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Geometrical characteristics of a machine's parts can be selected by reliability criterion. The parameters (thickness, length, cross-section area) can be chosen so that the total reliability of machine is equal to the required value. If the height dimension h of a spring corresponds to the required reliability, the reliability of the safety lock consisting of 5 springs is smaller. In this example the geometrical parameter is the height. This means that the variation coefficient defines the reliability of a "spring-pin" set and, together with other geometric characteristics, the reliability of the whole lock.

The condition of failure is defined at the stress coordinate axis. To transform the known statistical data for force at the axis, a special geometrical coefficient ki is used. Variation coefficients for force and stress are equal if the cross-sectional area is constant. The last formula relates the geometry coefficient and the reliability index.

There is linear dependence between the cross-sectional area A and the reliability index for the bar if all other variables are constant.

The dependence "reliability index - thickness" is also linear for the pipe.

A restriction can be specified for maximum deflection, linear displacement or angle of twist. For such a restriction, the dependence between the geometrical coefficient and the reliability index is also linear. There are one-sided (A and B) and two-sided (C) restrictions. The last condition, C or two-sided restrictions, is more severe.

The table shows values of geometrical coefficients for different loading conditions. The geometrical coefficient is the ratio of Stress / Force for tension if the strength criterion is used.

To meet a rigidity criterion for a cantilevered beam, the reliability index b is inversely proportional to L3. With beam length increase the reliability for the rigidity criterion decreases.

Dispersion in the size of wire (the variation coefficient) is reflected in dispersion of inner stress or inner force. The bigger dispersion of the force, the smaller the reliability index b.

The yield strength and cross-sectional area are independent variables. The bigger the dispersion, the smaller the reliability of the rivet.



A dictionary definition of risk is "to expose to the chance of injury or loss". For technical systems risk is a measure of probability of failures or accidents during a given period of time and/or measure of their consequences. The consequences can be social (severe injury or fatalities), technical (out of service), economical (financial losses) or ecological (environment contamination). The basic formula for risk is:

Risk = Probability of failure * Consequences.


r = F * C

The risk is measured in units of consequence (losses), for example, money, persons, etc. For a complex structure a risk table is used.

The total risk for a system is the sum of the risks of its elements (failure scenarios). Engineers often use the frequency or failure rate, [year-1] instead of probability of failure in the formula for risk assessment.

There two levels of acceptable (1) and unacceptable (2) risks. There are three ranges of risks: negligibly small, reasonable losses, unacceptable losses. Acceptable risk of severe injury in Europe is 10-6 1/year. The range of real risks is 10-4 to 10-10 year-1. The most dangerous objects are chemical plants, oil refineries and transportation systems.

A parameter such as risk is important in comparative analysis. Nuclear power plants are unique engineering structures, there are few similarities between any two plants. The summed service time of all nuclear reactors in the world does not exceed 106 years. Unfortunately, there have been human losses at nuclear power plants.

A probabilistic safety analysis of a nuclear power plant piping system can be made by corresponding classes of consequences.

With time, the probability of failure and the remaining cost of a machine affects the risk measure. Situations where the probability is equal to 0 or 1 are certainty situations - not risky situations. Risk changes with time for the machine with shown parameters.

The figure shows a map of a plant site with areas of risk of severe injury to a person per year. The information helps to choose the location for personnel.

Risk of failure can be decreased to ALARA (As Low As Reasonably Achievable) level by accounting for all economical aspects of a new plant. Risk of failure can be decreased to ALAPA (As Low As Practically Achievable) level. For most complex technical systems, risk of financial losses due to structural element failure can not be decreased to zero.




All failures can be categorized according to frequency of occurence and consequence of failure. The figure shows the classification of consequence levels for chemical plants. Frequency and consequence levels can be described by a table of safety classes where A is the most dangerous state and D is the safest state.

Besides levels of consequence and frequency, the numerical values of the parameters define zones of acceptable and unacceptable risks.

For complex engineering systems, for nuclear power plant piping for example, the classification can made by consequence and failure mechanism. This classification defines priority in the identifying of inspection regions. The highest priority is given to large leakages in pipe, nozzles, and welds with severe consequence to the safety of the plant and personal - A.

Safety classes can be categorized by stress state (from unidirectional to three-dimensional + residual stress), stress / yield strength ratio, materials or temperature. Safety class of a structure is the minimum rating after considering all the factors.

The risk - based design consists of 9 steps. Identification of failure scenarios is the first step of the analysis.


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