## What are Stress and Load?

Stress is defined as force per unit area. When there is localization of huge stresses in mechanical components, due to irregularities present in components and sudden changes in cross-section is known as stress concentration. For example, groves, keyways, screw threads, oil holes, splines etc. are irregularities.

Loads whose magnitude or direction or both changes with time when same load act repeatedly applied is known as fatigue load. This can be due to additional notches and holes in tension member, variation in properties of material such as internal cracks, cavities in welds, inclusion etc., sudden changes in section, discontinuities in the components.

Basically, we observe these three models when it comes to fatigue load

- Fluctuating stress or alternating stress
- Repeated stress
- Reversed stress

## Concept

Fluctuating stress a varies in a sinusoidal manner with respect to time. It has some mean value as well as amplitude value of stress. The value fluctuates between maximum and minimum stress. The nature of stress could be tensile or compressive or mix of both type of stress. From Figure 1, observe that it has a maximum stress peak σ_{max} ,Minimum stress peak σ_{max} , Mean stress value σ_{max}.

## Parameters Used in Design

Means stress: ${\sigma}_{m}=\frac{{\sigma}_{\mathrm{max}}+{\sigma}_{\mathrm{min}}}{2}$

Stress amplitude: ${\sigma}_{a}=\frac{{\sigma}_{\mathrm{max}}-{\sigma}_{\mathrm{min}}}{2}$

Stress range: ${\sigma}_{r}={\sigma}_{\mathrm{max}}-{\sigma}_{\mathrm{min}}$

Stress concentration factor is defined as ratio of the highest value of actual stress nearby irregularities to that of nominal stress derived from the minimum cross section. It is applicable to ideal materials which are homogenous, isotropic and elastic.

${K}_{t}=\frac{{\sigma}_{\mathrm{max}}}{{\sigma}_{o}}=\frac{{\tau}_{\mathrm{max}}}{{\tau}_{o}}$Fatigue stress concentration factor is defined as the ratio of maximum stress induced in the notched specimen to that of the stress in notch free specimen, it can also be defined as the ratio of endurance limit of a notch free specimen to that of the endurance limit of a notched specimen. It is applicable to actual materials and depends upon grain size of material.

Surface finish factor: It consider the surface finish level and geometric irregularities on the surface which acts as stress zones and leads to stress concentration. It is given by following equation

${K}_{a}=a{({S}_{ut})}^{b}$Size factor: It takes in account the reduction in endurance limit due to increase in size of component.

Reliability factor: It depends on what level of reliability the designer is considering in the design of component. When the reliability factor is 1, the reliability is 50 %. To ensure this, the stress amplitude should be made less than the given value of endurance limit.

## Different Case Scenario under Observation

Stress concentration because of hole in a component:

In this type of shape, the stress at the joints away from the hole is practically uniform and the maximum stress is induced at the edge of the hole.

${\sigma}_{\mathrm{max}}=\sigma \left(1+\frac{2a}{b}\right)$**Notch Sensitivity**

Notch sensitivity is the chance of a material to get affected to the effects of stress rising in fatigue loading.

Notch sensitivity factor is defined as, ratio of actual stress increase above the nominal stress to that of theoretical stress increase above the nominal stress.

Let actual stress = ${K}_{f}{\sigma}_{o}$

Theoretical stress = ${K}_{t}{\sigma}_{o}$

Notch sensitivity, $\begin{array}{c}q=\frac{{K}_{f}{\sigma}_{o}-{\sigma}_{o}}{{K}_{t}{\sigma}_{o}-{\sigma}_{o}}\\ =\frac{{K}_{f}-1}{{K}_{t}-1}\end{array}$

**Note:**

- K
_{t}depends on shape, size of discontinuity and type of load condition and its orientation. - K
_{f }is independent of material behaviour. - K
_{f}depends on the shape, size of orientation and type of loading condition and material of component. - Notch sensitivity, q depends of material of the component.
- If q = 0, then K
_{f}= 1, this implies that the material is insensitive to stress concentration or notch. - If q = 1, then K
_{f}= K_{t}, this means that the material of the component is highly sensitive to notch. - The q = 1, implies that worst design.

