Volume 7, Issue 2

Reliability Edge Home

The demand for products with high reliability and low manufacturing costs is an ever increasing one in recent times. Inception of a product that beats competition in terms of both reliability and cost requires a design strategy that steers the engineering design process toward higher reliability from the very beginning. Engineering Design By Reliability (EDBR) is a methodology used to produce components that meet a certain reliability goal by designing reliability directly into the components.

In this methodology, all design parameters are taken to be random variables. All parameters that define the failure governing stress acting on a component during its mission are taken to be distributed variables and all parameters defining the failure governing strength exhibited by the component during its mission are also taken as distributed. As a result, the failure governing stress and strength are obtained as distributions using various techniques of synthesizing these distributions. These are then coupled mathematically to obtain the reliability of the component, which is then compared to the target reliability. If the obtained reliability is equal to or greater than the target reliability then the design is accepted; otherwise, the design is iterated until the specified reliability requirement is met.

Advantages of EDBR Over Conventional Design Methods
EDBR is far superior to conventional design methods, where mean values of the design variables are used to arrive at a single value of the nominal stress, which is then modified by a variety of multiplicative single-valued factors. A single value of the nominal strength of the material to be used for the component is obtained from available design books or material property handbooks and multiplied by a number of single-valued factors to arrive at the minimum failure governing strength. Components resulting from such conventional design methods are either over-safe, leading to wastage of resources in terms of material, manufacturing and operational costs; or unsafe, leading to unexpected failures that result in higher warranty costs, loss of production and losses in terms of time and resources spent in redesigning the component. On the other hand, EDBR pins down the exact reliability of the component by taking all design parameters to be distributed, including the variability of operating loads, variations in geometric dimensions and variations in material properties. Thus the uncertainty associated with conventionally designed components is avoided by the EDBR methodology.

Obtaining the Distribution of a Function of Random Variables
Obtaining the failure governing stress and the failure governing strength distributions are the two most important steps in EDBR. These steps involve obtaining the distribution of a function of random variables. Various techniques are available to achieve this, including binary synthesis of distributions, generation of system moments and Monte Carlo simulation. Details of each method are available in Ref. [1].

The binary synthesis of distributions method assumes a normal distribution for all random variables of the desired function and also assumes a normal distribution for the resulting function distribution. The method uses the mean and standard deviation of all the random variables in a function to arrive at the mean and standard deviation of the function. This method may not provide accurate results for cases where a normal distribution assumption does not apply. The generation of system moments method involves obtaining the first four moments of the function whose distribution is sought. Then the Pearson distribution approximation is used to arrive at the distribution. The Monte Carlo simulation method involves the use of simulation to arrive at the desired distribution of the function of random variables. The accuracy of results obtained here depends largely on the number of simulations used. Non-repeatability of results is another minor drawback of this method. However, with increased complexity of problems and availability of software tools, simulation is emerging as the method of choice for many analysts.

Failure Governing Stress Distribution
To obtain the failure governing stress distribution, data in the form of distributions of all the parameters involved in the calculation of stress is needed. The formulations involved in obtaining the stress remain the same as those used in conventional design methods, and the first step is to obtain the stress function based on the applicable failure governing criterion. Then this function is substituted with the distributions of all the design variables involved, and any one of the previously discussed methods of synthesizing distributions is used to arrive at the failure governing stress distribution.

It must be realized that distributions of loads, geometric dimensions and any modifying factors used to obtain the stress distribution are component-specific and have to be determined through observation and experimentation as most of the time general use data for these may be either not available or not applicable. The distribution of loads, moments and torques acting on a component may be determined by attaching transducers to critical locations of the component where stress is expected to be the highest. Distributions of geometric dimensions may be determined by measurements on a sample of components and fitting the most appropriate distribution to the recorded observations. If no such observations are obtainable but a tolerance value is specified and a normal distribution can be assumed for the dimension of the component, then the base dimension can be taken as the mean of the normal distribution, and a sixth of the tolerance as the standard deviation, because the total tolerance is taken to equal six standard deviations. Multiplication factors used should be studied separately to obtain their distributions.

