Abstract
This paper considers the crowd loads encountered on cantilever grandstands and estimates these loads from response measurements. It considers two forms of loading: rhythmic loading which is usually encountered at pop concerts and non-rhythmic loading, usually encountered at football matches. The responses of several grandstands were monitored during pop concerts and the songs that produced the peak responses were identified. For two cantilever tiers at one stadium, the peak responses to four songs were considered. The measurements were processed to determine the accelerations corresponding to the first four Fourier coefficients (FCs) of the rhythmic load. These accelerations were used to determine the FCs of the load. The load model was then used for calculations on two other grandstands. The results were compared with measurements and a reasonable correlation obtained. For non-rhythmic loading the peak responses measured on three grandstands during seven football matches were considered. To represent a crowd jumping to its feet in response to a goal, a simple load model based on the instantaneous removal of the weight of the crowd was considered. Although the calculated displacements were reasonable, the peak accelerations were significantly underestimated on two grandstands.
Over the last decade the Building Research Establishment (BRE) has undertaken a number of investigations on cantilever grandstands. The initial report of this work examined structural characterisation, the peak accelerations measured at concerts and football matches and the changes in structural characteristics due to human–structure interaction.1 To focus upon two further items of interest, the experimental data from this work have been re-examined in the light of subsequent research on allied topics. This paper considers the estimation of crowd loads and a companion paper looks at serviceability evaluation.2
The experimental data used in this paper are primarily the vertical accelerations recorded at the front of the cantilever at the middle of the grandstand. The data were recorded for the complete event, either a concert or a football match. The grandstands are identified using the system adopted by Littler.1
The relevant guidance available to engineers in the UK concerning the dynamic response of grandstands has been issued by a joint Institution of Structural Engineers/Department for Transport, Local Government and the Regions/Department for Culture, Media and Sport working group,3 and is termed ‘interim’ guidance. The guidance considers the principal frequency of a new grandstand seating deck, suggesting that vertical frequencies should be greater than 3·5 Hz for an empty grandstand for normal, non-rhythmic loading, or greater than 6 Hz if pop concerts are to be held. For existing stands, a frequency between 3·0 and 3·5 Hz may be deemed satisfactory for sports viewing on the basis of past experience. It is recognised that this interim guidance is not the final solution, because some structures that do not meet these frequency criteria may perform perfectly well in service. The desired development is to provide a means of calculating a response so that each structure can be considered individually; but this requires a load model. Unfortunately there is a dearth of information concerning the loads produced by crowds on inclined decks and hence there is an urgent need for measured data. Unfortunately this is far from being a simple task.
In this paper an attempt is made to determine loads from the measured response of a number of grandstands. This type of evaluation has never been undertaken before, although a controlled evaluation of jumping loads on floors has been conducted at BRE.4 The first step in the process is to identify the type(s) of loading which generate the largest responses for pop concerts and sports events, and these are considered later under the headings ‘Rhythmic loading’ and ‘Non-rhythmic loading’.
For rhythmic loading the 6 Hz recommendation mentioned earlier arises from the concept that only the first two Fourier terms of the loads for rhythmic actions are significant for inclined seating decks. BS 63995 gives a limiting frequency of 8·4 Hz and this is concerned with the first three Fourier terms of rhythmic loads on structures like dance floors, but allows the alternative of calculating a structural response. It is therefore important to consider whether the third or higher Fourier components of the load are significant for inclined seating decks and this will be examined using the measurements.
Unfortunately the determination of loads from the measured response of a structure is not straightforward. In an experiment that was designed to measure loads induced by crowds jumping on floors,4 the following factors were considered to be important.
The test floor was selected on the basis that its structural characteristics would be reasonably simple for use in the analysis (i.e. a fundamental mode with a relatively high frequency well separated from other modes).
The crowd size and actions were selected, controlled and repeated.
Both displacements and accelerations were monitored.
In contrast, for the grandstands that were monitored, the dynamic characteristics were less amenable to analysis, the crowd reactions were uncontrolled and only accelerations were measured. Hence the analysis of the data inevitably includes a number of assumptions and simplifications that have a significant effect on the results. These assumptions are unavoidable if a range of grandstands is to be examined. Nevertheless, the data measured on the grandstands do represent true crowd behaviour rather than an artificially choreographed scenario.
