Engineering Structures 24 (2002) 1085–1090www.elsevier.com/locate/engstruct

Dynamic characteristics of cantilever grandstand roofs

C.W. Letchforda,∗, R.O. Denoonb, G. Johnsonc, A. Mallam c

a Department of Civil Engineering, Wind Science and Engineering Center, Texas Tech. University, P.O. Box 4089-1023, Lubbock, TX 79409,USA

b Ove Arup and Partners Hong Kong Ltd, Hong Kong, People’s Republic of Chinac Department of Civil Engineering, University of Queensland, Brisbane, Australia

Received 18 April 2001; received in revised form 18 February 2002; accepted 27 February 2002

Abstract

Full-scale dynamic characteristics of four grandstand roofs are reported. First mode natural frequencies, mode shape, and totaldamping values were obtained from forced excitation. Comparisons with finite element models of the structural response are made.Design recommendations for damping are given. 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

Prediction of the dynamic response of structures isbecoming a familiar task for structural engineers andnowhere is this more common than in flexible structuressubjected to wind loading. Whereas the calculation ofnatural frequency and mode shape, for even complexstructures, can be accomplished somewhat routinely withfinite element packages, energy dissipation throughdamping remains a significant parameter eluding com-prehensive theoretical treatment. Indeed, Jeary and Ellis[1] suggest that an error band of±100% be used withdamping values employed at the design stage formany structures.

Grandstand roofs form a unique group of structuresfrom a wind loading perspective. Their extensive andwinglike forms often cause them to be wind- sensitiverequiring extensive wind tunnel studies to predict theirresponse [2–7]. The earlier studies [2–4] employed aero-elastic models that attempted to simulate the dynamiccharacteristics, including damping, of full-scale grand-stands. The latter [5–7] report on comprehensive windtunnel studies that employed rigid pressure tapped mod-els. While several references [8–10] give information ondamping for tall flexible structures, there appears to belittle or no information on the dynamic characteristics ordamping in particular, of grandstand roofs.

∗ Corresponding author.E-mail address: chris.letchford@coe.ttu.edu (C.W. Letchford).

0141-0296/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (02)00035-4

The purpose of this research project was to measurethe dynamic characteristics of a number of large spancantilever grandstand roofs and compare the results withfinite element models of the same. These experimentaldata would also provide information on the expecteddamping in grandstand roofs. In the following section,a brief review of damping measurements is presented.Section 3 describes the experimental procedure. Section4 presents the results of this study and makes compari-sons with finite element model predictions. The con-clusions and design recommendations arising from thisstudy are summarised in Section 5.

2. Damping in structures

It is well known that damping is the most uncertainparameter in the prediction of the dynamic response ofstructures. There are two principal reasons for this: thevariety of mechanisms contributing to damping, and thepaucity of reliable field data on damping characteristicsof structures. The sources of damping of structures arevaried, and not well understood. Among the major con-tributions to structural damping are material damping,frictional damping and soil-structure interaction.Material damping dissipates energy by micro-crackingwithin the structural matrix of the material. Frictionaldamping refers to damping caused by relative movementof structural components, such as movement withinbolted connections, or relative motion between structuralframeworks and internal partitions in buildings. Damp-

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ing from soil-structure interaction depends on both thefoundation type and the underlying soil characteristics.

In addition to structural damping, some structuresexhibit significant aerodynamic damping. Large positiveaerodynamic damping may be exhibited by structuressuch as large roofs, where large volumes of air are dis-placed by the structures’ motion. Aerodynamic dampingmay become significantly negative for other structuresat various wind speeds, leading to aeroelastic insta-bilities. Both structural and aerodynamic damping mayexhibit significant amplitude dependence.

There are two main techniques for determining damp-ing in full-scale structures. These are ambient or forcedexcitation. In the former case, ambient conditions areused to allow the structure to oscillate, usually as a resultof wind excitation, while the latter requires active exci-tation through some form of mechanical forcinginducing a resonant response. Littler [11] describes thesetwo methods in some detail.

