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Non-linear dynamic analysis of Collapsed
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XII International Conference on Structural Dynamics
Journal of Physics: Conference Series 2647 (2024) 162003
IOP Publishing
doi:10.1088/1742-6596/2647/16/162003
1
Non-linear dynamic analysis of Collapsed grandstand
T Xu1, W Meijers1, SJH Meijers1 and R Verlinde1
1Royal HaskoningDHV, George Hintzenweg 85, 3068 AX Rotterdam, the Nether-
lands
Abstract. On the 17th of October 2021 a grandstand of the Goffert stadium partially collapsed
while a crowd was jumping on it. It was concluded that the main reason of the collapse was
that the load of a jumping crowd exceeded the design load of 4 kN/m2, as specified in the NEN
6702. The load also exceeds the current design load of 5 kN/m2 NEN-EN 1991-1-1. A geomet-
rically and physically non-linear finite element (FEM) model was used to obtain the backbone
curve of the response of the grandstand. The push-down analysis was force-controlled to allow
the grandstand to rotate and deform freely, so that the failure mechanism could develop freely.
The jumping load was modelled with a Fourier series based on the contact ratio, the coordina-
tion, and the weight of the crowd. These three parameters were estimated with video footage
of the collapse. The mass of the people was roughly 3.5 kN/m2. An equivalent SDOF system
was set up based on analytical formulae and the backbone curve. With this model the non-linear
dynamic behaviour of the grandstand was assessed. Failure was found with the estimated jump-
ing load after roughly the same number of jumps as was seen on video.
1 INTRODUCTION
On the 17th of October 2021 a grandstand of the Goffert stadium in Nijmegen (Netherlands) partially
collapsed while a crowd was jumping on it, see Figure 1. The supporting fans of the visiting football
team were celebrating the victory together with the players. Most of the fans grouped in the first three
rows of the grandstand and jumped in the rhythm that was instructed by the players. The grandstand
was visibly and plastically deforming more after every coordinated jump of the crowd. After 14 jumps
a crack formed in the middle of the span, and the grandstand collapsed. Luckily there were only minor
injuries. The collapse was captured on video and can be found on YouTube [1].
It was concluded based on forensic engineering [2] that the main technical cause of the collapse
was that the load of a jumping crowd exceeded the design load of 4 kN/m2, as specified in the Eurocode.
In this paper the research into the effect of the non-linear dynamic behaviour of the grandstand is
discussed. First an overview is given of the collapsed element and how it is discretized to a SDOF
system. Then the dynamic load effect from the crowd and the non-linear behaviour of the grandstand
is determined. This is then used to obtain an equivalent non-linear SDOF system in which the collapse
of the grandstand is modelled.
XII International Conference on Structural Dynamics
Journal of Physics: Conference Series 2647 (2024) 162003
IOP Publishing
doi:10.1088/1742-6596/2647/16/162003
2
Figure 1. Collapsed grandstand
2 DESCRIPTION OF GRANDSTAND
The grandstand is a prefab concrete element spanning 8.73 m which is supported by thick concrete
beams 0.45 m wide. One element consists of three rows with additional stairs in the middle of the
element of 1.1 meter wide. Longitudinal reinforcement of ø11-150 is present in the plate part of the
grandstand. Total weight of the element is 14.227 kg which is roughly 6.1 kN/m2. The grandstand is
designed for a variable load of 4 kN/m2, which corresponds to a seated crowd according to the codes
at the time. This is lower than the maximum crowd loads mentioned in 4.2.
Original design drawings of the collapsed grandstand are shown in Figure 2 and Figure 3.
XII International Conference on Structural Dynamics
Journal of Physics: Conference Series 2647 (2024) 162003
IOP Publishing
doi:10.1088/1742-6596/2647/16/162003
3
Figure 2. Sideview of collapsed grandstand in the Goffert stadium according to design drawings.
Figure 3. Top view of collapsed grandstand element (bottom) in the Goffert stadium according to de-
sign drawings.