An important part of the process of analysing response data to determine loads is the characterisation of the structure. For the grandstands considered here, the dynamic characteristics (frequency, damping, stiffness and mode shape) were determined for each significant mode using forced vibration tests6 at the locations where the responses were recorded during the various events. However, different grandstands can exhibit significantly different characteristics and this needs to be recognised.
Cantilever grandstands are usually formed from a series of raker beams that support the seating deck. The deck is often precast concrete seating units, which provide the horizontal link between raker beams. The attachment of these precast units can be relatively flexible, often using locating pins on the raker beams, which allow the precast units to be located and levelled before the joint is sealed with flexible mastic. This leads to a family of modes of vibration throughout the grandstand with frequencies of a similar order, although there will also be many higher frequency modes. The mode shape for any one of the lower frequency modes usually exhibits the classical deformed shape of a cantilever along the line of the raker beam, but the mode shape along the length of the grandstand shows several nodes, often with adjacent raker beams moving anti-phase. This is a function of the method of construction of the grandstand. For load determination from this type of structure, locations where one well-defined mode was encountered were selected to simplify the analysis, and situations where many modes were encountered were used for subsequent verification.
An alternative method of constructing the seating deck is to provide connections that are fully fixed, or capable of transferring moments. This leads to a monolithic behaviour of the whole seating deck that is much easier to model and similar in nature to a uniform plate with equivalent support conditions. There will also be situations that exhibit characteristics between these two extremes.
For the initial analysis, data recorded on the two similar-sized tiers of one grandstand (stand A) during two concerts will be considered. The grandstand is 150 m long and 45 m high with three cantilever tiers. The upper and lower tiers have concrete raker beams supported at centres a little over 14 m apart but the tiers have different structural details. Both tiers were of the type where the seating units had flexible attachments to the raker beams. The upper and lower cantilever tiers both support ten rows of seats and each main section supports approximately 280 people. The middle tier, which is situated immediately in front of the hospitality boxes, is a steel structure supported at centres a little over 3·5 m apart and has two modes which are very close in frequency; hence this was not used in the analysis. The measurements on this stand are described in greater detail by Littler.7.
From the forced vibration tests the characteristics given in Table 1 were measured. The frequency and damping were obtained using two independent methods. The comparison of the results obtained from the two types of test therefore provides an important quality check.
|
| Best-fit curves from spectra | Decay | ||||
|---|---|---|---|---|---|
| Frequency: Hz | Damping: %crit. | Stiffness: MN/m | Frequency: Hz | Damping: %crit. | |
| Upper tier | 5·70 | 2·63 | 107·5 | 5·69 | 2·52 |
| Lower tier | 4·68 | 1·41 | 75·9 | 4·67 | 1·32 |
To determine loads from the measured responses, several assumptions are required to achieve the desired goal. However, it is first necessary to identify the data to be used in the analysis. As the largest responses dominate both safety and serviceability evaluations, these are the obvious situations on which to focus. The data recorded for each complete event were split into a number of contiguous records each containing 8192 data points. This is similar to the procedure used for serviceability evaluation,2 although for the load estimation considered herein the frequency-weighting filter is not applied.
For the upper tier the data for the first concert were split into 133 records. The largest recorded responses occur between records 109 and 119. These records cover some particularly popular songs where 90% of the audience were standing, clapping and swaying and there was perceptible motion of the tier. Figure 1 shows the acceleration time history for record 115, and this is a typical example of the crowd reacting reasonably energetically to a song. The frequency composition of this record was determined by calculating the autospectrum using a fast Fourier transform procedure. This procedure works efficiently using highly composite numbers, which is why 8192 (213) data points were selected for each record. The part of the autospectrum between 0 and 8 Hz is shown in Fig. 2 and three definite peaks can be identified at 2·2, 4·4 and 6·6 Hz. Song 25, which induced this response, is one of the best known songs by the artist. It has a beat frequency of approximately 2·2 Hz, and the peaks in the autospectrum occur at the beat frequency and two and three times that frequency, which is characteristic of rhythmic actions.
