Reliability and Maintenance Assessement From Statistics to Probability – Part 2 (2)
In the first part of this article in MaintWorld 1/2013, we described how to assess reliability in mechanical field by conceiving probabilistic models of processes, in this second part we describe three practical examples of this process.
Example 1:
Cantilever Beam
As an example of time independent process is the case of a cantilever beam subjected to bending force. Calculations have been performed by means of the analytic model and of a Finite Element software tool. The Reliability evaluation takes into account the rated values and the tolerances of the physical and geometrical characteristic of the beam and the rated value and the tolerance of the load. Figure 1 (A) describes the layout and the input values and Figure 1 (B) shows the results and the comparison between the various approaches are reported. In this case, the probability density function (pdf) of the stress is resulted to be Lognormal and the pdf of the strength is of the Weibull type. The area in common between the two pdf’s, zoomed in the right figure, represents the unreliability.
Example 2:
High Speed Railway Suspension
Reliability calculation of a High Speed Railway suspension is an example of a time-dependent process; the layout of the suspension is shown in Figure 2 (A).
Figure 1. (a) Cantilever Beam scheme, input variables.
Figure 1. (b) Reliability Calculation by means probabilistic approaches.
FIGURE 2.
(a) Layout of a High Speed Railway suspension;
(b) Residual resistance in the most stressed elements;
(c) Hazard Rate (failure/year) of the weakest element;
(d) Reliability Calculation of the structure.
The deterministic calculation of the structure was performed by means of a Finite Element method by the customer itself and the results are shown in Figure 2 (B) and Figure 2 (C). The probabilistic evaluations were performed sing “Relysoft” tool, solving the Paris-Erdogan law under the following hypothesis:
- Surface treatment: machined
- pdf of aluminum alloy 600A–AlMgSi0.7: Weibull
- Tolerances of geometric dimensions: 2 %, corresponding to 4 standard deviations
- Maximum load 10 % of the rated value corresponding to 4 standard deviations
- pdf of the input parameters: Normal.
The hazard rates have been calculated as a ratio between pdf(t) and R(t). The probability assess- ment puts in evidence the weakness of elements over time, as shown in Figure 2 (D).
FIGURE 3. (a) Layout of the fastening of the insulator to the rail; (b) Histogram and pdf of MTBM of the Jaw.
Example 3:
Third Rail as Electrical Supply
Jaw is a device that fastens the insulator to the rail and this example evaluates the reliability over time of the jaw structure. The assessment concerns the fatigue phenomenon and was performed under the following hypothesis:
- Surface treatment: as forged
- Material: Cast Carbon Steel
- Strength pdf: Weibull
- Stress pdf: Lognormal
- Stress concentration coefficient: Kc = 2.5; cycle per hour: Cy = 576
- Tolerances of geometric dimensions: 0.5 %, corresponding to 4 standard deviation
- Tolerance of the load: 10 %, corresponding to 4 standard deviation.
The layout and the model of the rail are shown in Figure 3 (A) and the pdf of the meantime to maintenance in Figure 3 (B), the subtended area gives the probability of maintenance need for each instant in time.
»»References ›› [1] MIL-HDBK-217- F2- N2, “Reliability Prediction of Electronic Equipment”. ›› [2] Ken Blakely, “Using Design Sensitivity for Statistical Response Analysis”, The MacNeal Schwendler Corporation, CA, 1975 ›› [3] Naval Surface Warfare Center – Carderock Division, “Handbook of Reliability Prediction Procedure for Mechanical Equipment”, CARDEROCK DIV, NSWC-07, SEPT 2007 ›› [4] Richard E. Barlow, “A Bayes Explanation of an apparent Failure Rate Paradox”, IEEE Trans. On Reliability, Vol, R-34, NO. 2, June 1985 ›› [5] S.R.Calabro, B.G.Horowitz, “Evaluation Equipment Reliability using Techniques based on kappa square statistics”, IEEE Trans. on Power Apparatus and Systems, Vol. PAS-100, N 8, Aug. 1981. ›› [6] NTIS# PB2003–01115–SSC–420, “FAILU RE DEFINITION FOR STRUC TURAL RELIABILITY ASSESSMENT”, Ship Structure Committee – 2002 ›› [7] Ministero delle Infrastrutture e dei Trasporti, “TESTO UNICO , NORME TECNICHE PER LE CO STRUZIONI” ›› [8] AD813574,“Reliability Prediction – Mechanical Stress/Strength Interference”, Technical Report , RADC–TR–66–710, March 1967 ›› [9] RAC, Reliability Analysis Center, .DoD, NPRD ,“Non Electronic Parts Reliability Data”, 1995 ›› [10] R. Paggi, G.Mariotti, & alias “An Integrated Approach to RCM”, 17th International Conference of the Israel Society for Quality, Gerusalem, 16–20 Nov 2008 page 65 ›› [11] R. Paggi, V.Bisti,. & alias “Posterior Probability Density Function in Simulation Processes via Bayesian Approach”, 18th International Conference of the Israel Society for Quality, Gerusalem, 15–19 Nov 2010 ›› [12] R. Paggi, A. Paggi & alias, “Probabilistic Approach to the design through the use of Nastran”, 2011 Italian Users Conference, Torino 12–13 Oct, 2011.
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