Fluctuating Stress
Introduction
Some machine parts are subjected to static loading. Since many of the machine parts (such as axles, shafts, crankshafts, connecting rods, springs, pinion etc.) are subjected to variable or alternating loads (also known as fluctuating or fatigue loads), therefore we shall discuss, in this chapter, the variable or alternating stresses.. For example in below figure the fiber on the surface of a rotating shaft subjected to a bending load, undergoes both tension and compression for each revolution of the shaft.
Any fiber on the shaft is therefore subjected to fluctuating stresses. Machine elements subjected to fluctuating stresses usually fail at stress levels much below their ultimate strength and in many cases below the yield point of the material too. These failures occur due to very large number of stress cycle and are known as fatigue failure. These failures usually begin with a small crack which may develop at the points of discontinuity, an existing subsurface crack or surface faults. Once a crack is developed it propagates with the increase in stress cycle finally leading to failure of the component by fracture. There are mainly two characteristics of this kind of failures:
(a) Progressive development of crack.
(b) Sudden fracture without any warning since yielding is practically absent. Fatigue failures are influenced by
(i) Nature and magnitude of the stress cycle.
(ii) Endurance limit.
(iii) Stress concentration.
(iv) Surface characteristics.
These factors are therefore interdependent. For example, by grinding and polishing, case hardening or coating a surface, the endurance limit may be improved. For machined steel endurance limit is approximately half the ultimate tensile stress.
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Stress cycle
A typical stress cycle is shown in below figure using standard specimen. The maximum, minimum, mean and variable stresses are indicated. The mean and variable stresses are given by
In order to find the mean stress for completely reversed cycle the mean stress is zero. The stress verses time diagram for fluctuating stress having values σmin and σmaxis shown in fluctuating stress 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 fluctuating stress fig:
Endurance limit
Figure shows the rotating beam arrangement along with the specimen.
Fig. A typical rotating beam arrangement.
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The loading is such that there is a constant bending moment over the specimen length and the bending stress is greatest at the center where the section is smallest. The arrangement gives pure bending and avoids transverse shear since bending moment is constant over the length. Large number of tests with varying bending loads are carried out to find the number of cycles to fail. A typical plot of reversed stress (S) against number of cycles to fail (N) is shown in below figure. The zone below 103 cycles is considered as low cycle fatigue, zonebetween 103and 106 cycles is high cycle fatigue with finite life and beyond 106 cycles, the zone is considered to be high cycle fatigue with infinite life.
Fig. A schematic plot of reversed stress (S) against number of cycles to fail(N) for steel.
Effect of Loading on Endurance Limit—Load Factor
The endurance limit (σe) of a material as determined by the rotating beam method is for reversed bending load. There are many machine members which are subjected to loads other than reversed bending loads. Thus the endurance limit will also be different for different types of loading. The endurance limit depending upon the type of loading may be modified as discussed below:
Let
Kb = Load correction factor for the reversed or rotating bending load. Its value is usually taken as unity.
Ka = Load correction factor for the reversed axial load. Its value may be taken as 0.8.
Ks = Load correction factor for the reversed torsional or shear load. Its value may be taken as 0.55 for ductile materials and 0.8 for brittle materials.
∴Endurance limit for reversed bending load, σeb = σe.Kb= σe ...( Kb = 1)
Endurance limit for reversed axial load, σea = σe.Ka
and endurance limit for reversed torsional or shear load, τe = σe.Ks
Effect of Surface Finish on Endurance Limit—Surface Finish Factor
When a machine member is subjected to variable loads, the endurance limit of the material for that member depends upon the surface conditions. When the surface finish factor is known, then the endurance limit for the material of the machine member may be obtained by multiplying the endurance limit and the surface finish factor. We see that for a mirror polished material, the surface finish factor is unity. In other words, the endurance limit for mirror polished material is maximum and it goes on reducing due to surface condition.
Let Ksur= Surface finish factor.
∴Endurance limit,
σe1= σeb.Ksur = σe.Kb.Ksur= σe.Ksur ...( Kb = 1)
...(For reversed bending load)
= σea.Ksur= σe.Ka.Ksur ...(For reversed axial load)
= τe.Ksur = σe.Ks.Ksur ...(For reversed torsional or shear load)
Note : The surface finish factor for non-ferrous metals may be taken as unity.
Effect of Size on Endurance Limit—Size Factor
A little consideration will show that if the size of the standard specimen is increased, then the endurance limit of the material will decrease. This is due to the fact that a longer specimen will have more defects than a smaller one.