Fig. 1. Stand A record 115—unfiltered acceleration
Fig. 2. Stand A record 115—autospectrum
The autospectra from records 109 to 119 were then determined and are shown on a three-dimensional plot, which is termed a waterfall diagram (Fig. 3). The characteristic responses at the specific frequencies can be seen, but there is a change of frequency with different songs. For example record 109 was for song 23 with a beat frequency of 2·09 Hz. Records 111–113 are when song 24 was being played which has a beat frequency of 1·95 Hz, 114–116 were for song 25 and 117–119 were for the next song with a beat frequency of 2·27 Hz. This shows the nature of the loading and provides the key to analysing this type of data.
Fig. 3. Stand A—upper tier, first concert
Another waterfall diagram is shown in Fig. 4. This covers the same songs at the same concert, but was measured on the lower tier. The data were recorded using different equipment to that used on the upper tier, and adopted slightly different sampling frequencies and therefore record times and record numbers. It can be appreciated that the two waterfall diagrams show the peaks at the frequencies of the music (and whole number multiples thereof) but the magnitudes of the spectral response are quite different. This is due to dynamic amplification with a near resonant response of the lower tier when song 25 was being played (records 72–74); the second multiple of the beat frequency 4·46 Hz being close to the natural frequency of the structure 4·68 Hz.
Fig. 4. Stand A—lower tier, first concert
From the previous section, it is clear that for the rhythmic loading encountered at concerts the characteristic response is seen at the beat frequency of the music and whole number multiples thereof, and that this can result in a resonant response if the load frequency is close to the structure's resonance frequency. This is similar to the situation for jumping loads, but there are two major differences. First, for the concerts the largest responses were noted when the largest proportion of the crowd was reacting to the music, which was approximately 90% of the people dancing or moving; therefore these loads will be significantly lower than for a situation where everyone is jumping. Second, as contact is maintained between the structure and the people, and not everyone is moving, the interaction between the people and the structure is important, with a key item likely to be a significant increase in damping.8
The unknown parameters are the Fourier coefficients (FCs) and the phase lags.
The experimental assessment of jumping loads4 derived the variation in FCs with crowd size. The next few sections consider a similar derivation for the loads encountered at concerts; however there are a few significant differences for the reasons mentioned in the introduction.
For the data on stand A, four songs were selected which gave the largest responses at the two concerts. These were song 24 (beat frequency of 1·95 Hz) and song 25 (2·23 Hz) for the first concert, and song 17 (1·97 Hz) and song 20 for the second concert. Song 20 had a beat frequency of 2·82 Hz, which is quite a high frequency for dancing; hence the crowd motion was primarily at 1·41 Hz and this will be taken as the excitation frequency. For each of the songs there was a response close to resonance of one of the tiers for one of the Fourier components of the load, although the near resonance response of the particular component is not necessarily dominant. These were the second Fourier component of song 25 on the lower tier, the third component of songs 24 and 17 on the upper tier and the fourth component of song 20 on the upper tier. Figure 5 shows a waterfall diagram where resonance is excited by the fourth Fourier component of the load.
Fig. 5. Stand A—upper tier, first concert
Hence, for each of the eight records, that is two songs at two concerts recorded on two tiers, the average peak accelerations generated by each of the first four Fourier components of the loads were determined. These are given later along with the corresponding calculated values.
To estimate the FCs of the loads from the measurements a number of assumptions are required.
The acceleration can be estimated reasonably using the principal mode of vibration. This is a significant simplification although each of the grandstand tiers used in the evaluation had a well-defined principal mode of vibration.
For the generalised load each raker beam can be treated in isolation and therefore attracts a proportionate number of people. The generalised load is then the weight of people (in Newtons) multiplied by the mode shape factor (using the mode shape measured along the raker beam). This pragmatic approach allows the number of people to be defined and the general load to be determined. An analysis of a whole grandstand of this form of construction is conceivable (and would be appropriate if the stand behaved monolithically—see section 2) but there are many variables that complicate this approach; for example, the complex mode shapes and the phase variation of people reacting to music as a result of their distance from the musical source.9
The variation of FCs with crowd size follows the same relationships found for jumping loads measured on floors.4 This adopts the relationships determined for various group sizes jumping in controlled tests,4 primarily because this is another form of rhythmic loading and secondly because no other data are available and it is inevitable that lack of coordination within a crowd will effectively attenuate the loads.