There are significant difficulties in predicting dampingfrom random data using the two most commonly appliedtechniques: autocorrelation decay and random decrementanalysis. Autocorrelation decay requires self-stationarityof the data as a preliminary requirement. This is oftenvery hard to achieve during wind-excited response. Thelonger the sample size, the less likely it is to achieveself-stationarity. However, with shorter sample sizes,methods of assessing self-stationarity do themselvesbecome unreliable [12]. The random decrement methoddescribed by Jeary [13], but originally proposed by Cole[14], sought to overcome the requirements of large self-stationary samples by using many short samples and hasthe advantage of allowing determination of amplitudedependence. However, the number of samples requiredis large. Jeary [15] suggested 1000 as an optimum num-ber samples, while other researchers have suggestedminimum sample requirements between 400 and 2000[16,17]. It can be concluded from the work of Fallah[12] that in order to obtain a reliable estimate of dampingin a short period of time, the only available technique isforced excitation.

At a meeting on structural damping in Japan under theauspices of the International Wind Engineering Forum,a comprehensive discussion of structural damping wasreported [18]. There, sources of damping, measurementtechniques, modelling techniques and a plethora of full-scale measurements were given. Tamura, Suda andSasaki [19] report more recent full-scale measurementsof structural damping. Generally, these measurementspertain to the along-wind vibration of tall buildings, withsome reporting of measurements on long span bridgedecks. No work appears to have been undertaken ongrandstand roofs, where, although the crosswindresponse is akin to bridge decks, it is significantly affec-ted by the presence of under roof blockage.

3. Experimental arrangement

3.1. Full scale measurements

Four grandstand roofs in Brisbane, Australia wereselected for this study. The grandstand characteristics aresummarised in Table 1 and illustrated in Fig. 1. The twostraight cantilever roof designs date from the late 1970swhile the back stayed cantilevers date from the1990s.Truss members in each cantilever had welded connec-tions. For the backstays, the Suncorp Stadium roof hadpin connections, while the Ballymore East roof hadwelded connections.

The forced vibration technique was employed toobtain the dynamic characteristics of these grandstands.Four Allied Signal QA650 servo-accelerometers werelocated at various positions over the roof. Measurementswere obtained both along and across the tips of the canti-levers to obtain longitudinal and lateral dynamic charac-teristics of the roofs. Apart from Ballymore West, thecentre cantilever of each roof was excited. The acceler-ometers were connected to custom-built signal con-ditioning equipment (ASC650V2) capable of verticaloffset; signal amplification (1–900) and band pass fil-tering. A 7-channel Teac FM tape recorder and a Yoko-gawa oscillographic recorder then recorded the acceler-ometer outputs. The chart recorder allowed immediatedata interpretation while the tape recorder data was usedfor subsequent data analysis. A typical acceleration traceis shown in Fig. 2.

In the monitoring procedure, firstly an initial vibrationsignal was obtained, usually from ambient wind, fromwhich an estimate of the first mode natural frequencycould be made. Assisted by a metronome, five peoplewould then bounce in time at the estimated first modenatural frequency near the tip of the cantilever while theaccelerometer response monitored. It was quickly poss-ible to zero-in on the first mode natural frequency andexcite the cantilevers. The weight of the five exciterswas approximately 300 kg, much less than the weightsupported by each truss (Table 1) and therefore wouldnot change the dynamic characteristics of the structuralsystem under investigation. After about 20 cycles theforced excitation ceased and the vibration allowed todecay. The results were then interpreted for natural fre-quency, mode shape—from relative amplitudes, anddamping—from logarithmic decrement of the decayingaccelerometer trace. At the beginning and end of eachgrandstand test, the four accelerometers were co-locatedand excited so that a calibration signal could be obtained.The tests on the four grandstands roofs took place overtwo days in mild March weather in Brisbane. Windspeeds were low. Each test took about 3 h, includingset-up and removal. A description of the experimentalarrangement may be found in [20].

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Table 1Grandstand roof characteristics

Grandstand Roof type Cantilever span L (m) Cantilever spacing Roof length W (m) Truss massa (Tonnes)(m)

ANZ Stadium West Straight 37.8 9.6 124.8 22.5Ballymore West Straight 17.15 7.62 99.1 2.71Ballymore East Over-roof backstays 25.9 9.5 104.5 5.56–6.69b

Suncorp Stadium West Over-roof backstays 30 10 140 8.00

a Includes mass of supported roof.b Represent range of possible member wall thicknesses.

