As demand increases for robust products with high reliability and low manufacturing and operating costs, probabilistic design methods are being widely employed to design reliability directly into products (e.g., through Design for Reliability (DFR) initiatives, as described previously in this issue). The purpose of this article is to demonstrate the methodology of reliability analysis based on physics of failure by using the example of fatigue failure of crankshafts. The same methodology can be applied with minor modifications to other mechanical, electrical and electronic components to predict life under fatigue and it also can be generalized for other failure modes. ## Fatigue FailuresFatigue failures are failures caused in components under the action of fluctuating loads. They are estimated to be responsible for 90% of all metallic failures since loads on the components usually are not constant but instead vary with time. Fatigue failures occur when components are subjected to a large number of cycles of the applied stress. With fatigue, components fail under stress values much below the ultimate strength of the material and often even below the yield strength. What makes fatigue failures even more dangerous is the fact that they occur suddenly, without warning. The failure begins with a minute crack that is so small that it may not be detected by non-destructive methods such as X-ray inspection. The crack may get initiated by internal cracks in the component or irregularities in manufacturing. Once a crack has formed, it propagates rapidly under the effect of stress concentration until the stressed area decreases so much that it leads to a sudden failure. Fatigue is a complex phenomenon that is affected by a number of factors, such as the surface finish of the component, environmental effects, heat treatments, presence of stress concentration factors, etc. It is therefore important to carefully analyze components subjected to fluctuating loads so that the desired reliability can be built into these components and over-designed or under-designed components can be avoided. ## Fatigue TestingOne of the most common tests for components susceptible
to fatigue failures involves running specimens to failure at a constant
stress amplitude. Each specimen is loaded in four-point bending and the
stress on the specimen is cycled by rotating the specimen. The number of
cycles to failure is noted for the stress value. The test is then repeated
for different values of stress. It has been found that the cycles to
failure,
where
## Limitations of the S-N CurveAlthough S-N curves are widely used in
relation to fatigue failures, they have a number of shortcomings. The first
drawback is that the curves are obtained by testing standard specimens under
laboratory conditions. Thus the tests are unable to capture the variation of
actual components due to factors such as manufacturing defects. The tests
also ignore the actual use conditions of the components, which are usually
different from the test conditions. Another major drawback is that S-N
curves ignore the probabilistic nature of fatigue failures. As demonstrated
in Figure 2, the cycles to failure corresponding to a certain stress value
may vary considerably even when carefully machined specimens out of the same
lot of material are used. The reduction factors used to account for this are
an inaccurate way of dealing with this variation. A better approach is to
use
## Example of Crankshaft Fatigue FailuresConsider the case of the fatigue failure of the crankshaft shown in Figure 3, which has failed in pure bending. Assume that the manufacturer wants to be able to accurately predict the reliability of the crankshafts after one year of service at a stress of 20 ksi.
To obtain accurate results, it is decided
to randomly select crankshafts from different batches and conduct tests on
these units. It is also decided to use accelerated testing to reduce testing
time. From Eqn. (1), it can be noted that the
relationship between stress,
From the comparison of Eqns. (2) and (3), we have
It is also known from the theory of fatigue failures that, at each stress, the cycles to failure are lognormally distributed. Thus an ALTA model with the inverse power law life-stress relationship and a lognormal distribution can be used to represent the S-N-P curve. 30 crankshafts are randomly selected and assigned equally to three accelerated stress levels of 39.1 ksi, 63.1 ksi and 79.4 ksi. Table 1 shows the cycles to failure obtained from these tests. The data of Table 1 are entered into
the ALTA software to obtain the S-N-P curve for the fatigue failure of the
crankshafts (see Figure 4). The inverse power law - lognormal model with the
parameters of
Table 1: Cycles to failure obtained from accelerated testing at three stress
levels
The fitted ALTA model can now be used to obtain the distribution of cycles to failure at the stress level of 20 ksi. This is done by setting the "use stress" as 20 ksi in ALTA. Figure 5 shows the distribution of the cycles to failure obtained in this manner.