3 NON-LINEAR BEHAVIOUR OF GRANDSTAND
The capacity and behaviour of the grandstand is determined by a force controlled non-linear push-
down analysis. It was chosen to do it force controlled so that the grandstand is free to deform and rotate,
and the correct failure mechanism is assessed.
A FEM model is made with volume elements and embedded reinforcement in DIANA FEA, see
Figure 4. Material properties are based on averages of material tests from the collapsed element. The
grandstand is supported by friction interfaces.
With the model the force-displacement diagram in Figure 5 is obtained. Some key moments in the
diagram are expressed as a factor of the static crowd load (3.5 kN/m2) in Table 1. At a factor of 1.5
XII International Conference on Structural Dynamics
Journal of Physics: Conference Series 2647 (2024) 162003
IOP Publishing
doi:10.1088/1742-6596/2647/16/162003
4
cracks occur next to the stairs of the grandstand, and the system becomes non-linear. Due to redistri-
bution of the forces most of the reinforcement is utilized at the cost of large plastic deformation.
The failure load is given a bandwidth because the model becomes unstable close to failure in
DIANA. In addition, this bandwidth accounts for the uncertainty in material properties, spread in ma-
terial properties and local imperfections or damages that were not included in the model.
Table 1. Moments in non-linear behaviour expressed as load factor of the static crowd load
Moment
Description
Load factor
Permanent load
Load of the grandstand
0
Static variable load
Static load of the passive crowd + permanent load
1
Cracking
First significant cracking of cross section
1.5
Yielding
Yielding of first reinforcement bar
1.9
Failure (in DIANA)
Divergence in DIANA, capacity reached
3.2
Figure 4. Geometry of grandstand-element, isometric view in DIANA FEA
Figure 5. Force-displacement curve (blue line) of grandstand element including failure load (red line)
with upper and lower limit (dotted red lines)
XII International Conference on Structural Dynamics
Journal of Physics: Conference Series 2647 (2024) 162003
IOP Publishing
doi:10.1088/1742-6596/2647/16/162003
5
4 DYNAMIC LOAD OF JUMPING CROWD
Crowds can generate significant higher dynamic loads when jumping compared to the static loads, as
is evident from the collapse of the grandstand in this paper. Based on photos and videos of the collapse,
there were 93 people on the first three rows of the grandstand. Assuming a weight 85 kg per person,
this equates to an average static load of 3.5 kN/m2 on the grandstand.
4.1 Literature
The dynamic load of jumping crowds has been researched experimentally and numerically by various
researchers, best known is the research by B.R. Ellis [3] [4] [5]. Most commonly the dynamic load is
expressed as a Fourier series, see below. The magnitude of the dynamic load is mostly determined by:
• Density of the crowd
• Intensity of jump
• Coordination of the group
(1)
With:
Weight of crowd [N]
Jumping frequency [Hz]
Number of Fourier term [-]
Coefficient of hth term [-]
Phase shift of hth term [rad]
In Figure 6 the dynamic load of a crowd is expressed as the factor between the static load and the
maximum dynamic load, which will be called the jump factor. The jump factor for crowds varies be-
tween 1.7 and 3.6. For the casus in this paper this results in a dynamic load between 5.95 and 12.6
kN/m2.
Figure 6. Jump factors for individuals according to [4] with 6 Fourier-terms ( = 6) and product of
jump factor and coordination factor for crowds based on [6].
XII International Conference on Structural Dynamics
Journal of Physics: Conference Series 2647 (2024) 162003
IOP Publishing
doi:10.1088/1742-6596/2647/16/162003
6
4.2 Eurocode and ISO
In table NB.1 - 6.2 of the Dutch national annex of NEN-EN 1991-1-1 a load of 5 kN/m2 is given for
crowds. This load is based on research done in 1973 and 1976 by Sentler and Paloheimo [7]. In this
research a maximum natural crowd density of 2.5 kN/m2 was determined, resulting in a maximum load
of 5.5 kN/m2. For smaller crowds jump factors up to 5 were found.
Guideline ISO10137 [6] is focused on vibrations and comfort but specifies a more detailed dynamic
load for a jumping crowd compared to NEN-EN 1991-1-1. The same Fourier series is used as in eq 1
although different coefficients are used, Figure 7 shows the jump factor according to ISO10137. The
guideline specifies a maximum of 6 people per m2 and a coordination factor between 0.4 and 0.67, this
results in a maximum dynamic load of 10.2 kN/m2.
