STRESS DUE TO VARIABLE LOADING CONDITIONS: Completely Reversed or Cyclic Stresses, Fatigue and Endurance Limit

STRESS DUE TO VARIABLE LOADING CONDITIONS

A few machine parts are subjected to static loading. Since many of the machine parts (such as axles, shafts, crankshafts, connecting rods, springs, pinion teeth etc.) are subjected to variable or alternating loads (also known as fluctuating or fatigue loads).

Completely Reversed or Cyclic Stresses

Consider a rotating beam of circular cross-section and carrying a load W, as shown in the figure. This load induces stresses in the beam which are cyclic in nature. A little consideration will show that the upper fibres of the beam (i.e. at point A) are under compressive stress and the lower fibres (i.e. at point B) are under tensile stress. After half a revolution, the point B occupies the position of point A and the point A occupies the position of point B. Thus the point B is now under compressive stress and the point A under tensile stress. The speed of variation of these stresses depends upon the speed of the beam. From above we see that for each revolution of the beam, the stresses are reversed from compressive to tensile. The stresses which vary from one value of compressive to the same value of tensile or vice versa, are known as completely reversed or cyclic stresses.


Fatigue and Endurance Limit

It has been found experimentally that when a material is subjected to repeated stresses, it fails at stresses below the yield point stresses. Such type of failure of a material is known as fatigue. The failure is caused by means of a progressive crack formation which are usually fine and of microscopic size. The failure may occur even without any prior indication. The fatigue of material is effected by the size of the component, relative magnitude of static and fluctuating loads and the number of load reversals. In order to study the effect of fatigue of a material, a rotating mirror beam method is used. In this method, a standard mirror polished specimen, as shown in Fig. , is rotated in a fatigue testing machine while the specimen is loaded in bending. 

As the specimen rotates, the bending stress at the upper fibres varies from maximum compressive to maximum tensile while the bending stress at the lower fibres varies from maximum tensile to maximum compressive. In other words, the specimen is subjected to a completely reversed stress cycle. This is represented by a time-stress diagram as shown in Fig.  

A record is kept of the number of cycles required to produce failure at a given stress, and the results are plotted in stress-cycle curve as shown in Fig.  A little consideration will show that if the stress is kept below a certain value as shown by dotted line in Fig. , the material will not fail whatever may be the number of cycles. This stress, as represented by dotted line, is known as endurance or fatigue limit (σe). It is defined as maximum value of the completely reversed bending stress which a polished standard specimen can withstand without failure, for infinite number of cycles (usually 107 cycles).It may be noted that the term endurance limit is used for reversed bending only while for other types of loading, the term endurance strength may be used when referring the fatigue strength of the material. It may be defined as the safe maximum stress which can be applied to the machine part working under actual conditions. 

We have seen that when a machine member is subjected to a completely reversed stress, the maximum stress in tension is equal to the maximum stress in compression as shown in Fig. . In actual practice, many machine members undergo different range of stress than the completely reversed stress. The stress verses time diagram for fluctuating stress having values σmin and σmax is shown in Fig. . The variable stress, in general, may be considered as a combination of steady (or mean or average) stress and a completely reversed stress component σv. The following relations are derived from Fig.:


Standard specimen




completely reversed stress
completely reversed stress


Endurance/fatigue limit




fluctuating stress
fluctuating stress


EFFECT OF MISCELLANEOUS FACTORS ON ENDURANCE LIMIT

EFFECT OF MISCELLANEOUS FACTORS ON ENDURANCE LIMIT



Stress Concentration

Whenever a machine component changes the shape of its cross-section, the simple stress
distribution no longer holds good and the neighborhood of the discontinuity is different. This
irregularity in the stress distribution caused by abrupt changes of form is called stress
concentration. It occurs for all kinds of stresses in the presence of fillets, notches, holes,
keyways, splines, surface roughness or scratches etc. In order to understand fully the idea of
stress concentration, consider a member with different cross-section under a tensile load as
shown in Figure . A little consideration will show that the nominal stress in the right and left
hand sides will be uniform but in the region where the cross-section is changing, a redistribution
of the force within the member must take place. The material near the edges is
stressed considerably higher than the average value. The maximum stress occurs at some
point on the fillet and is directed parallel to the boundary at that point.



stress concentration
stress concentration


Theoretical Stress Concentration Factor (Kt)

The theoretical or form stress concentration factor is defined as the ratio of the maximum
stress in a member (at a notch or a fillet) to the nominal stress at the same section based upon
net area. Mathematically, theoretical or form stress concentration factor,

Theoretical Stress Concentration Factor (Kt)

The value of Kt depends upon the material and geometry of the part.

