Analysis of Engineering Uncertainties

1. randomness - uncertainty due to inherently random nature.
   
2. fuzziness - uncertainty caused by that the object is too complicated to understand, or by insufficient knowledge.
   
3. ambiguity - uncertainty contained in natural language.
   
4. vagueness - uncertainty included in, for instance, image processing.
   
5. imprecision - uncertainty due to lack of information.
   
6. generality - uncertainty due to multi-meanings or multi-interpretations for the object.
   

Among these uncertainties, the most essential and important is unquestionably the randomness which most of researches focuss upon.

As for failure phenomena, they will have inherent statistical properties of temporal as well as spatial nature. In fact, experimental data on strength or life of the material have been reported to have a wide variety of scatter even under the same loading conditions. The major reasons for these variabilities could be pointed out as follows [Ref (8)]:

1. thermal motion or thermal fluctuation of composed atoms.
   
2. spatial distribution of atomic voids and micro/macro cracks.
   
3. comparatively wide range of inhomogeneity such as imperfect mixture of composed materials, existence of abnormal liers, differences in cooling speed, cold press and moisture contents, etc..
   
4. difference in sizes and dimensions of the specimen.
   

Causes of the variability of strength of the material thus range fairly widely from the atomic or molecular level to the macroscopic level. These factors may differ in each location of the material and hence cannot be considered deterministic. The analysis of the phenomena in the order of atom or molecule requires a statistical dynamic approach. However, it appears that this kind of approach has not yet been so well established as to give a sufficient physical interpretation of actual failure phenomena.

The material will involve, to a degree, the aforementioned inhomogeneities at the time of its production. In addition, in its practical use, for example, in any experiment, fabrication of the material into specimens and their handling are hardly carried out in the same conditions for every specimen. This may cause additional variability.

In this respect, the probabilistic and statistical treatment must be introduced in dealing with uncertainties not only of the applied load but also of strength of the material.

As for the uncertainties associated with strength of the material, those of fatigue strength play a crucial role since most of practical structures will often fail through the fatigue crack propagation due to the repetition of randomly applied load. Therefore, the stochastic modelling of the fatigue crack growth becomes of vital importance in the reliability-based design. Major uncertainties associated with fatigue crack growth are schematically represented in Fig.1.1. The first major uncertainty is based upon the randomness of the applied load since service loading is usually random. The second is the uncertainty caused by spatially distributed random crack propagation resistance due to lack of material homogeneity, since the actual materials used in structural components will inevitably involve, to a degree, non-homogeneous micro-structures which will tend to cause spatial randomness of the propagation resistance. The third is the uncertainty associated with the size and the number of initial flaws involved in the material.

Concerned with a rational treatment of the abovementioned engineering uncertainties, the structural systems reliability theory also plays a crucial role in the probabilistic evaluation of structural safety and reliability.

Fig.1. Schematic Pepresentation of Major Uncertainties Associated with Fatigue Crack Growth Process.

Unlike the systems reliability theory in the field of electronic engineering, the structural systems reliability theory must conquer difficulties involved in dynamic phenomena such as the interference between applied load and strength of the material, and hence the relationship between the componental failure and the function loss of a structural system should be understood clearly. In this connection, the structural systems reliability theory must deal with the following major items and a variety of related papers have been published [Ref (7), (9)-(12)]:

1. treatment of strength of a structural element or a structural system as stochastic phenomena.
   
2. modelling of the load applied to a structural system as a random variable or a stochastic process.
   
3. probabilistic evaluation of the failure of a structural element or a structural system caused by the interference between applied loads and strength of the material.
   
4. reliability-based design method of a structural system.
   

In this respect, statistical consideration becomes of crucial and indispensable importance since failure phenomena should to be investigated clearly with a good understanding of engineering uncertainties associated with strength of the material as well as randomly applied load. The stochastic modelling of failure phenomena is also discussed, which leads to essential and useful statistical models in the reliability-based design of structures.

Further, the concept of TTFF (Time To First Failure) plays an important role and the design safety factor based upon TTFF concept should be considered. In the ordinary fatigue design, the so-called allowable design life is usually obtained by dividing the mean life of the material by the safety factor whose value is not less than unity. Although this safety factor has a function to increase reliability of the design in some way, the study of reliability against failure of the whole system would be performed more rationally with the aid of the concept of TTFF. In this sense, the safety factor based on TTFF becomes of crucial importance. This could be a solution to an engineering problem of determining the design safe life to assure the given level of structural reliability. In other words, the relationship is clarified based upon TTFF concept between the design safety factor of the whole structural system and that of a member.

(Detailed discussion on this subject will be continued sometime in the very near future. Sorry for this incompletion.)