From the forced vibration tests the modal stiffness and structural frequency are available and from the measured response the load frequency is known. Therefore the values of rn,p can be selected to match the measurements and ζ can also be estimated as it affects the near resonance situations far more than those away from resonance—that is, a subset of the data.
For each of the first four FCs of the loads there are eight measurements (see s. 3·4). Of these 32 measurements, four are for near-resonant excitation, one is for the second FC, two for the third FC one for the fourth FC. It is these four near-resonant terms that are of prime importance for the damping estimation, although other terms are also affected to a lesser degree.
It was initially assumed that the damping was that measured on the empty stand, with the knowledge that this will be less than the damping of the occupied stand due to human–structure interaction. Each Fourier term was considered in turn and for a range of FCs the accelerations were determined for the various songs. In this first iteration the four near-resonant terms were ignored. The calculations and measurements were compared using a least squares procedure and the FC selected which best predicts the average response.
Damping was then determined, with the knowledge that it would have the greatest effect on the four near-resonant components. Therefore the damping, which was assumed to be the same for both occupied tiers, was increased from that of the empty stand to obtain the best fit with the four measurements. It is to be noted that this is now the damping of the human–structure system, which for the cases examined has a large proportion of the crowd moving. It is to be expected that the damping would increase still further if the crowd remained stationary.
The FCs were then recalculated using all of the measurements and the new damping value. The selected FCs and their variation with crowd size are given in Table 2. The damping value that was determined was 16%. The accelerations that were calculated are given in Table 3 together with the measured values.
|
| Fourier term | Fourier coefficient, r | Adjustment for number in crowd |
|---|---|---|
| 1 | 0·42 | p−0·082 |
| 2 | 0·087 | p−0·24 |
| 3 | 0·017 | p−0·31 |
| 4 | 0·016 | p−0·5 |
Note: Where p is the number of people in the group.
|
| Record | Response to FC1 | Response to FC2 | Response to FC3 | Response to FC4 | ||
|---|---|---|---|---|---|---|
| Upper tier (frequency 5·70 Hz) | C2 song 20 | Measurement | 0·078 | 0·21 | 0·22 | 0·67 |
| 1·41 Hz | Calculation | 0·24 | 0·25 | 0·17 | 0·44 | |
| C1 song 24 | Measurement | 1·51 | 0·33 | 0·41 | 0·17 | |
| 1·95 Hz | Calculation | 0·49 | 0·63 | 0·48 | 0·27 | |
| C2 song 17 | Measurement | 0·70 | 0·69 | 0·35 | 0·17 | |
| 1·97 Hz | Calculation | 0·51 | 0·66 | 0·48 | 0·27 | |
| C1 song 25 | Measurement | 0·93 | 0·91 | 0·66 | 0·20 | |
| 2·23 Hz | Calculation | 0·67 | 1·03 | 0·39 | 0·23 | |
| Lower tier (frequency 4·68 Hz) | C2 song 20 | Measurement | 0·24 | 0·42 | 1·36 | 0·46 |
| 1·41 Hz | Calculation | 0·36 | 0·41 | 0·35 | 0·33 | |
| C1 song 24 | Measurement | 1·36 | 1·39 | 0·34 | 0·23 | |
| 1·95 Hz | Calculation | 0·75 | 1·28 | 0·32 | 0·21 | |
| C2 song 17 | Measurement | 0·64 | 1·72 | 0·44 | 0·48 | |
| 1·97 Hz | Calculation | 0·76 | 1·33 | 0·32 | 0·20 | |
| C1 song 25 | Measurement | 0·58 | 3·64 | 0·31 | 0·26 | |
| 2·23 Hz | Calculation | 1·04 | 2·13 | 0·26 | 0·18 |
The spread of the experimental values can be considerable, and is best illustrated by the response to the songs 24 and 17, which had similar beat frequencies at the two concerts (1·95 and 1·97 Hz respectively) and where the crowd actions were somewhat similar. For both tiers song 24 yielded a significantly larger response to the first Fourier component of the load, while song 17 produced a larger response to the second Fourier component of the load. These indicate that the different rhythmic actions occurred in response to the different songs, which suggests that calculations using any one load model will, at best, be approximate.
