Bike Frame Design

(www.slwotwitch.com)
by Kim Blair PhD
Director, Center for Sports Innovation
Massachusetts Institute of Technology

FRAME DESIGN 101 (4.14.00)
FRAME DESIGN 102 (4.19.00)
FRAME DESIGN 201 (6.7.00)

FRAME DESIGN 101

If you’re like me, you buy a new bike, then spend all your time riding it, and not looking at new bikes. A couple years down the road, the interest in seeing the new toys returns, and you realize that you’ve apparently missed a revolution in design. Certainly this was true with materials. Used to be, everything was steel. Then along came aluminum, carbon fiber and titanium. The latest innovation is in using frame tubes that aren’t round, or are very large. This article will talk about some of the basic structural issues that govern tube shapes, both the oversized tubes used for aluminum frames, and the new odd shapes appearing on your local bike showroom. (Aerodynamics are also a factor pushing this design, but won’t be covered in this article.)

Before we get into this, let me state up front that I am making many major simplifications here for the purposes of explanation. This information is only intended to provide non-engineers an appreciation of some of the concepts pushing the new frame designs.

This is part 1 of a series of articles. The series topics will be as follows:

  • Stress and Strain: What happens when you put a force on a bike frame component?
  • Materials Issues: Issues that result from using various frame materials
  • Tube Geometry: Now that we understand the effect of applying forces on frame components of different materials, we can start playing with the geometry of the tubes to get target performance needs.

• Stress and Strain

Two terms used to describe the way an external load effects a bike component are stress and strain. Like your body, too much stress or strain on a bike is harmful. Unlike your body, if you apply too large of load to the bike causing excessive stress (or strain), the bike won’t heal with a couple weeks of rest. Thus, the bicycle designers carefully design bike frame parts to handle the loads during the life of a frame.


FIGURE 1: AXIAL LOADING OF A TUBE.

Stress is a measure of how the intensity of an applied load is distributed over a given section of a component. Figure 1 shows an axial (along the axis of the tube) applied load (P) on a tube. For this simple case, distributing the applied load over the cross-sectional area results in a stress of

The Greek letter sigma, to the left of the equation, is used to denote stress. Stress is expressed in terms of lbs/in2 (psi). As an example, let's say I stand a bike tube on end, on the ground, and I am able to perfectly balance my body weight on the tube by balancing on one foot. Furthermore, assume the tube is 1" in outside diameter, with a wall thickness of 0.1", which results in a cross sectional area of 0.283 in2. (The area of a circle is &Mac185;r2, where r is the radius of the circle.) Assume I weigh 160 lbs. Thus, the stress in the tube resulting from my balancing on it is 45 psi. Note that we have yet to mention anything about the material of the tube.

When I stand on the tube, the load applied results in the tube being compressed. You may not see it, because it is a small displacement, but it does become compressed. This deformation of the tube is called strain, and is expressed as

At this point, it is clear that stress and strain are likely related.

• Modulus of Elasticity

Let’s continue with our bike tube, and do an experiment. If we were to add varying weights (or load) to the tube and measure the resulting deflection (this requires very sensitive measuring equipment), we could chart the loads and associated deflections, connect the values and end up with something that looks like Figure 2. The first thing we notice is that we get a straight line. Thus, if we stand on the tube, then get off the tube, we expect the tube to return to its original length. This is true up to a point. If, in fact, we apply enough of a load, the tube yields, and permanently deforms. It will never return to its original shape.



FIGURE 2: STRESS-STRAIN PLOT RESULTING FROM AXIAL LOADING OF A TUBE

Frame design rule: The frame components must operate within this linear stress-strain range.

So what about the strain? Designing using strain as a parameter is certainly done as well. If you ever hear about track cyclists who claim they can make the crank arms hit the frame on a hard sprint, they are talking about strain. The frame has stretched enough to allow this to happen. It is likely that the frame has not yielded at this point, and will return to its original shape.

Frame design concept: The amount of deflection (or strain) in a frame may be the driving factor in frame design, as a frame will deflect to the point that it hampers performance, long before there is a material failure.

So far, we have been talked about the concepts of stress and strain resulting from applying loads to a bike frame. Next time, we will bring in the effect of using various materials.



