynamic stability in turbulent air flow, preliminary studies were carried out to examine how to generate a boundary-layer turbulence, also 1/100 in scale, simulating the case with natural wind prevailing at the bridge construction site. As an aid in generating the simulated boundary layer turbulence, it was actually decided to use spires and roughness blocks in alternately different combinations.

The simulation process in the wind tunnel consisted, so to say, of trial and error, with reference to empirical formula. The boundary layer turbulence characteristics targeted to reproduce were, taking into account as basic requirements, the following parameters specified by the code requirement for this bridge; power exponent of vertical mean wind speed distribution α = 1/8 and intensity of along wind turbulence at the girder height I_u = 10% and 5%.

In designing the spire, a pilot scheme was conducted using spires at the wind tunnel, 1/16 in scale, to decide on approximate size and shape to which to fabricate the spires for use in the large wind tunnel.

Consequently, it was decided that generating a turbulent flow exceeding 3 m in boundary layer thickness would require the use of spires about as tall as the height (4 m) of the wind tunnel ceiling and that to ensure the spire structural integrity, each spire be fixed at its top to the ceiling.

In the process of generating the turbulent flow, it was necessary to add the use of brick-blocks placed on the floor surface to get the required level (10%) of intensity. Finally, judging the first priority to be to simulate the intensity of turbulence at the girder height, the following two types of turbulent flow were provided to use for the test; I_u = 9.5% (I_w = 6.8%) and I_u = 6.5 %(Iw =5.5%), where the integral scale of along wind L^x_u was about 1 m long, the decay factor of correlation K^x_u = 5 ̃12 and power spectrum plotting in shape agreed fairly well with that of natural wind prescribed for the code requirement.

It was found that the integral scale was slightly smaller than required, and some other parameters also slightly differed from the required. However, the difference was considered not so large and adjustable for the evaluation of test results through the correction by analytical calculations.

Full Aeroelastic Bridge Model Simulation:

In order to verify the detailed three-dimensional behaviors of a long-span truss stiffened suspension bridge, a lot of efforts to realize a good performance on structural as well as aerodynamic properties must be done as for the similarity of the model in configuration, mass distribution, stiffness distribution, damping capacity, aerodynamic interference between each element and so on. In this case of the Akashi Kaikyo Bridge, the geometric scale of full aeroelastic bridge model was chosen 1/100, and various analytical studies and preliminary model tests were carried out in advance.

Modeling of the Stiffening Truss Girder:

The stiffening girder model is shown in Fig. 4. The model is composed of rigid blocks of four truss panels connected by V-shaped springs, which represent expansion and contraction of the girder chords resulting in the simulation of vertical and lateral rigidities, while torsional stiffness is simulated by the out-of-plane bending of each member of a V-shaped spring, which means disparity of the chord axis.

By directing the four V-shaped springs toward the center of the truss section, their large axial stiffness can guarantee a constant shear center of the cross section. In order to simulate the mass, polar moment of inertia and the center of gravity at the same time, the mass distribution of the model cross section must be analogous to that of the actual bridge.

This requirement is satisfied by using light, rigid composite material (0.3 mm carbon fiber plate and foaming polystrol core) for the truss members, and an aluminum plate for the road deck floor. The other parts of the road deck and auxiliary facilities are made of a wood whose specific mass density is very small.

The structural and aerodynamic performance of the stiffening girder model was investigated before assembling it to the full model. The result showed the discrepancies of bending stiffness and torsional stiffness, unavoidable nonlinearity taking place in the higher range of design wind speed though, with required values were within 10%, and those of mass, polar moment of inertia and center of gravity were within 2%. The sectional model tests, conducted with/without the V-shaped springs, showed only a minor effect on the aerodynamic properties of the cross section.

Modeling of the Main Tower:

The stiffness of the tower has a significant effect on the entire behavior of the bridge. This modeling was carefully studied, and the tower model shown in Fig. 5 was finally selected. The model skeleton was consisted of a frame of steel bars on which a wood replica was mounted to simulate the geometric shape.