Let Ksz= Size factor.
∴Endurance limit,
σe2= σe1 × Ksz ...(Considering surface finish factor also)
= σeb.Ksur.Ksz= σe.Kb.Ksur.Ksz= σe.Ksur.Ksz (Kb = 1)
= σea.Ksur.Ksz= σe.Ka.Ksur.Ksz ...(For reversed axial load)
= τe.Ksur.Ksz= σe.Ks.Ksur.Ksz ... (For reversed torsional or shear load)
Effect of Miscellaneous Factors on Endurance Limit
In addition to the surface finish factor (Ksur), size factor (Ksz) and load factors Kb, Ka and Ks, there are many other factors such as reliability factor (Kr), temperature factor (Kt), impact factor (Ki) etc. which has effect on the endurance limit of a material. Considering all these factors, the endurance limit may be determined by using the following expressions :
1. For the reversed bending load, endurance limit,
σ'e= σeb.Ksur.Ksz.Kr.Kt.Ki
2. For the reversed axial load, endurance limit,
σ'e= σea.Ksur.Ksz.Kr.Kt.Ki
3. For the reversed torsional or shear load, endurance limit,
σ'e= τe .Ksur.Ksz.Kr.Kt.Ki
In solving problems, if the value of any of the above factors is not known, it may be taken as unity.
Relation Between Endurance Limit and Ultimate Tensile Strength
For steel, σe= 0.5 σu;
For cast steel, σe = 0.4 σu ;
For cast iron, σe = 0.35 σu ;
For non-ferrous metals and alloys, σe = 0.3 σu
Factor of Safety for Fatigue Loading
When a component is subjected to fatigue loading, the endurance limit is the criterion for failure. Therefore, the factor of safety should be based on endurance limit. Mathematically,
Stress Concentration:
Whenever a machine component changes the shape of its cross-section, the simple stress distribution no longer holds good and the neighbourhood 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 below Fig. 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 re-distribution 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.
Theoretical or Form Stress Concentration Factor:
The stress concentration is based on either the photo-elastic analysis using a circular polariscope or theoretical or finite element analysis method. 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,
The value of Kt depends upon the material and geometry of the part.
Imp. Note:
1. In static loading, stress concentration in ductile materials is not so serious as in brittle materials.
2. In cyclic loading, stress concentration in ductile materials is always serious because the ductility of the material is not effective in relieving the concentration of stress caused by cracks, flaws, surface roughness, or any sharp discontinuity in the geometrical form of the member.
3. In brittle materials, cracks may appear at these local concentrations of stress which will increase the stress over the rest of the section. It is, therefore, necessary that in designing parts of brittle materials subjected to both static load as well as fluctuating load.
Stress Concentration due to Holes and Notches:
Consider a plate with transverse elliptical hole and subjected to a tensile load as shown in Fig. (a). We see from the stress-distribution that the stress at the point away from the hole is practically uniform and the maximum stress will be induced at the edge of the hole. The maximum stress is given by
and the theoretical stress concentration factor,
When a/b is large, the ellipse approaches a crack transverse to the load and the value of Kt becomes very large. When a/b is small, the ellipse approaches a longitudinal slit [as shown in Fig. (b)] and the increase in stress is small. When the hole is circular as shown in Fig. (c), then a/b = 1 and the maximum stress is three times the nominal value.
Methods of Reducing Stress Concentration:
We have already discussed that, whenever there is a change in cross-section, such as shoulders, holes, notches or keyways and where there is an interference fit between a hub or bearing race and a shaft, then stress concentration results. The presence of stress concentration cannot be totally eliminated but it may be reduced to some extent. A device or concept that is useful in assisting a design engineer to visualize the presence of stress concentration and how it may be mitigated is that of stress flow lines, as shown in Fig. The mitigation of stress concentration means that the stress flow lines shall maintain their spacing as far as possible.
Some examples:
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,
Notch Sensitivity:
The notch sensitivity ‘q’ for fatigue loading can now be defined in terms of Kf and the theoretical stress concentration factor Kt and this is given by
Fatigue strength formulations:
Fatigue strength experiments have been carried out over a wide range of stress variations in both tension and compression below figure shows a schematic diagram of experimental plots of variable stress against mean stress and Gerber, Goodman and Soderberg lines. But the following are important from the subject point of view:
1. Goodman method 2. Soderberg method.
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