Methods of Reducing Stress Concentration

1. By providing fillets as shown in Fig

Fillets to improve stress concentration


2. By providing notches shown in Fig.


Notches to improve stress concentration
Notches to improve stress concentration



Fatigue Stress Concentration Factor

When a machine member is subjected to cyclic or fatigue loading, the value of fatigue stress
concentration factor shall be applied instead of theoretical stress concentration factor. Since the determination of fatigue stress concentration factor is not an easy task, therefore from experimental tests it is defined as

Fatigue stress concentration factor,

Fatigue stress concentration factor,


Notch Sensitivity

In cyclic loading, the effect of the notch or the fillet is usually less than predicted by the use of the theoretical factors as discussed before. The difference depends upon the stress gradient in the region of the stress concentration and on the hardness of the material. The term notch sensitivity is applied to this behaviour. It may be defined as the degree to which the theoretical effect of stress concentration is actually reached. The stress gradient depends mainly on the radius of the notch, hole or fillet and on the grain size of the material.

When the notch sensitivity factor q is used in cyclic loading, then fatigue stress concentration
factor may be obtained from the following relations:


Notch Sensitivity


Notch Sensitivity

Combined Steady and Variable Stress

The failure points from fatigue tests made with different steels and combinations of mean and variable stresses are plotted in Fig.  as functions of variable stress (σv) and mean stress (σm).

The most significant observation is that, in general, the failure point is little related to the mean stress when it is compressive but is very much a function of the mean stress when it is tensile. In practice, this means that fatigue failures are rare when the mean stress is compressive (or negative). Therefore, the greater emphasis must be given to the combination of a variable stress and a steady (or mean) tensile stress.


Combined mean and variable stress
Combined mean and variable stress



There are several ways in which problems involving this combination of stresses may be solved, but the following are important from the subject point of view :

1. Goodman method, and 
2. Soderberg method.

Goodman Method for Combination of Stresses

A straight line connecting the endurance limit (σe) and the ultimate strength (σu), as shown by line AB in Fig., follows the suggestion of Goodman. A Goodman line is used when the design is based on ultimate strength and may be used for ductile or brittle materials.


Goodman Method for Combination of Stresses


σu is called Goodman’s failure stress line. If a suitable factor of safety (F.S.) is applied to endurance limit and ultimate strength, a safe stress line CD may be drawn parallel to the line AB. Let us consider a design point P on the line CD. Now from similar triangles COD and PQD,


Goodman Method for Combination of Stresses

This expression does not include the effect of stress concentration. It may be noted that for
ductile materials, the stress concentration may be ignored under steady loads. Since many machine and structural parts that are subjected to fatigue loads contain regions of high stress concentration, therefore equation (i) must be altered to include this effect. In such cases, the fatigue stress concentration factor (Kf) is used to multiply the variable stress (σv). The equation (i) may now be written as



Goodman Method for Combination of Stresses

Considering the load factor, surface finish factor and size factor, the equation (ii) may be
written as

Goodman Method for Combination of Stresses


Soderberg Method for Combination of Stresses

A straight line connecting the endurance limit (σe) and the yield strength (σy), as shown by the line AB in Fig. , follows the suggestion of Soderberg line. This line is used when the design is based on yield strength. the line AB connecting σe and σy, as shown in Fig. , is called Soderberg’s failure stress line. If a suitable factor of safety (F.S.) is applied to the endurance limit and yield strength, a safe stress line CD may be drawn parallel to the line AB. Let us consider a design point P on the line CD. Now from similar triangles COD and PQD,



Soderberg Method for Combination of Stresses





Soderberg Method for Combination of Stresses



Soderberg Method for Combination of Stresses


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