The elastic modeling was based on how to select the bending rigidity appropriately for the skeleton members, as a result of which the longitudinal rigidity of the members was significantly greater than the simulation requirement. The diagonal members were disconnected at the intersection portions in order to reduce the unfavorable contribution to the tower stiffness by those rigidity.

With the tower in free standing condition, the static deformation and natural vibration properties of the model showed a good simulation of the prototype. However, a FEM structural analysis for the entire bridge model revealed that the non-simulated longitudinal rigidity of the members, especially that of tower post members, caused a little difference in the torsional displacement and natural frequency from those calculated by using the actual bridge dimensions.

Modeling of the Main Cable:

In modeling the cable, simulation of mass, longitudinal rigidity and aerostatic drag was required at the same time. Since there was no material that satisfied such a requirement, a separate weight model shown in Fig. 6 was chosen. To obtain the proper longitudinal rigidity as well as sufficient rupture strength, stainless steel woven wire (1.2 mm Ø, 1×19) was used.

The weight, configuration and its mounting rate were determined from the mass and aerostatic drag similarity. Regarding drag simulation, the effect of Reynolds’ number and three-dimensional flow at the edge of the cylindrical weight was investigated by the wind tunnel test on the cable model.

Results of the test indicated that the model had the proper overall drag coefficient effect. It was found out in the sectional wind tunnel tests that the aerodynamic properties were affected by the existence of the cables, and this influence could not be ignored unless the distance between the cables and the girder became nearly the same to the girder height.

Considering the aerodynamic interference effects on the entire behavior of the bridge, it was chosen to use a geometrically simulated, mass also simulated though, cable for the center portion of the main span.

After assembling each part and element to the full suspension bridge model, the properties of static deformation and natural vibration were examined and assured of its performance. As for the static deformation with vertical, lateral and torsional loadings acted respectively, analytical calculations were compared with measured data, which showed rather good agreement.

Natural frequency, vibration mode shape and structural damping are quite essential and important dynamic properties. On the bridge model, the various modes of vibration were excited initially by an electric exciter, and from subsequent decaying vibration, the above dynamic properties were measured.

Table 1 shows the natural frequency and damping of various modes. The frequencies of vertical and horizontal modes were generally found to agree rather well with those calculated on the prototype bridge. Only the frequency of the first symmetric torsional mode was slightly higher than that of the actual bridge.

With regard to structural damping, the logarithmic decrements of torsional modes were, in particular, slightly higher than those required in the code, although the decrements of vertical and horizontal modes were comparable with the required.

However, those discrepancies were judged not so great and then adjustable in the final stage of interpretation to the actual bridge on the targeted aerodynamic behaviors. Investigating these measured properties of the full model, the performance of the model was considered rather well.

Full Model Wind Tunnel Test Results:

Primary test results will be described to show how the full bridge model behaved in a smooth flow as well as in the prescribed turbulent flows. For the better understanding of basic characteristics of wind-induced deformation and flutter instability, first of all, the test started with a smooth flow.

The tests were conducted on a couple of types of cross section, paying attention to the effects of such stabilizing devices as installed under and/or on the road deck at its center. With the rise of wind speed, the truss girder was laterally deflected and twisted by wind loads.

Each component of static displacement is such as illustrated in Fig. 7. Because of large drag force on a truss stiffening girder, the horizontal displacement reached nearly the scale of road deck width at the midpoint of main span under the equivalent design wind speed.

Along with greater horizontal displacement, the large torsional one also took place at the same time as much as -4 degrees, windward-down or negative to the wind action at the midpoint, accompanied with vertical displacement lifted. It can be seen that such large deformation was primarily caused by large drag force on the truss girder.

The drag directly made the horizontal displacement to take place. As for the torsional displacement, a kind of coupled moment effect, caused by a combination of drag force acting at the center of the truss girder and hanger reaction force at the top of it which acted in the opposite direction, played a primary role.

