Optimization of Extended Warranty Cost for Multi-Component Products with Failure Interaction

In practice, the assumption of failure independence between components is seldom valid, especially for those complex systems with complicated failure mechanism. Users can decide whether to purchase extended warranty (EW) at the end of basic warranty, and there are many factors that influence this decision, such as product reliability and EW price. In order to solve the problem of EW pricing for multi-component systems with failure interaction reasonably, considering the failure interaction characteristics between components of the multi-component systems, under the condition of type II failure interaction, this paper constructed a dependent failure rate model and developed a EW cost model of two-component systems. Thus, after optimizing the preventive maintenance (PM) strategy, this paper obtained the optimal PM interval when EW cost is the lowest, which is a win-win strategy to reduce the EW price for manufacturers and users under the premise of ensuring the manufacturer's profit demand. Finally, the validity of the model was verified by a numerical example and sensitivity analysis for important parameters was presented.

Effect of a heterogeneity on tensile failure: interaction between fractures in a limestone

<p>Not all rocks are perfect. Frequently heterogeneities will be present, either in the form of pre-existing fractures, or in the form of sealed fractures. To date, investigation of sample heterogeneity, specifically tensile strength and strength anisotropy has focused on layered rocks, such as shales, sandstones and gneisses. Data is lacking on the effect of single planar heterogeneities, such as pre-existing fractures or stylolites, even though these frequently occur in geo-energy settings.</p><p>We have performed Brazilian Disc tests on limestone samples containing planar heterogeneities, investigating Brazilian test Strength (BtS) and the effects of orientation on strength. We used prefractured Indiana limestone to represent a planar heterogeneity without cohesion and Treuchtlinger Marmor samples with central stylolites to represent a planar heterogeneity of unknown strength (as an example of a sealed fracture). The planar discontinuity was set at different rotation angles of approximately 0–20–30–45–60–90⁰, where 90⁰ (steep angle) is parallel to the principal loading direction, and 0⁰ (low angle) to the horizontal axis of the sample. All experiments were filmed, and where possible Particle Image Velocimetry was used to determine internal particle motion. Moreover, we used a 2D Comsol model in which we simplified the stylolite surface as a sinusoid. The model was used to qualitatively determine how i) a different period of the sinusoid and ii) relative strength of sinusoid/matrix affect the results.</p><p>Our results show that all imperfect samples are weaker than intact samples. The 2D Comsol model indicates that the qualitative results remain unaffected by changing the period (assumed to be representative of roughness) of the cohesive heterogeneity, nor by the relative strength contrast: the location of the first fracture remains unaffected. For both heterogeneity types, the fracture patterns can be divided into four categories, with two clear endmembers, and a more diffusive subdivision in between.</p><p>For a cohesion-less heterogeneity:</p><ul><li>steep angles lead to frictional sliding along the interface, and only a small hypothesized permeability increase.</li> <li>Intermediate angles lead to a combination of tensile failure of the matrix and sliding along the interface, where for steeper angles more new fractures form which follow the path of the existing fracture.</li> <li>Low angles lead to closure of the old fracture and new tensile failure.</li> </ul><p>For a cohesive heterogeneity of unknown cohesion:</p><ul><li>Steep angles lead to intensive failure of the heterogeneous zone, attributed to the presence of a stress concentrator.</li> <li>Intermediate angles lead to partial failure along the heterogeneous zone, and the formation of new fractures in the matrix, potentially instigated by mode II failure to accommodate motion.</li> <li>Low angles lead to the formation of a new fracture plus opening within the heterogeneous zone.</li> </ul><p>These results imply that hydrofracture (i.e. creating tensile stresses) of a stylolite-rich zone will lead to more fractures than fractures in a homogeneous zone, where the orientation of the stylolites and bedding will control the orientation of the permeable pathways.</p>

Effect of a heterogeneity on tensile failure: interaction between fractures in a limestone

