Design during fluctuating stresses, Lecture notes of Machine Design

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Prepared by: Dr. Gagandeep Bhardwaj, Assistant Professor, MED Email: gagandeep.med@thapar.edu Contact No. 8954388548

FLUCTUATING STRESSES

• The external forces acting on a machine component were assumed to be static. In many applications, the components are subjected to forces, which are not static, but vary in magnitude with respect to time. The stresses induced due to such forces are called fluctuating stresses.
• The pattern of stress variation is irregular and unpredictable, as in case of stresses due to vibrations.
• For the purpose of design analysis, simple models for stress–time relationships are used. The most popular model for stress–time relationship is the sine curve.
• There are three types of mathematical models for cyclic stresses—fluctuating or alternating stresses, repeated stresses and reversed stresses.

INTRODUCTION

TTPES OF CYCLIC STRESSES

• The reversed stress varies in a sinusoidal manner with respect to time, but it has zero mean stress. In this case, half portion of the cycle consists of tensile stress and the remaining half of compressive stress.
• There is a complete reversal from tension to compression between these two halves and therefore, the mean stress is zero.
• σ (^) max. and σmin. are maximum and minimum stresses, while σm and σa are called mean stress and stress amplitude respectively

FATIGUE FAILURE

Shear and Fatigue Failure of Wire: (a) Shearing of Wire; (b) Bending of Wire; (c) Unbending of Wire

• The first method is to cut the wire by applying equal and opposite forces by left and right hand.
• The second method consists of alternatively bending and unbending the wire for few cycles. The wire can be cut very easily in few cycles of bending and unbending. This is a fatigue failure and the magnitude of stress required to fracture is very low.
• In other words, there is decreased resistance of material to cyclic stresses. Fatigue failure is defined as time delayed fracture under cyclic loading.

ENDURANCE LIMIT

• The fatigue or endurance limit of a material is defined as the maximum amplitude of completely reversed stress that the standard specimen can sustain for an unlimited number of cycles without fatigue failure.
• Since the fatigue test cannot be conducted for unlimited or infinite number of cycles, 106 cycles is considered as a sufficient number of cycles to define the endurance limit.
• The fatigue life is defined as the number of stress cycles that the standard specimen can complete during the test before the appearance of the first fatigue crack.

Rotating Beam Subjected to Bending Moment: (a) Beam, (b) Stress Cycle at Point A

Specimen for Fatigue Test

In the laboratory, the endurance limit is determined by means of a rotating beam machine developed by R.R. Moore.

ENDURANCE LIMIT

15/02/2019 Dr. Gagandeep Bhardwaj, AP MED, TIET S-N curve for steel 10

The S–N curve (Wohler diagram) is the graphical representation of stress amplitude (Sf) versus the number of stress cycles (N) before the fatigue failure on a log-log graph paper. The S–N curve becomes asymptotic at 106 cycles, which indicates the stress amplitude corresponding to infinite number of stress cycles. The magnitude of this stress amplitude at 10^6 cycles represents the endurance limit of the material.

ENDURANCE LIMIT

Factors affecting the endurance Limit:

• Size of the component;
• Shape of component;
• The surface finish, temperature and the notch sensitivity of the material;

LOW CYCLE AND HIGH CYCLE FATIGUE

Low and high cycle fatigue

There are two regions of this curve namely, low-cycle fatigue and high-cycle fatigue.

Any fatigue failure when the number of stress cycles are less than 1000, is called low-cycle fatigue.

Any fatigue failure when the number of stress cycles are more than 1000, is called high-cycle fatigue.

Components subjected to high- cycle fatigue are designed on the basis of endurance limit stress. The S–N curves, Soderberg lines, Gerber lines or Goodman diagrams are used in the design of such components.

NOTCH SENSITIVITY

• The material is brittle then there will be more notch sensitivity (Can quickly getting notches due to the cyclic loading or impact loading).
• As the material is Ductile, the notch sensitivity is less (not so quick in getting the notches due to the cyclic loading or impact loading).

The above equation can be rearranged as:

NOTCH SENSITIVITY

• When the material has no sensitivity to notches, q = 0 and Kf = 1
• When the material is fully sensitive to notches, q = 1 and Kf = Kt

Figure

Figure

• The endurance limit of a component is different from the endurance limit of a rotating beam specimen due to a number of factors.
• The difference arises due to the fact that there are standard specifications and working conditions for the rotating beam specimen, while the actual components have different specifications and work under different conditions.
• Different modifying factors are used in practice to account for this difference. These factors are, sometimes, called derating factors.
• The four factors used for reducing the endurance limit of rotating beam specimen are:

ENDURANCE LIMIT (APPROX. ESTIMATION)

Surface finish factor : The ultimate tensile strength is also a parameter affecting the surface finish factor. High strength materials are more sensitive to stress concentration introduced by surface irregularities.

ENDURANCE LIMIT (APPROX. ESTIMATION)

Size Factor : The endurance limit, therefore, reduces with increasing the size of the component. The size factor Kb takes into account the reduction in endurance limit due to increase in the size of the component

REVERSED STRESSES

DESIGN FOR FINITE AND INFINITE LIFE

The design problems for completely reversed stresses are further divided into two groups:

• Design for infinite life, and
• Design for finite life.

Case I : When the component is to be designed for infinite life, the endurance limit becomes the criterion of failure. The amplitude stress induced in such components should be lower than the endurance limit in order to withstand the infinite number of cycles.

REVERSED STRESSES

DESIGN FOR FINITE AND INFINITE LIFE

Case II : When the component is to be designed for finite life, the S–N curve as shown in Figure can be used. The curve is valid for steels. It consists of a straight line AB drawn from (0.9 Sut) at 10^3 cycles to (Se) at 10^6 cycles on a log-log paper. Se is the corrected endurance limits in reversed bending. The design procedure for such problems is as follows:

• Locate the point A with coordinates [3, log 10 (0.9 Sut )] since log 10 (10^3 ) = 3.
• Locate the point B with coordinates [6, log 10 ( Se )] since log 10 (10^6 ) = 6.
• Join AB , which is used as a criterion of failure for finite-life problems.
• Depending upon the life N of the component, draw a vertical line passing through log 10 ( N ) on the abscissa. This line intersects AB at point F.
• Draw a line FE parallel to the abscissa. The ordinate at the point E , i.e. log 10 ( Sf ) , gives the fatigue strength corresponding to N cycles.