The system reliability of a sample truss structure was investigated using two methods of system reliability analysis. First, the system reliability analysis was performed using the β-unzipping
method for both of the cases of normal and non-normal random variables. Next, the branch and bound method was used to evaluate the system reliability of the truss structure only for the case of non-normal random variables. Both methods present ways of generating the failure sequences, and were used to create a parallel-series model for the whole structural system. In general, through both methods, it was observed that there are certain elements that are more critical and play an important role in the system reliability of the structure. Accordingly, by looking at the failure elements forming the parallel-series model, it is seen that elements 7, 10, 5, 6, and 1 are quite influential in the system reliability of the structure. This fact is verified by both the β-unzipping method as well as the branch and bound approach since both of the methods are probabilistic methods that use the reliability evaluation results to search for the most important failure sequences. Nevertheless, other components of the structure are to some extent influential in the system reliability. The inclusion of less critical elements such as elements 8, 9, 3 and 4 can be accommodated by choosing higher unzipping intervals in the β-unzipping method or by further expansion of the structure failure tree in order to search for more failure paths in the branch and bound search approach.
Through comparing the different failure trees and parallel-series models developed for both of the methodologies, it is possible to pinpoint the differences and similarities of the branch and bound and the β-unzipping methods. Both of the aforementioned system reliability methods are similar in a sense that they are both probabilistic methods. The objective of both methodologies is to identify and generate the failure mechanisms (failure sequences or failure paths) based on a stochastic evaluation of the structure, where the same methodology can be used for the modelling and reliability evaluation of each damaged state of the system. Moreover, it is possible to use both of the methodologies to form a parallel-series model for the whole system as well as the same approach to form the reliability bounds for the system reliability assessment such as Ditlevsen bounds. Nevertheless, it is clear from the failure trees generated for the β-unzipping method and the branch and bound method that for the formation of similar number of failure sequences the two methods are slightly different. In the β-unzipping method the part of the failure tree branching out from failure states S1 and S5 were ignored whereas the failure tree resulted from the branch and bound search includes this failure states which leads to the generation of failure sequences 1-5-3, 1-2-5, 1-5-7,1-10-5 and 5-2-1, 5-2-7, 5-10-7, respectively. The failure states S1 and S5 were ignored in the β-unzipping method because the unzipping interval at level one was not large enough to include these damaged states. Conversely, in the branch and bound search, it is possible to go back to a different level of the system and generate different failure paths. In general the following distinctions can be made between these two methods:
1. The β-unzipping method is dependent on the level where the evaluation is performed, and once a certain damaged state is excluded from the failure path generation process, it is not possible to include it again. On the other hand, the branch and bound method shifts between different damaged levels in order to generate the failure paths.
failure path generation and damaged levels are evaluated in turn, but in the branch and bound method, it is necessary to shift between different levels which makes the method more difficult to program.
3. The β-unzipping method is highly dependent on the unzipping intervals where these in- tervals are selected completely arbitrarily. In contrast, the branch and bound method is entirely independent of any chosen interval and is solely based on the reliability evaluation results of different damaged states (nodes in the failure tree).
4. The failure path generation in the β-unzipping method is based on the definition of different damaged levels. For instance, the definition of system reliability evaluation at level 3 is the formation of a parallel-series model for the system which is composed of parallel triples of failure elements. The methodology is somewhat unclear regarding the failure mechanisms happening at an earlier level such as the failure sequence of elements 5 and 7 in the structure of Figure 7.1. This failure sequence corresponds to the system reliability evaluation at level two; however, due to the degree of redundancy of the structure, failure sequences at level 3 were also identified. A solution to this inconsistency can be the so-called system reliability evaluation at mechanism level. Nonetheless, this can make the computerisation of the method significantly complex.
The branch and bound method is, however, based on the identification of terminal nodes where a terminal node is any node referring to the collapse of the system. It should be noted that for the sake of comparison the terminal nodes of states S7,5 and S5,7 were ignored in the system reliability evaluation in Section 7.3.1. If these failure mechanisms are also included in the system reliability analysis, the following parallel-series model can be used. This parallel series model is shown in Figure 7.45.
10 7 5 7 2 5 2 7 5 10 1 5 5 10 7 1 10 5 2 1 5 5 2 7 5 2 1 1 2 5 1 5 7 1 5 3 7 10 5 7 5 2 5
Figure 7.45: parallel-series model for the system reliability
As is seen in Figure 7.45, in the parallel-series model elements 5 and 7 are also included. The Ditlevsen bounds for this model is shown below where P {S7,5} = 7.2950 × 10−3 and P {S5,7} = 6.5232 × 10−3.
7.3211 × 10−3 ≤ PBB
fsys ≤ 9.7433 × 10
−2
The inclusion of the sequence of elements 5 and 7 doesn’t change the lower bound of the Ditlevsen bounds for the system since the lower bound is the maximum of failure probabilities of the elements of the series system and the minimum belongs to the sequence of element 5,7,10. But it does change the upper bound of the system. It should also be noted that in this case the system reliability doesn’t change since the lower bound of the system bounds didn’t change (the system reliability is equal to the lower Ditlevsen bound due to high correlation between the
safety margins of the structural elements in this case).
In short, both of the methods appear to be efficient in system reliability calculation. The β- unzipping method is easier to program, and this makes it easier to choose bigger intervals for the system reliability calculation which consequently makes it possible to include a bigger number of critical failure elements that might be ignored in the case of smaller intervals. Therefore, sufficient accuracy for the method can be provided. the branch and bound method could also be used for the system reliability, but the computer program developed for this method is only used to model different damaged states and the rest of the procedure is performed manually which can become rather complex. Therefore, it is recommended that a β-unzipping method with large intervals be used instead of the branch and bound method.
Chapter
8
Conclusions and Recommendations
8.1
General
Chapter 8 is dedicated to the conclusions and recommendations for the further study in this area of research. The thesis was mainly focused on a fully computerised stochastic evaluation of truss structures. Throughout this study common methods of component and system level reliability analysis were investigated where computer algorithms and programs were developed based on these methods. Clearly, it follows the objective of providing a completely stochastic finite element environment for a computerised system and component level evaluation of truss structures.