*6.1. Strength-Stress Model and Reliability Calculations*

The reliability of an engineering product deals with the undesirable events or failures during its service life. It can be precisely defined as the reliability of a part is the probability that, when operated under defined set of conditions, the part will perform its intended function adequately for a specified interval of time [19]. It is an established fact that apparently identical parts operating under similar conditions fail at different points in time. This brings about a need to describe failure phenomena in probabilistic terms and therefore, fundamental aspects of reliability heavily rely on concepts from probability.

The classic strength-limited design suggests the strength should be greater than the stress. A design factor is always added to cover the uncertainties. If strength and stress distributions are known, the reliability of a part can be determined using interference theory presented in [8]. For a strength-limited design, let the density function for the strength is f1 and that for stress is f2, the reliability function will be a joint probability function, where

$$\begin{aligned} \text{P}(\text{S} > \sigma) &= \text{P}[\text{S} - \sigma > 0] = \text{R} \\ \text{R} &= \int\_{-\infty}^{\infty} \text{f}\_1 \text{ (S)} \left[ \int\_{S}^{\infty} \text{f}\_2 \left( \sigma \right) \text{d}\sigma \right] \text{dS} \end{aligned} \tag{3}$$

where, S is the significant strength and σ is the significant load-induced stress. The task for a given design is to ensure that S > σ. Based on fatigue life of specimens obtained at different stress levels in Section 5, the reliability of cast specimens is estimated using this model in this work.

Reliability computations are done for two scenarios: (i) Time-independent loadinduced stress and (ii) Time-dependent load-induced stress. Four different load-induced stress values are selected based on the expected loading conditions on steel castings, i.e., 79 MPa, 87 Mpa, 96 Mpa and 104 Mpa. FE-safe combines the variability in both material fatigue strength and applied loading (if any), to calculate the probability of failure for a specified life. For time-independent load-induced stress case, the reliability computations are based on normally distributed stress and Weibull distributed strength. The details of failure rate calculations are presented in [8].

The reliability analysis for time-dependent load-induced stress provides a more conservative estimate of component performance during service life. The strength-stress interference theory is also applicable for this scenario, but load-induced stress cannot be modeled through normal distribution. For this reason, Fe-safe could not be used for reliability computations in this case. Instead, analytical methods proposed by Samar et al. [20]

are used, which models both the strength and the stress through Weibull distribution. The probability density function of strength S and stress σ distributions are given by:

$$\text{If } \mathbf{f}\_1(\mathbf{S}) = \frac{\boldsymbol{\upbeta}\_{\mathbf{S}}}{\boldsymbol{\upTheta}\_{\mathbf{S}}} \left(\frac{\mathbf{S}}{\boldsymbol{\upTheta}\_{\mathbf{S}}}\right)^{\boldsymbol{\upbeta}\_{\mathbf{S}} - 1}. \exp\left(-\frac{\mathbf{S}}{\boldsymbol{\upTheta}\_{\mathbf{S}}}\right)^{\boldsymbol{\upbeta}\_{\mathbf{S}}}\tag{4}$$

$$f\_2(\sigma) = \frac{\wp\_{\sigma}}{\Theta\_{\sigma}} \left(\frac{\sigma}{\Theta\_{\sigma}}\right)^{\beta\_{\sigma}-1}. \exp\left(-\frac{\sigma}{\Theta\_{\sigma}}\right)^{\beta\_{\sigma}}\tag{5}$$

And the resultant reliability function is similar to Equation (3). The change in loadinduced stress with time can be modeled through Rayleigh distribution, which is a special case of Weibull distribution with shape parameter β equal to 2. If βS = 2βσ, then the reliability analysis is based on Weibull distributed strength and Rayleigh distributed loadinduced stress [8]. Using the results presented by Samar et al. [20] and with βS = 2βσ, the reliability function is given by

$$\mathcal{R} = \mathcal{P}(\mathcal{S} > \sigma) = \frac{\theta\_{\mathcal{S}}}{\theta\_{\sigma}} \sqrt{\pi}. \exp\left(\frac{1}{4} \left(\frac{\theta\_{\mathcal{S}}}{\theta\_{\sigma}}\right)^{2}\right). \left\{1 - \phi\left[\frac{1}{\sqrt{2}} \cdot \left(\frac{\theta\_{\mathcal{S}}}{\theta\_{\sigma}}\right)\right] \right\} \tag{6}$$

Hence, the reliability can be estimated against the ratio of scale parameters, i.e., θS θσ for the targeted lives. Here, the θS θσ ratio is approximated to be similar to that of the Sσ ratios for the targeted lives.
