The simultaneous influence of several random quantities can be studied by the Latin hypercube sampling method (LHS). The values of distribution functions of each quantity are distributed uniformly in the interval (0; 1) and these values of all variables are randomly combined. This method yields statistical characteristics with less simulation experiments than the Monte Carlo method. In this chapter, the creation of the randomized input values is explained.
- Monte Carlo method
- Latin Hypercube Sampling
- probabilistic transformation
Permanent effort exists to make the components and constructions more reliable and with longer life. Higher reliability can be achieved by better design and more ample dimensioning of load-carrying parts and by better maintenance. However, all these cost money, and one can ask: How is reliability related to the costs? Does an optimum reliability exist from the costs’ point of view? How can it be found? This chapter is devoted to the following issues: (1) optimum time for the renewal of objects with gradual deterioration, (2) optimum dimensions of the cross-section of load-carrying components that can fail suddenly (e.g. due to overloading) or due to fatigue or similar processes, (3) optimum probability of failure, and (4) cost-based optimum strategy of the maintenance and renewal of a group of objects.
1. Optimum time for the renewal of deteriorating objects
Examples of deteriorating objects are machines, cars, bridges, cutting tools, pumps, or aircrafts. Basically, there are three kinds of costs related to these objects: (1) purchase costs
Note: The concept of probable failure-related costs
The optimum lifetime from the economic point of view is that the object attains during its life the minimum costs per unit of the demanded production or service [e.g. per 1 km for a vehicle, per one machined part in the case of a cutting tool, per time unit of service (e.g. year) of a bridge, etc.]. The lifetime means the time until the demolition, complete overhaul, or replacement by a new one. Equation (1) can be rewritten as
The time course of individual components and of the total unit costs is depicted schematically in Figure 2. One can see that the total unit costs
The optimum condition for renewal must be understood in a wider sense. From a mathematical point of view, the optimum is exactly the time, for which d
2. Optimum dimensioning of load-carrying components — sudden failure
Sudden failures occur due to a “weak” component or overloading from external cause (e.g. snow, wind, flood, or error in operation). The failure can be a fracture, permanent deformation, or collapse of a structure. Failure by overloading can occur at any time. If it happens before the end of the assumed service life, the system must be put back into its original state either by repairing the failed component or by replacing it by a new one. More ample dimensioning of the load-carrying parts means higher safety and lower probability of failure due to overloading, but also higher price of the component. One can thus seek such dimensions, which will guarantee the best combination of high reliability and low costs.
The influence of the size of the cross-section on the probability of failure and total costs was studied by Menčík . The main results for a bar loaded by tensile force are shown in Figures 3 to 5. The nondimensional scales express the ratio of individual values (
Typical is the fast increase of costs with the decreasing cross-section at the left part of the diagram. For small cross-sections, the probability of failure increases and the failure-related costs are no more negligible. The cost growth after the minimum has been reached is usually much slower. The analysis for various cases has also revealed that the economically optimal failure probability is sometimes relatively high (e.g. 10–3 or even more). This is acceptable if the costs caused by a failure are low. For higher failure losses, the optimum failure probability will be lower. It is thus important that all possible losses be included into
Based on the same data as above, Figure 4 shows the total costs
These examples assumed that the cross-section area can change continuously. The reality is often more complex: the cross-section of standardized components changes stepwise, and so also the price. The computations should thus be done for all possible nominal cross-sections and arrangements, and the variant with minimum total costs can be chosen.
3. Optimum dimensioning of components with stress-enhanced deterioration
Examples of gradual deterioration are fatigue or wear of metallic parts as well as corrosion, creep at increased temperature, or carbonation of concrete. In these cases, the failure occurs after some time of operation depending on the load. Thus, the costs must be related to the time to failure or replacement of the component
The time to failure depends on the load effect
The load effect
and can be ensured by proper dimensioning (
The relationships between cross-section size, costs, and time to failure may be used for studying the influence of the cross-section size on the cost-effectiveness. Figures 5 and 6 show the development of unit costs and time to failure as the function of the diameter of a shaft loaded in tension (Fig. 5) and bending (Fig. 6a,b). Also, these diagrams can be plotted in non-dimensional coordinates.
