Coil Windings: Hoop Stress Analysis

Hello! Welcome back to our course on high-field magnet design.

In the previous lesson, we established that a current flowing through a solenoid winding generates a powerful, outward-directed radial force density, $f_{r} = j_{θ} B_{z} r^$ , which we can approximate as an effective magnetic pressure, $p_{m} = B^{2} / (2 μ_{0})$ . This force tries to burst the coil apart.

Today, we transition from calculating the forces to understanding their consequences. Our learning outcome is to apply thin-shell and thick-cylinder analysis to calculate the hoop stress distribution in coil windings. This is a fundamental skill in magnet engineering, allowing us to predict whether a coil will survive the immense forces it generates. We will translate the abstract concept of magnetic force into the tangible, material-specific quantity of stress ( $σ$ ).

1. A First Approximation: The Thin-Shell Model

The simplest way to model the stress in a coil is to treat the entire winding pack as a single, thin-walled cylindrical shell. This model is useful for a quick first estimate and is reasonably accurate for coils where the winding thickness, $t$ , is small compared to its radius, $r$ (a common rule of thumb is when the diameter-to-thickness ratio $D / t > 20$ ).

To understand the core idea of hoop stress, the following video provides a clear and intuitive derivation.

Understanding Pressure Vessels

In 'Understanding Pressure Vessels' by The Efficient Engineer, the derivation for hoop stress in a thin-walled cylinder is presented. This will serve as the foundation for our first model.

Please watch from 1:34 to 3:15. Focus on how the internal pressure creates an outward force that must be balanced by the tensile stress in the cylinder's walls.

As the video shows, for a cylinder under internal pressure $P$ , the hoop stress $σ_{θ}$ is:

$σ_{θ} = \frac{P r}{t}$

To apply this to our magnet, we replace the fluid pressure $P$ with the magnetic pressure $p_{m}$ we derived in the last lesson. Let's use the magnetic field at the inner bore, $B_{i}$ , to calculate this pressure, so $P \approx p_{m} = B_{i}^{2} / (2 μ_{0})$ . Here, $r$ is the inner radius $r_{i}$ and $t$ is the total winding thickness $t = r_{o} - r_{i}$ .

Thus, the thin-shell approximation for the average hoop stress in a solenoid is:

$σ_{θ} \approx \frac{B _{i}^{2}}{2 μ _{0}} \frac{r _{i}}{r _{o} - r _{i}}$

A variation of this is the "floating shell" model, which is particularly relevant for coils where the windings are not perfectly bonded together. In this model, each individual turn of the conductor is treated as its own thin shell, supporting only the force generated by the current within it. The detailed resource Stress analysis of tape-wound magnet coils by Jesse Hord (which we will refer to as the NBS paper) derives this in Section 2.6.

2. The Reality of High-Field Coils: Thick-Cylinder Analysis

Most high-field magnets are not "thin." The radial build of the winding is significant compared to its bore size, so the $D / t > 20$ condition is rarely met. In these "thick-walled" cases, two key assumptions of the thin-shell model break down:

Hoop stress is not uniform: The stress is highest at the inner radius and decreases as you move outward.
Radial stress is not negligible: The pressure between adjacent layers creates a significant compressive stress in the radial direction, $σ_{r}$ .

This distinction is critical for accurate design. To understand it better, let's turn to a more detailed tutorial.

Stress Analysis: Thick Walled Pressure Vessels, Press & Shrink Fits (4 of 17)

This video, 'Stress Analysis: Thick Walled Pressure Vessels', clearly explains why thick-walled analysis is necessary and presents the governing equations, known as Lamé's equations.

Please watch from 46:55 to 1:05:23. The video will cover: The difference between thin and thick-walled vessels and the importance of radial stress. The general Lamé equations for hoop (\sigma_T) and radial (\sigma_r) stress. The simplification of these equations for the common case of internal pressure only, and the resulting stress distribution.