For analytical prediction of those displacements, it was confirmed that a three-dimensional frame model method of load increment, considering the structural geometric nonlinearity due to the deformation as well as the variation of aerostatic forces on the girder twisted along the bridge axis, was necessary to get a good agreement with the test results.

The twisted girder means the occurrence of additional relative angle of the girder to uniform wind action which really varied along the bridge axis. In fact, then, negative torsional displacement might result in the raise of onset wind speed of flutter instability, because the specified cross section of the truss girder came up with more stable aerodynamic characteristics at the negative angle of attack of flow incidence, looking at the results of sectional model wind tunnel tests carried out in advance.

A coupled flutter instability occurred at higher wind speed in the full model test, although it was somewhat different from that estimated from the sectional model test. It can be seen from Fig. 8 that the observed coupled flutter had a particular mode shape of vibration with predominant torsional and vertical motions slightly accompanied by horizontal one.

Fig. 9. shows a time history record at the time of flutter onset at the midpoint of main span, from which it can be understood that there were pretty large phase differences between each component of motion. Looking at the change of the center of rotation along the bridge axis, assumed at a point during one period, it can be also found out that the flutter mode shape was quite a complicated, three-dimensional one.

The full model tests were conducted on a couple of cross sections:

(a) The basic section of a truss stiffening girder originally designed, and

(b) The improved sections with stabilizing devices against flutter.

The angles of flow incidence were implemented in a horizontal wind as well as a blow-up one of about 2.7 deg. at the girder height, because flutter instability was found quite susceptible to the angle of attack of air flow.

Fig. 10 shows the onset wind speeds of flutter instability in the basic section and the improved one with stabilizing devices installed on entire spans, including those of sectional models as well. As for the angles of attack in the full model, a range of varied relative angle over the entire spans is indicated because the flutter at specified wind speed was accompanied by large torsional displacement caused by wind loads.

It can be found from sectional model tests that:

1) The basic section was unstable in flutter, the onset wind speed of which was lower than the code requirement, although it was getting more stable with angle of attack reduced to the negative; and

2) The improved section was satisfied with the code requirement by installing such stabilizing devices as shown in Fig. 3, the onset wind speed of which was also slightly increasing in the negative angle of attack.

Looking at onset wind speeds in the full model tests, not to speak of the improved section, those of the basic section were also higher than required in the code, although actually comparable with the prescribed after correcting some avoidable inconsistency in the test similarity of full model such as a slightly higher realization of the lowest torsional natural frequency.

There was no difference in the basic section practically seen, while a reversal in the improved one, between two sets of angle of attack in 0 and 2.7 deg. Such complicated increase of onset wind speed in the full model was obviously caused by a combined result of higher occurrence in the negative angle of attack, characteristic of this truss-stiffened bridge girder, and large negative torsional displacement found in the mid-span at higher wind speed as mentioned above. This means that the study on aerodynamic instability should be carried out with consideration of static wind-induced displacement at the same time.

For the better understanding of basic characteristics of the coupled flutter instability observed in the full model, of course in the sectional model as well, the change of aerodynamic damping and frequency with the rise of wind speed was measured at a specified amplitude from the damped vibration after excitation in case of positive damping, while from the divergent vibration developed from the rest in case of negative damping, at specified wind speeds.

Fig. 11 (a) presents one of those measurements for the basic section in a smooth flow set at 0 deg. angle. The aerodynamic damping in logarithmic decrement can be traced in a slightly varied curve with the increase of wind speed, and dropping abruptly toward the negative in higher wind speed than about 7 m/s. The wind speed of 8.4 m/s (84 m/s for the actual bridge), in this case, is a critical one of flutter onset where the damping changes from positive to negative.