FRAME DESIGN 102

This is part 2 of a series of articles. The series topics will be as follows:

  • Stress and Strain: What happens when you put a force on a bike frame component?
  • Materials Issues: Issues that result from using various frame materials
  • Tube Geometry: Now that we understand the effect of applying forces on frame components of different materials, we can start playing with the geometry of the tubes to target specific performance needs.

• Review of 101

In Part 1, I defined the concept of stress, strain, and the modulus of elasticity. Recall the example of standing on a bicycle tube. Your body weight supplies the load, which results in the tube feeling a stress, and a strain. We discussed how the relationship between stress and strain is linear, and that relation is called the modulus of elasticity of a material. Finally, we noted that one designs a frame so that all stresses remain in this linear region, or below the material’s Yield Strength.

• Frame Materials

In a very simplified view, metal frame materials are made as follows. An alloy is made by mixing together various metallic elements. For example, steel is actually a mix of iron (which is why it rusts) and other metals such as chromium, nickel, and molybdenum. (Remember Crome-moly frames?). Aluminum is alloyed with copper, manganese, silicon, and magnesium. Finally, a titanium frame is actually an alloy consisting of titanium, aluminum and vanadium. Once the desired ratio of elements is selected, the metal is formed by melting, mixing and then cooling the mix. Once that process is complete, the material may be further heat treated to obtain the exact desired properties. This entire process is quite complicated. For the purpose of our story, recognize that for each frame type (e.g. steel, aluminum, and titanium) there can be a wide array of alloys.

(I have purposely avoided composite materials in this article. While an excellent frame material, its inherent properties preclude carbon-fiber composites from being included in any article titled Bike Frame Design 101.)

• Material Properties

For our discussion, we will focus on three material properties:

  • the modulus of elasticity (its "springiness"),
  • the yield strength of the material (where it fails), and
  • the density, which determines the weight of a tube.

The following table lists these three values for common types of frame alloys described above.

MATERIAL MODULUS OF ELASTICITIY (psi) YIELD STRENGTH (psi) DENSITY lb/in(3)
STEEL 30,000,000 60,000 - 150,000 0.285
ALUMINUM 10,000,000 20,000 - 70,000 0.100
TITANIUM 16,500,000 115,000 - 140,000 0.161

When building frames (or any component for that matter) the designer needs to work with all of the material properties to optimize the characteristics of the frame. Note that the density and modulus are nearly the same for all alloys. The yield strength, however, varies widely dependent on the alloying elements and heat treatment process. Note that a high yield strength allows one to form thinner tubes, which of course reduces weight. The trade-off is typically cost. More processes nearly always means a higher cost.

• Let's Build a Tube

To illustrate how the material properties effect design, lets make three identical tubes, one of steel, one of aluminum and one of titanium. Assume we select a 20" long, round tube with a 1" outside diameter and a tube thickness of 0.02". (These are the rough dimensions of a "normal-size" steel top-tube. For each tube we will determine the weight, and the stress and deflection for a given applied load.

First thing we need to calculate is the cross-sectional area of the tube. For our round tube with a 1" diameter and 0.02" wall thickness, the area is 0.062 square inches. The weight of the tube can be calculated using the density of the material, the tube area, and its length. Thus for steel, the weight of the tube is W = 0.285*0.062*20 = 0.35 lbs. (Results for all three tubes will be summarized later.)

In Part 1, we noted that we need to be concerned with both the stress and the deflection caused by an applied load. Let’s assume an applied load, P of 2000 lbs: big number, we don’t want our frame to break. Further, let’s assume we are interested in the condition of axial loading, show in Figure 1 of Part 1. For this condition, the stress in the tube is the load, P divided by the area of the tube, l2000/0.062 = 32,300 psi.

We can use known engineering relationships to calculate the deflection caused by the load. For the case of axial loading, the deflection in the tube is d = (load * length)/(area * Young’s Modulus). For the steel tube, d = (2000*20)/(0.062*30,000,000) = 0.021 in. The results for each material are summarized below:

MATERIAL TUBE WEIGHT (lbs) STRESS (psi) YIELD STRENGTH (psi) DEFLECTION
STEEL .35 32,300 60,000 - 150,000 0.022
ALUMINUM .12 32,300 20,000 - 70,000 0.065
TITANIUM .20 32,300 115,000 - 140,000 0.039

As we expect, the table shows that the steel tube is the heaviest, followed by the titanium tube and the aluminum tube.