Failure Governing Strength Distribution
Distributional strength data is required for implementation of the EDBR methodology. Distributional material properties such as yield and ultimate strength, Young's modulus, Poisson's ratio, bulk modulus, cycles-to-failure or stress-to-failure data in fatigue, and coefficient of thermal expansion, may be required as per the application and operation environment of the component. These data are usually obtained from well planned and organized tests and tabulated for future use so that the tests do not need to be repeated if material properties of the component remain unchanged in future designs or if the same material is used under identical conditions on a different component design. A number of distributions can be fitted to the strength data obtained from tests. Fortunately, repeated tests and physical background provide an indication regarding the best distribution for a particular type of strength data. For instance, usually the normal distribution represents the static strength data while the lognormal distribution represents the cycles-to-failure distribution. However, when sufficient and trustworthy data is available, the best fit distribution should be used even if the distribution does not match the ones mentioned previously.

Estimation of Component Reliability
Once the failure governing stress and strength distributions are available, the unreliability or reliability of the component can be obtained by using the stress-strength distribution interference approach. Under this approach the unreliability Q of the component is given by the probability that the failure governing stress s exceeds the failure governing strength S. Mathematically this can be expressed as:

Where:

f*(s) is called the failure function. It can be seen that Q is the cumulative function of f*(s). However f*(s) is not a probability density function but simply a failure function and the area under f*(s) is generally less than 1.

It should be noted that the unreliability of the component is not given by all of the area of interference of the stress and strength distributions. The area representing the unreliability may be a part of the area of the interference and at times may lie outside the overlap of the two distributions. For instance, if a stress distribution is normal with a mean value of 50 units and standard deviation of 15 units, while the corresponding failure governing strength distribution is normal with a mean of 75 units and standard deviation of 5 units, then the area bound by the failure function representing the unreliability of the component is as shown in Figure 1.

Figure 1: Stress-Strength distribution interference

The reliability R of the component can be obtained as the difference between 1 and the unreliability value obtained from the above expression of Q. Alternatively, the reliability of the component can be calculated as the probability that strength S is greater than all possible values of stress s, which in mathematical terms is expressed as:

Where:

f' *(s) is called the survival function. Once the reliability of the component is known, it can be compared to the target reliability. If the reliability value equals or exceeds the reliability target then the component design is a successful one. If the value of R obtained above is lower than the target value then the design of the component is modified and the design process is repeated after incorporating changes in the design parameters that lead to a higher probability of mission success for the component. The procedure is iterated until the reliability goal is met. In the remainder of this article, a simple example illustrates the EDBR methodology using ReliaSoft’s Weibull++ software.

Example
Steel rods with a diameter of 0.5 ? 0.015 in. made of AISI 4340 cold drawn and annealed steel are to be used in an application where the load is purely tensile. Investigations of the application reveal that the load is not constant in magnitude. Typical load values are recorded and presented in Table 1. Ultimate tensile strength data from previous strength tests on 50 specimens of AISI 4340 steel rods, cold drawn and annealed, are made available to the design engineers, as shown in Table 2. The design engineers want to estimate the probability of failure of these rods due to fracture and see if they meet the desired goal of no more than 10% failures.  

Table 1: Typical load values (in lbs) for the application

Table 2: Ultimate tensile strength data (in psi) of 50 specimens of AISI 4340 steel rods. Ref [1]

Stress Formulation

Stress formulation in the EDBR methodology is the same as in conventional design methods. For the present example, stress s is simply:

where s represents the stress, F represents the tensile load on the rod and d represents the rod's diameter.

Determination of the Stress Distribution

In Eqn. (5), there are two design variables: the diameter d and the load F. Both of these variables are distributed. The appropriate distributions and the respective parameters of the distributions for both the variables need to be determined.

Assuming that the diameters of the rods 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 0.5. The standard deviation, which is taken to be a sixth of the total tolerance value, is 0.005. The distribution for the rod diameter d can be represented as N(0.5, 0.005).