Figure 6 shows the first 10 s of the acceleration record shown in Fig. 1, after it had been subjected to a 10 Hz low-pass filter. To calculate the corresponding acceleration time history using the FCs given in Table 2, it is necessary to adopt values for the phase lags. The phase lags were obtained using a trial and error approach. This involved comparing the time history, which was filtered to leave only the first two FCs, with calculations that considered only two Fourier terms and adjusting the second-phase angle to provide a reasonable correlation. The exercise was then repeated considering the first three Fourier terms, and then the first four. This gave the first four phase lags of π/6, π/2, 5π/6, −5π/6. Figure 7 shows the corresponding calculated response using the load model and the measured characteristics of the upper tier with a damping value of 16%. It can be seen that the calculations provide a reasonable estimation of the average measured response. However, the crucial test occurs when this load model is used on different stands.
Fig. 6. Stand A record 115—after 10 Hz filter.
Fig. 7. Mathcad calculation for record 115
Only two other stands, where concerts had been recorded, provided sufficient information for analysis; but neither case was ideal. Stand D exhibited a well-defined principal mode, but here the crowd was quite subdued as the concert was during a period of national mourning. The largest response occurred with 90% of the audience dancing or standing and clapping. This has been analysed using the load model developed earlier and the measured and calculated results are given in Table 4. The calculations yield larger accelerations than the measurements, but it should be remembered that this was a particularly subdued concert.
|
| Record | Response to FC1 | Response to FC2 | Response to FC3 | Response to FC4 | ||
|---|---|---|---|---|---|---|
| Stand D | Measurement: %g | 0·20 | 0·72 | 0·91 | 0·32 | |
| 2·17 Hz | Calculation | 0·94 | 1·13 | 0·98 | 0·69 | |
| Stand J | Measurement: %g | 0·26 | 0·30 | 0·37 | 0·20 | |
| Concert 1 | 2·20 Hz | Calculation | 0·21 | 0·26 | 0·22 | 0·15 |
| Stand J | Measurement: %g | 0·10 | 0·55 | 0·23 | 0·39 | |
| Concert 1 | 1·92 Hz | Calculation | 0·16 | 0·17 | 0·14 | 0·19 |
| Stand J | Measurement: %g | 0·09 | 0·36 | 0·22 | 0·37 | |
| Concert 2 | 1·85 Hz | Calculation | 0·15 | 0·16 | 0·12 | 0·21 |
On stand J the forced vibration test identified five modes between 6·6 and 7·5 Hz, but yielded some useful crowd measurements at two concerts. Three peak responses at the two concerts were selected and the accelerations relating to the first four Fourier components of the load determined. For the calculations a multi-degree-of-freedom model was used, adopting the measured frequencies and stiffnesses but with each mode having a damping value of 16%. For the first concert, the song with a beat frequency of 2·20 Hz, relates to 80% of the audience standing and dancing, with some jumping. The song at 1·92 Hz had 90% of the audience dancing. In both cases movement was felt. For the second concert the largest response was for a song with a beat frequency of 1·85 Hz which had 95% of the audience dancing, with some jumping to the beat. The results are given in Table 4. For these three songs the calculations underestimate the response for the second, third and fourth Fourier terms.
The first four FCs derived for the grandstands were 0·42, 0·087, 0·017 and 0·016 with their variation with crowd size aligning with that given in Table 2. The equivalent coefficients determined from an experimental evaluation of jumping loads on a floor were 1·61, 0·94, 0·44 and 0·12. Also the effects of human–structure interaction provide a large increase in damping which would not be true if everyone was jumping. This suggests that the structural response at the concerts is much smaller than would be encountered for synchronised jumping.
The Canadian code provides guidance for dynamic loads for a variety of situations.10 For dancing it gives just the first FC which is 0·5, which does not vary with the crowd size. For loads at lively concerts, where fixed seating is present, it recommends two FCs, which are 0·25 and 0·05. This is for a situation where jumping is not encountered, hence equivalent to the concerts considered in this paper. If, for comparison, the coefficients for the upper tier of stand A are considered, with the cantilevered raker beam carrying 280 people, then from Table 2 the first two Fourier coefficients are 0·42 × 280−0·082 = 0·265 and 0·087 × 280−0·24 = 0·022.