It doesn’t matter which material you use, the stress resulting from the applied load is the same. But, it’s not the actual value of the stress that is important, rather how that value compares with the yield strength for the particular alloy used in the tube. Thus, depending on the aluminum alloy selected for this tube, the tube may have yielded (permanently deformed). Components are typically designed to operate well below their yield, so for this loading, the aluminum tube would not be acceptable.

Finally, as we stated in Part 1, the deflection of the tube may be the design criteria of most importance. As seen in the above table, the deflection for steel is lowest, followed by titanium, and then aluminum.

What does all this mean to the designer? It means that if we want to control the deflection of the tube, or make sure our aluminum tube doesn’t fail, we need to change something. Assuming we want to build frames and not design new metal alloys, our choice is thus limited to changing the tube geometry, which I will discuss in Part 3.



FRAME DESIGN 201

This is part 3 of a series of articles. The series topics will be as follows:

  • Stress and Strain: What happens when you put a force on a bike frame component?
  • Materials Issues: Issues that result from using various frame materials
  • Tube Geometry: Now that we understand the effect of applying forces on frame components of different materials, we can start playing with the geometry of the tubes to target specific performance needs.


Review of Parts 1 and 2

In Part 1 I defined the concept of stress, strain, and the modulus of elasticity. In Part 2 I described how the modulus of elasticity is a function of a given material. It was shown that Aluminum is more compliant (bends easier) than steel or titanium. The density of the material was also discussed, aluminum being the lightest for a fixed tube size. However, it was shown that for a given loading condition, the aluminum tube may fail, while the steel and titanium tubes work fine. But, wait, we want the reduced weight of aluminum, without resorting to high-cost titanium. What can we do?

Goals of Tube Design

The goal of the tube designer is actually quite simple, at least in principle. Use the least amount of material possible to get the required strength of the tube set. In reality, this is actually a very difficult design optimization problem. The designer not only needs to understand the loading conditions for the bike, but also must understand a plethora of manufacturing techniques and relative costs. The perfect design may not be manufacturable at a price point that is acceptable by the consumer.

Wall Thickness and Tube Diameter

Wall thickness and tube diameter are the easiest variables to adjust. From a manufacturing perspective, it’s quite easy to draw round tubing. Recall the example tube from Part 2: a 20" long, round tube with a 1" outside diameter and a tube thickness of 0.02". When we applied the 2000 lb axial load, the deflection of the aluminum tube was nearly 3 times that of the steel tube. In addition, depending on the particular aluminum alloy used, it may have failed.

Let’s do a simple redesign of the aluminum tube. We’ll increase the outside diameter to 1.5" (we’re building a fat tube aluminum frame) and increase the wall thickness to 0.03". If we repeat the calculations for stress, deflection and tube weight, we find that the new design has a deflection similar to that of the steel tube, a stress level about half that of the original design, and a weight midway between the steel and titanium tube. The numbers are shown in the following table.

MEASURE DENOM. FAT ALUM ALUM TITAN STEEL
LENGTH INCHES 20 20 20 20
OUTSIDE DIAM. INCHES 1.5 1 1 1
THICKNESS INCHES .03 .02 .02 .02
APPLIED LOAD LBS 2000 2000 2000 2000
TUBE WEIGHT LBS .277 .123 .197 .351
STRESS PSI 14436 32481 32481 32481
DEFLECTION INCHES .029 .065 .039 .022

Tube Inertia

At the outset of this series of articles, I promised to explain why many bike frames have such odd shaped tubes. Well, we are nearly there. Just a couple more basic engineering concepts are needed. The first of these is inertia. In this context, the inertia is the distribution of the material about the cross section of the tube, calculated about a selected point. Some basic concepts: (a) more material results in a larger value of inertia, and (b) the further the material is from the reference point, the larger the inertia. Obviously, tube shape can have a large effect on these properties.