<p>Not all rocks are perfect. Frequently heterogeneities will be present, either in the form of pre-existing fractures, or in the form of sealed fractures. Tensile strength and strength anisotropy of rocks has been investigated for strongly layered rocks, such as shales, sandstones and gneisses, but data is lacking on the effect of single planar heterogeneities, such as pre-existing fractures or stylolites. We have performed Brazilian Disc tests on limestone samples containing pre-existing fractures and stylolites, investigating Brazilian test Strength (BtS) and fracture orientation. We used Indiana limestone samples, pre-fractured with the Brazilian Disc method, and Treuchtlinger Marmor samples which contained central stylolites. All experiments were filmed. The planar discontinuity was set at different rotation angles of approximately 0–20–30–45–60–90⁰, where 90⁰ is parallel to the principal loading direction, and 0⁰ to the horizontal axis of the sample. Pre-fracturing Indiana limestone samples results in a cohesion-less planar discontinuity, whereas the stylolites in the Treuchtlinger Marmor samples are discontinuities which have some strength.</p><p>The results show that our imperfect samples with a planar discontinuity are always weaker than an intact sample. For the Indiana limestone, with a cohesion-less interface, there is 10 to 75% of weakening, which is angle-dependent. Once the angle is 30 or lower there is no influence from the initial fracture for the orientation of the new fracture. The stress-displacement pattern followed the expectation for Brazilian Disc testing. However, in the samples with a stylolite, strength is isotropic and between 25 and 65% of the strength of an intact sample. For all cases several new cracks appeared, of which the orientation is influenced by the orientation of the stylolite. The fracture pattern and associated stress drops are more complex for high angles. Interestingly, in the samples with stylolites, always more than one fracture was formed, whereas in the samples with a cohesionless interface usually only one new fracture formed, which for natural settings suggests a potential for higher fracture density when hydrofracturing a stylolite-rich interval.</p><p>A second difference between these datasets is the amplitude of the pre-existing interface. The effect of amplitude will be qualitatively investigated with a 2D Comsol model, to investigate the location of the first fracture occurring, which can then be compared to the camera data of the experiments.</p>

A failure interaction model for multicomponent repairable systems

Modeling of stochastic dependency among components in a repairable system is still a challenging task when dealing with the maintenance of multicomponent systems. With the help of stochastic dependency information, failure of a component brings attention to the components having strong interactions with the failed component. With this information, one can plan the maintenance of components in a better way. Since a change in failure probability of a component (due to deterioration or failure of a component in a given time interval) influences the failure probabilities of other components in the system, therefore, in this article, we consider probability of failure to represent the state of the component to model the stochastic dependency among components. We apply the Bayesian belief network to model such scenario of dependency among the components and present two case studies to compute various probabilities. In the first study, expert elicitation is being used, whereas the time between failure of the components is used in the second case to calculate failure probabilities. To illustrate the applicability of the proposed approach, one case study for each is presented. The first case study takes the case of an army truck through expert elicitation approach whereas the second case study deals with a rolling mill gearbox whose time between failure of components was available.

Availability and maintenance modeling for a two-component system with dependent failures over a finite time horizon

This article studies the availability and optimal maintenance policy for a two-component system with failure interaction over a finite time horizon. Failure of component 1 is soft and can only be detected by inspections. Failure of component 2 is hard and self-announcing. Each hard failure acts as a shock to the first component and increases its hazard rate. Periodic and opportunistic inspections (offered by failures of component 2) are used to reveal the failure of component 1 and followed by replacement decisions. Furthermore, age-based preventive replacement is performed for component 1. Under this maintenance policy, a recursive method is developed to obtain the availability measure of component 1. Furthermore, the total maintenance cost of component 1 over a finite time horizon is analyzed. The objective is to find the optimal maintenance policy for component 1 such that the expected total cost is minimized. A case study on electrical distribution system is provided to validate the effectiveness of the adopted approach.