Calculations of elements of machine design

It is the custom among most firms engaged in the designing of machinery to settle upon certain stresses as proper for given materials in given classes of work. These stresses are chosen as the result of many years of experience on their own part, or of observation of the successful experience of others, and so long as the quality of the material remains unchanged, and the service does not vary in character, the method is eminently satisfactory.

Progress, however, brings up new service, for which precedent is lacking, and materials of different qualities, either better or cheaper, for which the safe working stresses have not been determined, and the designer is compelled to determine the stress proper for the work in hand by using a so-called "factor of safety" The name "factor of safety" is misleading for several reasons. In the first place, it is not a factor at all, from a mathematical point of view, but is in its use a divisor, and in its derivation a product. In order to obtain the safe working stress, we divide the ultimate strength of the material by the proper "factor of safety," and in order to obtain this factor of safety we multiply together several factors, which, in turn, depend upon the qualities of the material, and the conditions of service. So our factor of safety is both a product and a divisor, but it is not a factor. Then again, we infer, naturally, that with a factor of twelve, say, we could increase the load upon a machine member to twelve times its ordinary amount before rupture would occur, when, as a matter of fact, this is not so, at least not in a machine with moving parts, sometimes under load, and sometimes not subjected to working stresses. Still more dangerous conditions are met with when the parts are subjected to load first in one direction, and then in the other, or to shocks or sudden loading and unloading. The margin of safety is, therefore, apparent, not real, and we will hereafter call the quantity we are dealing with the "apparent factor of safety," for the name factor is too firmly fixed in our minds to easily throw it off.

The apparent factor of safety, as has been intimated, is the product of four factors, which for the purpose of our discussion, we will designate as factors a, b, c, and d. Factors & and c, as will appear later, may be, and often are, 1, but none the less they must always be considered and given their proper values. Designating the apparent factor of safety by F, we have then

F = a * b * c * d.

The first of these factors, a, is the ratio of the ultimate strength of the material to its elastic limit. By the elastic limit we do not mean the yield point, but the true elastic limit within which the material is, in so far as we can discover, perfectly elastic, and takes no permanent set. There are several reasons for keeping the working stress within this limit, the two most important being: First, that the material will rupture if strained repeatedly beyond this limit; and second, that the form and dimensions of the piece would be destroyed under the same circumstances. If a piece of wire be bent backward and forward in a vise, we all know that it will soon break. And no matter how little we bend it, provided only that we bend it sufficiently to prevent it from entirely recovering its straightness, it will still break if we continue the operation long enough. And similarly, if the axle of a car, the piston rod of an engine, or whatever piece we choose, be strained time after time beyond its limit of elasticity, no matter how little, it will inevitably break. Or suppose, as is the case with a boiler, that the load is only a steady and unremitting pressure. The yielding of the material will open up the seams, allowing leakage. It will throw the strains upon the shorter braces more then upon the others, thus rupturing them in detail. It is absolutely necessary, therefore, excepting in very exceptional cases, that we limit our working stress to less than the elastic limit of the material.

Among French designers it is customary to deal entirely with the elastic limit of the material, instead of the ultimate strength, and with such a procedure no such factor as we have been discussing would ever appear in the make-up of our apparent factor of safety. Although this method is rational enough, it is not customary outside of France, because many of the materials we use, notably cast iron, and sometimes wrought iron and hard steels, have no definite elastic limit. In any case where the elastic limit is unknown or ill-defined, we arbitrarily assume it to be one-half the ultimate -strength, and factor a becomes 2. For nickel-steel and oil-tempered forgings the elastic limit becomes two-thirds of the ultimate strength, or even more, and the factor is accordingly reduced to 1%.

