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Frame Design |
Posted by: Admin on Wednesday, January 21, 2004 - 04:28 PM
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Frame Design considerations
FRAME DESIGN
If you're like me, you buy a new bike, and 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 bikes returns and you realise that you've apparently missed a
revolution in design.
Certainly this is always true with materials; everything used to be steel,
and then along came aluminium, carbon fibre and titanium. The latest innovation
is in using frame tubes that aren't round, or are very large. This article will
discuss some of the basic structural issues that govern tube shapes, both
the oversized tubes used for aluminium frames, and the new odd shapes appearing
in your local bike shop near you.
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 new frame designs.
Section 1
In Section one the 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 & Strain
The two terms Stress and Strain are used to describe the way an external load effect's a bikes design features.
Like your body, too much stress or strain on a bike is harmful.
Unlike your body, if you apply too large aload to the bike causing
excessive stress (or strain), the bike won't heal with a couple weeks of rest, you have totalled it.
Thus, bicycle frame designers have to carefully design bike frame parts to handle the
loads during the life of a frame, this includes abuse loading. Stress is a measure of how the intensity of
an applied load is distributed over a given section of a component.
The Greek letter sigma, to the left of the equation, is used to denote stress. Stress
is expressed in terms of lbs/in2. 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. Assume I weigh 160 lbs.
Thus, the stress in the tube resulting
from my balancing on it is 45 pounds/square inch. 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 1. 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 1: stress-strain plot resulting from axial loading of a tube.
Frame design rule: The frame components must operate within this linear stress-strain range.
Strain
Designing using strain as a parameter is certainly a requirement. 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.
Section 2
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.
In 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 the
design of a frame is such 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,
chrome-moly. Aluminium is alloyed with copper, manganese, silicon, and magnesium.
Finally, a titanium frame is actually an alloy consisting of titanium, aluminium
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, recognise
that for each frame type (e.g. steel, aluminium, 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-fibre composites).
Material Properties
For our discussion, we will focus
on three material properties:
Modulus of elasticity (its "springiness")
Yield strength of the material (where it fails)
Density, which determines the weight of a tube
The table 1 lists these three values for common
types of frame alloys described in section 2.
| Material |
Modulus of Elasticity(psi) |
Yield Strength(psi) |
Density(ilb/in3) |
| Steel |
30,000,000 |
60,000-150,000 |
0.285 |
| Al |
10,000,000 |
20,000-70,000 |
0.100 |
| Ti; |
16,500,000 |
115,000-140,000 |
0.161 |
Table 1
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 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.
Consider Tube configuration
To illustrate how the material properties effect design, lets make three identical
tubes, one of steel, one of aluminium 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 the first section, 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. Further,
let's assume we are interested in the condition of axial loading. For this condition, the stress
in the tube is the load, P divided by the area of the tube, l2000/0.062 = 32,300
p/square inch.
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 summarised
in table 2:
| Material |
Tube weight |
Stress (psi) |
Yiels Strength (psi) |
Density (in) |
| Steel |
0.35 |
32,000 |
60,000-150,000 |
0.022 |
| Al |
0.12 |
32,000 |
20,000-70,000 |
0.065 |
| Ti |
0.20 |
32,000 |
115,000-140,000 |
0.039 |
Table 2
As we expect, the table shows that the steel tube is the heaviest, followed by the titanium tube
and then the aluminium 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 aluminium 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 aluminium tube would
not be acceptable.
Finally, as we stated in the first section, 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 aluminium.
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 aluminium 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.
Section 3
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 sections 1 and 2
In the first section I defined the concept
of stress, strain, and the modulus of elasticity. In second I described how
the modulus of elasticity is a function of a given material. It was shown that
aluminium is more compliant (bends easier) than steel or titanium. The density
of the material was also discussed, aluminium being the lightest for a fixed
tube size. However, it was shown that for a given loading condition, the aluminium
tube may fail, while the steel and titanium tubes work fine. But, wait, we want
the reduced weight of aluminium, 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 aluminium tube was nearly 3 times that of the steel tube. In addition,
depending on the particular aluminium alloy used, it may have failed. Let's
do a simple redesign of the aluminium tube. We'll increase the outside diameter
to 1.5" (we're building a fat tube aluminium 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 table 3.
| Measure |
Dimension |
Fat Al Tube |
Normal Al tube |
Normal Ti Tube |
Normal Steel Tube |
| Length |
inches |
20 |
20 |
20 |
20 |
| O/D |
inches |
1.5 |
1 |
1 |
1 |
| W/T |
inches |
1.5 |
1 |
1 |
1 |
| Applied Load |
lbs |
2000 |
2000 |
2000 |
2000 |
| Tube weight |
lbs |
0.277 |
0.123 |
0.197 |
0.351 |
| Stress |
psi |
14436 |
32481 |
32481 |
32481 |
| Deflection |
inches |
0.029 |
0.065 |
0.039 |
0.022 |
Table 3.
I mentioned that I would 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:
More material results in a larger value of inertia
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 * [O/D (4) - I/D(4)]
Where O/D is the outside diameter, and I/D 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 aluminium 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 grabbing the end of a very pliable tube and bending it. 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 centre 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 |
Dimension |
Fat Al |
Normal Al |
Normal Ti |
Normal Steel |
| Length |
inches |
20 |
20 |
20 |
20 |
| O/D |
inches |
1.5 |
1 |
1 |
1 |
| W/T |
1.5 |
1 |
1 |
1 |
1 |
| Inertia |
inches(4) |
0.0374 |
0.00739 |
0.00739 |
0.00739 |
| Applied moment |
In/lbs |
2000 |
2000 |
2000 |
2000 |
| Bending Stress |
psi |
40066 |
135223 |
135223 |
135223 |
| Curvature |
inches |
9.34 |
0.55 |
0.90 |
1.64 |
Table 4.
Finally, Tube Shape
Now that we understand bending and inertia, we can discuss tube geometry.
Lets review the cross sectional section of a rectangular 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 = (HO/WO (3) - (HI/WI (3))/12.
Where :
"O" refers to the outside dimensions
Note that W and H are relative to the applied moment.
The following table shows the inertia, stress
and curvature results for an aluminium 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.
| Measure |
Dimension |
Tube vertical |
Tube horizontal |
| W |
inches |
1 |
2 |
| H |
inches |
2 |
1 |
| W/T |
inches |
0.02 |
0.02 |
| Inertia |
lb(4) |
0.064 |
0.002 |
| Applied load |
in/lbs |
2000 |
2000 |
| Bensing Stress |
psi |
31102 |
45127 |
| Curvature |
inches |
322 |
111 |
Table 5.
Consider your bike's down tube. Where it meets the head tube,
you would imagine that the largest bending loads occur as a result of the head
tube pulling on the tube in the vertical direction. Conversely, at the bottom
bracket, you'd imagine the largest bending (torsion) is the result of the pedalling
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 affected by changing
the tube's material.
The tube's material dictates how much stress the tube
can endure without failure. In other words, an aluminium 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.
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