REPEATED MECHANICAL LOADINGS: FATIGUE BASIC AND TUTORIALS

REPEATED MECHANICAL LOADINGS: FATIGUE BASIC INFORMATION
Effects Of Repeated Loadings

In the preceding sections we have considered the behavior of a test specimen subjected to an axial loading. We recall that, if the maximum stress in the specimen does not exceed the elastic limit of the material, the specimen returns to its initial condition when the load is removed.

You might conclude that a given loading may be repeated many times, provided that the stresses remain in the elastic range. Such a conclusion is correct for loadings repeated a few dozen or even a few hundred times.

However, as you will see, it is not correct when loadings are repeated thousands or millions of times. In such cases, rupture will occur at a stress much lower than the static breaking strength; this phenomenon is known as fatigue. A fatigue failure is of a brittle nature, even for materials that are normally ductile.

Fatigue must be considered in the design of all structural and machine components that are subjected to repeated or to fluctuating loads. The number of loading cycles that may be expected during the useful life of a component varies greatly.

For example, a beam supporting an industrial crane may be loaded as many as two million times in 25 years (about 300 loadings per working day), an automobile crankshaft will be loaded about half a billion times if the automobile is driven 200,000 miles, and an individual turbine blade may be loaded several hundred billion times during its lifetime.

Some loadings are of a fluctuating nature. For example, the passage of traffic over a bridge will cause stress levels that will fluctuate about the stress level due to the weight of the bridge. A more severe condition occurs when a complete reversal of the load occurs during the loading cycle.

The stresses in the axle of a railroad car, for example, are completely reversed after each half-revolution of the wheel. The number of loading cycles required to cause the failure of a specimen through repeated successive loadings and reverse loadings may be determined experimentally for any given maximum stress level.

If a series of tests is conducted, using different maximum stress levels, the resulting data may be plotted as a s-n curve. For each test, the maximum stress s is plotted as an ordinate and the number of cycles n as an abscissa; because of the large number of cycles required for rupture, the cycles n are plotted on a logarithmic scale.

A typical s-n curve for steel is shown in Fig. 2.16. We note that, if the applied maximum stress is high, relatively few cycles are required to cause rupture. As the magnitude of the maximum stress is reduced, the number of cycles required to cause rupture increases, until a stress, known as the endurance limit, is reached.

The endurance limit is the stress for which failure does not occur, even for an indefinitely large number of loading cycles. For a low-carbon steel, such as structural steel, the endurance limdecrease as the number of loading cycles is increased. For such metals, one defines the fatigue limit as the stress corresponding to failure after a specified number of loading cycles, such as 500 million.

Examination of test specimens, of shafts, of springs, and of other components that have failed in fatigue shows that the failure was initiated at a microscopic crack or at some similar imperfection. At each loading, the crack was very slightly enlarged.

During successive loading cycles, the crack propagated through the material until the amount of undamaged material was insufficient to carry the maximum load, and an abrupt, brittle failure occurred.

Because fatigue failure may be initiated at any crack or imperfection, the surface condition of a specimen has an important effect on the value of the endurance limit obtained in testing. The endurance limit for machined and polished specimens is higher than for rolled or forged components, or for components that are corroded.

In applications in or near seawater, or in other applications where corrosion is expected, a reduction of up to 50% in the endurance limit can be expected.is about one-half of the ultimate strength of the steel. For nonferrous metals, such as aluminum and copper, a typical s-n curve (Fig. 2.16) shows that the stress at failure continues to decrease as the number of loading cycles is increased. For such metals, one defines the fatigue limit as the stress corresponding to failure after a specified number of loading cycles, such as 500 million.

AXIAL LOADING; NORMAL STRESS BASICS AND TUTORIALS

AXIAL LOADING; NORMAL STRESS TUTORIALS
What Is Axial Loading? What Is Stress?

The deformation caused in a body by external forces or other actions generally varies from one point to another, i.e., it is not homogeneous. In fact, a homogeneous deformation is rare. It occurs, for example, in a body with isostatic supports under a uniform temperature variation or in a slender member under constant axial force.

Rod BC of the example considered in the preceding section is a two-force member and, therefore, the forces FBC and F'BC acting on its ends B and C (Fig. 1.5) are directed along the axis of the rod. We say that the rod is under axial loading.

An actual example of structural members under axial loading is provided by the members of the bridge truss shown in Photo 1.1.

Returning to rod BC of Fig. 1.5, we recall that the section we passed through the rod to determine the internal force in the rod and the corresponding stress was perpendicular to the axis of the rod; the internal force was therefore normal to the plane of the section (Fig. 1.7) and the corresponding stress is described as a normal stress.

Thus, formula (1.5) gives us the normal stress in a member under axial loading:

σ =P/A 

We should also note that, in formula (1.5), s is obtained by dividing the magnitude P of the resultant of the internal forces distributed over the cross section by the area A of the cross section; it represents, therefore, the average value of the stress over the cross section, rather than the stress at a specific point of the cross section.

To define the stress at a given point Q of the cross section, we should consider a small area DA. Dividing the magnitude of DF by DA, we obtain the average value of the stress over DA. Letting DA approach zero, we obtain the stress at point Q:

σ = lim dF/dA      as dA approaches infinity (1.6)

In general, the value obtained for the stress s at a given point Q of the section is different from the value of the average stress given by formula (1.5), and s is found to vary across the section. In a slender rod subjected to equal and opposite concentrated loads P and P' , this variation is small in a section away from the points of application of the concentrated loads, but it is quite noticeable in the neighborhood of these 

It follows from Eq. (1.6) that the magnitude of the resultant of the distributed internal forces is

∫dF = ∫σ dA     lower limit = A

But the conditions of equilibrium of each of the portions of rod require that this magnitude be equal to the magnitude P of the concentrated loads. We have, therefore,

P = ∫dF = ∫σ dA    lower limit = A

which means that the volume under each of the stress surfaces must be equal to the magnitude P of the loads. This, however, is the only information that we can derive from our knowledge of statics, regarding the distribution of normal stresses in the various sections of the rod. 

The actual distribution of stresses in any given section is statically indeterminate. To learn more about this distribution, it is necessary to consider the deformations resulting from the particular mode of application of the loads at the ends of the rod.

In practice, it will be assumed that the distribution of normal stresses in an axially loaded member is uniform, except in the immediate vicinity of the points of application of the loads. The value s of the stress is then equal to save and can be obtained from formula (1.5). 

However, we should realize that, when we assume a uniform distribution of stresses in the section, i.e., when we assume that the internal forces are uniformly distributed across the section, it follows from elementary statics† that the resultant P of the internal forces must be applied at the centroid C of the section. 

This means that a uniform distribution of stress is possible only if the line of action of the concentrated loads P and P' passes through the centroid of the section considered. This type of loading is called centric loading and will be assumed to take place in all straight two-force members found in trusses and pin-connected structures, such as the one considered in Fig. 1.1. 

However, if a two-force member is loaded axially, but eccentrically we find from the conditions of equilibrium of the portion of member that the internal forces in a given section must be equivalent to a force P applied at the centroid of the section and a couple M of moment M = Pd. The distribution of forces—and, thus, the corresponding distribution of stresses—cannot be uniform. Nor can the distribution of stresses be symmetric.