The inertia of a circular tube can be found by I = 0.049 * [OD(4)- ID(4)], where OD is the outside diameter, and ID is the inside diameter. For our 1" round tube with a 0.02" wall thickness, the inertia is 0.0074 in(4). For our fat aluminum tube, (OD = 1.5", thickness = 0.03") the inertia is 0.0374 in(4).

Bending of Tubes

The other concept we need to consider is bending, in particular bending of the tube. Imagine a grabbing the end of a very pliable tube and bending it. This is illustrated in the following figure, where M is the applied moment (or bending load).



Once again, we are concerned with the stress and deflection resulting from bending the tube. The stress can be calculated by S = M*y/I where M is the applied moment, y is the distance from the center of the tube and I is the inertia. Note that this equation tells us that the maximum stress is in the outer surface of the tube (where y is the largest).

Deflection is also of interest. For this case, we measure deflection as a radius of curvature. Thus, a large radius of curvature means a smaller deflection. The radius of curvature is found from R = E*I/M, where E is the Young’s Modulus (a material property), and I and M are defined as above.

The following table shows the calculations for our various tubes. As we expect, the stress is the same for all of the same sized tubes. However, for the fat Al tube, the stress is reduced by over 4 times. The radius of curvature is different for each tube, as would be expected, since this depends on tube material and geometry. Notice that the fat Al tube has the highest radius of curvature, meaning that it deflects the least for pure bending.

MEASURE DENOM. FAT ALUM ALUM TITAN STEEL
LENGTH INCHES 20 20 20 20
OUTSIDE DIAM. INCHES 1.5 1 1 1
THICKNESS INCHES .03 .02 .02 .02
INERTIA IN.(4) .0374 .00739 .00739 .00739
APPLIED MOMENT IN-LBS 2000 2000 2000 2000
BENDING STRESS PSI 40066 135223 135223 135223
CURVATURE INCHES 9.34 .55 .90 1.64

Finally, Tube Shape

Now that we understand bending and inertia, we can discuss tube geometry. Lets make a really ugly bike tube. (Very useful for demonstrating this concept.) It will be 1" x 2" rectangular cross section, of wall thickness 0.02" and 20" long. Inertia of a hollow square tube is calculated as I = (boho(3)- bihi(3))/12. Here the o’s refer to the outside dimensions, and the i’s the inside dimensions, b is the base, and h is the height. Note that b and h are relative to the applied moment. Thus, in the figure below, we can see that h is always defined as in the plane of the moment.

The following table shows the inertia, stress and curvature results for an aluminum tube, one standing upright and one lying on its side. What one observes is that you want the long side of the tube to be in line with the loading.

DENOM. ALUM UPRIGHT ALUM SIDEWAYS
B INCHES 1 2
H INCHES 2 1
THICKNESS INCHES .02 .02
INERTIA IN.(4) .064 .022
APPLIED MOMENT IN-LBS 2000 2000
BENDING STRESS PSI 31102 45127
CURVATURE INCHES 322 111

Consider your bike’s down tube. Where it meets the head tube, you’d imagine that the largest bending loads occur as a result of the head tube pulling on the tube in the upright direction. You’d want your square tube in the upright position as shown above. Conversely, at the bottom bracket, you’d imagine the largest bending is the result of the pedaling motion. Thus, you’d want your tube in the horizontal position. Certainly, some manufacturers have utilized this principal in their tube sets. You can find bike frames with tubes that vary from tall and thin at the head tube, to wide and fat at the bottom bracket.

Summary

First, let me again reiterate that many simplifying assumptions have been made for the purposes of this article. The design of bike frames is a complex process, combining various amounts of engineering analysis, tradition and technical wizardry. That being said, I think I can make a few points that may help you appreciate the designs you see on your next visit to your local bike shop.

The stress in a tube is a function of the tube shape and applied loading. The stress seen by a tube will not be effected by changing the tube’s material.

The tube’s material dictates how much stress the tube can endure without failure. In other words, an aluminum tube may fail, where a steel tube will not. Both experience the same stress, but steel is stronger.

For a given loading condition that exceeds the yield stress of the tube, you have to change some combination of the tube geometry or tube material. (This combines the above two points.)

The deflection in a tube is a function of the tube’s material and geometry and the applied loading. Thus material choice makes a large difference in the deflection characteristics of the frame.