Machine Design : Design for Static , Dynamic Fatigue Loading

(b). Fluctuating stresses: These vary from a minimum value to maximum value and these are of same nature, either tensile or compressive, are called as fluctuating stresses.

Repeated stresses: These are the stresses whose magnitude vary from zero to a certain maximum value.

Alternating stresses: The loading in which the stresses vary from a minimum value to maximum value with opposite nature and these stresses are known as alternating stresses.

Fatigue failure:

• Materials fail under fluctuating stresses at a stress magnitude which is lower than the ultimate tensile strength of the material and sometimes it is even lower than the yield strength.
• It has also been found that the magnitude of the stress causing failure decreases as the number of stress cycles increase.
• Fatigue failure is known as time delayed fracture under cyclic loading.
• The fatigue failure depends upon a number of factors, such as number of cycles, mean stress, stress amplitude, stress concentration, residual stresses, corrosion, size of the component and creep.

Endurance limit:

• The fatigue or endurance limit is the maximum amplitude of completely reversed stress which the standard specimen can be subjected to an unlimited number of cycles 10⁶ without fatigue failure.
• It is used for reversed bending only.

Fatigue life: 

• The number of stress cycles completed by a standard test specimen before the first fatigue crack appearance, is called as Fatigue life. The results of this test are plotted by means of an S-N curve.
  
• The graphical representation of stress amplitude (S_f) vs. the number of stress cycles (N) before the fatigue failure on a log-log graph paper is known as S-N Curve.
• The S-N curve becomes asymptotic at 10⁶ cycles for ferrous materials like steels and it shows the stress amplitude corresponding to infinite number of stress cycles. The endurance limit of the material is the magnitude of this stress amplitude corresponding to the 10⁶ cycles.
• For non-ferrous metals like aluminium alloys, the S-N curve slopes gradually even after 10⁶ These materials do not exhibit a distinct value of the endurance limit in a true sense.

Low cycle and High cycle fatigue:  Two regions of S-N curve are termed as low-cycle fatigue and high-cycle fatigue:

• Low-cycle fatigue: It is the fatigue failure for which the number of stress cycles < 1000. The basis of design for the low cycle fatigue components is ultimate tensile strength or yield strength with a suitable factor of safety.
• High cycle Fatigue: It is the fatigue failure for which the number of stress cycles > 1000. The basis of design for the high cycle fatigue components is endurance limit stress. S-N curves such as Soderberg, Gerber or Goodman criteria's are followed for the design of such components.

Reversed stresses: Design for finite and Infinite life:

Case 1: When the component is to be designed for infinite life, the endurance limit becomes the criterion of failure.

Case 2: When the component is to be designed for finite life, the S-N curve can be used. The curve is valid for steels. It has a straight-line AB drawn between 0.9S_ut at 10³ cycles to S_e at 10⁶ cycles on a log-log paper.

• Locate the point A and B with coordinates [3, log₁₀(0.9S_ut)] since log₁₀(10³) = 3 & [6, log₁₀(S_e)] since log₁₀(10⁶)= 6, respectively.
• Now Join the line points A and B and now it is used as a criterion of failure for finite-life problems.
• Corresponding to life (N) of the component, make a vertical line from log₁₀(N) on the abscissa ( showing intersection at F) and then draw a line FE parallel to the abscissa. The obtained ordinate at E i.e. log₁₀(S_f), shows the fatigue strength corresponding to N cycles.

Stress concentration: 

• The localization of high stresses caused by the irregularities present in the component and abrupt changes of the cross-section is termed as stress concentration.  
• A stress concentration factor (K_t) is used to consider the effect of stress concentration and it helps in finding the localized stresses:
  
• In static loading, stress concentration in ductile material is not so serious because in ductile material local deformation or yielding takes place which reduces the concentration.
• The effect of stress concentration factor under static loading is more serious for brittle material because they don’t permit any yielding.
• The effect of stress concentration under fatigue loading is more serious for both the materials.

Causes of stress concentration:

(a). Variation is properties of materials: It may be due to the following reasons:

• Internal cracks and flaws like blow holes
• Cavities in welds
• Air holes in steel components
• Non-metallic or foreign inclusions

(b). Load Application:

• The engagement of the meshing teeth of the driving and the driven gear.
• The cam and the follower contact.
• The balls and the races contact of ball bearing.

(c). Discontinuities in the component:

• The machine characteristic such as oil holes, keyways and splines provided, and screw threads cause the discontinuities in the component cross section area.
• For a plate with transverse elliptical hole and subjected to a tensile load as shown in fig. The theoretical stress concentration factor is given as:
  
• Machining scratches: Machining scratches, stamp marks or inspection marks are surfaces irregularities, which causes stress concentration.

Reduction of stress concentration:

• Single notch results in a high degree of stress concentration. The severity of stress concentration can be reduced by providing multiple notches. The severity of stress concentration can be reduced by providing multiple drilled holes along with notches.

• The figure shows stepped bar subject to bending moment and it can be seen that stress line changes rapidly along sudden enlargement. The severity of stress concentration can be reduced by providing undercut and notches.

Notch sensitivity (q):

It is defined as the susceptibility of a material to succumb to the damaging effects of stress raising notches in fatigue loading:

Failure Criteria under fatigue loading:

Soderberg criterion: A straight line joining S_e (on the ordinate) to S_yt (on the abscissa). Here f_s is the factor of safety:

Goodman Criterion: A straight line joining S_e (on the ordinate) to S_ut (on the abscissa). Here f_s the factor of safety, then:

Gerber criterion: A parabolic curve joining S_e (on the ordinate) to S_ut (on the abscissa). Here f_s is the factor of safety:

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