## What is meant by fluctuating loads?

Machine elements are subjected to varieties of loads, some components are subjected to static loads, while some machine components are subjected to fluctuating loads, whose load magnitude tends to fluctuate. The components of a machine, when rotating at a high speed, are subjected to a high degree of load, which fluctuates from a high value to a low value. For the machine elements under the action of static loads, static failure theories are applied to know the safe and hazardous working conditions and regions. However, most of the machine elements are subjected to variable or fluctuating stresses, due to the nature of load that fluctuates from high magnitude to low magnitude. Also, the nature of the loads is repetitive. For instance, shafts, bearings, cams and followers, and so on.

According to Linear elastic fracture mechanics, there are three stages of fatigue failure.

- Stage 1 (Development of one or more micro-cracks)
- Stage 2 (Crack propagation)
- Stage 3 (Fracture)

Fatigue failure is mainly due to crack formation and propagation. Fatigue cracks usually initiate at regions of high-stress concentration.

## Introduction to fatigue loads

Fatigue loading is a phenomenon associated with variable loading or cyclic loading that leads to the formation of stresses that tend to fluctuate on their magnitudes. In other words, their value changes from maximum stress to minimum stress repeatedly. Most of the mechanical components experience fatigue due to a change in the direction or magnitude of loading conditions.

When a mechanical component is under fatigue loading, the components fail at stresses, which are generally under the yield point of that component. The fatigue of a material is generally affected by the size of the component, the magnitude of static and fluctuating loads, the number of stress cycles, and shape and irregularities present in the components.

## Types of fatigue stresses

The different types of fatigue stresses are summarized below:

**Fluctuating fatigue stress:**These stresses fluctuate from a minimum value to a maximum value. They are tensile and compressive.**Repeated fatigue stress:**The stress cycle is such that the minimum stress magnitude is zero and goes all the way up to a maximum stress value. The values fluctuate between zero to a maximum value.**Completely reversed fatigue stresses:**These kinds of stresses fluctuate from one value of compressive to another value of tensile and vice-versa.**Alternating fatigue stress:**These stresses fluctuate from a minimum value to a maximum value, but are opposite in nature.

## Stress concentration factor

When the continuity of a component’s cross-section changes suddenly, or the surface of the component contains a large number of irregularities, stresses that are induced in the components localize in those areas more. Hence that particular area is said to be in high-stress concentration.

To consider the effects of stress concentration and to determine localized stresses, a factor known as the stress concentration factor is used extensively. It is denoted by ${K}_{t}$. Stress concentration is the ratio of the maximum value of actual stress to the value of nominal stress, Mathematically, it is represented by the equation,

${K}_{t}=\frac{{\sigma}_{max}}{{\sigma}_{\circ}}=\frac{{\tau}_{max}}{{\tau}_{\circ}}$

Where, ${\sigma}_{\circ}$and ${\tau}_{\circ}$ are the stresses derived by elementary concepts, and ${\sigma}_{\mathrm{max}}$ and ${\tau}_{\mathrm{max}}$ are the maximum values of localized principal stresses and shear stresses at the region of discontinuities. The letter $t$ denotes the theoretical stress concentration factor which depends on the geometry of the component.

There are various causes of stress concentration factors such as

**Variation in properties of the materials **

Homogeneity of a machine component is a primary assumption made in designing components. However in actual practice, there are certain amounts of irregularities and roughness present in the machine components that are induced due to internal cracks and flaws, cavities caused during welding (especially in fillet welds), air holes in the components, caused primarily due to casting, and foreign inclusions.

**Load applications that tend to fluctuate**

Machine components are usually subjected to forces, these forces are generally at a point or over a small area of interest. Since the area is small, the pressure induced by the forces is large in magnitude. This results in the deformation of the component and generates stress concentrations. Some of the practical example for this scenario are, contact between the gears which remains meshed, contact between cam and follower, contact between ball and races of a rolling element bearing, contact between rail and wheel, and so on.

## Notch sensitivity

Notch sensitivity is the sensitivity of a component due to notches, or geometric discontinuities. Notch sensitivity is influenced by notch geometry.