As the video explains, for a thick-walled cylinder of inner radius $r_{i}$ and outer radius $r_{o}$ subjected only to an internal pressure $P_{i}$ , the hoop stress $σ_{θ}$ and radial stress $σ_{r}$ at any radius $r$ are given by:

σ_{θ} (r) = \frac{P _{i} r _{i}^{2}}{r _{o}^{2} - r _{i}^{2}} (1 + \frac{r _{o}^{2}}{r ^{2}})

σ_{r} (r) = \frac{P _{i} r _{i}^{2}}{r _{o}^{2} - r _{i}^{2}} (1 - \frac{r _{o}^{2}}{r ^{2}})

Notice a few key features:

The hoop stress $σ_{θ}$ is always tensile (positive). It is maximum at the inner radius ( $r = r_{i}$ ).
The radial stress $σ_{r}$ is always compressive (negative). Its magnitude is maximum ( $- P_{i}$ ) at the inner radius ( $r = r_{i}$ ) and drops to zero at the free outer surface ( $r = r_{o}$ ).

Stresses in Thick-Walled Cylinders — Summary of forces and stress distributions in a thick-walled cylinder under internal pressure. The graphs clearly show that hoop stress ($\sigma_t$) is maximum at the inner radius, while radial stress ($\sigma_r$) is compressive and also maximal in magnitude at the inner surface, decreasing to zero at the outer surface.

The Bitter Method: A Practical Application

A widely used and very effective engineering approximation, known as the Bitter method, directly applies this thick-walled pressure vessel model to a solenoid. It assumes the coil acts like a thick cylinder subjected to an internal pressure equal to the magnetic pressure at the bore.

Using the equations above with $P_{i} = B_{i}^{2} / (2 μ_{0})$ , we get the Bitter-Lame equations for magnet stress:

σ_{θ, max} = σ_{θ} (r_{i}) = \frac{B _{i}^{2}}{2 μ _{0}} \frac{r _{o}^{2} + r _{i}^{2}}{r _{o}^{2} - r _{i}^{2}}

This formula is a cornerstone of preliminary magnet design. It provides a much more realistic estimate of the peak stress at the inner winding than the simple thin-shell model.

3. A More Rigorous Approach: Body Force Analysis

The pressure vessel analogy is powerful, but it's not perfectly accurate. The Lorentz force in a magnet is a body force, distributed throughout the entire volume of the winding, not just a pressure applied at the inner surface. A more complete analysis accounts for this.

This problem was solved analytically by assuming the winding is a homogeneous, isotropic elastic material. Given your physics and math background, you will appreciate seeing the structure of the solution.

Lecture #4 – Magnetic Forces and Stresses

The MIT OCW lecture notes 'Magnetic Forces and Stresses' present the full analytical solution for stresses in a solenoid, treating the Lorentz force as a distributed body force. This provides the most accurate analytical model.

Please read pages 5-11 of this document. Focus on the following: Page 6: See how the problem is set up, moving from the simple 'independent turns' approximation to a coupled problem involving radial stress. Pages 7-8: Note the governing differential equation (Eq. 5a) and the general solution (Eq. 6). The constants are determined by boundary conditions (usually zero radial stress at the inner and outer surfaces). Page 9: Study the final stress equations (7a and 7b). These are the key results. Pages 10-11: Read the 'Comments' and 'Conclusions' to understand the physical implications, especially how compressive radial stress can actually be beneficial.

The key insight from this more rigorous model is that the interaction between turns generates a compressive radial stress $σ_{r}$ throughout the winding. This internal compression helps counteract the tension from the hoop stress. As a result, for many common coil geometries, the peak hoop stress is actually lower than predicted by the simpler Bitter method. For very "fat" coils ( $r_{o} ≫ r_{i}$ ), this can even reverse, and radial stress can become tensile, which is dangerous.

