On the 17th of October 2021, a concrete grandstand element collapsed at the Goffert stadium in the Netherlands after a crowd jumped on it for 8 seconds. An investigation by an engineering consultancy firm concluded that the design loads in the building code are too low, raising questions about the safety of grandstand elements. Currently, no method to assess the reliability of grandstand elements exists.

This thesis presents a state-of-the-art method to determine the reliability of concrete grandstand elements. The reliability is assessed by performing a non-linear dynamical analysis. The element is modelled as a non-linear single-degree-of-freedom system. Excitation signals are synthetically generated by consulting the literature and by analyzing a data set of jumping crowds. A bi-linear force- displacement relationship based on technical drawings of the collapsed grandstand element is adopted and extended by a model uncertainty parameter which accounts for both the non-linearity of the analysis and the uncertainty related to the dynamical basis of the analysis. The reliability is determined through a Monte Carlo simulation: almost 100,000 simulations can be performed per assessment.

Out of the 100,000 simulations, 0 failed. This result is not in line with what happened; one failed out of only a few elements. This gives rise to two different investigations. On the one hand, the failure of the Goffert stadium grandstand elements has to be explained. On the other hand, the lifetime reliability of a grandstand element has to be determined.

The first assessment indicated that if the element’s resistance conforms to the technical drawings, there is no cause for concern regarding its reliability. Measurements on 23 other grandstand elements in the same stadium showed a high variation in the concrete cover. The collapsed element was, therefore, also likely subjected to a high variation in the concrete cover. To understand the influence of an increased concrete cover on the reliability, two additional analyses were performed, where the post-yielding resistance of the structure was slightly reduced. This resulted in an increase in the probability of failure, which indicates that this parameter plays a crucial role in determining the reliability of grandstand elements. These points combined make it more plausible that the element failed because the concrete cover was larger than intended rather than the design loads being too low, as concluded by the engineering firm.

When investigating the lifetime reliability of grandstand elements in general, no collapse is expected after 8 seconds but rather after 30 seconds or even longer durations. Therefore, two additional analyses were performed where signals of longer durations (120 seconds and 300 seconds) excited the system. In these analyses, it is assumed that the resistance of the grandstand element conforms to the technical drawings. Signs of a converged reliability were perceived as 120- and 300-second excitation signals led to a probability of failure of the same order of magnitude, indicating that a steady-state solution is obtained after 120 seconds of jumping. In that case, the lifetime reliability of a grandstand element would be equal to the 120-second reliability. The corresponding reliability passes the lifetime reliability requirements for existing structures in consequence class 2 (for a reference period of 15 years).

While the results presented are estimations, and a larger sample size is needed for a converged probability of failure, the proposed method provides a valuable framework for assessing the reliability of grandstand elements. This method can easily be extended to any grandstand element by changing the model parameters, and the true reliability of grandstand elements can be assessed by performing more (1-10 million) simulations.

This thesis presents a novel methodology to determine the reliability of cantilever grandstands under dynamic crowd loading. No method currently exists to evaluate this, despite the occurrence of significant vibrations in such structures, for example in the Feyenoord Stadium in Rotterdam. These vibrations can negatively affect both the safety and serviceability of such structures, so a method able to evaluate both is desirable. The methodology consists of four components: a model of a grandstand, a model of the dynamic crowd load, a failure criterion, and a reliability analysis. Of these, the first and third component are case-specific, as a wide range of different grandstands and different criteria could be considered. In this thesis, the methodology is applied on a case study based on the Feyenoord Stadium, with a model of the trusses beneath the main grandstand of this stadium, and a criterion based on the intensity of the vibrations of the upper cantilever of this grandstand, a serviceability issue. The second component represents the most novel aspect of the methodology: a model describing the dynamic crowd load, defined in the frequency domain, and generally applicable regardless of the structure being considered. The core of this model is a parameterisation of the amplitude spectrum of the load applied by a group of jumping spectators. The parameters describing the spectrum are considered as stochastic variables, with distributions fitted using samples from a state-of-the-art database of measured loads. As opposed to the first and third components, this load model should be generally applicable.The parameters of the load model, together with other stochastic variables in the grandstand model, form the input of a Limit State Function. The output of this function determines whether the structure loaded by the crowd meets a certain criterion, and forms the basis of a reliability analysis together with the distribution of the stochastic variables. The probability of \emph{not} meeting the criterion, the failure probability, is calculated through a number of reliability methods in the final component of the methodology. These methods should be generally applicable as well, though which methods fits the best could still depend on the case being considered. This methodology was successfully applied on the case study, and a failure probability was found with two different methods. One of these, Crude Monte Carlo, allowed for a more fluid view of the concept of `failure' when a serviceability criterion based on a subjective limit is considered. The other method, SDARS, returned a slightly larger failure probability, in a much shorter runtime than Monte Carlo. This method is therefore more fit for safety criteria with a small (expected) failure probability, for which Monte Carlo would require an unreasonably long runtime. In addition, the wider applicability of the methodology has been investigated. An important requirement for this is flexibility with regards to the applied reliability methods. Two other methods, Directional Sampling and SDARS, were also attempted but did not yield usable results. This is caused by the manner in which phase angles are defined in the frequency-domain model. Random values are drawn, which causes noise to appear in the limit state function: a constant input does not lead to a constant output. Being able to use different reliability methods is an important requirement for applying the methodology in a wider variety of cases. In order to apply these methods, the phase angles need to be defined in a manner which does not introduce noise. Another important requirement is the ability to consider different failure criteria. While the criterion applied in the case study could be evaluated in the frequency domain, many others will require the results to be transformed to the time domain. Applying this transformation to the frequency-domain results of the models currently does not yield usable results, which is likely also caused by the definition of the phase angles in the load. In conclusion, while the methodology is applicable on the case study, the definition of the phase angles requires attention before it can be applied on other cases.

Structures are considered safe,when they comply with minimum reliability requirementswhich are expressed by a minimum reliability index ¯ in the Eurocodes (EN1990). However, a complete reliability assessment for every structure would be very time-consuming. Therefore in the Eurocode a partial factor approach is utilized. In this research it is investigated how the reliability of the main bearing structure of a dynamically sensitive building designed within the Eurocode framework can be assessed in a full-probabilistic way for global response to wind loading including both size and dynamic effects. The global response of buildings to wind loading as well as uncertainties in this wind load have been the subject of many research, but few reference has been made to the reliability of structures by linking both aspects. Current research provides
this link and investigates the reliability of these buildings with respect to the target reliability in EN1990. For the purpose of this research the methods were developed for alongwind response at foundation level of the main bearing structure for slender high-rise buildings with a concrete core bearing structure.