The flutter occurred at the wind speed over 8.4 m/s was, in fact, of a coupled type showing a particular mode shape. The torsional component was practically the same in shape as the first symmetric natural mode at no wind speed, while the vertical component was so complicated in shape that it was not similar to any of vertical natural modes.

Actually, those components of flutter mode shape should be also influenced by the specified displacement due to wind loads. As for the change of frequency specified in a particular mode branch of the first symmetric torsion, as indicated in Fig. 11 (b), natural frequency at no wind speed was reduced with the rise of wind speed and changed to about 90% of it at the flutter onset. This is typically because of stiffness effects of aerodynamic unsteady forces on this specified truss girder.

Following the smooth flow test, a couple of tests were carried out in the prescribed boundary layer turbulent flows. As expected, quite three- dimensional random oscillations took place, sometimes showing snake-like motions of a kind of wave propagation on the girder under statically deflected and twisted displacements caused by mean wind loads as in a smooth flow.

Fig. 12 presents typical buffeting torsional responses measured at the midpoint of main span at two wind speeds, in which the random fluctuation of amplitude can be seen around the mean displacement.

Taking the statistical properties averaged over 1 min. (10 min. for the actual bridge) measured at each wind speed, those of mean, root mean square, peak(maximum and minimum) values are scattered as plotted in Fig. 13.

With the increase of wind speed, the scattemess of peak and root mean square becomes larger, resulting to make us recognize the possibility of flutter onset in the turbulent flow at about 8 m/s and much higher mean wind speed, quite close to that in a smooth flow. Significant random responses were similarly observed in vertical and horizontal motions as well.

Actually, in comparison among a couple of gust effects observed in the tests, the horizontal buffeting response was pretty smaller than expected by conventional spectral analysis for individual motion, while the vertical and torsional responses were rather larger. As for one of reasons why the measured responses were different from the calculated on the basis of conventional spectral analysis, it may be likely caused by their complicated, three-dimensional fashion.

Three-Dimensional Flutter Instability Analysis:

It is well known that a calculation of critical wind speed for heaving bending- torsion coupled flutter has been often made as a kind of effective index to identify wind-resistibility of long-span bridges, as shown in Fig. 1, on assumption of flat plate unsteady aerodynamic lift and pitching moment forces.

In this calculation, a combination of one heaving-bending and one torsional natural modes, usually the lowest ones in similar mode shape, has been used-to assume the flutter mode shape. This fact may be originated in conventional approach to the flutter problem of aircraft wings by means of a rigid sectional model of two-degrees-freedom of heaving and rotation, and then allow the use of sectional model mounted two-dimensionally in the tests on the bridge studies as well.

However, looking at a complicated, three-dimensional mode shape of coupled flutter instability found in the present full model test, it can be seen that a single combination of the lowest two modes is not necessarily correct to describe the actual behavior of a full model, and also only a sectional model wind tunnel test of similar single combination is not good enough to verify the wind resistibility of a specified bridge.

In order to analyze such a complicated, three-dimensional flutter instability on the deflected and twisted truss girder as observed actually in the full model test, a new method was investigated by the use of an equation motion formulated by applying the unsteady aerodynamic forces, or derivatives, to a three-dimensional structural frame model, as illustrated in Fig. 14, of the entire bridge structure deformed under specified wind loads.

The equation of motion is described as follows:

Where {u}: Deformed displacement vector = [u_s ф_s v_s u_c v_c v_h]^T,

[M]: Deformed mass matrix,

[C]: Deformed structural damping matrix,

[K]: Deformed stiffness matrix,

and [F]: Unsteady aerodynamic force matrices which are defined as functions of the reduced frequency k=cub/U, measured experimentally in sectional model wind tunnel tests under specified harmonic motions on particular configurations of the girder at specified angles of attack and with or without the simulated cables equipped.