The NBS paper provides alternative derivations under plane stress (Section 2.2) and plane strain (Section 2.3) assumptions, which are excellent for further reading. Figure 9 in that paper powerfully compares the results from all the models we've discussed: floating shell, Bitter (thick wall), plane stress, and plane strain.

Test your understanding!

Consider a pulsed coil with an inner radius $r_{i} = 25$ mm, an outer radius $r_{o} = 75$ mm, and a peak central field of $B_{i} = 60$ T.

Calculate the maximum hoop stress using the Bitter method. ( $μ_{0} = 4 π \times 1 0^{- 7}$ H/m).
Is this coil "thin" or "thick" according to the $D / t > 20$ rule of thumb? Does this justify using a thick-cylinder model?

Show answer

Bitter Method Calculation:
The formula for maximum hoop stress is:
$σ_{θ, max} = \frac{B _{i}^{2}}{2 μ _{0}} \frac{r _{o}^{2} + r _{i}^{2}}{r _{o}^{2} - r _{i}^{2}}$
First, calculate the magnetic pressure:
$P_{i} = \frac{B _{i}^{2}}{2 μ _{0}} = \frac{( 60 T ) ^{2}}{2 \cdot ( 4 π \times 1 0 ^{- 7} H/m )} \approx 1.432 \times 1 0^{9} Pa = 1432 MPa$
Next, calculate the geometric factor:
$\frac{r _{o}^{2} + r _{i}^{2}}{r _{o}^{2} - r _{i}^{2}} = \frac{( 7 5 ^{2} + 2 5 ^{2} )}{( 7 5 ^{2} - 2 5 ^{2} )} = \frac{5625 + 625}{5625 - 625} = \frac{6250}{5000} = 1.25$
Finally, calculate the stress:
$σ_{θ, max} = 1432 MPa \times 1.25 = 1790 MPa$
This is an enormous stress, far beyond the yield strength of even the strongest steels, highlighting why reinforcement is critical.
Thin vs. Thick:
- Inner diameter $D_{i} = 2 \times r_{i} = 50$ mm.
- Thickness $t = r_{o} - r_{i} = 75 - 25 = 50$ mm.
- Ratio $D_{i} / t = 50/50 = 1$ .
  Since $1 ≪ 20$ , this is definitively a thick-walled cylinder. The use of a thick-cylinder model like the Bitter method is not only justified but essential. A thin-shell model would be highly inaccurate.

Conclusion

Today we have bridged the gap between electromagnetic forces and mechanical stress. We have developed a toolbox of analytical models to estimate the hoop stress in a solenoid, which is often the life-limiting factor in a high-field magnet.

Key Takeaways:

The thin-shell model ( $σ_{θ} \approx p_{m} \frac{r _{i}}{t}$ ) provides a simple but often inaccurate estimate for stress.
Thick-cylinder analysis is necessary for most high-field magnets. It accounts for the non-uniform distribution of stress and the presence of radial stresses.
The Bitter method is a robust and widely used application of thick-cylinder theory, modeling the coil as a pipe with an internal pressure of $B_{i}^{2} / (2 μ_{0})$ .
The most rigorous models treat the Lorentz force as a distributed body force, revealing that internal compressive radial stress can help reduce the peak hoop stress.

Next Lesson Preview:
We calculated a peak stress of nearly 1800 MPa in our exercise. No unreinforced conductor can withstand this. So, how do we build a coil that doesn't explode? In our next lesson, we will investigate radial reinforcement strategies, exploring how high-strength composite materials like Zylon and fiberglass are used to contain the immense hoop stresses we learned to calculate today. We will also touch on concepts like interference fits, which you saw introduced in the context of shrink-fits in the video resources.

Recommended resources

1. A First Approximation: The Thin-Shell Model

2. The Reality of High-Field Coils: Thick-Cylinder Analysis

The Bitter Method: A Practical Application

3. A More Rigorous Approach: Body Force Analysis

Conclusion