Although these do not align exactly with the Canadian values they are of a similar order. However, it is important to remember that the FCs have been selected to predict an average response and not an upper bound solution.
The response recorded at football matches is quite different to that seen at concerts. Figure 8 shows the acceleration–time history for the largest response measured during one game. The maximum response was recorded when the crowd jumped to its feet in response to a goal scored by the home team in front of the stand. This is a typical example of an incident that produces the peak responses at this type of event. In contrast, Fig. 1 shows a measurement at a concert where rhythmic loading is encountered.
Fig. 8. Stand F—peak response of record 137 (unfiltered)
A waterfall diagram (Fig. 9) shows the response for the events around the maximum response given in Fig. 8. This is different in nature to the concerts because the response is not concentrated at discrete frequencies, but shows a much broader spectral content, albeit the peak response can be seen at the resonance frequency of the stand. It can be appreciated that this does not relate to a form of loading which would generate resonance, but rather a forced response as the crowd jump to their feet. At football matches the peak responses are usually isolated incidents whereas the loading is far more repetitive at concerts, hence the load models are different for the two cases.
Fig. 9. Stand F—spectra around main response
The acceleration would decay at a rate determined by the damping. This concentrates on the initial event that causes the crowd to jump and neglects any subsequent movement. However, the grandstand will have many modes of vibration and the impulsive load will excite several modes. Hence the peak responses seen in the spectral domain will relate primarily to the structural frequencies and will not be a function of the load as seen for rhythmic loading. When many modes are excited it will usually be the lower frequency modes that are critical for displacement evaluation, but as will be shown the higher modes can have a significant influence on accelerations.
Recordings of accelerations have been made at three stadia during football matches, and several games were recorded at two of the stadia. In this section the analysis will focus on the peak responses which have been measured. Because large accelerations at relatively high frequencies were recorded on two grandstands, some frequency-weighted accelerations2 are also given in which the higher frequency content is attenuated. The un-weighted accelerations are those that should result from calculations; the frequency weighting procedure takes account of the frequency of these accelerations which is important for serviceability evaluation.
For this form of loading, the dynamic displacements will also be examined and these were determined from the measured accelerations using a double integration procedure. It is acknowledged that numerical integration is not always the most stable of procedures. However, as it can be assumed that there is no overall difference in displacement before and after the event that produced the peak response, a high-pass filter (set below the frequency of the fundamental mode) can be used to remove any trends due to numerical instability.
First consider the grandstand on which the data shown in Figs 8 and 9 were recorded. This stand exhibited monolithic behaviour of the whole structure—that is, the raker beams were rigidly connected. Using the equations in 4·2 the peak acceleration and displacement can be calculated and then compared with the measured peak acceleration and displacement determined by double integration of the acceleration record.
The calculated response for the fundamental mode on stand F gave an acceleration of 0·546 m/s2 and a displacement of 1·94 mm. Theoretically the response in the second mode will be zero as the monitoring position is at a nodal point. If the third mode response is considered this gives an acceleration of 0·081 m/s2 and a displacement of 0·18 mm. The peak acceleration measured at the three matches was 1·69 m/s2 (with a frequency-weighted value of 0·945 m/s2). The peak frequency weighted accelerations at the other two matches were 0·726 and 0·776 m/s2. If a more accurate estimation of the peak acceleration is required it would be necessary to consider many more modes. In contrast, for displacement it is the lower frequency modes that are dominant. The displacement, determined by double integration, at the match that encountered the largest acceleration, was 2·03 mm, and this is close to the calculated value. Naturally this does not yield a time-history like that seen in Fig. 8, as not all people will jump instantaneously and there will be considerable movement following the initial jump. Neither will it give the spread of energy seen in the spectrum.
At the other grandstand considered, the measured responses were somewhat more complicated. Although the forced vibration tests determined some global modes for the stand (similar to the previous structure), it was also clear that local modes were significant. In this example the responses measured at four locations on the front of the cantilevered stand are considered.
Consider the incident that induced the peak acceleration at one game. The acceleration autospectra determined for the four measurement positions each have a different frequency composition as can be appreciated from Fig. 10. The peak accelerations, peak-weighted accelerations and the frequencies with the largest spectral peaks are given in Table 5.