Figure 7. Load of jumping individual with a mass Q, graph from [6]
4.3 Measurements of different jump loads
After the collapse the load effect of jumping crowds were measured in several stadiums in the Nether-
lands. This was done by counting the amount of people present and measuring the deformations of
grandstands during football matches. In Table 2 the results of these measurements are given. The meas-
ured jump factors are in line with the literature. With the measured density of people, this results in
higher loads than specified in the Dutch annex of NEN-EN-1991-1-1.
Table 2. Results of measurements of jumping crowd loads in 5 stadia in the Netherlands
Name of stadium
Type of grandstand
Static weight
of crowd
[kN/m2]
Total dynamic
load- effect
[kN/m2]
Dynamic
jump factor
[-]
Galgenwaard
Sitting
1.7
3.8
2.2
Arena
Safe standing
1.8
4.0
2.2
FC Groningen
Sitting
2.5
6.2
2.5
AFAS
Safe standing
2.3
4.1
1.8
Abe Lenstra
Standing
3.4
8.5
2.5
5 EQUIVALENT NON-LINEAR SDOF-SYSTEM
Constructions can be schematized as mass-spring-damper systems, the simplest form is the 1-mass-
spring system (single degree of freedom system, SDOF) [8]. The dynamic response of the linear equiv-
alent SDOF system of the grandstand is described in 5.1. in 5.2 the non-linear behaviour is added. The
properties of the SDOF are mentioned in the table below. For the linear dynamic analysis, the jump
XII International Conference on Structural Dynamics
Journal of Physics: Conference Series 2647 (2024) 162003
IOP Publishing
doi:10.1088/1742-6596/2647/16/162003
7
load of the crowd is described as a half-sine load while for the non-linear dynamic analysis the jump
load according to chapter 4.1 is used.
Table 3. Relevant properties of SDOF system
Property
Value
Modal mass
10.500 kg
Eigenfrequency
10 Hz
Damping
4%
Weight of crowd
77.500 N
5.1 Linear dynamic analysis
The displacement of the grandstand due to a half-sine load is described by:
(2)
With:
(3)
(4)
(5)
(6)
Eigenfrequency [rad/s]
Damping [-]
Load frequency [Hz]
Time [s]
Equivalent spring stiffness [N/m]
Equivalent load [N]
The analytical expressions above are solved numerically for a rhythmic load of 2 Hz and eigenfrequen-
cies of 10 Hz (left) and 5 Hz (right) in the figure below. Using the properties of grandstand, this leads
to dynamic amplification factors of 1.55 and 2.33 respectively. This includes a jump factor of 2.4
(medium coordination of group).
XII International Conference on Structural Dynamics
Journal of Physics: Conference Series 2647 (2024) 162003
IOP Publishing
doi:10.1088/1742-6596/2647/16/162003
8
Figure 8. Linear elastic response of SDOF system with an eigenfrequency of 10 Hz (left) and 5 Hz
(right).
5.2 Non-linear dynamic analysis
Based on the push-over curve the stiffness in the SDOF-system is varied, since at higher forces the
stiffness of the SDOF is reduced. A secant stiffness is assumed during unloading and reloading. Again,
the system is solved numerically, with Newmark time integration and timesteps of 3.38 ms and New-
ton-Raphson iterations. In the figure below the displacements are shown for seven different jump loads
on the grandstand with an initial stiffness of 10 Hz. Failure is reached based on the push-over curve
for the largest jump load (contact ratio 0.33 and high coordination). Note the explosive character of the
deformations for this jump load. In Figure 10 the energy balance of the calculation is shown.
Figure 9. Non-linear dynamic response for 7 different jump loads of crowds.
5.3 Dynamic amplification factor
At the failure point of the SDOF-system the internal force increased by a factor 1.7 compared to the
static system while the deformations increased by a factor 8. The dynamic amplification of the load of
the crowd is 3.1.
