Realistic Reserves

© Lev Gelimson

Any strength theory should consider not only critical (limiting, ultimate) but also noncritical (nonlimiting, intermediate) stress states (processes). Every critical strength criterion (not obligatorily a general one) not only describes the corresponding limiting surface but also should express an equivalent stress as a certain function of the principal stresses. But each principal stress is a certain function of the initial data in a problem to be solved. In strength problems, these data usually concern the shape and sizes of a solid, the mechanical and other properties of its material, and the description of loading.

If the stress state at a natural solid's point under consideration is not critical, it is necessary to introduce a certain measure of the safety of the state as compared with the closest critical one. The sole well-known measure is a safety factor as the ratio of a limiting stress to an equivalent stress. The safety factor in every triaxial stress state is equal to 1 if and only if the state is critical. This measure is often satisfactory in the case of so-called simple loading when all the principal stresses at every point of a solid under consideration are directly proportional to one common parameter depending on time. In other words, such a parameter has to exist, which holds in particular in the case of one load and a uniaxial stress that can be such a parameter itself. This circumstance is natural because the above safety factor definition has been originally proposed just in the case of a uniaxial stress state. But applying a usual safety factor in the general case of a complex stress state caused by different loads can result in ambiguity of values of a safety factor and in inadmissible errors by its determination. Indeed, experimental data on material strength concern only critical states in the sole limiting surface.

But this usual safety factor does not show by what maximum number each of the input data (here the principal stresses) in a problem can be multiplied or divided independently of one another so that, by the worst (most dangerous) realizable combination of such modified values of the input data, the stress state (in general, in nonhomogeneous stress state, at the most dangerous point of the solid) becomes critical. It is especially dangerous that in such a case a usual safety factor gives values much greater than realistic ones. Let us consider two examples.

Example 1: The principal stresses 250 MPa, 240 MPa, 210 MPa; the limiting one 235 MPa.

The usual safety factor by the theory of maximum shearing stresses is

235 MPa / (250 MPa - 210 MPa) = 5.9

whereas

5.9 * 250 MPa - 210 MPa / 5.9 = 1439 MPa

and thus much greater than 235 MPa.

Even in uniaxial stress under two opposite loads the same can hold.

Example 2: A bar with strengths 100 MPa in tension and 800 MPa in compression is contracted and stretched by two pairs of forces independent from each other and causing the stresses

-500 MPa + 400 MPa = -100 MPa.

The usual safety factor is

800 MPa / 100 MPa = 8

whereas

-500 MPa * 8 + 400 MPa / 8 = -3950 MPa

(much more dangerous than -800 MPa) and

-500 MPa / 8 + 400 MPa * 8 = 3137.5 MPa

(much more dangerous than 100 MPa).

So a well-known safety factor cannot show maximum tolerable errors in determining the initial data (the principal stresses, their separate components, loads, sizes, etc.) in a problem. Therefore, a usual safety factor is not sufficient to properly solve strength problems and other ones.

Thoroughly analyzing the structures of general strength criteria as fundamental laws of nature allows to found a certain doctrine for determining reserve factors as measures of tolerable errors in determining the input data not only in strength of materials but also in any other problem.

The cardinal ideas to realistically determine the reserve of a system under consideration are:

introducing individual reserves in a problem;

worst-case combining values of input and output parameters in their ranges determined by their reserves.

In other words, the reserves of the original parameters are separately taken into account, each of these reserves being expressed via a common additional number. It is obtained from the condition that, by the worst realizable combination of the values of these parameters arbitrarily modified within the bounds determined by the corresponding reserves, the state of at least one element of the system becomes limiting, no element of it being in an overlimiting state. This is a further generalization of the universalization methods for critical state criteria.

Those criteria realize the idea of separating the reserve factors for diverse arguments in a certain function. Indeed, a relative (reduced) equivalent stress is an inverse result reserve (safety) factor. The reduced principal stresses are the inverse individual reserve (safety) factors (for the corresponding three uniaxial stress states) having certain signs (if loading is static) or vectorial directions (if loading is variable). Thus each general critical state (process) criterion gives an expression of a result reserve (safety) factor as a certain function of the three individual reserve (safety) factors. But such criteria even in this generalized interpretation concern critical states (processes) only, such as elasticity, yield, or failure, of a material. Therefore, this separation idea needs its further development, generalization, and extension as applied to arbitrary stress states (processes) including noncritical ones.

The additive approach to obtaining reserves develops, generalizes, and extends the relative error whereas the multiplicative approach develops, generalizes, and extends the reserve factor.

Often it is possible and useful, to express some or all individual reserves through a common reserve extra introduced. Then its value can be determined by the most dangerous realizable combination of those data. Finally, all the individual reserves are determined as corresponding functions of that common reserve and really show what combinations of the initial data are tolerable.

In the simplest case of the equal reserves of all initial parameters, in example 1 we obtain the additive reserve 1.423 and the multiplicative reserve 1.5 instead of the above usual safety factor 5.9.

In example 2 we obtain the multiplicative reserve 1.25 and 2 in tension and compression, respectively, instead of the above usual safety factor 8.

These realistic reserves are significantly less than the usual ones too optimistic.

For formulae and details, see the scientific website of the author and especially his works under "Strength".