Those unsteady aerodynamic forces, or derivatives, proportional to displacement, velocity and acceleration of motion, the last is usually neglected though, are written in the form:

where L,M and D are lift, pitching moment and drag forces respectively, u, ф, and v are vertical, torsional and horizontal displacements respectively and the subscripts s, c and h mean stiffening girder, main cable and hanger cable respectively. The reduced frequency k= ωb/U is defined in a harmonic motion of u =Xexp(iωt), where co is the circular frequency, b half width of the girder and U wind speed.

Making the complex eigenvalue analysis for Eq. (2) with a specified reduced frequency k, a set of complex frequencies ω= ω_R + iωi and complex mode vectors X=X_R+iXi are directly obtained as much as the degrees of freedom of the three- dimensional structural frame model, from which the aerodynamic damping decrement can be derived in the form δ_u = ω₁/√ω_R²+ ω₁² at a specified wind speed U= ω_Rb/k, along with the complex mode shape.

Carrying out similar calculation by changing the reduced frequency over an appropriate range concerned, the change of aerodynamic damping and frequency with the rise of wind speed is to be described in the form of such root traces as shown in Fig. 15, connected with adjacent points of similar mode shape at different wind speeds.

Among a lot of significant branches of those root traces, some branches illustrate a particular change of aerodynamic damping from the positive to the negative sign where the zero crossing means a critical wind speed of flutter instability in the specified mode shape. The branch to give the lowest critical wind speed must be of the present concern.

Along with the direct analysis on Eq. (2), an approximate method is available by assuming its flutter mode shape by means of a combination of natural mode shapes at the deflected stage in specified wind speed or the non-deflected stage of zero wind speed. It can be seen that the combined number of natural mode shapes may be appropriately limited to make an easier analysis in case of measured data of flutter instability in hand as a target, while not so simply in general case without a target in the beginning of calculation.

Meanwhile, applying a simplified method to analyze the present flutter behavior observed in the full model wind tunnel test, it was expected for the first time that a good agreement might be seen between measured and calculated aerodynamic damping changes, on the past understanding of coupled flutter where the vibration was to be governed by coupled unsteady lift and pitching moment due to vertical and torsional motions of the stiffening girder, and also by some additional unsteady forces such as drag due to horizontal motion of the girder as well as drag and lift due to horizontal and vertical motions of the main cables, all boxed in Eq. (3).

In fact, as shown in Fig. 11 (a), the agreement was pretty good in lower wind speed range up to about 7 m/s. However, the larger changes in higher wind speed range beyond it could not be successfully followed by this idea. Further investigations have been done to find out a key factor to present the abrupt drop of aerodynamic damping to the negative in the higher wind speed, resulting in a discovery of significant contribution of unsteady drag forces due to torsional and vertical motions of the girder, indicated by dotted and solid lines in Eq. (3).

That is, looking at a rather large change of averaged static drag with the increase of negative angle of attack, it was recognized that the effect of unsteady drag due to change of torsional displacement was not so small, particularly on the windward-down twisted stiffening girder around the center of main span in higher wind speed as mentioned in the preceding.

In reality, applying those additional terms of unsteady force, the targeted abrupt drop of aerodynamic damping could be realized in the calculation as attached also in Fig. 11 (a), still certain difference found from the measured though. As for the change of frequency with the rise of wind speed, there happened also better agreement with the measured as shown in Fig. 11 (b).

Such a complicated behavior of flutter instability might be caused by the truss stiffening girder itself in this case, because the drag force on it was essentially large. It is concluded that the effect of drag on coupled flutter was one of new findings from the full model wind tunnel tests.

Analysis of Gust Response Behaved Three- Dimensionally:

The random gust responses caused by turbulence of natural wind take place at any time on any structural system. Analytical description of those gust responses is normally done by conventional spectral approach in the frequency domain, making use of appropriately determined power spectrum, correlation functions of turbulence and aerodynamic admittances along with the aerodynamic damping defined by the quasi-steady assumption, particularly only for a single-dimensional motion concerned.

