Some definitions to get us started:

• Density: the measure of mass per unit volume.
• Stress: force per unit area.
• Strain: the amount a material deforms as a result of stress, written as change in length divided by initial length.
• Strength: the ability of a material to resist applied loads [1]. Getting more specific than just strength, yield strength is the amount of stress which causes the material to deform past a point where it would spring back to its original size. Ultimate tensile strength is the amount of stress which causes the material to break.
• Stiffness: the ability of a material to resist deflection [2]. A more stiff material will be less "affected" by force, while a less stiff material will be more "affected" by force.
• Elastic Modulus (Young's modulus): the ratio of stress (force per unit area) to strain (the amount a material deforms as a result of stress).
• Tension Test: a test used to determine key material properties like strength (ultimate and yield strength) and stiffness (Young's modulus).
• Material Properties: properties inherent to the material. These include: density, elongation, fatigue limit, hardness, stiffness, shear strength, tensile strength, and toughness [3].
• Isotropic: Metals (steel, titanium, aluminum) are called isotropic materials because they exhibit the same material properties in all orientations.
• Anisotropic: Wood is called an anisotropic material, meaning it has different material properties in different directions. For this analysis, I will be evaluating wood along it's strongest direction (the longitudinal axis, or "with the grain"), which is also the direction of the wood grain in our bikes.

White Ash and Black Walnut come in a distant last place with densities of 0.60 g/cm³and 0.55 g/cm³ respectively [8]. They are 13 times less dense than steel. Is this a bad thing? Not at all because this allows us to design the frame with much thicker walls that are buckling, torsion, and impact resistant. Just like titanium and aluminum, wood is (and must be) designed with thicker walls that increase strength and stiffness.

Density and wall thickness adds another extremely important constraint on dense metal tubes, which I will cover in the "Geometric Stiffness" section. This is where wood has the ability to truly perform as a frame building material.

So hold on, why are you making bikes out of wood? At first glance, we shouldn't be. But let's take a deeper look. The test used to obtain these results (called a tension test) is testing the material's raw/inherent strength and stiffness, meaning it's maximum load bearing capacity in a small cross-sectional area [9]. So far so good. The problem with only looking at the raw strength and stiffness of a material begins when assuming larger numbers mean stronger and stiffer frames. This is not necessarily true. There is nothing wrong with inherent strength and stiffness numbers, but often people will look at these properties and instantly conclude that wood can never truly be strong or stiff.

In frame design (and engineering design in general), I need to be sensitive to the application that the materials are being used in. This means there is more to frame design than raw/inherent strength and stiffness numbers alone, however impressive or unimpressive they may be. Material density needs to be balanced with material strength and stiffness. This will show us frame strength and stiffness in a new light that makes sense in frame design. This concept is not a new one; titanium and aluminum bicycle tube manufacturers know full well that titanium and aluminum are not as strong as steel, but are also less dense than steel, so these tubes have greater wall thicknesses to compensate and create a strong, stiff frame. They're playing the same game that I'm playing to make a light, strong, and stiff frame. Below is a quick teaser on how material density is balanced with material strength and stiffness. More to come on this later in the analysis.

The next section graphs what I just introduced, which is strength and stiffness relative to material density. These metrics, called strength efficiency and stiffness efficiency, are the metrics that really matter in designing a wood frame.

Strength Efficiency

By factoring in material density, we are led to something called strength efficiency: the strength of a material relative to its density. Put another way: how strong the material is, pound for pound. White Ash has a strength efficiency of 171.67 MPa/g/cm³ and Black Walnut  has an efficiency of 183.64, both of which handsomely beat out steel (138.82) in terms of strength efficiency.

Stiffness Efficiency

By factoring in material density, we are led to something called stiffness efficiency: the stiffness of a material relative to its density. Put another way: how stiff the material is, pound for pound. White Ash has a stiffness efficiency of 20.0 GPa/g/cm³ and Black Walnut has an efficiency of 21.1. Both fair much better in terms of stiffness efficiency in comparison to steel (26.6), but still fall short. 

So why would I consider strength and stiffness efficiency? I can use more of a less dense material (like wood) and end up with approximately the same strength, stiffness, and frame weight as steel. Pound for pound, they have similar strength and stiffness results.