## Usage DistributionThe manufacturer realizes that different customers will
use the crankshaft differently for the one-year period. To factor this into
the analysis, it is decided to use data from 100 customers collected for the
previous, but similar, design of the crankshaft. It is expected that the use
of the crankshaft for the present design under test will be similar to the
experience for the previous design. ReliaSoft's Weibull++ software is used
to fit a distribution to the cycles used per year for the 100 customers. The
resulting usage distribution is the generalized gamma distribution with the
parameters of
## Calculation of ReliabilityThe fatigue failure of the crankshafts at the end of one
year will not occur as long as the cycles used in one year are less than the
value of the critical cycles to failure at a stress of 20 ksi. However, it
must be noted that both the cycles used per year and the critical cycles at
the stress of 20 ksi are random variables. To obtain the probability that
the cycles to failure used will exceed the critical cycles at 20 ksi, the
manufacturer needs to compare the distribution of the critical cycles to
failure at 20 ksi (lognormal distribution with parameters of
Although it seems that this analysis is thorough, since it takes into consideration the variation in the crankshafts by testing a random sample and also accounts for the variation in customer usage, it can be further improved. An important point missed in the initial Weibull++ analysis is that the manufacturer assumes that the field stress is constant at 20 ksi. However the stress experienced by the crankshafts will depend upon the loads in the field and the geometrical variation introduced by manufacturing. The next section discusses an analysis approach that accounts for this variation. ## Field Stress DistributionThe failure of the crankshafts in the present case occurs in bending. The bending stress equation applicable here is:
where The manufacturer is able to obtain the distribution for
the bending moment based on the data collected for the previous crankshaft
design. Usually such data are collected by attaching appropriate sensors to
a few units and recording observations over the component's life. The
distribution of the bending moment for the present case is found to be the
three-parameter Weibull distribution with parameters of
The distribution for the diameter is obtained using the tolerance achieved during manufacturing. The tolerance for the diameter section of interest is 1 ± 0.015 inch. Assuming that the diameters are normally distributed, the parameters of the corresponding normal distribution can be determined from the given base dimension and tolerance value. Thus the mean of the assumed normal distribution, which is taken to be the base dimension, is 1 inch. The standard deviation, which is taken to be a sixth of the total tolerance, is 0.005 inch. Once the distributions of all the variables in the stress equation are known, the distribution of the stress can be obtained using Monte Carlo simulation in Weibull++. Weibull++ generates a number of random values for the bending moment and diameter based on their respective distributions. Then using Eqn. (4), each pair of these random values is used to obtain a value of the bending stress and a distribution is fitted to all such values. (For details see reference [2].) Figure 8 shows the stress distribution obtained in this way.
## RENO ModelTo accurately predict the reliability of the crankshafts after one year of service, an advanced model is required that takes into account the S-N-P distribution for the fatigue failure of the crankshafts, the usage distribution for the crankshafts and the distribution of the field stress experienced by the crankshafts. Such a model can be created easily in ReliaSoft's RENO software, as shown next.
The basis of the RENO flowchart simulation (presented in Figure 9) is the fact that any crankshaft will not fail in a year as long as the cycles used do not exceed the critical cycles at the field stress level. The RENO model can be used to run a number of simulations to predict the
required reliability. During each simulation, the RENO model generates a
random bending moment value,
The results from the RENO model can be extended to obtain a plot of unreliability at the end of each year of service for a desired time period. This is done by using the option of sensitivity analysis shown in Figure 10. The RENO model applies the usage distribution for one year of service to other periods by simply multiplying the random usage cycles generated by the time period. For example, while calculating the probability of failure of the crankshafts at the end of two years, the generated random value of the cycles used is multiplied by a factor of two.
## ConclusionIn this article we have shown how principles of mechanical design, reliability and customer usage data can be combined effectively to obtain accurate reliability predictions and to design reliability directly into components that are subject to complex failure mechanisms. The combined use of different software tools offered by ReliaSoft makes such analyses easy and efficient. ## References[1] Kececioglu, D.B., [2] ReliaSoft,
"Engineering Design by Reliability," [3] ReliaSoft, [4] Shigley, J.E. and Mischke,
C.R., |