The notch sensitivity is defined as the ratio of increase in actual stress over nominal stress to increase theoretical stress over nominal stress.

Mathematically, the expression can be written as $q=\frac{{K}_{f}-1}{{K}_{t}-1}$

Where ${K}_{f}$ = Fatigue stress concentration factor

${K}_{t}$ = Theoretical stress concentration factor

The fatigue stress concentration factor is defined as the ratio of the endurance limit of the notch-free specimen to the endurance limit of the notched specimen.

## Endurance limit

The endurance limit ${\sigma}_{v}$, of a material is defined as the maximum amplitude of completely reversed stresses that the component can withstand for an unlimited number of load cycles without the fatigue failure. ${10}^{6}$ cycles are considered to be an ideal load cycle to determine the endurance limit of the component.

The term fatigue is another term used along with endurance limit. Results of the fatigue tests are plotted by the means of an S-N curve. In the S-N curve, the abscissa is the number of cycles, N, and the ordinate is the applied stress, S. The S-N curve is a graphical representation of stress amplitude ${\sigma}_{a}$ versus the number of stress cycles $N$. For ferrous materials such as steel, the S-N curve becomes asymptotic at ${10}^{6}$.

## Soderberg lines

When a machine component is subjected to fluctuating stresses, there forms a mean stress ${\sigma}_{m}$ and stress amplitude ${\sigma}_{a}$. It has been observed that ${\sigma}_{m}$ has an effect on the fatigue failure of the component when the component undergoes functionality with an alternating component. The magnitude of ${\sigma}_{m}$ and ${\sigma}_{a}$ depends on the magnitudes of maximum and minimum forces on the component.

In the Soderberg line, the abscissa is the mean stress, and the ordinate is the stress amplitude. The design of shafts for strength involves certain additional calculations. The mean stress and stress amplitudes need to be calculated for different loads such as bending, torsion, and axial loads. When a machine component rotates, it is experienced by a high degree of alternating stress. Soderberg criteria is a design criterion for the design under variable loads. In the Soderberg criteria, the mean stress material property is the yield point.

The Soderberg relation is given as

$\frac{{\sigma}_{a}}{{\sigma}_{v}}+\frac{{\sigma}_{m}}{{\sigma}_{y}}=1$

where ${\sigma}_{y}$ is the yield stress.

## Context and Applications

The topic is extensively taught in graduate and post-graduate curriculums such as

- Master of science (Physics)
- Bachelor of science (Physics)
- Post-doctoral degree research
- Doctoral research

## Practice Problems

- Which of the following quantity is determined by the S-N curve?

- Endurance limit
- Yield strength
- Fatigue strength
- Fillet strength

Correct option- a

Explanation: S-N curve is a graphical representation that plots the results of a fatigue test, which can be used to estimate the endurance limit of the component.

2. The stress concentration factor is related to which of the following options?

- Discontinuities in the specimen
- Abrupt changes in cross-sectional areas
- Both a and b
- None of the above

Correct option- c

Explanation: The stress concentration factor speaks about the magnitude of stress concentration on a component. It depends on the material properties such as the number of discontinuities in the specimen, abrupt changes in the cross-section, and so on.

3. Which of the following correctly represents a fluctuating stress condition?

- A shaft under torsion
- Forces exerted by the hot molten fluid in the riser
- A beam under a point load
- None of these

Correct option- a

Explanation: When a shaft is under torsion, the surfaces of the shaft are under cyclic loading conditions. These loading conditions give rise to fluctuating stress conditions.

4. Which of the following failure does gives an indication before its occurance?

- Failure due to static loads
- Failure due to gradually applied loads
- Failure due to fluctuating loads
- None of these

Correct option- c

Explanation: Fluctuating loads induce fatigue stresses. These are repetitive in nature and are sudden in occurrence.

5. The riser, used in the casting process, is subjected to fluctuating stresses. True or false?

Correct answer: True

Explanation: The riser in the casting is used to hold the hot molten liquid. When the liquid is unto the riser, it is under expansion, so the tensile stresses act on the riser. When the riser is emptied, it begins to cool, so it undergoes contraction, which leads to the development of compressive forces.

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