Those gust responses observed in the full model wind tunnel tests were actually taking place quite three- dimensionally, and pretty smaller in horizontal motion, while rather larger in vertical and torsional motions than calculated by conventional spectral approach although a couple of factors associated with those responses were separately investigated, for instance, by the use of directly measured data on the site.

In order to reflect such three-dimensional responses as observed in the full model tests, a trial of time domain approach may be useful to analyze the entire random response at the same time, although quite complicated and troublesome in actual treatment.

The most simplified approach in the time domain normally starts to formulate appropriately following two basic variables; instantaneous relative wind speed as well as effective angle of attack which combines the wind speed fluctuations and the structural motions behaved three- dimensionally. On the basis of quasi-steady assumption, the aerodynamic forces can be expressed in the global directions for the entire structure at a particular instant time as shown in Fig. 16.

Where the aerodynamic drag, lift and pitching moment in the relative wind axes are given by:

The instantaneous relative wind speed, using velocities ż, ẏ, v and θ of horizontal, vertical and torsional motions of the girder along with a kind of distance mp assumed to describe the vertical effect of the last torsional motion θ(t).

In this simplified quasi-steady model of the forces concerned, by fitting those averaged static coefficients C_D, C_L and C_M with the polynomials of effective angle of attack and also simulating time series of wind fluctuations by an appropriate stochastic method, ARMA model, the three-dimensional random responses come to be available in the time integration.

Actually, in order to carry out a calculation, a measure of distance ml associated with torsional motion must be assumed appropriately, say to be equal to B/4, that is, taking account of the wind loads on the girder concentrated at the quarter chord point, although this is quite a gross assumption which may not realistically represent the location of loading of wind forces and may lead to an overestimate of the aerodynamic damping since there are indications that the value of must be frequency-dependent and vary with the entire bridge, because of those unsteady aerodynamic forces, or derivatives, really coupled as seen in Eq. (3) in three-dimensional fashion.

Fig. 17 presents one of comparisons among a couple of gust response analyses with the measured data in the full model wind tunnel tests mentioned in the preceding. It can be seen that the calculated results by present simplified approach in the time domain are much closer to the measured than the calculated by conventional spectral analysis for a single-dimensional motion concerned. In reality, this may be a result of introducing the effects of three-dimensional instantaneous coupled motion more inclusively into the evaluation of mode mass as well as mode damping.

A numerical approach to perform the three-dimensional response of a truss- stiffened suspension bridge under gusty wind seems quite effective, which focused on the time domain to look closely into the instantaneous coupled response considering the time-space variation in wind speeds and aerodynamic forces.

Actually, however, difficulties were encountered in expressing the wind-force- response relationship in the time domain so that much simplification had to be made. Therefore, further efforts to advance in the numerical prediction of gust response are needed, particularly in more realistic description of instantaneous aerodynamic forces, for instance, by the use of indicial functions to consider the effect of time delay between wind fluctuation, structural motion and corresponding aerodynamic force.

Such indicial functions in the time domain are to correspond to self-excited unsteady aerodynamic forces, or derivatives, caused by coupled structural motions as well as to aerodynamic admittances used to bring wind fluctuations to random exciting forces, which are all defined in the form of frequency-dependence.

Topographical Effects on Entire Bridge Responses:

The description of wind is normally dependent on site terrain and location of the structure concerned. As for the structure stretching over a long distance, the wind structure at one point may be different from that at another point because of surrounding topography. Those influences of topography on the responses of long-span bridges have been indicated on several occasions.

Some of general conclusions, derived from topographical wind tunnel testings as well as field observations of natural wind, are as follows:

(a) Decreases in the strength of mean wind are frequently accompanied by increase in the level of turbulence, which can increase the amplitude of gust response but decrease the static mean response and suppress the vortex excitation.

(b) By screening the wind from directions of a prevailing storm, the risk of severe winds or the safety of specified objects can be significantly altered.

(c) Investigations of site meteorology through the measurements of full scale wind conditions have proven quite useful in the calibration of topographical wind tunnel model studies.