The available load data can be used to determine the distribution of F using probability plotting, least squares or maximum likelihood estimation. The data can be sent to the Weibull++ software and the Distribution Wizard can be used to decide on the best distribution and calculate the parameters. Weibull++ gives the two-parameter Weibull distribution as the best fit distribution to the present load data. The parameters of this distribution are calculated as beta = 3.3435 and eta = 14278. Thus the distribution of load F can be represented as W(3.3435, 14278).

Once the distributions of all the design variables in the failure governing stress formulation are known, the distribution of the failure governing stress s can be obtained using any of the three methods of synthesizing distributions discussed previously. The Monte Carlo simulation method is illustrated here.

Monte Carlo Simulation

Weibull++ can be used to generate random numbers based on the distribution of the diameter d and the load F. These random numbers can then be used to calculate random stress values using equation Eqn.(5). A distribution can be fitted to these stress values to obtain the failure governing stress distribution.

For this example, a set of 1000 random numbers are generated based on the distribution N(0.5, 0.005) of diameter d. A set of 1000 random numbers are also generated based on the load F distribution W(3.3435,14278). These random numbers from the d and F distributions are substituted into Eqn. (5) to obtain a set of 1000 stress values. Figure 2 shows the generated diameter and load values as well the stress calculation. A distribution is then fitted to the calculated stress values to obtain the failure governing stress distribution. The generalized gamma distribution G(11.1591, 0.3133, 0.7592) is obtained as the best fit distribution.

Figure 2: The Monte Carlo utility used to generate 1000 d and F values and the General Spreadsheet used to calculate the Stress (the first 25 rows are shown)

Determination of the Strength Distribution

A distribution is fitted to the ultimate tensile strength data of Table 2. As expected, the normal distribution gives the best fit to these strength data. The probability plot for the data is shown in Figure 3. The fitted failure governing strength distribution is obtained as N(103420, 2395.0019).

Figure 3: Normal probability plot of the ultimate tensile strength data of Table 2

Calculation of Reliability

The Weibull++ Stress-Strength Wizard is used to estimate the reliability of the steel rods once the failure governing stress distribution G(11.1591, 0.3133, 0.7592) and the failure governing strength distribution N(103420, 2395.0019) have been obtained. The reliability value is obtained as 95.45%. Figure 4 shows the Stress-Strength Wizard result and a plot of the distributions. It is clear that the desired goal of no more than 10% failures is met by the present design.

Figure 4: Stress-Strength plot and Wizard result tensile strength data of Table 2

Alternative Approach Using Monte Carlo Simulation
The Monte Carlo simulation approach described thus far involves a combination of simulation and analytical methods to obtain reliability using stress-strength interference. The Weibull++ software also offers an alternative approach that is based entirely on simulation. Using this approach, a number of random values representing the difference between strength S and stress s are obtained. The distribution for strength is again used here, but for stress, instead of a distribution, a random stress value is directly obtained based on Eqn. (5) and the distributions of the diameter d and load F.

After a number of random values representing the difference between strength and stress are obtained, a normal distribution is fitted to these points and the desired reliability is obtained as the probability that any of these values would be greater than 0. This approach is demonstrated in Figures 5 and 6. The result, 95.69%, is comparable to the one obtained with the other method.

Figure 5: Alternative Monte Carlo approach, Step 1 - Use Monte Carlo utility and user-defined function to generate values that represent the difference between S and s.

Figure 6: Alternative Monte Carlo approach, Step 2 - Fit a normal distribution, N(37093.7627, 21620.5030) and use QCP to calculate probability that generated S - s  values > 0.

Conclusion
This article demonstrates that the Engineering Design by Reliability methodology is a powerful tool in product design that can be used to minimize unexpected failures and control warranty costs. The application of EDBR was illustrated using a simple design example and ReliaSoft’s Weibull++ software.

References
[1] Kececioglu, D.B., Robust Engineering Design-By-Reliability With Emphasis on Mechanical Components and Structural Reliability, DEStech Publications, Lancaster, PA, 2003.
[2] Hahn, G. J. and Shapiro, S.S., Statistical Models in Engineering, John Wiley and Sons, New York, 1967.
[3] Stephenson, S., McCoy, T. and Thomas, J., “Do You Have Enough Strength To Take the Stress?” Proceedings of the International Applied Reliability Symposium, 2005.