XII International Conference on Structural Dynamics
Journal of Physics: Conference Series 2647 (2024) 162003
IOP Publishing
doi:10.1088/1742-6596/2647/16/162003
9
For structural design the dynamic load is often simplified to an equivalent quasi-static load effect,
where the equivalent load results in the same deformations as the dynamic load. For non-linear dy-
namic systems this relation is not valid anymore as shown above.
Figure 10. Non-linear response of mass-spring system with initial eigenfrequency of 10 Hz with a
rhythmic jump load of the crowd according to [5] with a frequency of 2 Hz. Contact ratio of 0.33 and
high group coordination according to [6]
6 RESULTS FOR DIFFERENT JUMP LOADS
The non-linear calculation from the previous chapter is solved for different jumping frequencies in
Figure 11 below. In Figure 12 it is shown after how many seconds failure was reached for the different
jumping frequencies, note that calculations reaching 8 seconds did not fail. This shows how sensitive
and unpredictable non-linear dynamic calculations are.
Figure 11. Influence of different jumping frequencies on non-linear dynamic behaviour of the grand-
stand. Contact ratio of 0.35 and high group coordination.
XII International Conference on Structural Dynamics
Journal of Physics: Conference Series 2647 (2024) 162003
IOP Publishing
doi:10.1088/1742-6596/2647/16/162003
10
Figure 12. Time to failure for different jumping frequencies, calculations reaching 8 seconds com-
pleted the whole calculations. Contact ratio of 0.35 and high group coordination.
7 CONCLUSIONS
Based on the forensic engineering of the collapse of the grandstand-element in the Goffert stadium the
following conclusions are drawn:
Jumping crowds can cause larger dynamic loads than currently specified in the Dutch annex of
NEN-EN 1991-1-1, the design value in the code is 5 kN/m2. The estimated equivalent quasi-static load
on the collapsed grandstand was 2.8-3.2 times the static load of the crowd, roughly 10 kN/m2. This was
largely because the crowd was very dense and jumped in good coordination.
Measurements were done in similar existing stadia in the Netherlands, loads up to 8.5 kN/m2 and
jumping factors between 1.8 and 2.5 were determined. Similar values are found in current literature.
Converting non-linear dynamic behaviour into dynamic amplification factors to determine equiva-
lent quasi-static load effects based on deformation can lead to overestimation of the dynamic loads. In
the shown case a factor 8 was found, while the internal force only increased by a factor 1.7.
Non-linear behaviour of the structure can change the dynamic behaviour of the structure, leading to
an increase or decrease of the dynamic load. Although this effect is possible to analyze, it is difficult
and prone to mistakes. Therefore, structural design should aim to stay in the linear elastic regime of
the structure when dynamic loads or behaviour are expected.
REFERENCES
[1]
Footage of collapse of grandstand in goffert stadium,
https://www.youtube.com/watch?v=i5mzG-BH2o0.
[2]
Royal HaskoningDHV, "Onderzoek naar de technische oorzaken van het bezwijken van
het tribune-element van het Gofferstadion te Nijmegen," 2022.
XII International Conference on Structural Dynamics
Journal of Physics: Conference Series 2647 (2024) 162003
IOP Publishing
doi:10.1088/1742-6596/2647/16/162003
11
[3]
B. Ellis and T. Ji, "Loads generated by jumping crowds: experimental assessment," BRE
Centre for Structural Engineering, 2002.
[4]
B. Ellis and T. Ji, "Loads generated by jumping crowds: numerical modelling," The
Structural Engineer, pp. 35-40, 7 Sept. 2004.
[5]
B. Ellis. and et al, "The response of grandstands to dynamica crowd loads," ICE
Proceedings Structures and Buildings, vol. 140, pp. 355-365, 2004.
[6]
ISO, "Basis for design of structures - Serviceability of buildings and walkways against
vibrations, ISO 10137, 2nd ed.," Switzerland, 15 Nov. 2007.
[7]
TNO, " Achtergronden van de belastingen volgens concept NEN 6702," TNO, 1989.
[8]
Biggs, J.M., Introduction to structural dynamics, McGraw-Hill, 1964.