Strength distribution

Many methods for the evaluation of from equation ( Al.5.20) use moments of the dipole oscillator strength distribution (DOSD) defined, for molecule A, by... 

Having previously introduced the key methods to determine the important variables with respect to stress and strength distributions, the most acceptable way to predict mechanical component reliability is by applying SSI theory (Dhillon, 1980). SSI analysis is one of the oldest methods to assess structural reliability, and is the most commonly used method because of its simplicity, ease and economy (Murty and Naikan, 1997 Sundararajan and Witt, 1995). It is a practical engineering tool used for quantitatively predicting the reliability of mechanical components subjected to mechanical loading (Sadlon, 1993) and has been described as a simulative model of failure (Dasgupta and Pecht, 1991). 

The approaeh taken by Carter (1986, 1997) to determine the reliability when multiple load applieations are experieneed (equation 4.34) is first to present a Safety Margin, SM, a non-dimensional quantity to indieate the separation of the stress and strength distributions as given by ... 

This is essentially the eoupling equation for the ease when both stress and strength are a Normal distribution. A parameter to define the relative shapes of the stress and strength distributions is also presented, ealled the Loading Roughness, LR, given by ... 

Figure 4.30 Relative shape of loading stress and strength distributions for various loading roughnesses and arbitrary safety margin...

The numerieal solution of equation 4.35 is suffieient in most eases to provide a reasonable answer for reliability with multiple load applieations for any eom bination of loading stress and strength distribution (Freudenthal et al., 1966). 

Consider the situation where the loading stress on a eomponent is given as Z, A (350,40) MPa relating to a Normal distribution with a mean of /i = 350 MPa and standard deviation cr = 40 MPa. The strength distribution of the eomponent is A (500, 50) MPa. It is required to find the reliability for these eonditions using eaeh approaeh above, given that the load will be applied 1000 times during a defined duty eyele. 

Beeause the strength distribution is Normal, we ean determine the Standard Normal variate, z, as ... 

Figure 4.34 shows the reliability as a funetion of the number of load applieations, using the three approaehes deseribed to determine There is a large diserepaney between the reliability values ealeulated for n = 1000. Repeating the exereise for the same loading stress, A (350,40) MPa, but with a strength distribution of... 

The calculated loading stress, L, on a component is not only a function of applied load, but also the stress analysis technique used to find the stress, the geometry, and the failure theory used (Ullman, 1992). Using the variance equation, the parameters for the dimensional variation estimates and the applied load distribution, a statistical failure theory can then be formulated to determine the stress distribution, f L). This is then used in the SSI analysis to determine the probability of failure together with material strength distribution f S). 

In designing the eon-rod, we wish to ensure that the pin will fail, in the ease of an overload, in preferenee to the eon-rod. To realize this, the mean values of their individual strength distributions are to be set apart by a margin to ensure this requirement. In this way, the probability of eon-rod failure will beeome insignifieant to that of the pin. The foree to shear the pin in an overload situation is a funetion of the ultimate shear strength, t, of the material. The relationship between the ultimate tensile and shear properties for steel is (Green, 1992) ... 

Suppose again that both the stress and strength distributions of interest are of the Normal type, where the loading stress is given as L A (350,40) MPa and the strength distribution is S A (500, 50) MPa. The Normal distribution eannot be used with the integral transform method, but ean be approximated by the 3-parameter Weibull distribution where the CDF is in elosed form. It was determined above that the loading stress parameters for the 3-parameter Weibull distribution were ... 

Vemia, A. K. and Murty, A. S. R. 1989 A Reliability Design Procedure for Arbitrary Stress-Strength Distributions. Reliability Engineering and System Safety, 26, 363-367. 

The effect of varying the mean pore size and number of pores on the predicted strength distributions of grade H-451 graphite are shown in Fig. 25. In this instance, both the mean size (SJ and number (N) of pores per unit volume was varied in the "SIFTING" code. This was necessary because the density was held constant, i.e., the pore fraction remains unchanged, and therefore a decrease of the mean pore size must be accompanied by an increase in the number of pores per unit... 

The environmental effects on the failure rate may be modeled using Arrhenius or pow ei laws. In some cases it may be necessary to model the failure rate using the technique of overlapping -.iress/ strength distributions (Haugen, 1972). 

Here Z is the charge of the projectile with velocity v. In order to calculate stopping powers for atomic and molecular targets with reliability, however, one must choose a one-electron basis set appropriate for calculation of the generalized oscillator strength distribution (GOSD). The development of reasonable criteria for the choice of a reliable basis for such calculations is the concern of this paper. 

Thus the Bethe sum rule is fulfilled exactly in the RPA at all values of the momentum transferred, provided that a complete basis set is used. Therefore, as in the case of the TRK sum rule when optical transition properties (q = 0) are considered, we expect that the BSR sum rule will be useful in evaluating basis set completeness when generalized oscillator strength distributions are calculated, for example for use in calculating stopping cross sections. It should be noted [12] that the completeness of the computational basis set is dependent on q, and thus care needs be taken to evaluate the BSR at various values of q. 

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