Fig. 10. Autospectra for four positions on stand H, match 2, record 225: (a) truss 8/9; (b) truss 7; (c) 8; (d) 6/7
|
| Recording channel | Acceleration: m/s2 | Frequency-weighted acceleration: m/s2 | Frequencies of spectral peaks in order of magnitudes: Hz |
|---|---|---|---|
| 0 | 1·05 | 0·567 | 8·85, 17·42, 26·37, 43·46, 50·39 |
| 2 | 0·806 | 0·349 | 50·39, 26·37, 8·85, 43·46, 66·86 |
| 4 | 0·797 | 0·272 | 39·43, 49·28 |
| 6 | 1·33 | 0·298 | 50·39, 26·37, 8·85, 66·91, 43·46 |
When the displacements and their autospectra were determined it was found that the main response was around the lower frequencies. The results are given in Table 6.
|
| Recording channel | Position | Displacement: mm | Frequencies of spectral peaks in order of magnitudes: Hz |
|---|---|---|---|
| 0 | Between truss 8 & 9 | 0·404 | 8·85, 2·29, 2·60, 3·59 |
| 2 | On truss 8 | 0·316 | 2·29, 2·60, 3·59 |
| 4 | On truss 7 | 0·309 | 2·29, 2·52, 3·59 |
| 6 | Between truss 6 & 7 | 0·356 | 2·25, 2·52, 3·55 |
What appears to be happening is that the initial jump of the spectators in reaction to an incident excites all modes of vibration, including any local modes. For the displacements the lower-frequency modes are usually dominant, but for (un-weighted) accelerations the local modes are significant, at least in this case. These higher-frequency modes certainly complicate the situation. For the first mode the calculations for truss 8 (channel 2) gave an acceleration of 0·064 m/s2 and displacement 0·239 mm. It can be seen that although the displacement is reasonable the acceleration shows a considerable underestimation reflecting the significant contribution from higher frequency modes that have not been considered in the calculations.
The final grandstand monitored during a football match showed a dominant acceleration response at 56 Hz, again probably related to a local mode. This stand exhibited the type of behaviour described for the grandstands considered at the concerts. The peak acceleration was 5·15 m/s2 but the frequency weighted acceleration was 0·56 m/s2. The displacements determined from the acceleration records gave a peak value of 0·45 mm. Given the characteristics of the 56-Hz mode the acceleration due to the crowd jumping could be calculated, but this does not seem to be a viable approach for wider applications.
This paper has examined a relatively complex topic and the solutions that are given are only likely to be one step on the way to more robust solutions in the future. However, the analysis of the data recorded on a number of grandstands does illustrate some important points.
Two types of load are considered, namely those resulting from rhythmic and non-rhythmic actions. The rhythmic actions are typically encountered when the crowd responds to music, usually at pop concerts, with the musical beat helping to synchronise the crowd movement. The non-rhythmic actions occur when a crowd jumps to its feet in response to an incident, typically a goal at a football match. It is perhaps stating the obvious, but at a football match if the crowd started to respond in a rhythmic fashion, perhaps as a result of music being played, then the response would be similar to that seen at the concerts. However, at concerts there is a far greater risk of a crowd responding rhythmically to music.
For rhythmic actions the response will be at the load frequency and integer multiples thereof, and this is clearly seen when an autospectrum of the recorded rhythmic response is examined. In Fig. 2 the response to the first three FCs of the load can be identified, and in Fig. 5 the response to the first four FCs is apparent. From Fig. 5 it can be appreciated that the response is dominated by the fourth FC, which in this case was a resonant response. In the introduction it was mentioned that current recommendations for a minimum frequency of seating decks subject to synchronised loads was selected to avoid resonance from the first two FCs of the load. This does not imply that the third and higher Fourier terms do not exist, but suggests that even for the resonance situation they will not result in excessive vibrations and this seems to align reasonably with the measurements considered herein. It is important to recognise that the largest responses examined have been for structures where approximately 90% of the audience are standing and moving in response to music (but with few people jumping) and this is a far less severe load condition than if everyone was jumping.
