Once harvested the lumbar vertebrae (L1 and L2) were cut through the discs so that the specimens then had flat superior and inferior surfaces. Also, the spinous processes were removed resulting in specimens in the form of short cylinders having curved perimeters as depicted in Figure 1, where the principal dimensions are shown.

We used a dial caliper to measure the five dimensions of Figure 1 for the 18 specimens used in the testing (9 L1 and 9 L2).

After taking these measurements we placed the specimens in shallow plastic flat-dishes containing a liquid cold-curing resin. Upon hardening the specimens were inverted and the other end (superior end) was placed in hardening resin dishes with care taken to keep the superior/inferior dish surfaces parallel. Figure 2 shows a sketch of the resulting vertebral specimen constructs.

Next, we placed each of the 18 constructs in an Instron tester and compressed them to failure (large displacement with minimal load increase). During the compression we recorded the load/displacement values up to failure and then the failure load itself.

As with the vertebrae, the distal radii (once harvested) were cut perpendicular to their axes with the proximal and distal cutting planes being parallel. The bone tissue in the cross-sections was found to be near annular ellipses as represented in Figure 3, where the pertinent dimensions are also show. The specimens then have the approximate shape of a truncated annular elliptical conical segment.

Next, we placed the nine radii constructs in the Instron machine and compressed them to failure. As with the vertebrae, we recorded the load/displacement values up to failure and then the failure load itself.

Table 1 lists the measurements of the geometric parameters shown in Figure 1 for the L1 and L2 vertebrae.

Table 2 lists the failure loads and the slopes of the nearly linear load/displacement relations obtained during the compression testings, for all 18 vertebral specimens.

Tables 3 and 4 list the measurements of the parameters of Figure 3 for the distal and proximal ends of the right radii specimens, and also in Table 4 the specimen lengths.

Finally, Table 5 lists the failure loads and the slopes of the nearly linear load/displacement relations obtained during compression testing.

To determine the vertebral strengths we needed to know their cross-section areas. To this end we used the geometry represented in Figure 1 together with the measured data listed in Table 1. To facilitate this we divided the vertebral cross-section region into three parts as in Figure 4 with circle segments representing the ends and the central region being a near rectangle.

Using standard handbook mensuration formulas [4] for the segment areas we were able to estimate the vertebral cross-section areas in terms of the measured dimensions of Figure 3. From the summary analysis of the Appendix we obtained the following area algorithm:

Let t be defined as:

Next, let r be defined as:

Then the cross-section area A of a vertebral specimen is approximately:

Table 1. Vertebral geometric measurements¹ (see Figure 1).

¹Measurements are in inches (in) with millimeters (mm) in parentheses.

Table 2. Failure loads and stiffnesses of the 18 vertebral constructs.

¹Loads are in pounds (lb) with kiloNewtons (kN) in parentheses. ²Stiffnesses are in pounds per inch (lb/in) with kiloNewton per meter (kN/m) in parentheses.

where a, b, c, and e are shown in Figure 1. Once we have values for the cross-section areas, we can readily determine the specimen strength using elementary strength of materials formulas: Specifically, for a uniform cross-section member with length ℓ and cross-section area A, and subjected to a compressive load P as in Figure 5, the shortening δ of the member is simply:

where E is the elastic modulus (see Beer and Johnston [6]).

The stress and the strain ε are then defined as

Table 3. Distal end geometric measurements^*.

^*Subscripts d for distal Measurements are in inches (in) with millimeters (mm) in parentheses.

Table 4. Proximal end geometric measurements^* (see Figure 3).

^*Subscripts p for proximal. Measurements are in inches (in) with millimeters (mm) in parentheses.

Table 5. Failure loads and stiffnesses of the 9 right radii (wrist) constructs.

¹Loads are in pounds (lb) with kiloNewtons (kN) in parentheses. ²Stiffnesses are in pounds per inch (lb/in) with kiloNewtons per meter (kN/m) in parentheses.

and           (5)

with the “strength” being the maximum stress value occurring at the failure load, or at the collapse of the specimen member.

The elastic modulus E is a measure of the stiffness of the specimen. It is simply the slope k of the linear portion of a graphical representation between the stress σ and the strain ε, or equivalently, between the load P and the displacement δ. That is, from Eqs.4 and 5 we have

or (6)

Where Δ is a finite difference.

The strength and the stiffness E are thus measures of the structural integrity of the vertebral specimens and thus can be used for comparison evaluations.

As with the vertebrae, we also needed the cross-section areas of the distal radii, to evaluate their strengths. To this end, consider again the sketch of the distal radii bone specimens shown in Figure 3. Although the specimens have a longitudinal taper, the measurements recorded in Tables 4 and 5 show that the taper is slight, so that for deformation and stress calculations it is reasonable to simply assume a linear cross-section area change.

The bone content of the radii cross-sections are nearly annular ellipses. Therefore, with an ellipse area being:, with a and b being semi-major and semi-minor axes, the distal and proximal cross-section areas are:

The average bone cross-section is then

Consider again the side view of the radius specimen as in Figure 3 and as shown again in Figure 6 identifying the end areas and with an axial coordinate y. Then we see that the cross-section area A(y) along the axis of the specimen is:

Beer and Johnston [6] state that the shortening δ of the tapered specimen is given by the formula:

By substituting for A from Eq.9 we find δ to be:

In the course of the loading of the specimen (before collapse) we found the loading and the deformation to be linearly related. That is,

where, as with the vertebrae, the stiffness constant k may be identified as the slope of the newly linear load/displacement relation.

Table 6 lists the cross-section areas of the L1 and L2 specimens, calculated using the formulas of Eqs.1-3, and the resulting strengths and elastic moduli.

We used the failure load data of Table 2 and the computed areas to obtain the strengths of the vertebrae. Also, by using the stiffnesses listed in Table 2 we obtained the elastic moduli using Eq.6.

The radii strengths and elastic moduli can be computed using the failure loads and stiffness values listed in Table 6, the measured geometric dimensions of Tables 3 and 4, and then Eqs.7, 8 and 13, to determine the numerical results. To this end, it is helpful to initially determine the cross-section area parameters of Eqs.8 and 13.

In view of Eq.8, it is helpful to compute some area ratios needed for determining the elastic modulus. Also, it is useful to have the product of the construct height h (see Figure 3) and the stiffness k of Table 5. To this end Table 7 lists the distal, proximal and average areas and area ratios and stiffness-height products of the bone content of the various radii.

Finally, using failure loads listed in Table 5 and the parameters of Table 7, we obtain the radii strengths and elastic moduli as listed in Table 8.