## Criteria Used for Design Components

There are various criteria used to separate safe design and fail design. There are various graphs plots such as Gerber line, Soderberg line, Goodman line which are plot between stress amplitude on y axis and mean stress on x axis. The line distinguishes between safe zone and failure zone.

Gerber line: It is a parabolic curve joining S_{e} on the ordinate to S_{ut} on the abscissa.

Soderberg line: A straight line joining ${\sigma}_{e}$ on the ordinate to ${\sigma}_{yt}$on the abscissa.

Goodman Line: A straight line joining ${\sigma}_{e}$ on the ordinate to the ${\sigma}_{ut}$on the abscissa.

## Design Methods

Gerber Methods: Gerber parabola is used when design is based on ultimate strength. It is used for ductile material.

It gave relation as $\frac{1}{F.S.}={\left(\frac{{f}_{m}}{{f}_{u}}\right)}^{2}F.S+\frac{{f}_{v}{K}_{f}}{{f}_{e}}$

Where F.S is factor of safety, f_{e }is fatigue strength corresponding to the case of complete reversal (f_{m} = 0), f_{ut} is static ultimate strength corresponding to f_{v} =0.

Goodman Method: It is used to design on basis of ultimate strength and can be used for ductile or brittle material. Line which connects f_{e} and f_{u} is called goodman’s failure stress line.

It is given by following condition.

$\frac{1}{F.S.}={\left(\frac{{f}_{m}}{{f}_{u}}\right)}^{}+\frac{{f}_{v}{K}_{f}}{{f}_{e}{K}_{sur}{K}_{sz}}$Here the load, surface finish and size factors are also considered. If their value is not mentioned it is assumed to be 1.

Soderberg Method:

It is a straight-line connecting endurance limit and yield strength. It is used while designing on yield strength. It is given by following expression. It is particularly used for ductile material. For a reversed shear loading.

$\frac{1}{F.S.}={\left(\frac{{f}_{m}}{{f}_{y}}\right)}^{}+\frac{{f}_{v}{K}_{f}}{{f}_{eb}{K}_{sur}{K}_{SZ}}$Corresponding load factor, surface finish and size factor are used above, if not given it is considered as 1.

SN Diagram: It is developed by joining the 0.9S_{ut} at 1000 cycles and S_{e} at 10^{6} cycles by a straight line on a log S – log N graph. It shown as below:

## Example

Problem: Given that for a mechanical component, the bending stress varies in from 100 MPa to 300 MPa, the ultimate strength in tension is 700 MPa. Yield strength in tension is 500 MPa and endurance strength is 350 MPa.

Determine the factor of safety using Goodman criteria.

Solution:

Given, ${\sigma}_{max}$=300 MPa

${\sigma}_{min}$ = 100 Mpa

Mean stress: ${\sigma}_{min}$ =( ${\sigma}_{max}$+${\sigma}_{min}$)/2 = 200 MPa

Stress amplitude: ${\sigma}_{a}$= $\left({\sigma}_{max}-{\sigma}_{min}\right)/2$ = 100 MPa

Endurance stress :${\sigma}_{e}$ = 350 MPa

Yield strength :${\sigma}_{yt}$= 500 MPa

Ultimate strength: ${\sigma}_{ut}$= 700 MPa

Using good man criteria we have,

$\frac{1}{F.S.}={\left(\frac{{f}_{m}}{{f}_{u}}\right)}^{}+\frac{{f}_{v}{K}_{f}}{{f}_{e}{K}_{sur}{K}_{sz}}$Since surface, size factor, load factor not given, these values are put to 1.

$\frac{1}{F.S.}={\left(\frac{{f}_{m}}{{f}_{u}}\right)}^{}+\frac{{f}_{V}}{{f}_{e}}$

$\frac{1}{F.S}=\frac{200}{700}+\frac{100}{350}$

F.S = 1.75

## Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

- Bachelors in Engineering/Technology
- Masters in Engineering/Technology

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