# FATIGUE FAILURE OF A SPRING: WÖHLER AND GOODMAN DIAGRAMS

We continue on the topic of spring failure, talking about **statistical tools** (in graph / table form), which can be used for a preventive check of the spring geometry, to assess the risk of fatigue failure: **Wöhler and Goodman diagrams.**

Let’s focus on the use of springs in *dynamic regime*. We can say *dynamic regime* when the number of cycles of use exceeds 10,000 within the life cycle of a spring. The fatigue verification tables start at 105 cycles.

At the moment** there is no physical-mathematical theory that constitutes a predictive model for the fatigue verification**. In the absence of a model, we rely on physical tests, the results of which are systematically organized with the aid of statistical tools.

In other words, **empirical curves** have been obtained over the years, based on the repetition of hundreds / thousands of fatigue tests, on springs with known stress, varying the stroke (difference between *L2* and *L1*) and noting the number of cycles at which the failure occurred.

## Wöhler diagram

Assuming the homogeneity of the batch of tested springs, both from a dimensional point of view and the mechanical characteristics of the wire, a common trend should emerge regarding the number of cycles at which the spring breaks.

The distribution of the number of cycles under the same conditions is practically normal as the mean and variance can be estimated on the basis of the sample data.

Using the** inferential statistic**, we define a value which significantly (in a statistical sense) includes 95% of the detected values. That value indicates the **fatigue life** for a specific material between two defined stresses (initial at *L1* and final at *L2*).

The tests are repeated by increasing the stress at *L2* and keeping the stress fixed at *L1*.

It is not possible to test for all the stresses continuously. We choose an incremental interval between one test batch and the other and we linearly interpolate between the two values.

Representing the information obtained in a graph with a logarithmic scale (x-axis), we obtain an **S-diagram**, the so-called Wöhler diagram.

The figure below shows a *Wöhler diagram* for a carbon spring steel, assuming a fixed stress at *L1* of *100 Mpa* and increasing the stress at *L2* over *900 Mpa*.

You will notice that at *700 Mpa* practically the spring has an infinite duration, the curve flattens, while as the stress at *L2* increases, we observe a decrease in the cycles at which the break occurs.

## Goodman diagram

The *Wöhler diagram* is a **less practical** tool for the purpose of theoretical verification.

**By combining the results** of multiple *Wöhler diagrams*, it is possible to obtain **a graph that is definitively more manageable**: we are talking about the *Goodman diagram*.

The* Goodman diagram* has a trapezoidal shape, bounded below by the bisector of the quadrant and above by a line that defines the fatigue strength limit. The figure is defined with a horizontal line representing the **yield stress limit**.

The initial stress (at *L1*) is reported on the x-axis and the final stress (at *L2*) on the y-axis.

The use of the *Goodman diagram* is relatively simple, given the stresses at *L1* and *L2*, the point is identified on the Cartesian plane, using the stresses as coordinates.

If the point falls within the trapezium area, **the spring is checked under stress**, if it is outside it is not. It is reasonable to expect fatigue failure before reaching the number of cycles indicated on the graph.

In the figure shown 3 almost parallel upper straight lines are visible. These lines identify the safety area for 10^{5 }– 10^{6 }– 10^{7 }cycles, in a decreasing sense. The top line is for 10^{5 } cycles and the bottom line is 10^{7 } cycles. It is intuitive that longer durations are obtained by decreasing the stress at *L2*.

For this reason, if we expect a dynamic use of the spring, **a preliminary fatigue check is required**, using the appropriate *Goodman diagram*, in relation to the material with which the spring is to be produced.

An unverified spring is a candidate for **premature failure** for reasons related to the phenomenon of fatigue.

Si laurea in ingegneria elettronica al Politecnico di Milano nel 1992. Dal 2000 lavora al **Mollificio Valli** come responsabile tecnico commerciale.

Acquisisce negli anni una consolidata esperienza nell’ambito del calcolo e degli aspetti tecnici legati alla produzione delle molle.

Da sempre appassionato di matematica e statistica, ha avuto modo di applicare le sue conoscenze nelle tecniche statistiche di controllo, negli aspetti metrologici e in generale in ambito pratico nei casi di problem solving e miglioramento continuo.