The second factor, b, appearing in our equation is one depending upon the character of the stress produced within the material. The experiments of Wohler, conducted by him between the years 1859 and 1870 at the instance of the Prussian government, on the effects of repeated stresses, confirmed a fact already well known, namely, that the repeated application of a load which would produce a stress less than the ultimate strength of a material would often rupture it. But they did more. They showed the exact relation between, the variation of the load and the breaking strength of the material under that variation. The investigation was subsequently extended by Weyrauch to cover the entire possible range of variation. Out of the mass of experimental data so obtained a rather complicated formula was deduced, giving the relation between the variation of the load (or rather the stress it produced), the strength of the material under the given conditions (which is generally known as the "carrying strength" of the material) and the ultimate strength. To Prof. J. B. Johnson, we believe, is due the credit of substituting for this formula a much simpler and more manageable one, which perhaps represents the actual facts with almost equal accuracy.

The third factor, c, entering into our equation, depends upon the manner in which the load is applied to the piece. A load suddenly applied to a machine member produces twice the stress within that member that the same load would produce if gradually applied. When the load is gradually applied, the stress in the member gradually increases, until finally, when the full load is applied, the total stress in the member corresponds to this full load. When, however, the load is suddenly applied, the stress is at , first zero, but very swiftly increases. Since both the load and the stress act through whatever slight distance the piece yields, the product of the average total stress into this distance must equal the product of the load into this same distance. In order that the average stress should equal the load, it is necessary that the maximum value of the stress should equal twice the load. In recognition of this fact, we introduce the factor c = 2 into our equation when the load is suddenly applied.

The last factor, d, in our equation, we might call the "factor of ignorance." All the other factors have provided against known contingencies; this provides against the unknown. It commonly varies in value between 1% and 3, although occasionally it becomes as great as 10. It provides against excessive or accidental overload, against unexpectedly severe service, against unreliable or imperfect materials, and against all unforeseen contingencies of manufacture or operation.

When we can compute the load exactly, when we know what kind of a load it will be, steady or variable, impulsive or gradual in its application, when we know that this load will not be likely to be increased, that our material is reliable, that failure will not result disastrously, or even that our piece for some reason must be small or light, this factor will be reduced to its lowest limit, 1%.

The conditions of service in some degree determine this factor. When a machine is to be placed in the hands of unskilled labor, when it is to receive hard knocks or rough treatment, the factor must be made larger. When it will be profitable to overload a machine by increasing its work or its speed in such a way as to throw unusual strains upon it, we are obliged to discount the probability of this being done by increasing this factor. Or again, when life or property would be endangered by the failure of the piece we are designing, this factor must be made larger in recognition of the fact. Thus, while it is 1% to 2 in most ordinary steel constructions, it is rarely less than 2 1 / for a better grade of steel in a boiler. Even if property were not in danger of destruction, and the failure of the piece would simply result in considerable loss in output or wages, as in the case of the stoppage of a factory, it is best to increase this factor somewhat.

The reliability of the material in a great measure determines the value of this factor. For instance, in all cases where it would be 1% for mild steel, it is made 2 for cast iron. It will be larger for those materials subject to internal strains, for instance for complicated castings, heavy forgings, hardened steel, and the like. It will be larger for those materials more easily injured by improper and unskillful handling, unless we know that the work will be done by skilled and careful workmen. It will be larger for those materials subject to hidden defects, such as internal flaws in forgings, spongy places in castings, etc. It will be smaller for ductile and larger for brittle materials. It will be smaller as we are- sure that our piece has received uniform treatment, and as the tests we have give more uniform results and more accurate indications of the real strength and quality of the piece itself.

Of all these factors that we have been considering, the last one alone has an element of chance or judgment in it, except when we make an allowance for shock. In fixing it, the designer must depend on his judgment, guided by the general rules laid down.

Someone may ask at this point, why, if we introduce a factor for the elastic limit, do we also introduce a factor for repeated loads? It may be argued that if we keep the stress within the elastic limit, no harm will be done, no matter how often the load be repeated, and they are right. However, with a dead load acting upon a piece and straining it to its elastic limit, we have as a margin of safety the difference between its elastic limit and its ultimate strength. But when the load is a repeated load, of the same amount as before, the piece has no margin of safety, unless its section be increased, and it does not have the same margin of safety as it had in the first place, until its section is doubled.