# Determine the thickness of resistance bands

Resistance bands have been used since the early 1900’s as an exercise tool, and are now also widely used as a rehabilitation tool to strengthen (injured) muscles and joints. They became popular due to their affordability and convenience. Just like weight training, they can increase muscle strength, endurance and flexibility. When designing a resistance band, it is important to determine several aspects, like what material to use, and how long, thick and wide to make the bands. Figure 1. Commercially available resistance bands. Figure 2. Chest exercise performed with resistance band.

The question we will consider here is what the appropriate dimensions are of resistance bands so that they are able to provide the appropriate resistance forces. It is important to be able to produce several bands that gradually increase the force needed to stretch them for several reasons. For example, strength varies between individuals and between different muscle groups within an individual so need different resistance levels are needed, and people (re)building strength after an injury need to be able to gradually rebuild with different resistance levels. Here, we will focus on circular resistance bands, and therefore the specific variable that will be determined is the radius of the resistance band.

A resistance band is usually made of natural rubber, as this provides enough strength while still stretching with the user’s movements.  Natural rubber behaves as a linear elastic material, with a Young’s Modulus (E) between 1 and 5 MPa. As you can see in Figure 3, the Young’s Modulus is a measure of stiffness for a material’s elastic regime. It represents the ratio of tensile stress over strain, where tensile stress is the tension force applied on a surface area and strain is the amount of deformation (amount of stretch of the resistance band).

We will simplify the problem by assuming this is a simple an exercise as shown in Figure 2, and the resistance felt is only coming from the band itself. We also simplify the situation by assuming a constant position of the stretching band in front of the user’s body, so that the increase in resistance felt is for the same muscle group, and coming from the band properties only. See Figure 4 for a free body diagram of the resistance band. Figure 4. Free body diagram showing the forces acting upon the resistance band when performing the exercise shown in Figure 2.

Assume:

• Natural rubber behaves as a linear elastic material, such that E = σ/ε
• Tensile stress,  σ = F/A
• Strain, ε = ΔL/L
• Circular resistance band with
• Length, L
• surface area, A=π*R2 (where R=radius)
• Stress applied is below yield strength

We will estimate the following values:

• E = 2.5 MPa = 0.0025 N/mm2  (average value (3))
• ε=1.5 and ε=3 (we determine the force needed to stretch the band 150% and 300%)
• Resistance force/force applied to band: 1lb, 3lb, 7lb, 9lb, 10lb, which equals 4.4N, 13.3 N, 22.2N, 31.1N, 40N, 44.5N.

We can now solve for the radius (so the thickness) of the band, using the following steps:

1. E = σ/ε
2. Rewrite σ = E*ε
3. Plug in the values for E and ε=1.5 you get σ = 0.00375 N/mm2
4. Since σ=F/A, we can rewrite this as A = F/0.00375
5. We can now plug in the different values for F and find the surface area, A
6. Since A=π*R2, we can solve for R using R=sqrt(A/π)
7. Similarly, for ε=3 you obtain values for R by using A = F/0.0075

Plugging in these values in the equations gives the following results:

 F (lb) F(N) Equation [ε=1.5] A (mm2) [ε=1.5] R (mm) [ε=1.5] to stretch 150% R(mm) [ε=3] to stretch 300% 1 4.4 A = 4.4/0.00375 1173 25 14 3 13.3 A = 13.3/0.00375 3547 34 24 5 22.2 A = 22.2/0.00375 5920 43 31 7 31.1 A = 31.1/0.00375 8293 51 36 9 40 A = 40/0.00375 10666 58 41 10 44.5 A = 44.5/0.00375 11867 61 43

According to the above calculations, the radius of a resistance band made of natural rubber could range from 25 mm to 61 mm in order to create resistance forces ranging from 1 to 10 lb for stretching 150%, and 14 mm to 43 mm for stretching 300%. Intuitively these values seem high. I have personally never seen tubular resistance bands with a radius much larger than 7mm, but according to the calculations above twice as large diameter (14mm) is needed only to create a 1lb force to stretch the band 300%. If we compare these values to the forces created by the commercially available Thera-Band®, a resistance band made of natural rubber with a radius around 40 mm would create similar forces as the red band (which is the second strongest of the 5 colored bands).

It seems fair to say that the solved values are not reasonable, considering the forces produced by existing Thera-Band® and the dimensions of commercially available tubular resistance bands. I think the solution here is limited due to several simplifications. We assumed that the force to stretch the band is applied to the end of the tubular stretching band. However, if a band is directly grabbed by a person, it is a more complex situation as he/she will probably use their hands to grab the band over a larger surface than estimated above. It seems like the resistance band cannot be modeled as a tubular piece of solid material on which a stress is applied to the end surfaces, as shown in Figure 4. Another aspect that is not taken into account here, is that the different resistance bands can also be altered in design in “how hollow vs. solid” they are, in other words how dense the natural rubber is packed. The more dense it is packed, the more resistance it will provide as more fibers in the band need to be stretched.

In conclusion, the approach provided above does not seem to provide reasonable answers to the question what the radius of tubular resistance bands should have in order to provide certain resistance levels. The model seems to simplified, and might therefore not be appropriate to apply when designing resistance bands.