 Next I would like to talk about the dependency of value of SHM on system characteristics. So we, okay, so now I understand the issue here with this thing. There's another practical thing we could announce is that it is very important for the reimbursement that you sign a list, an attendance list, and we have to do this daily. So anyone who arrived later today, please sign in the lunch break the list so that we can reimburse you. Okay, now we have a working battery, but, okay, there's something wrong with the screen here. Okay, I need to look at this screen then. Okay, sorry. Okay, let's start with the decision theory. We have seen very similar things today already, the value of information theory developed by Riefer and Schleifer. So that is something we find in the book of Riefer and Schleifer. So this is the very basic concept of modeling experiments, outcomes, acts as you call it. So that's action, structural integrity management, action for instance, and the state spaces and this is associated to choices or decisions and chances. So this is a probabilistic outcome for instance. And also mentioned already today much more explicit than it is written here. We have an extensive and normal form analysis. So this is how we are calculating this decision tree from the left to the right or from the right to the left. Very close to what we see here is this extended decision tree for quantifying the value of structural health monitoring. So this is the basic choice we are modeling to do SHM or to do no SHM, so this and that. And the choice of our experiment is then already here incorporated. So we take two steps here. We have the SHM strategy with its outcome, that's a chance, then we again have the decision rule on what to do with this information and the possible outcomes. We then have the choice of an adaptive actions, we have the chance of its outcome and we have the life cycle performance leading to the life cycle benefit B1. And if we do not do SHM then we still have this decision tree for the structural integrity management to quantify the life cycle benefit B0 and the value of structural health monitoring is then B1 minus B0. I will skip this due to time limitations. Let's come to the structural performance model. So here we can have a model for the system structural performance. This is written rather general. We can describe the failure domain with this limit state function, very general formulated. We have model uncertainty MR multiplied with the resistance over time of the system minus model uncertainty multiplied with the system load and the resistance over time may reduce. The deterioration, I've been concentrating my work so far on fatigue. So we can have an SN limit state function and then we can calibrate fracture mechanics limit state function to that and we can model the expected stress ranges again here with the model uncertainties and that's a viable formulation for the expected stress ranges. And now we can cover these two models. So the component resistance reduces over time due to deterioration. So this is this expression and the deterioration we can describe utilizing our fracture mechanics fatigue model where we have the crack sizes and here we also introduce the components diameter so that maybe circular hollow sections and resistance, oh no it's not the diameter it's the wall thickness of course. So that's not limited to circular hollow sections. So that's the crack size to wall thickness ratio here that develops over time and we have a resistance reduction factor RR and that can be adjusted for each component in dependency of the cross-sectional areas and the cross-sectional properties of the component. I'm coming out through the SHM strategies. So we are modeling the SHM strategies in a way that we are working with the realizations of the model uncertainties. So this is something which comes when we are measuring the structures already there and our model uncertainties with which the structure was designed have been realizing. So this is what we are using to model a preposterior monitoring outcome. So here I'm concentrating on load monitoring. So this is the probability of system failure given realization of the load monitoring model uncertainty here. The same goes to the monitoring of fatigue. Here I can rewrite the model uncertainties in the model uncertainty for the loading model, for the far field stress, for the hotspot stress and for the weight quality and I'm just working with the realizations of the load monitoring model uncertainty. And we are also accounting for the measurement, the SHM uncertainty, you. Okay, there's something, okay. The service life benefits we are calculating with the expected inspection costs, expected repair costs, the component deterioration risks, they are here and the structural system failure risks, they are written here outside the sum over the components. So the life cycle benefit doing SHM is then calculated with the same, basically the same expressions, but here we have also the costs of structural health monitoring and the other components are modified because we are utilizing the SHM information here. And the decision rule we are utilizing is reliability based inspection and repair planning with the adaptive actions inspect inspection and repair and we have a normalized cost model. And I think also on one paper here for the ICAS 12 where we have implemented the, also the reliability based inspection and repair planning. Let's come to an example. Let's consider a Daniel's system. So this is characterized by an assembly of components which are subjected to load. And the behavior of the individual components can be either ductile or brittle. Daniel's system has similar characteristics like parallel system, but it also has, and that's the important point, mechanical justification whereas purely parallel system doesn't have it. And the behavior of a brittle system is a little similar to a serious system. So yes, this gives the, I think this is a relatively important or normalized cost model. So component failure will be, has the value of one system failure, the value of 100. And then we have the cost of system, SHM system, investment, installation and operation. We have a discount rate, so the costs are discounted to the time of the decision. So this is what the discount rate does. And then we can here quantify the value of structure, health monitoring for our, for the example. And this here on the left side is a ductile Daniel's system. Here we have a brittle Daniel's system. And here we have the probability of component deterioration failure thresholds for the structural integrity management. So this is the threshold which is kept throughout the life cycle. And when a component reaches this threshold, then an inspection event takes place. This was quantified with the correlation of 0.5 between the resistances of the Daniel's system and with a correlation coefficient of 0.6 between the damages in the Daniel's system. And the component probability of failure is 1.0 times 10 to the power of minus 2. So we see with a, with a decreasing. So this is logarithmic the other way around. So decreasing probability of failure threshold, decreasing value of structure, health monitoring both for the ductile and the brittle Daniel's system. Let's look at two system characteristics. One is the correlation between the resistances from 0 to 1, here the same. And we have here a ductile Daniel's system and here a brittle Daniel's system. So with the increasing correlation, the value of structural health monitoring increases for all probability of components, deterioration failure thresholds for all these different thresholds, the value of SHM increases with increasing correlation. We have the opposite picture for a brittle Daniel's system here with a decreasing coefficient of correlation, with increasing correlation the value of SHM decreases. What is the explanation for that? Oh, I'm running out of time. The explanation for that is that with the ductile Daniel's system the reliability decreases when the correlation of the resistance increases. But as I'm only considering load monitoring, so the uncertainty reduction stays constant, but the resistance of the system changes due to the changing correlation. So and that's why we have here an increasing value of structural health monitoring and here a decreasing because for a brittle Daniel's system if the correlation increases towards the end of the correlation being around here in this area, the system reliability significantly increases. And again, this is caused by the resistance, but which is independent here mainly because we are considering only load monitoring. We have a similar, well no, we don't have a similar picture maybe here, but if we look at the correlation of the deterioration, we see here a similar picture and here a similar picture to the previous one. Well this happens only, so the deterioration correlation has only influence if we have a rather high probability of air threshold because then the deterioration affects the system reliability. Very short the conclusions, I think the main conclusions I've just elaborated. So very important here what we see, what we have seen in the first slide where we quantified the value of structural health monitoring, it is dominated by the system reliability and by the system consequences. Because you have seen the system consequences of air are very high, there are 100, which is the usual case for our structural engineering systems. Okay, thank you for your attention.