The sitting of a very long-span cable-stayed bridge, the Tatara Bridge, was recognized to be relative to significant topographic features because it was to be constructed spanning over a strait between an island and another, both standing in excess of 400 m above sea level.

With the wind of the direction in which the bridge was hidden behind the islands, for instance, the flow might probably make a sharp turn in the direction of bridge axis, resulting in the possibility of deflected wind loads or increased gust responses on the bridge posing more seriously under the construction stage that was still much unstable structurally as well.

To make a rough survey of wind pattern prevailing at the bridge site, a 1/2000-scale relief model wind tunnel tests, representing the topography of a region within the radius of 6 km, were conducted in advance, with due reference to vertical distribution of wind speeds and turbulence intensities measured at observation towers on the shore.

On evaluating the aerodynamic stability of this very long-span cable-stayed bridge, it involves the consideration of structural and aerodynamic problems, the former being the problem of the stay cables and deck or pylons vibrating in mutual interference and the latter that of particular site wind pattern resulted from the local topography.

In addition, the wind resistance is warranted not only after completion but also during erection. Actually, however, it was very difficult to devise a workable wind tunnel test plan capable of satisfying all the requirements at the same time to build a scale bridge as well as topography models.

In other words, while producing the stay cable structural properties (weight distribution or tension) precisely requires preferably a largest possible model over 1/70 in scale, the scale of boundary layer turbulent flow to be generated in the large wind tunnel, mentioned in the preceding, requires the model to measure 1/200 and under in scale.

Therefore, in this case, it has been judged that those contradictory requirements should be met by carrying out two different kinds of wind tunnel test, one using a large bridge model with an emphasis on precise simulation of structural properties and the other using a small model with an emphasis on necessary production of surrounding wind pattern influenced by topography. Between both results taken in two models, there must be careful investigations and arguments to be interpreted.

Concluding Remarks:

Focusing on how to approach to full model wind tunnel testings and how to get better findings from those tests, some of governing factors interactions and influenced behaviors of wind effects on very long-span bridges are illustrated and discussed with particular reference to the Akashi Kaikyo Bridge and partly to the Tatara Bridge, for which construction is well under way.

Recent other wind tunnel testings by means of full model of very long-span bridges can be referred to in a couple of cases such as the Great Belt East Bridge, the Messina Straits Bridge and the Normandie Bridge. It goes without saying that individual circumstances should have been taken into account in carrying out those full model wind tunnel testings in the sense of the choice of targeted behaviors under wind, the similarity of entire structural systems concerned and their dimensions, the reliability of test results and their analytical explanations, and so on.

In process of practical experiments on the wind-induced behaviors of bridge structures, generally speaking, the sectional model wind tunnel tests are still effective in verifying and measuring the single-dimensional vibrations and aerodynamic forces. Particularly, it is quite so in case of finding out desirable configurations of bridge deck proposals.

As described in the preceding, however, the full model wind tunnel tests are quite essential in those cases of possibility of such as complicated three-dimensional behaviors, interacted between associated factors and also under turbulent flows.

In this sense, it is an important key point to prospect their possibility in advance by appropriate means. Further efforts are needed to rebuild more effective approach for this purpose by expanding the ideas of conventional sectional model wind tunnel testings. Of course, a detailed full model testing must be conducted appropriately for a kind of calibration.

Nowadays, a tool of numerical computation is quite powerful in analytical explanation of wind tunnel test results. Even so, the computer model actually realized is only a model assumed in certain definition and appropriate size, which often results in a cause of some inconsistency with the full model in wind tunnel, although the full model also has its own defect and limitation in modeling.

In the argument of a triangular relationship of those three models, here including the prototype structure as a kind of conceptual model because it is not existing in reality at the stage of investigation before its completion, it is quite natural to keep a careful insight on the comparison and interpretation among three in order to minimize a likely inconsistency taking place essentially.