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.

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.

Lately structural safety is an often discussed subject in The Netherlands. Recent collapses have made people question if our buildings are actually safe enough. Research shows that the individual probability of death due to structural failure is below the limit of 10-5 per year. However, it is unknown if every single building is safe enough as well. The main cause of structural failures are human errors (90% of the collapses). However, the Dutch Building Decree does not mention the unidentified accidental actions (the type the human errors belong to) as those an engineer has to take into account when designing the building structure. The Eurocode does mention unidentified accidental actions and states that for consequence class 3 buildings a risk assessment has to be done to check which accidental actions are hazardous and if the total risk to a building is acceptable. However, the Eurocode does not give a maximum value of an acceptable risk in their risk assessment method. So currently when engineers perform a risk assessment, they cannot verify whether the risks of accidental actions on their designed structure are acceptable according to the regulations. This thesis conducts research for a new method for performing a risk assessment in a quantitative way. First it investigates what the advantages and disadvantages of existing methods for performing a risk assessment are. It appears that the main step missing in existing methods is the verification, based on a quantified limit, whether the risks of accidental actions are acceptable. Based on the known methods, a new protocol for performing a risk assessment is proposed. This protocol includes a determined limit for the probability an accidental action occurs and causes a certain failure mechanism and consequence. This limit is based on the maximum probability the Eurocode gives for failures with a consequence indicated as consequence class 1, 2 and 3 (CC1, CC2 and CC3). By dividing the accidental actions that can occur and their failure mechanisms into the same consequence classes, the estimated probability and the limit of the Eurocode can be compared. In this way the known consequence classes from the Eurocode are used in another way than engineers are used to. Furthermore research on past structural failures has been investigated to establish whether it can be used to predict the probability a certain accidental action causes a CC1, CC2 or CC3 consequence. Not enough research is done to fully trust on these outcomes; in the protocol proposed in this thesis the engineer will have to estimate a part of the probabilities himself, with the help of a classification table. The principal advantage of performing a risk assessment on accidental actions in a quantitative way with a prescribed maximum probability per consequence class, is that it is made clear whether a building fulfills the requirements or not. This can be helpful for communication with other involved parties. However, it will never be 100% clear what the limit should be, because estimating the risks is based on the engineer’s estimations and outcomes may differ as they depend on the person performing the risk assessment. This means a hard line, below which the risks are acceptable and above the risks are considered too high, cannot be drawn. An area in between is needed in which it is up to the engineer and client to decide whether or not they find the risks acceptable.

The analytical expressions for the ultimate load bearing capacity of the RC structures do not provide the generalized notion of ultimate load bearing capacity, which can be obtained through nonlinear finite element analysis (NLFEA). In order to obtain an accurate estimate of failure probability of a RC structure it is necessary to use NLFEA based limit state function in a reliability analysis. However, there is a relative lack of NLFEA based reliability analysis efforts in the literature. Whatever efforts there are, none of them explicitly attempts to account for the uncertainty introduced by the NLFEA model, called modeling uncertainty, in the reliability analysis. Nor has there been much effort to study the impact of the numerical noise from NLFEA on the accuracy and efficiency of the reliability algorithms. Since the run time of each NLFEA is high, for a NLFEA based reliability analysis to be practically feasible it is imperative that the reliability algorithm is efficient and capable of handling different kinds of limit state functions (with multiple failure modes, for example). Keeping this in mind two adaptive response surface based methods, directional adaptive response surface method (DARS) and adaptive Kriging Monte Carlo simulation (AK-MCS), are selected based on the preliminary literature survey, for the investigation of NLFEA based reliability analysis of RC structures. The key objective of this thesis is to study the strengths and limitations of these two algorithms for RC structures and make necessary modifications in the DARS algorithm to make it more suitable for the reliability analysis of RC structure. A NLFEA solution strategy is formulated for RC beams and the modeling uncertainty is quantified based on 53 experimental results. Three RC beams, are selected as demonstrative cases. One of these beams fails in shear, another in bending and the last one can switch in failure modes between shear and bending. Based on these three beams it is demonstrated in this thesis that there is pronounced numerical noise in the NLFEA predicted bearing capacity whenever the beams fail is shear failure mechanism. Whereas for the bending failure mechanism the NLFEA solution strategy produces a much more smooth capacity prediction. Clear indications are found to the effect that the shear failure mechanism is more sensitive to certain choices adopted in the NLFEA solution strategy. @en