Fig. 1. Roof trusses for the four grandstands.

Fig. 2. Acceleration time history from Ballymore East grandstand,showing scaled maximum peak-to-peak tip displacement.

3.2. Finite element models

Engineering drawings were made available for threeof the four grandstands and the structural details of thefourth were obtained from site measurements. As shownin Table 1, two of the grandstand roofs were straightcantilever trusses, while the other two had over-roofbackstays supporting trusses. Fig. 1 illustrates the struc-tural form of each roof. Two finite element structural

analysis packages were employed to predict the dynamiccharacteristics of the grandstand roofs. These wereSPACE GASS 8.00d for Windows and MICROSTRAN5.0 for DOS. Initially the cantilevers were modelled as2D structures with purlin and roofing loads applied attruss nodes. Member connections were assumed to befixed as most joints were welded. 2D computer model-ling was undertaken prior to full-scale measurements.

4. Results and discussion

4.1. Natural frequencies

The measured first mode natural frequencies and thosepredicted from 2D finite element models using SPACEGASS and MICROSTRAN are compared in Table 2. Itis seen that apart from the Suncorp Stadium Grandstand,the simple 2D computer models predict the natural fre-quency quite well. The 16% overestimation by the mod-els for the Suncorp stadium, indicate a stiffer cantileverroof than that found in the field. This may be because thefixity assumed at the supports is less than that actuallyachieved, as the 2D computer model did not include theconcrete supporting structure, only the steelwork. How-ever, a similar situation exists for the other stayed roof,Ballymore East, and there the computer models predictthe natural frequency well.

Table 2Comparison of measured and predicted first mode natural frequencies

Grandstand Natural frequency (Hz)

Measured Predicted 2D Predicted 2DSPACE GASS MICROSTRAN

ANZ Stadium West 1.89 1.81 1.81Ballymore West 1.93 1.99 2.02Ballymore East 1.98 1.99 2.01Suncorp Stadium 1.75 2.03 2.02West

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Fig. 3. Measured and predicted first mode shape for straight cantil-ever grandstand roofs.

4.2. Mode shapes

Mode shapes for first mode oscillation were obtainedby comparing the amplitude of responses of the acceler-ometers along each cantilever. This was done over fiveseparate trials of forced oscillation. Fig. 3 compares themeasured and predicted mode shapes for the straightcantilever roofs and Fig. 4 shows the same for the stayedroofs. The mode shapes are expressed as ratio of acceler-ation (z-displacement) to that at the tip of the cantilever(Z). The standard deviation in mode shape (z/Z) rangedfrom 0.01 to 0.08 and ±1 standard deviation error barsare shown on the figures. It is seen that the measuredmode shapes are generally within experimental error ofthe predicted values, although the measurements indicategreater deviation for the stayed cantilevers than thestraight cantilevers.

The mean measured mode shape along each cantileverroof was fitted to a power law of the form

�zZ� � �x

L�n

where z is the vertical acceleration (displacement) nor-malised by the tip acceleration (displacement), Z, and xthe distance along the cantilever truss normalised by thecantilever span L. The power law indices (n) were all

Fig. 4. Measured and predicted first mode shape for stayed cantilevergrandstand roofs.

Fig. 5. Mode shape of cantilever leading edge excited at the firstnatural frequency of the individual cantilevers.

close to 1.5, which approximates the theoretical modeshape for uniform cantilevers.

By re-positioning the accelerometers along the leadingedge and exciting the central cantilever as before, the3D mode shape for the first mode cantilever oscillationwas obtained. This experiment was conducted on thethree larger roofs; ANZ, Ballymore East and Suncorp.Fig. 5 shows the measured leading edge mode shapes.The lateral dimension has been normalised by the widthof the roof, W. Also shown in the figure is the fourthmode shape predicted by a 3D SPACE GASS model ofthe Ballymore East roof, the only roof studied numeri-cally in 3D. This mode had a natural frequency of 2.38Hz. The lower mode shapes and frequencies of the 3DSPACE GASS model for this roof are detailed in Table3. The extension to 3D analysis and the consequentincrease in number of members did not reproduce thesimple cantilever mode frequency (1.99 Hz) obtained inthe 2D analysis. Indeed, the lowest frequency was a hori-zontal oscillation in the y-direction. It is also apparentthat the three grandstand roofs excited here at the lowestvertical natural frequency developed full wavelengthoscillations along the roof leading edge, rather than the

