Identify

Barbells have been used for strength training for centuries, and the basic design of those used today was invented in 1928, yet they remain one of the most popular and effective exercise tools out there. From the main power lifts of bench, squat, and deadlift, to the olympic lifts of clean, jerk, and snatch, and limitless other movements, a barbell can be used to target any muscle group to improve strength and power. However, it must retain its shape. Through countless loading cycles, years of use, and sometimes extreme bending stresses, a barbell needs to be ready to be picked up and used again right away, and that means it cannot yield, or permanently bend – this would make it more difficult to use, change its motion patterns, and put it at risk of breaking. While typical use of a barbell for most people would not push it to its mechanical limits (Figure 1), those who compete in weightlifting often place so much weight on the ends of the bar that it indeed bends very much (Figure 2).  A barbell must be constructed of the proper material to withstand the loads it is placed under and bend without becoming permanently bent – or, in engineering terms, deform elastically but not plastically.

Figure 1. Use of a loaded barbell to perform a deadlift

Figure 2. Use of an extremely heavily loaded barbell to perform a deadlift

Formulate

When it comes to competition weightlifting, there are actually different dimensions and specifications required of barbells used for different lifts – read about it here. I decided to focus on a barbell for deadlifting because it’s the movement that can be done with the most weight and is not dynamic like olympic lifts. I borrowed dimensions from the most commonly-used barbell for deadlifting, the Texas 7-1/2″ Bar (Figure 3).

Figure 3. Dimensions of the most commonly-used barbell made for deadlifting

I also decided to design for preventing yield failure rather than fatigue failure because it is a more pressing design concern; it would make more sense to constrain for yielding and optimize for fatigue life rather than the other way around.

The world record deadlift is 500 kg (1,102.3 lbs) by Eddie Hall, so I used a weight of 453.6 kg (1,000 lbs), as events involving more weight than this are so infrequent that yielding in that case would not be of particular concern. This weight is divided into two evenly distributed loads at the ends of the bar, treated as a point load at the center of the distribution, while the opposing forces act where the hands would be placed (I assumed this to be the middle of the knurled portion as seen in Figure 3) [Figure 4].

Figure 4. Lifting of a barbell designed as a beam deflection problem

However, the problem can be simplified to fit a common pattern of loading/support (Figure 5), allowing for a few simple hand calculations to find the stress in the bar. This requires ignoring the weight of the bar itself (which, because of its even distribution and relative lightness, is not crucial anyway) and placing the loads at the very ends of the bar. In the end these assumptions will skew the estimate towards a slightly higher stress, giving an even safer design constraint.

Figure 5. Beam ends overhanging supports & two equal loads applied at symmetrical locations – http://www.engineersedge.com/beam_bending/beam_bending7.htm

Solve

By calculating the bar’s moment of inertia, the distance from the neutral axis, and the section modulus of the cross section of the beam, the maximum bending stress can be found to be 587 MPa (Figure 6).

Figure 6. Simplified representation of a loaded, held barbell and calculation of stress

Therefore, the barbell must be made of a material with a yield strength greater than 587 MPa. A look at a plot of materials’ yield strengths shows that metals, ceramics, and composites are all possibilities (Figure 7).

Figure 7. A plot of different materials’ yield strengths compared to their densities (from the text Materials Engineering, Science, Process and Design by Ashby et. al, 2007)

Metals make the most sense, however, because of their density and ductility. Composites’ light weight means they would be difficult, or impossible, to make into regulation-weighted-and-dimensioned barbells. Ceramics are also very brittle, meaning they break before bending at all; it is usually safer for a product to give warning before breaking, in the form of bending, making a ductile metal a better choice. Given its cost compared to titanium alloys, steel is easily the best choice for a barbell.

There is a dizzying amount of different steel mixtures and grades, but based on searching through tables and information sheets such as this and this, it is a safe bet that molybdenum-alloyed steels (steel alloy 4140/4340, yield strength 655/852 MPa) , cold worked austenitic stainless steels (stainless steel grade 301/304/310, yield strength 470-1310 MPa), and martensitic stainless steels (stainless steel grade 410/420/431, yield strength 415-1895 MPa) are all appropriate choices for a barbell that would not suffer permanent deformation even under the most weight a human has ever (dead)lifted.