**Design - BEng 2 - DSGN 215**

**Statistical Considerations.**

Variability in materials, in manufacturing processes and in expected loads is inevitable and needs to be considered during the design process.

**Variability in materials**

Only a small number of metals have been extensively tested to provide statistical information about the spread of their properties. In the majority of cases the properties have a normal or 'Gaussian' distribution which can be defined by the mean and the standard deviation. However work on a titanium alloy revealed a highly skewed distribution of tensile strengths.

E values of different samples of the medium carbon SAE 4340 steel were found to vary by + or - 3%.

The UTS of samples from finished rolled SAE 1020 structural steel was found to have a standard deviation of 3% of the mean UTS.

By exerting close control over rolling / annealing schedules or over tempering temperatures, tailoring processing to individual 'heats' - tolerance bands can be reduced somewhat - but at additional costs.

**Variability in component size**

Similar considerations apply when producing parts to a specific size. Generally by choosing an additional process, accuracy can be improved - but at a rapidly increasing cost.

It is therefore essential that the widest tolerance band that is acceptable is specified to keep cost down. Information about tolerances and how they are applied in design is available in many text books.

**Variability in loading**

Virtually all products designed by mechanical engineers will be subjected to fluctuating loads. In cases where the load is fluctuating in a regular sinusoidal manner, the mean stress and stress range in the component can be calculated and checked with a modified Goodman diagram. Problems arise when loads are of a statistical nature, eg wind, wave and vehicle suspension loads, being three common ones. Over the past 30 years the automotive industry has spent a lot of time and money recording and analysing vehicle suspension loads with the result that they now have a good idea of the different load spectra that different types of vehicle suspensions (and consequently the vehicle structures and their occupants) will face. Where this data is not available, estimates need to be made. The data or estimates are then converted into equivalent numbers of cycles of specified amplitude. The cumulative fatigue damage that these stress cycles cause can then be assessed with the Palmgren - Miner theory (Miner's rule). This can be written as:

(n1/N1) + (n2/N2) +..+ (ni/Ni) = C

where n is the number of cycles of a particular stress applied and N is the life corresponding to that stress. The constant C is usually in the range 0.7 to 2.2 and a value of 1 is often used.

Further reading - 'Mechanical Engineering Design', by J E Shigley .., chapter 7.

Return to Module Introduction.

David J Grieve, updated 21st July 2004, original: September 1998.