Because the response to rhythmic excitation occurs at the excitation frequency and whole number multiples thereof, it is possible to process the acceleration measurements using a band-pass filtering procedure to examine the response to each Fourier component individually, and this simplifies the analysis greatly. However, to determine loads, it is necessary to make a number of significant assumptions. This allows the grandstand response to be calculated, and although the comparison between observation and calculation is reasonable, it is not perfect. However, as the crowd actions are uncontrolled, it is clear that a perfect correlation is unlikely. The crowd actions observed at pop concerts are less severe than those that would be encountered if the crowd were all jumping, and this is reflected by the FCs that have been derived. It is pertinent to note that the largest responses occurred when a near-resonant situation was encountered. The largest observed responses were when the frequency of the second Fourier component of the load was near to a structure's principal frequency but even this situation did not cause undue concern within the crowd. No grandstand was monitored during a concert where there was the possibility of the first Fourier term generating a resonant response, and in fact this is a situation to be avoided.
If the calculations are repeated using the load model for jumping then the response of most of the grandstands would be found to be unacceptable, and this aligns with the results from a number of tests where jumping tests have been undertaken. It is therefore very important to define the appropriate type of loading for design, together with a reasonable knowledge of the load factors of FCs to be used in the model.
Non-rhythmic loading is less easy to define, but a relatively simple model can be postulated where the entire crowd jumps at the same time, and therefore the crowd's weight is effectively removed from the structure instantaneously. This model can be used to determine peak accelerations and displacements, although it consistently underestimates accelerations when high-frequency modes are not included in the calculation. These high-frequency accelerations are unlikely to be significant for serviceability evaluation. The model does not determine a response-time history like those measured and it does not replicate the crowd action following their initial reaction to the noteworthy incident. This tends to be a less severe load case than the rhythmic loading observed at concerts. It is of interest to note that in this situation the use of added damping on a grandstand would have little effect on the peak response as it is not a form of resonant excitation.
The current guidance for sports viewing or for situations where non-rhythmic actions will be encountered suggests a minimum vertical frequency of the seating deck of 3 or 3·5 Hz. As the loading is not a repetitive dynamic action and resonance is not an issue, the key factor controlling displacement is stiffness not frequency. However, the specification of a minimum frequency is reasonable, because if rhythmic action was encountered and the first FC of the load was at the structure's principal frequency, then unacceptable motion would result.
Currently there is concern regarding the safety and serviceability of cantilever grandstands subject to dynamic crowd loading11 and it is desirable to have appropriate load models for design and analysis. To help the development of realistic load models it is useful to evaluate what loads are encountered in practice. The objective of the work described in this paper was to determine loads from the back analysis of measurements made on cantilever grandstands.
The response of several cantilever grandstands has been measured for a number of concerts and football matches. To convert the measured response to loads requires the characterisation of the grandstand, and the dynamic properties used for this analysis are those that were measured using forced vibration tests. It is still necessary to carefully select appropriate grandstands for analysis, primarily to avoid unnecessary complications that would arise with some structures. But even for grandstands with relatively well-defined characteristics some assumptions/simplifications are required to determine loads. These assumptions are critical to the load evaluation.
The load models fall into two categories.
Rhythmic loading encountered at concerts. The peak loads were encountered when approximately 90% of the audience were standing and responding to music. This is clearly a less severe load scenario than if everyone was jumping. Whether the situation could be encountered where the whole crowd is jumping in response to music is debatable.
Non-rhythmic loading encountered at football matches. These loads are less amenable to analysis, and the simple load model suggested cannot replicate the actual grandstand response although it may be reasonable for evaluating peak displacements. The high accelerations at relatively high frequencies that have been recorded on several grandstands are not a major serviceability concern as human perception of vibrations varies with frequency; however, it would be a difficult task to calculate these accelerations with any accuracy.
The analysis presented in this paper clearly shows the nature of the loads. The loads that have been determined are for the largest responses encountered, but the crowd actions can vary enormously. Rhythmic loading is the most severe situation as resonance can occur and indeed it is the resonant situations that are critical. Even when the loads are derived for a range of events the selection of the load models for design/analysis requires consideration of the largest loads that could be encountered rather than those seen at a few events. The determination of design loads and appropriate safety factors is an interesting and challenging task for the future.
7. ACKNOWLEDGEMENTS
The authors would like to thank the UK Office of the Deputy Prime Minister for sponsoring this work.
