An Inexpensive Semi-Variable-Span Bend

Fixture for Rat and Mouse Femurs

Nick Musser

ME 497 – Independent Study

Professor Amy Wagoner Johnson

June 3, 2008

INTRODUCTION

To obtain the biomechanical characteristics of bone, a series of standard material

testing techniques could be employed including: compression, tension, torsion, and bending.

For long cylindrical bones, such as the femur, compression and tension tests may prove

difficult to accurately complete. Although, submitting a specimen with such a shape to a bend

test is often less difficult and more appropriate. Biomechanical data for many species can be

found utilizing the aforementioned techniques. Generally, the experimental setup is scaled up

or down depending on the size of the sample. However, difficulty may arise when testing

very large or very small bones. For example, when testing mouse or rat femurs special

attention must be given to the design of the bend fixture due to the extremely small size of the

bone. Parameters such as the radius of the support pins, testing span, and precision of the

fixture must allow for accurate results and fall within the machining capability of the

manufacturer. The main intent of this project was to create a bend fixture to test rat and

mouse femurs which could be manufactured for less than eight hundred dollars.

As mentioned earlier, bend tests are often performed on bones because bend tests are

much more accepting of irregular cross sections. There are two types of bend tests generally

used: the three point bend test and the four point bend test. The four point test utilizes two

bottom supports and two points of contact on top of the specimen. An identical force diagram

for a four point test can be seen in Figure 1. In contrast, the three point bend test only has one

point of contact on the top and is identical to the setup in Figure 2. Each test has benefits and

fallbacks which must be considered to determine which test would be most appropriate. First,

the four point bend test offers the advantage of a section of constant internal moment. Using

the equations of beam theory, the moment can be calculated at any point in the beam. Figure

3 is an arbitrary example distribution for a four point bend test. Notice the flat line between

the points of contact for the two forces; the flat line indicates a constant internal moment.

However, in order to assume a moment distribution such as the one in Figure 3 the two top

forces must be exactly the same. This is easy to ensure for regular shaped objects but may be

nearly impossible for objects such as a femur. For simplicity, the four point bend test is rarely

used for femurs; rather, the simple and easily repeated three point test is used for such bones

[2]. An arbitrary example moment distribution for a three point bend test appears in Figure 4.

Notice that the maximum moment occurs at the point of contact of the top force. As long as

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the top force is known in a three point bend test the results can be calculated with higher

confidence in their accuracy than in the four point test.

Often, the mechanical characteristics of interest are ultimate stress and elastic

modulus. Employing beam theory, simple relations for ultimate stress, σ, and elastic

modulus, E, can be found for a three point bend test. Respectively, these appear in Equations

(1) and (2) [1].

(1)

(2)

where F is the force exerted by the testing machine, L is the span between the two bottom

supports, c is the distance from the centerline of the bone to the outer surface, I is the second

moment of inertia, and d is the displacement of the point of contact at the instant of failure

[1].

When conducting a bend test it is important that the radius of the support pins be an

appropriate size. For an ideal bend test the force the pins exert onto the femur would be

applied through a line. Intuitively, one would choose a very small radius to achieve a perfect

line load. However, when the radius of the support pin is extremely small, very large

unwanted contact stresses would be present at the point of contact. As a result, the ideal line

load is not an appropriate testing configuration. When the radius of the pins is increased the

force is applied over an area instead of a line and the contact stress decreases, but this is not

ideal. The radius of the pins must be optimized so that the stress concentration at the points of

contact is minimized and the area of contact is minimized.

METHODS

The radius of the support pin influences the contact stress applied to the femur which,

if significant, could influence the test results. Contact stresses were calculated using

Engineer’s Toolbox, software created by ENGRASP, Inc. The pin and the femur were

modeled as perpendicular cylinders because this is the orientation they will have during the

test. Two separate analyses were completed, one for rat femurs and one for mouse femurs.

The radius of the mouse femur and the radius of the rat femur were chosen based on

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previously observed averages. The mechanical properties of mouse and rat femurs were

assumed to be the same and estimated based on experimental results of a similar test and

appear in Table 1. Also in Table 1 are the mechanical properties for aluminum, a common

building material. Forces of 50N and 150N were applied to the mouse and rat models,

respectively. The radius of the pin was the variable. For the rat and mouse femur three

contact stress analyses were conducted each with a different pin size.

Due to the fact that mouse and rat femurs are small, high precision will be needed

during the machining process to ensure accurate data. By utilizing the finite element software

ANSYS, a mouse femur was modeled and analyzed to estimate the appropriate level of

precision needed. Once more, suitable and common mechanical properties of bone were used

during the finite element analysis. To begin, the bone was displaced by a fixed amount at its

center and the stress was noted. Next, the point of displacement was shifted 250µm away

from the center and the stress was recorded. Finally, the point of displacement was shifted

again so that it was 500µm from the center. The simulation was carried out and the stress was

recorded.