Table 3Predicted 3D mode shapes and frequencies for Ballymore East usingSPACE GASS

Mode Natural Mode shapefrequency(Hz)

1 1.88 Horizontal in plan2 2.06 Symmetric vertical half wavelength along

leading edge3 2.16 Anti-symmetric vertical half wavelength along

leading edge4 2.38 Symmetric vertical full wavelength along

leading edge-see Fig. 5

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Fig. 6. Damping as a function of amplitude for straight cantileverroofs.

half wavelength of the lower modes two and three pre-dicted by the numerical model.

4.3. Damping

Total damping (mechanical + aerodynamic) in thegrandstand roofs was obtained by estimating the logar-ithmic decrement over decaying oscillations. To deter-mine whether there was any amplitude dependence inthe damping estimates, the amplitudes at each end ofthe log decrement portions were averaged and then non-dimensionalised by the span of the respective cantilever.Fig. 6 shows the results of damping (as a % of critical)as a function of the normalised amplitude of oscillationfor the straight cantilevers, while Fig. 7 shows the samefor the stayed cantilevers.

The straight cantilever roofs have higher total damp-ing (~2% of critical) than the stayed roofs (~1%) whileapart from the longest and heaviest cantilever (ANZStadium), there appears to be little amplitude depen-dence for damping in the range of motions studied here.It might be expected that the aerodynamic dampingwould be similar for these structures, since their overallsize, natural frequencies and mode shapes are similar.Increased structural damping is expected to account forthe higher total damping in the ANZ stadium roof, as itis fully clad on both its upper and lower chords. The

Fig. 7. Damping as a function of amplitude for stayed cantileverroofs.

other grandstand roofs only have roof-decking spaningover purlins, themselves supported off the cantilevertrusses. The accelerations (and displacements) generatedhere were barely at serviceability levels, but the totaldamping measured was above that recommended forstructural damping in steel structures at serviceabilityloads, (0.5–1%) [9]. Values of total damping in latticesteel towers obtained by the same method [8] rangedfrom 0.5–1%. The additional damping here is likelyassociated with radiant damping, as the large planar roofsurfaces, surrounded by air, offer the opportunity forenhanced damping due to the motion of the roof. Underwind conditions leading to vertical movement of theroof, it could be anticipated that there would be signifi-cant damping associated with this motion, which may beconsidered as aerodynamic damping. In addition, there isunlikely to be any aerodynamic instability arising atsmall amplitudes as the net uplift on grandstand roofswas found to be almost independent of roof pitch in therange ±7° from horizontal [5].

5. Conclusions

The dynamic characteristics of four large cantilevergrandstand roofs were obtained from forced vibrationfield studies. All cantilevers had welded steel trusses,with two roofs acting as straight cantilevers while theother two had over-roof backstays.

Natural frequency, 2D and 3D mode shapes for thefirst mode oscillation and total damping estimates wereobtained. These were compared with computer predic-tions. It was found that natural frequencies and 2D modeshapes were reasonably well predicted, although this wasless so for cantilever roofs with backstays, and this mayhave been associated with only modelling the steel canti-levers and not the supporting concrete structure. A 3Dmodel of one roof produced a similar 3D mode shapeto that observed during the forced vibration study, how-ever the prediction was for the fourth mode at a highernatural frequency to that observed.

Welded steel trusses themselves offer little structuraldamping, but the extensive roofing sheets and enhancedaerodynamic damping raised the fraction of criticaldamping at small amplitude motion to above 1% and upto 2.5% for two of the roofs. The straight cantileverroofs, although possessing similar natural frequenciesand mode shapes to the back-stayed roofs, had signifi-cantly higher damping.

Acknowledgements

Engineering drawings of the grandstands were kindlyprovided by the Brisbane City Council, Connell Wagnerand Alexander, Browne and Cambridge. The permission

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of the grandstand owners to undertake this research andthe assistance of Chris Mans, Martin Sommerville, Dani-elle Ferrari and Anna Stewart in exciting the roofs isgratefully acknowledged.

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