That's great, you think. The wood frame can be as strong as steel, titanium, and aluminum. Now can the frame stiffness of a wood bike and steel bike be similar? Looking at stiffness efficiency alone, no. But it has the potential to be, yes. That potential yes means a frame designer has one more extremely important principle to consider, geometric stiffness, which will be discussed next.

Geometric Stiffness (Cross-Sectional Shape), Wood and Metal

There is one engineering equation that helps shed light onto why the cross-sectional shape of the frame's tubes are so critical. That is the Polar Moment of Inertia, Iz (a measure of a tube's ability to resist torsion). The formula for a hollow circular cross section is as follows:

Iz = π (D⁴ - d⁴) / 32     where D is the outside diameter, and d is the inside diameter [11].

This engineering equation may be slightly intimidating. The essential thing to note is that diameter is raised to the 4th power. As diameter increases, the tube's resistance to bending and torsion increases dramatically. For example, if we double the diameter of a tube from 1 inch to 2 inches (see picture), we increase the torsional stiffness 8X. The increased diameter of a tube holds extreme stiffness potential. 

Putting it all together

Steel has material properties that are highest in stiffness and highest in density. In order to make a reasonably light frame, minimal amounts of steel must be used. No problem; steel is strong and stiff, right? Yes. So, let's just make the walls really thin and keep the overall weight down. That's what manufacturers like Columbus, True Temper, Reynolds, etc. already do. These tubes are designed to be as light as possible by taking advantage of a large diameter with thin walls, while balancing being thick enough to resist buckling, torsion and impact. Steel, however, can only be made so thin before it becomes useless under any load or force, thus it is limited in how large the diameter can be, and how thin the walls can be. An ideal material is one where you can retain wall thickness (thus keeping buckling, torsion and impact in check) while taking advantage of large diameter tubes which increases geometric stiffness. Titanium and aluminum (and wood) do this.

Below is an illustration of what this looks like when analyzing the cross-sectional area of a steel, titanium, aluminum and wood down tube.

Below is a simple comparison of steel, titanium, aluminum, and wood. Each material's cross-section weighs approximately the same amount (within 9%). Titanium, aluminum, and wood each have a percent difference in area relative to steel. I.e. titanium has 32.8% more area than steel. Titanium, aluminum, and wood also each have a percent difference in strength relative to steel in these configurations. I.e. titanium is 10.4% stronger in tension than steel in its current configuration.

This strength calculation was simple: the ultimate tensile strength (which is the stress in tension it can endure before breaking) of steel is 1080 MPa (highlighted in a previous section). The cross-sectional area of steel is 70.7mm², which is a commonly sized down tube on a steel road bike. Using the formula: stress = force / area, the maximum force this steel tube can endure in tension is 76356 N, or 17166 pounds. Likewise, the ultimate tensile strength of titanium is 810 MPa. The cross-sectional area is 105.2mm², which is a commonly sized down tube on a titanium road bike. The maximum force this titanium tube can endure in tension is 85212 N, or 19156 pounds. This is a 10.4% increase over the steel tube from our example.

Taking material density into account is good news for wood. Wood is 13 times less dense than steel, so we can use 13 times the amount of wood for our cross-sectional area (92% more area than steel) and attain impressive strength and stiffness results because of this increased amount of wood and because of geometric stiffness, all while keeping weight nearly equal to steel, titanium, and aluminum [10]. Not only does the cross sectional area increase, but I am able to increase the diameter of the tubes to leverage geometric stiffness as well as making the wall thickness of the frame 5mm to ensure resistance to buckling, torsion and impact.

A Comparison of Steel and Wood in Deflection

Taking everything we've learned from above in terms of material density, material stiffness, and geometric stiffness, we're now ready to visit the simplified case study from before of steel and wood in bending. Below are the results of a wood down tube and steel down tube in bending using Finite Element Analysis (FEA). FEA is a computer based simulation tool engineers use early on in the design process to validate designs for strength, stiffness, and other parameters. Keeping the total length the same (400mm) and the total tube weight the same (.456 lbs for steel and .452 lbs for wood), the deflection of the tubes are nearly identical (0.87mm for steel and 0.84mm for wood) when a 22 lb force is applied to the free end (while the other end is fixed) Low density is turning out to be a really good thing, allowing us to leverage geometric stiffness because it is 8X more effective in stiffening tubes than the material property of stiffness alone. All while staying lightweight.