RESULTS AND DISCUSSION

Contact stresses were calculated for several different support pin sizes to determine

which radius would be appropriate for the rat and mouse femur bend fixture. Table 2

summarizes the results for a few different pin sizes. First, notice that the contact stresses are

quite high. The elevated stresses can be attributed to the small contact area, a result also listed

in Table 2. Observe that the contact stress decreases as the radius of the pin is increased, a

result of the increasing contact area. However, it is not a simple inverse relationship. In other

words, the contact stress is not halved if the pin radius is doubled. In reality, the contact stress

decreases only slightly if the radius of the pin is doubled. Secondly, the contact stresses

observed for the rat femur are higher than that of the mouse because the ratio of the femur

radius to the pin radius is larger for the rat than it is for the mouse. In other words, the greater

the difference in the radii the greater the contact stresses.

To promote similarity it was decided that the radius of the support pins should be

chosen such that the contact stress experienced by the rat femur would be approximately the

same as that experienced by the mouse femur. A pin diameter of 4mm was chosen for the rat

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femur bend fixture which yields a contact stress of 2.0 GPa. In response, a pin diameter of

1mm was chosen for mouse bend fixture to generate a contact stress near 2.0 GPa.

Due to the atypical size of the bend fixture the precision to which the fixture was

manufactured was a significant design variable. To determine the precision needed, a finite

element analysis was conducted. The results of the analysis were presented as a slide show

and appear in the appendix. Included in the slide show is the internal stress for three different

contact point locations in addition to the percent error compared to the centered configuration.

Shifting the contact point by 250µm and 500µm generated an error of 1.56% and 6.25%,

respectively. Consequently, the machining tolerance was set to 250µm which is within the

capabilities of the machinist and kept the fixture under budget. However, it should be noted

that fixture is most likely more precise than 250µm because of the very few cuts needed to

machine it.

The final design appears in Figures 5-7. There were several constraints which affected

the final design. To begin, the bend fixture needed to fit onto an existing testing machine. A

portion of the testing machine appears as the transparent black part in the figures. The fixture

is inserted into the machine and a pin is then placed through the hole in the fixture. A nut

then tightens down onto the machine to lock the fixture in place. Notice that the machine has

two holes oriented perpendicular to each other. This allows the entire fixture to be rotated 90°

by simply placing the pin through the other hole. Taking advantage of this, the upper block

(appears light grey in the figures) was designed with two different spans for the rat femurs:

15mm and 20mm. Each span is accessible by rotating the fixture to the correct orientation

and gluing the support pins in the correct grooves. Having two different spans is beneficial

because one may wish to test male rat femurs at a longer span than the female femurs, for

example. The two smaller grooves in the upper block, which are 8mm apart, are intended for

the support pins used with the mouse femurs.

According to the test engineer, the fixture needed to be able to rotate so perfect

alignment with the upper ram could be achieved. That is, the side surfaces of the upper ram

and the fixture need to be parallel. To accomplish this, a threaded hole in the fixture allows

the fixture to be rotated to the correct position and then tightened against the lock nut (appears

purple in the figures).

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In order to cut costs, the ram already available from the testing lab will be used.

However, the contact radius of the ram is too small for rat femurs; the contact stresses would

be too large. To achieve a larger contact radius for the rat femurs, a large pin was machined

so that it could be attached to the ram as seen in Figure 5. The original radius on the ram is

not too small for the mouse femurs, so the adaptive pin is not needed.

In the beginning of the design process it was determined that having a variable-span

bend fixture would be beneficial. With this in mind, a model was created of a bend fixture

which would allow the investigator to choose any span he/she desired by sliding the supports

along a dovetail groove. Figure 8 depicts the model of the variable-span fixture. In the end,

despite its potential benefits, the variable-span fixture would be too expensive to manufacture

due to its complex parts. Thus, this design was abandoned.

CONCLUSIONS

This bend fixture was designed to perform bend tests on rat and mouse femurs. To cut

down on costs the design needed to be simple, but it still needed to provide the few different

spans. The different side lengths of the upper block allowed for the different spans to be

achieved. In addition, by using a simple lock nut, the upper block could be positioned and

locked at any angle permitting it to be aligned to the ram. By utilizing components already

provided by the testing lab and by designing simple easy-to-machine parts, the bend fixture

was created for an amount well under eight hundred dollars.

REFERENCES

[1] T. Jamsa, P. Jalovaara, Z. Peng et al., “Comparison of three-point bending test and

peripheral quantitative computed tomography analysis in the evaluation of the strength of

mouse femur and tibia,” Bone, vol. 23, no. 2, pp. 155-161, Aug, 1998.

[2] C. H. Turner, D. B. Burr, “Basic Biomechanical Measurements of Bone: A Tutorial,”

Bone, vol. 14, pp. 595-608, 1993.

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TABLES AND FIGURES

Table 1: The assumed physical characteristics for the femurs and pins during the contact stress analysis.

Table 2: Results of contact stress analysis.

Figure 1: Four point bend test equivalent force diagram.

Figure 2: Three point bend test equivalent force diagram.

P

A

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Figure 3: An arbitrary internal moment distribution for a four point bend test.

Figure 4: An arbitrary internal moment distribution for a three point bend test.

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Figure 5: View of assembled bend fixture.

Figure 6: View of assembled bend fixture.

Testing Machine

Ram

Upper Block

Adaptive Pin

Support Pins

Upper Lock Nut

Lower Lock Nut

Washer

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Figure 7: Exploded view of bend fixture.

Figure 8: Variable-span bend fixture.

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APPENDIX

Slide 1

Slide 2

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Slide 3

Slide 4