Several important conclusions can be drawn from this analysis. An optimum with minimum unit costs exists only in some cases. Sometimes, it lies outside the suitable interval of service times or acceptable dimensions of the cross-section. An important role is played by the kind of loading and fatigue exponent
Because of the many factors involved, it is impossible to formulate simple universal rules for finding the optimum size. A more practical way is to model the situation for admissible ranges of the input quantities. A graphical representation is very useful. Sometimes, it becomes obvious that the unit costs vary only monotonously or insignificantly in the possible range of input variables, which can make the choice of optimum dimensions easier.
4. Cost-based maintenance optimization of long-life objects
Examples of such objects are bridges, cooling towers, locomotives, or heavy machines in power plants. Two cases can be distinguished: maintenance optimization of a single object, and of a group of similar objects, such as bridges in a railway or road network.
Various strategies can be used for organizing the maintenance and repairs (e.g. no repair until the replacement, or regular maintenance plus small repairs), which would enable a longer time of operation until replacement. The optimization consists of comparing the costs
In an ideal case, the optimum variant for maintenance of each object in the network is found as described above. In real life, however, various constraints must often be respected. Very important is the amount of money available for maintenance and repairs in the individual years. It could happen that more bridges should be reconstructed simultaneously than the budget would permit, whereas, in other years, the working capacities for repairs would not be fully used. Thus, it is necessary to calculate the cost components for every bridge and maintenance variant for every year in the considered time interval, in which the renewal should be optimized (e.g. 20 years). These costs can be written into a table with the columns corresponding to years and the rows to individual bridges and maintenance variants (Fig. 8) and compared with the money available. A comparison of all variants reveals the best strategy.
In addition to the demand of uniform flow of money for maintenance and repairs, several other factors must be considered when prioritizing the objects (e.g. bridges) for repairs:
The condition of the stock as a whole (the worst objects are of special interest);
The rate of deterioration, as some objects can be in a better condition but deteriorate faster;
The importance of the object in the whole network or in some region.
Step 1. Condition rating prioritization. All objects are ranked according to their condition as revealed by inspections. Only those with condition worse than a certain value will be considered for the maintenance in the near future. This preselection significantly reduces the number of candidates for further steps.
Step 2. Object importance prioritization. The role of individual objects in the network is considered. One bridge can be in a worse condition than another bridge. However, if it lies on an unimportant road, whereas the other is on a main road, the latter one will be repaired preferably.
Step 3. Optimization of money allocation. The possible maintenance strategies are compared with respect to the available money and the work capacities. Then, the strategy is chosen, which is economically most favorable for the stock in the longer period.
5. Time value of money
In long-term planning, one must be aware of the difference between spending money today and in the future. Due to interests, the value of (suitably deposited) money gradually increases, so that the today’s value
However, due to inflation, the prices of material, components, and work gradually grow, and the gain from postponing an investment is smaller. If the inflation rate is close to the interest rate, the profit can be negligible, and the standardization (9) is not necessary for the comparison of individual variants.
Menčík J. Optimum-Reliability Dimensioning: Cost-Based Approach. In: 3rd International Colloquium ASRANet 2006. Glasgow, 10 – 12th July 2006. Universities of Glasgow and Strathclyde, ASRANet Book of abstracts, p. 29, full text: CD-ROM. ISBN: 0-9553550-0-4.
Bień J, Elfgren L, Olofsson J, editors. Sustainable bridges. Wroclav: Dolnoślaskie Wydawnictvo Edukacyjne; 2007. 490 p.
Menčík J, Rudolf P. Optimisation of maintenance strategy in a bridge network. In: Proc. Int. Conf.: Reliability and Diagnostics of Transport Structures and Means 2005 (J. Menčík, editor). Pardubice: University of Pardubice; 2005. p. 224 – 231.
Menčík J. Cost-based design and renewal strategy for long-life structures. In: Life-Cycle Civil Engineering (F. Biondini and D.M. Frangopol, editors), Varenna, 10-14 June 2008, Leiden: CRC Press (Taylor and Francis); 2008, p. 385 – 390. ISBN 978-0-415-46857-2.