This chapter should be seen as a brief introduction to the different lessons. The different concepts introduced here will be explained in more details in the following chapters.

Overview > Fatigue

Historical background

Picture I.1: Versailles rail accident, 8 May 1842, Meudon, Paris (source English Wikipedia, this file has been identified as being free of known restrictions under copyright law, including all related and neighboring rights).
Picture I.2: Train axle subjected to shear loading inducing a cyclic flexural loading.

In the middle of the 19th century with the generalisation of the use of steam engines, especially for trains, engineers had to face a new kind of problem never really encountered before: some trains were suffering of severe damage, although the loading remains in the design range. One of the most famous accidents, the Versailles rail accident illustrated in Picture I.1, is due to the rail axle failure of the leading locomotive. However at that time the failure could not be explained.

One of the first engineers to investigate this issue was the German August Wöhler around 1860. His analysis led to the following conclusions:

His conclusion was that the incidents were due to the fact that a train axle is submitted to a cyclic loading/unloading. Indeed an axle is submitted to a shear load due the weight of the train car, which in turns induces a bending moment in the axle, see Picture I.2. As the axle is rotating, a point of the axle isl alternatively submitted to compression and to traction when lying above or below the neutral axis, see Picture I.3. Due to this cyclic loading, the axle experiences fatigue and the life of the structure is limited. Based on this observation Wöhler has developed a theory based on an empirical approach: the total life theory.

Total life theory

Picture I.3: Characterization of a cyclic loading.

Based on experiments Wöhler found that the life of a structure submitted to a cyclic loading depends on two values characterizing this loading, see Picture I.3:

Alternatively the cycle, and thus the life of the structure, is characterized by any two of the following values, see Picture I.3:

Under particular environmental conditions such as high humidity, high temperature ... one also needs to consider the parameters related to the creep effect:

Once those values are known the limit cycles number (or life) in terms of the loading can be predicted using one of the two total life approaches summarized as:

Stress life approach

Picture I.4: Example of a Wöhler curve for a given material and a cyclic loading characterized by $\sigma_m = 0$.

This approach can be used for structures experiencing essentially elastic deformations. By applying different cyclic loading on samples made of a given material, Wöhler has experimentally extracted curves predicting the limit cycles number $N_f$ (or life) in terms of the loading parameters: $\sigma_a = 0$ and $\sigma_m = 0$. An example of such a curve obtained for $\sigma_m = 0$ is illustrated in Picture I.4. It is important to note that a Wöhler curve is obtained

For some materials there exists an endurance limit $\sigma_e$ defined as the stress state below which the structure can handle at least 107cycles. An order of magnitude for $\sigma_e$ is [0.35-0.5] of $\sigma_\text{TS}$ (tensile strength of the material), see Picture I.4.

Thus for a given structure, one generally knows the material while the loading ($\sigma_a$ and $\sigma_m$) can be computed at strain concentration zones, in which case the Wöhler curves directly give the limit cycles number $N_f$ of the structure.

One practial way of using the Wöhler curves is the recourse to the Basquin law (1910), which describes the curves for $\sigma_m=0$ as:

\begin{equation} \frac{\Delta \sigma}{2} = \sigma_a = \sigma'_f (2N_f)^b, \label{eq:basquin}\end{equation}


Note that all these parameters are determined from experiments, which are time consuming as for a single material they require multiple tests, each of one implying millions of cycles.

Strain life approach

This approach is meant to be used for structures suffering locally plastic deformations. The Manson-Coffin law (1952) can be applied for identical cycles:

\begin{equation} \frac{\Delta \bar{\varepsilon}^p}{2} = \varepsilon'_f (2N_f)^c, \label{eq:mansonCoffin}\end{equation}


Limits of the total life approach

Picture I.5: De Havilland 106 Comet 1 (This is photograph ATP 18376C from the collections of the Imperial War Museums. This artistic work created by the United Kingdom Government is in the public domain)

The total life approach is a purely empirical approach, which does not take into account the microscopic origin of fatigue. It has been successfully used for many years to design structures up to the middle of the 20th century, time at which its limits dramatically appeared with the arising of the first pressurized jetliner: the De Havilland 106 Comet 1 airplane whose first flight occured in 1952, see Picture I.5. This jetliner was able to transport 36 passengers in a pressurized cabin at 0.58 atm. Due to the pressurization, the fuselage experiences cyclic loading, and the airplane was designed for 10,000 cycles.

Picture I.6: The recovered (shaded) parts of the wreckage of G-ALYP and the site (arrowed) of the failure. (Picture adapted by Ian Dunster 17:44, 3 February 2006 (UTC) from an illustration in Air Disasters by Stanley Stewart - Arrow Books (UK) 1986/89 - ISBN 0-09-956200-6. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License , Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.)

However, this plane suffered from design issues:

Picture I.7: Fuselage fragment of de Havilland Comet G-ALYP which crashed January 10, 1954. This fragment, retrieved from the bottom of the Mediterranean Sea, was determined to be the original cause of the crash as it tore loose. Picture -5 October 2009- by Krelnik licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Nevertheless, although the mode of the failure was identified, the origin was still unknown. Indeed the fuselage has been fatigue designed to sustain 10,000 cycles. Moreover a fuselage has been tested at the design stage and has sustained these 10,000 cycles. So how could one explain the in service failure for a much lower cycles number?

Picture I.8: Round windows on the Comet 4. Image by Klaus Nahr from Germany licensed under the Creative Commons Attribution-Share Alike 2.0 Generic license.

The direct outcomes of the Comet investigation were